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Chern class

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In <a href="Mathematics" title="Mathematics">mathematics</a>, in particular in <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a>, <a href="Differential_geometry_and_topology" class="mw-redirect" title="Differential geometry and topology">differential geometry</a> and <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, the Chern classes are <a href="Characteristic_class" title="Characteristic class">characteristic classes</a> associated with <a href="Complex_vector_bundle" title="Complex vector bundle">complex</a> <a href="Vector_bundle" title="Vector bundle">vector bundles</a>. They have since become fundamental concepts in many branches of mathematics and physics, such as <a href="String_theory" title="String theory">string theory</a>, <a href="Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons theory</a>, <a href="Knot_theory" title="Knot theory">knot theory</a>, and <a href="Gromov%E2%80%93Witten_theory" class="mw-redirect" title="Gromov–Witten theory">Gromov–Witten invariants</a>.

Chern classes were introduced by <a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Shiing-Shen Chern</a> (<a href="#CITEREFChern1946">1946</a>).

<div class="mw-heading mw-heading2"><h2 id="Geometric_approach">Geometric approach</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Basic_idea_and_motivation">Basic idea and motivation</h3></div>

Chern classes are <a href="Characteristic_class" title="Characteristic class">characteristic classes</a>. They are <a href="Topological_invariant" class="mw-redirect" title="Topological invariant">topological invariants</a> associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true.

In topology, differential geometry, and algebraic geometry, it is often important to count how many <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> sections a vector bundle has. The Chern classes offer some information about this through, for instance, the <a href="Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a> and the <a href="Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index theorem</a>.

Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the <a href="Curvature_form" title="Curvature form">curvature form</a>.

<div class="mw-heading mw-heading3"><h3 id="Construction">Construction</h3></div>

There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class.

The original approach to Chern classes was via algebraic topology: the Chern classes arise via <a href="Homotopy_theory" title="Homotopy theory">homotopy theory</a> which provides a mapping associated with a vector bundle to a <a href="Classifying_space" title="Classifying space">classifying space</a> (an infinite <a href="Grassmannian" title="Grassmannian">Grassmannian</a> in this case). For any complex vector bundle V over a manifold M, there exists a map f from M to the classifying space such that the bundle V is equal to the pullback, by f, of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle. In turn, these universal Chern classes can be explicitly written down in terms of <a href="Schubert_cycle" class="mw-redirect" title="Schubert cycle">Schubert cycles</a>.

It can be shown that for any two maps f, g from M to the classifying space whose pullbacks are the same bundle V, the maps must be homotopic. Therefore, the pullback by either f or g of any universal Chern class to a cohomology class of M must be the same class. This shows that the Chern classes of V are well-defined.

Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his. The resulting theory is known as the <a href="Chern%E2%80%93Weil_theory" class="mw-redirect" title="Chern–Weil theory">Chern–Weil theory</a>.

There is also an approach of <a href="Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> showing that axiomatically one need only define the line bundle case.

Chern classes arise naturally in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely, <a href="Locally_free_sheaves" class="mw-redirect" title="Locally free sheaves">locally free sheaves</a>) over any nonsingular variety. Algebro-geometric Chern classes do not require the underlying field to have any special properties. In particular, the vector bundles need not necessarily be complex.

Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a <a href="Section_(category_theory)" title="Section (category theory)">section</a> of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (<a href="Hairy_ball_theorem" title="Hairy ball theorem">hairy ball theorem</a>). Although that is strictly speaking a question about a real vector bundle (the "hairs" on a ball are actually copies of the real line), there are generalizations in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields.

See <a href="Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons theory</a> for more discussion.

<div class="mw-heading mw-heading2"><h2 id="The_Chern_class_of_line_bundles">The Chern class of line bundles</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">For a sheaf theoretic description, see <a href="Exponential_sheaf_sequence" title="Exponential sheaf sequence">Exponential sheaf sequence</a>.</div>

(Let X be a topological space having the <a href="Homotopy#Homotopy_equivalence" title="Homotopy">homotopy type</a> of a <a href="CW_complex" title="CW complex">CW complex</a>.)

An important special case occurs when V is a <a href="Line_bundle" title="Line bundle">line bundle</a>. Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of X. As it is the top Chern class, it equals the <a href="Euler_class" title="Euler class">Euler class</a> of the bundle.

The first Chern class turns out to be a <a href="Complete_set_of_invariants" title="Complete set of invariants">complete invariant</a> with which to classify complex line bundles, topologically speaking. That is, there is a <a href="Bijection" title="Bijection">bijection</a> between the isomorphism classes of line bundles over X and the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2}(X;\mathbb {Z} )}">

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<annotation encoding="application/x-tex">{\displaystyle H^{2}(X;\mathbb {Z} )}</annotation>

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</math><img src="./b711408c4ef248563ebab8a86c8860d0b18239f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.531ex; height:3.176ex;" alt="{\displaystyle H^{2}(X;\mathbb {Z} )}" loading="lazy">, which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism (thus an isomorphism):

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(L\otimes L')=c_{1}(L)+c_{1}(L');}</annotation>

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the <a href="Tensor_product" title="Tensor product">tensor product</a> of complex line bundles corresponds to the addition in the second cohomology group.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of) <a href="Holomorphic_line_bundle" class="mw-redirect" title="Holomorphic line bundle">holomorphic line bundles</a> by <a href="Linear_equivalence" class="mw-redirect" title="Linear equivalence">linear equivalence</a> classes of <a href="Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisors</a>.

For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.

<div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Via_the_Chern–Weil_theory">Via the Chern–Weil theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Chern%E2%80%93Weil_theory" class="mw-redirect" title="Chern–Weil theory">Chern–Weil theory</a></div>

Given a complex <a href="Hermitian_metric" class="mw-redirect" title="Hermitian metric">hermitian</a> <a href="Vector_bundle" title="Vector bundle">vector bundle</a> V of <a href="Vector_bundle" title="Vector bundle">complex rank</a> n over a <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a> M, representatives of each Chern class (also called a Chern form) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}(V)}">

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</math><img src="./361c9460aef10f8a89b8b2da47fade6e65557c8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.692ex; height:2.843ex;" alt="{\displaystyle c_{k}(V)}" loading="lazy"> of V are given as the coefficients of the <a href="Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of the <a href="Curvature_form" title="Curvature form">curvature form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> of V.

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<annotation encoding="application/x-tex">{\displaystyle \det \left({\frac {it\Omega }{2\pi }}+I\right)=\sum _{k}c_{k}(V)t^{k}}</annotation>

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The determinant is over the ring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

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<annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation>

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</math><img src="./59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy"> matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The <a href="Curvature_form" title="Curvature form">curvature form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \Omega =d\omega +{\frac {1}{2}}[\omega ,\omega ]}</annotation>

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with ω the <a href="Connection_form" title="Connection form">connection form</a> and d the <a href="Exterior_derivative" title="Exterior derivative">exterior derivative</a>, or via the same expression in which ω is a <a href="Gauge_field" class="mw-redirect" title="Gauge field">gauge field</a> for the <a href="Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> of V. The scalar t is used here only as an <a href="Indeterminate_(variable)" class="mw-redirect" title="Indeterminate (variable)">indeterminate</a> to <a href="Generating_function" title="Generating function">generate</a> the sum from the determinant, and I denotes the n × n <a href="Identity_matrix" title="Identity matrix">identity matrix</a>.

To say that the expression given is a representative of the Chern class indicates that 'class' here means <a href="Up_to" title="Up to">up to</a> addition of an <a href="Exact_differential_form" class="mw-redirect" title="Exact differential form">exact differential form</a>. That is, Chern classes are <a href="Cohomology_class" class="mw-redirect" title="Cohomology class">cohomology classes</a> in the sense of <a href="De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a>. It can be shown that the cohomology classes of the Chern forms do not depend on the choice of connection in V.

If follows from the matrix identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {tr} (\ln(X))=\ln(\det(X))}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {tr} (\ln(X))=\ln(\det(X))}</annotation>

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</math><img src="./c4ddbb69490f4ba7c25988fe41fd6eed3d27cbd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.22ex; height:2.843ex;" alt="{\displaystyle \mathrm {tr} (\ln(X))=\ln(\det(X))}" loading="lazy"> that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(X)=\exp(\mathrm {tr} (\ln(X)))}">

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<annotation encoding="application/x-tex">{\displaystyle \det(X)=\exp(\mathrm {tr} (\ln(X)))}</annotation>

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</math><img src="./6449511546179926b1c6c2c257772824dfdd3023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.833ex; height:2.843ex;" alt="{\displaystyle \det(X)=\exp(\mathrm {tr} (\ln(X)))}" loading="lazy">. Now applying the <a href="Taylor_series" title="Taylor series">Maclaurin series</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(X+I)}">

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<annotation encoding="application/x-tex">{\displaystyle \ln(X+I)}</annotation>

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</math><img src="./16b3ca91de31e9c35cde9e4ec520e4d61ebc2e6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.741ex; height:2.843ex;" alt="{\displaystyle \ln(X+I)}" loading="lazy">, we get the following expression for the Chern forms:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k}c_{k}(V)t^{k}=\left[1+i{\frac {\mathrm {tr} (\Omega )}{2\pi }}t+{\frac {\mathrm {tr} (\Omega ^{2})-\mathrm {tr} (\Omega )^{2}}{8\pi ^{2}}}t^{2}+i{\frac {-2\mathrm {tr} (\Omega ^{3})+3\mathrm {tr} (\Omega ^{2})\mathrm {tr} (\Omega )-\mathrm {tr} (\Omega )^{3}}{48\pi ^{3}}}t^{3}+\cdots \right].}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Via_an_Euler_class">Via an Euler class</h3></div>

One can define a Chern class in terms of an Euler class. This is the approach in the book by Milnor and Stasheff, and emphasizes the role of an <a href="Orientation_of_a_vector_bundle" title="Orientation of a vector bundle">orientation of a vector bundle</a>.

The basic observation is that a <a href="Complex_vector_bundle" title="Complex vector bundle">complex vector bundle</a> comes with a canonical orientation, ultimately because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}</annotation>

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</math><img src="./1c81fb83c9a3c1639fe166eaa1c72180c64410b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.983ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}" loading="lazy"> is connected. Hence, one simply defines the top Chern class of the bundle to be its Euler class (the Euler class of the underlying real vector bundle) and handles lower Chern classes in an inductive fashion.

The precise construction is as follows. The idea is to do base change to get a bundle of one-less rank. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon E\to B}">

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon E\to B}</annotation>

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</math><img src="./188864d353771fd8e24d463c5946edd7db679f8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.176ex;" alt="{\displaystyle \pi \colon E\to B}" loading="lazy"> be a complex vector bundle over a <a href="Paracompact_space" title="Paracompact space">paracompact space</a> B. Thinking of B as being embedded in E as the zero section, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'=E\setminus B}">

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<annotation encoding="application/x-tex">{\displaystyle B'=E\setminus B}</annotation>

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</math><img src="./b5ef3a19c7bee2503e92dff3945bf422e9455205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.282ex; height:3.009ex;" alt="{\displaystyle B'=E\setminus B}" loading="lazy"> and define the new vector bundle:

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<annotation encoding="application/x-tex">{\displaystyle E'\to B'}</annotation>

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such that each fiber is the quotient of a fiber F of E by the line spanned by a nonzero vector v in F (a point of B′ is specified by a fiber F of E and a nonzero vector on F.)<a href="#cite_note-3">[3]</a> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E'}">

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<annotation encoding="application/x-tex">{\displaystyle E'}</annotation>

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</math><img src="./a574600572696493d48300245a45b8de0638ce21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle E'}" loading="lazy"> has rank one less than that of E. From the <a href="Gysin_sequence" class="mw-redirect" title="Gysin sequence">Gysin sequence</a> for the fiber bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi |_{B'}\colon B'\to B}">

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<annotation encoding="application/x-tex">{\displaystyle \pi |_{B'}\colon B'\to B}</annotation>

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</math><img src="./5244ab13007ee30718bd8ef1580158c1d9e8912a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.851ex; height:3.009ex;" alt="{\displaystyle \pi |_{B'}\colon B'\to B}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle \cdots \to \operatorname {H} ^{k}(B;\mathbb {Z} ){\overset {\pi |_{B'}^{*}}{\to }}\operatorname {H} ^{k}(B';\mathbb {Z} )\to \cdots ,}</annotation>

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we see that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi |_{B'}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi |_{B'}^{*}}</annotation>

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</math><img src="./c6640432ef0a3b73b5c4ad8950f20e0a18aa6c23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.99ex; height:3.009ex;" alt="{\displaystyle \pi |_{B'}^{*}}" loading="lazy"> is an isomorphism for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<2n-1}">

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<annotation encoding="application/x-tex">{\displaystyle k<2n-1}</annotation>

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</math><img src="./6a03278f5e7648a542c912fc1f7e4b38b2942d13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.87ex; height:2.343ex;" alt="{\displaystyle k<2n-1}" loading="lazy">. Let

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}(E)={\begin{cases}{\pi |_{B'}^{*}}^{-1}c_{k}(E')&k<n\\e(E_{\mathbb {R} })&k=n\\0&k>n\end{cases}}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{k}(E)={\begin{cases}{\pi |_{B'}^{*}}^{-1}c_{k}(E')&k<n\\e(E_{\mathbb {R} })&k=n\\0&k>n\end{cases}}}</annotation>

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It then takes some work to check the axioms of Chern classes are satisfied for this definition.

See also: <a href="Thom_space#The_Thom_isomorphism" title="Thom space">The Thom isomorphism</a>.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="The_complex_tangent_bundle_of_the_Riemann_sphere">The complex tangent bundle of the Riemann sphere</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {CP} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {CP} ^{1}}</annotation>

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</math><img src="./82c98c461bccbd4567716df0a6a2ff97bf9371af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.153ex; height:2.676ex;" alt="{\displaystyle \mathbb {CP} ^{1}}" loading="lazy"> be the <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>: 1-dimensional <a href="Complex_projective_space" title="Complex projective space">complex projective space</a>. Suppose that z is a <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> <a href="Manifold" title="Manifold">local coordinate</a> for the Riemann sphere. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=T\mathbb {CP} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle V=T\mathbb {CP} ^{1}}</annotation>

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</math><img src="./b1852467e7830fdb0c37256dbcc0123a4574e97a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.675ex; height:2.676ex;" alt="{\displaystyle V=T\mathbb {CP} ^{1}}" loading="lazy"> be the bundle of complex tangent vectors having the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\partial /\partial z}">

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<annotation encoding="application/x-tex">{\displaystyle a\partial /\partial z}</annotation>

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</math><img src="./2db4ebcc3fa920e0aa8f44df5290aedd515980f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.117ex; height:2.843ex;" alt="{\displaystyle a\partial /\partial z}" loading="lazy"> at each point, where a is a <a href="Complex_number" title="Complex number">complex number</a>. We prove the complex version of the <a href="Hairy_ball_theorem" title="Hairy ball theorem">hairy ball theorem</a>: V has no section which is everywhere nonzero.

For this, we need the following fact: the first Chern class of a trivial bundle is zero, i.e.,

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(\mathbb {CP} ^{1}\times \mathbb {C} )=0.}</annotation>

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This is evinced by the fact that a trivial bundle always admits a flat connection. So, we shall show that

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(V)\not =0.}</annotation>

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Consider the <a href="K%C3%A4hler_metric" class="mw-redirect" title="Kähler metric">Kähler metric</a>

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<annotation encoding="application/x-tex">{\displaystyle h={\frac {dzd{\bar {z}}}{(1+|z|^{2})^{2}}}.}</annotation>

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One readily shows that the curvature 2-form is given by

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<annotation encoding="application/x-tex">{\displaystyle \Omega ={\frac {2dz\wedge d{\bar {z}}}{(1+|z|^{2})^{2}}}.}</annotation>

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Furthermore, by the definition of the first Chern class

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<annotation encoding="application/x-tex">{\displaystyle c_{1}=\left[{\frac {i}{2\pi }}\operatorname {tr} \Omega \right].}</annotation>

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We must show that this cohomology class is non-zero. It suffices to compute its integral over the Riemann sphere:

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<annotation encoding="application/x-tex">{\displaystyle \int c_{1}={\frac {i}{\pi }}\int {\frac {dz\wedge d{\bar {z}}}{(1+|z|^{2})^{2}}}=2}</annotation>

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after switching to <a href="Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</a>. By <a href="Stokes'_theorem" title="Stokes' theorem">Stokes' theorem</a>, an <a href="Exact_form" class="mw-redirect" title="Exact form">exact form</a> would integrate to 0, so the cohomology class is nonzero.

This proves that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\mathbb {CP} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle T\mathbb {CP} ^{1}}</annotation>

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</math><img src="./e3ea251f381775dec1afad722d3b413491fe9698.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.789ex; height:2.676ex;" alt="{\displaystyle T\mathbb {CP} ^{1}}" loading="lazy"> is not a trivial vector bundle.

<div class="mw-heading mw-heading3"><h3 id="Complex_projective_space">Complex projective space</h3></div>

There is an exact sequence of sheaves/bundles:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle 0\to {\mathcal {O}}_{\mathbb {CP} ^{n}}\to {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)^{\oplus (n+1)}\to T\mathbb {CP} ^{n}\to 0}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}}</annotation>

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</math><img src="./0b5dc73c3707f1b9d5d3aaf56ae1b9ae54faf2ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.238ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}}" loading="lazy"> is the structure sheaf (i.e., the trivial line bundle), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)}</annotation>

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</math><img src="./caa29337d8832614d8fc103c4e13e0fb7ea8beac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.21ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)}" loading="lazy"> is <a href="Serre's_twisting_sheaf" class="mw-redirect" title="Serre's twisting sheaf">Serre's twisting sheaf</a> (i.e., the <a href="Hyperplane_bundle" class="mw-redirect" title="Hyperplane bundle">hyperplane bundle</a>) and the last nonzero term is the <a href="Tangent_sheaf" class="mw-redirect" title="Tangent sheaf">tangent sheaf</a>/bundle.

There are two ways to get the above sequence:

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<annotation encoding="application/x-tex">{\displaystyle z_{0},\ldots ,z_{n}}</annotation>

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</math><img src="./21ec9ae61dc75585e0eae0a51ba216ba7a4880f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.613ex; height:2.009ex;" alt="{\displaystyle z_{0},\ldots ,z_{n}}" loading="lazy"> be the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n+1},}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n+1},}</annotation>

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</math><img src="./7b5f8b5e894e9665936f67f6a7e2cdf8233082c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.644ex; height:3.009ex;" alt="{\displaystyle \mathbb {C} ^{n+1},}" loading="lazy"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon \mathbb {C} ^{n+1}\setminus \{0\}\to \mathbb {C} \mathbb {P} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon \mathbb {C} ^{n+1}\setminus \{0\}\to \mathbb {C} \mathbb {P} ^{n}}</annotation>

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</math><img src="./9e4ec46a25a3a2263681b829b660f5ff3b265c29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.976ex; height:3.176ex;" alt="{\displaystyle \pi \colon \mathbb {C} ^{n+1}\setminus \{0\}\to \mathbb {C} \mathbb {P} ^{n}}" loading="lazy"> be the canonical projection, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\mathbb {CP} ^{n}\setminus \{z_{0}=0\}}">

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<annotation encoding="application/x-tex">{\displaystyle U=\mathbb {CP} ^{n}\setminus \{z_{0}=0\}}</annotation>

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</math><img src="./7f7760a26834f333f1e3a1177af834ff2101b701.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.114ex; height:2.843ex;" alt="{\displaystyle U=\mathbb {CP} ^{n}\setminus \{z_{0}=0\}}" loading="lazy">. Then we have:

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<annotation encoding="application/x-tex">{\displaystyle \pi ^{*}d(z_{i}/z_{0})={z_{0}dz_{i}-z_{i}dz_{0} \over z_{0}^{2}},\,i\geq 1.}</annotation>

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In other words, the <a href="Cotangent_sheaf" title="Cotangent sheaf">cotangent sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{\mathbb {C} \mathbb {P} ^{n}}|_{U}}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega _{\mathbb {C} \mathbb {P} ^{n}}|_{U}}</annotation>

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</math><img src="./11571b96259e243550a5dda65848fdada1a05d11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.206ex; height:3.009ex;" alt="{\displaystyle \Omega _{\mathbb {C} \mathbb {P} ^{n}}|_{U}}" loading="lazy">, which is a free <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{U}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{U}}</annotation>

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</math><img src="./c9a9cef2217740befda1b31bb9a65ec7cb8a1d8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{U}}" loading="lazy">-module with basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(z_{i}/z_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle d(z_{i}/z_{0})}</annotation>

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</math><img src="./99c8b23eda348dca00fd932c2d4525910d04c4c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.204ex; height:2.843ex;" alt="{\displaystyle d(z_{i}/z_{0})}" loading="lazy">, fits into the exact sequence

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<annotation encoding="application/x-tex">{\displaystyle 0\to \Omega _{\mathbb {C} \mathbb {P} ^{n}}|_{U}{\overset {dz_{i}\mapsto e_{i}}{\to }}\oplus _{1}^{n+1}{\mathcal {O}}(-1)|_{U}{\overset {e_{i}\mapsto z_{i}}{\to }}{\mathcal {O}}_{U}\to 0,\,i\geq 0,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>

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</math><img src="./ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"> are the basis of the middle term. The same sequence is clearly then exact on the whole projective space and the dual of it is the aforementioned sequence.</li><li>Let L be a line in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n+1}}</annotation>

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</math><img src="./a6690b0bbf64200f073d88bdbc1bb4bc01de9b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.997ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{n+1}}" loading="lazy"> that passes through the origin. It is an exercise in <a href="Elementary_geometry" class="mw-redirect" title="Elementary geometry">elementary geometry</a> to see that the complex tangent space to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \mathbb {P} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \mathbb {P} ^{n}}</annotation>

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</math><img src="./5ce44e4dcf84518b8984ab1dc80ad6b9296ffc71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.317ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} \mathbb {P} ^{n}}" loading="lazy"> at the point L is naturally the set of linear maps from L to its complement. Thus, the tangent bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\mathbb {C} \mathbb {P} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle T\mathbb {C} \mathbb {P} ^{n}}</annotation>

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</math><img src="./aebed8808cc60cca78bbdfa0948dcb6bd9435aac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.343ex;" alt="{\displaystyle T\mathbb {C} \mathbb {P} ^{n}}" loading="lazy"> can be identified with the <a href="Hom_bundle" class="mw-redirect" title="Hom bundle">hom bundle</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} ({\mathcal {O}}(-1),\eta )}</annotation>

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where η is the vector bundle such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(-1)\oplus \eta ={\mathcal {O}}^{\oplus (n+1)}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(-1)\oplus \eta ={\mathcal {O}}^{\oplus (n+1)}}</annotation>

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</math><img src="./dd199db3441a5015ed97a51e8a3228a4d8d875ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.465ex; height:3.343ex;" alt="{\displaystyle {\mathcal {O}}(-1)\oplus \eta ={\mathcal {O}}^{\oplus (n+1)}}" loading="lazy">. It follows:

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<annotation encoding="application/x-tex">{\displaystyle T\mathbb {C} \mathbb {P} ^{n}\oplus {\mathcal {O}}=\operatorname {Hom} ({\mathcal {O}}(-1),\eta )\oplus \operatorname {Hom} ({\mathcal {O}}(-1),{\mathcal {O}}(-1))={\mathcal {O}}(1)^{\oplus (n+1)}.}</annotation>

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By the additivity of total Chern class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=1+c_{1}+c_{2}+\cdots }">

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<annotation encoding="application/x-tex">{\displaystyle c=1+c_{1}+c_{2}+\cdots }</annotation>

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</math><img src="./05a020c82e03678b68fe658daa278a4d967e9a66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.634ex; height:2.509ex;" alt="{\displaystyle c=1+c_{1}+c_{2}+\cdots }" loading="lazy"> (i.e., the Whitney sum formula),

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<annotation encoding="application/x-tex">{\displaystyle c(\mathbb {C} \mathbb {P} ^{n}){\overset {\mathrm {def} }{=}}c(T\mathbb {CP} ^{n})=c({\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(1))^{n+1}=(1+a)^{n+1},}</annotation>

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where a is the canonical generator of the cohomology group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2}(\mathbb {C} \mathbb {P} ^{n},\mathbb {Z} )}">

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<annotation encoding="application/x-tex">{\displaystyle H^{2}(\mathbb {C} \mathbb {P} ^{n},\mathbb {Z} )}</annotation>

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</math><img src="./54e4f770e9150129554dac4d2cc113f931cbc154.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.868ex; height:3.176ex;" alt="{\displaystyle H^{2}(\mathbb {C} \mathbb {P} ^{n},\mathbb {Z} )}" loading="lazy">; i.e., the negative of the first Chern class of the tautological line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(-1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(-1)}</annotation>

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</math><img src="./089df702289698274e002f0edaa516382e76e6f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.018ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(-1)}" loading="lazy"> (note: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(E^{*})=-c_{1}(E)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(E^{*})=-c_{1}(E)}</annotation>

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In particular, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle k\geq 0}</annotation>

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</math><img src="./79214ef55efadfb1d9c9b02252eb8a71cf6f8b6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.472ex; height:2.343ex;" alt="{\displaystyle k\geq 0}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle c_{k}(\mathbb {C} \mathbb {P} ^{n})={\binom {n+1}{k}}a^{k}.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Chern_polynomial">Chern polynomial</h2></div>

A Chern polynomial is a convenient way to handle Chern classes and related notions systematically. By definition, for a complex vector bundle E, the Chern polynomial ct of E is given by:

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E)=1+c_{1}(E)t+\cdots +c_{n}(E)t^{n}.}</annotation>

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This is not a new invariant: the formal variable t simply keeps track of the degree of ck(E).<a href="#cite_note-6">[6]</a> In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{t}(E)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E)}</annotation>

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</math><img src="./e12ccad5a78e754c041e08e36dddb4742cd809df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.418ex; height:2.843ex;" alt="{\displaystyle c_{t}(E)}" loading="lazy"> is completely determined by the total Chern class of E: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(E)=1+c_{1}(E)+\cdots +c_{n}(E)}">

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<annotation encoding="application/x-tex">{\displaystyle c(E)=1+c_{1}(E)+\cdots +c_{n}(E)}</annotation>

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</math><img src="./f66181c2ea77c357be80dc219792689f9e0246dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.553ex; height:2.843ex;" alt="{\displaystyle c(E)=1+c_{1}(E)+\cdots +c_{n}(E)}" loading="lazy"> and conversely.

The Whitney sum formula, one of the axioms of Chern classes (see below), says that ct is additive in the sense:

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E\oplus E')=c_{t}(E)c_{t}(E').}</annotation>

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Now, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=L_{1}\oplus \cdots \oplus L_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle E=L_{1}\oplus \cdots \oplus L_{n}}</annotation>

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</math><img src="./ff1ecea91db7c727e5443d412f18718af9673fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.717ex; height:2.509ex;" alt="{\displaystyle E=L_{1}\oplus \cdots \oplus L_{n}}" loading="lazy"> is a direct sum of (complex) line bundles, then it follows from the sum formula that:

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E)=(1+a_{1}(E)t)\cdots (1+a_{n}(E)t)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle a_{i}(E)=c_{1}(L_{i})}</annotation>

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</math><img src="./924a01931b88631ecd346a1bc32182676d58ad65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.966ex; height:2.843ex;" alt="{\displaystyle a_{i}(E)=c_{1}(L_{i})}" loading="lazy"> are the first Chern classes. The roots <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}(E)}">

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<annotation encoding="application/x-tex">{\displaystyle a_{i}(E)}</annotation>

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</math><img src="./94bd2bc6384606b26046d9e9fa5a3bc1724541ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.614ex; height:2.843ex;" alt="{\displaystyle a_{i}(E)}" loading="lazy">, called the Chern roots of E, determine the coefficients of the polynomial: i.e.,

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<annotation encoding="application/x-tex">{\displaystyle c_{k}(E)=\sigma _{k}(a_{1}(E),\ldots ,a_{n}(E))}</annotation>

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where σk are <a href="Elementary_symmetric_polynomials" class="mw-redirect" title="Elementary symmetric polynomials">elementary symmetric polynomials</a>. In other words, thinking of ai as formal variables, ck "are" σk. A basic fact on <a href="Symmetric_polynomial" title="Symmetric polynomial">symmetric polynomials</a> is that any symmetric polynomial in, say, ti's is a polynomial in elementary symmetric polynomials in ti's. Either by <a href="Splitting_principle" title="Splitting principle">splitting principle</a> or by ring theory, any Chern polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{t}(E)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E)}</annotation>

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</math><img src="./e12ccad5a78e754c041e08e36dddb4742cd809df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.418ex; height:2.843ex;" alt="{\displaystyle c_{t}(E)}" loading="lazy"> factorizes into linear factors after enlarging the cohomology ring; E need not be a direct sum of line bundles in the preceding discussion. The conclusion is

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</style><div class="block-indent" style="padding-left: 1.5em;">"One can evaluate any symmetric polynomial f at a complex vector bundle E by writing f as a polynomial in σk and then replacing σk by ck(E)."</div>

Example: We have polynomials sk

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<annotation encoding="application/x-tex">{\displaystyle t_{1}^{k}+\cdots +t_{n}^{k}=s_{k}(\sigma _{1}(t_{1},\ldots ,t_{n}),\ldots ,\sigma _{k}(t_{1},\ldots ,t_{n}))}</annotation>

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with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}=\sigma _{1},s_{2}=\sigma _{1}^{2}-2\sigma _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle s_{1}=\sigma _{1},s_{2}=\sigma _{1}^{2}-2\sigma _{2}}</annotation>

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</math><img src="./1127820190765bf4f9fb2618ecfd28453f8ffb9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.671ex; height:3.176ex;" alt="{\displaystyle s_{1}=\sigma _{1},s_{2}=\sigma _{1}^{2}-2\sigma _{2}}" loading="lazy"> and so on (cf. <a href="Newton's_identities#Expressing_power_sums_in_terms_of_elementary_symmetric_polynomials" title="Newton's identities">Newton's identities</a>). The sum

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ch} (E)=e^{a_{1}(E)}+\cdots +e^{a_{n}(E)}=\sum s_{k}(c_{1}(E),\ldots ,c_{n}(E))/k!}</annotation>

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is called the Chern character of E, whose first few terms are: (we drop E from writing.)

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ch} (E)=\operatorname {rk} +c_{1}+{\frac {1}{2}}(c_{1}^{2}-2c_{2})+{\frac {1}{6}}(c_{1}^{3}-3c_{1}c_{2}+3c_{3})+\cdots .}</annotation>

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Example: The <a href="Todd_class" title="Todd class">Todd class</a> of E is given by:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {td} (E)=\prod _{1}^{n}{a_{i} \over 1-e^{-a_{i}}}=1+{1 \over 2}c_{1}+{1 \over 12}(c_{1}^{2}+c_{2})+\cdots .}</annotation>

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Remark: The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let Gn be the <a href="Infinite_Grassmannian" class="mw-redirect" title="Infinite Grassmannian">infinite Grassmannian</a> of n-dimensional complex vector spaces. This space is equipped with a tautologous vector bundle of rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle E_{n}\to G_{n}}</annotation>

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</math><img src="./77ba7885aa8208773e07718182ade25aec9d97d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.593ex; height:2.509ex;" alt="{\displaystyle E_{n}\to G_{n}}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n}}">

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</math><img src="./f74402fbd65c1683d50670c7ecddc180b66fec1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.045ex; height:2.509ex;" alt="{\displaystyle G_{n}}" loading="lazy"> is called the <a href="Classifying_space" title="Classifying space">classifying space</a> for rank-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle f_{E}:X\to G_{n}}</annotation>

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such that the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}">

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</math><img src="./ad6b82f2a00af6c9efd4c16d4e99329605645c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> along <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{E}}">

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</math><img src="./3d89c6f721934c4eb29d905b8a11abe79a47f9aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle f_{E}}" loading="lazy"> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle f_{E}}</annotation>

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</math><img src="./3d89c6f721934c4eb29d905b8a11abe79a47f9aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle f_{E}}" loading="lazy"> is unique up to homotopy. <a href="Borel's_theorem" title="Borel's theorem">Borel's theorem</a> says the cohomology ring of Gn is exactly the ring of symmetric polynomials, which are polynomials in elementary symmetric polynomials σk; so, the pullback of fE reads:

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<annotation encoding="application/x-tex">{\displaystyle f_{E}^{*}:\mathbb {Z} [\sigma _{1},\ldots ,\sigma _{n}]\to H^{*}(X,\mathbb {Z} ).}</annotation>

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One then puts:

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<annotation encoding="application/x-tex">{\displaystyle c_{k}(E)=f_{E}^{*}(\sigma _{k}).}</annotation>

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Remark: Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }}</annotation>

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</math><img src="./ea2a2baceea3d6e1374781a8659784b625469bd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.131ex; height:3.176ex;" alt="{\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }}" loading="lazy"> be the contravariant functor that, to a CW complex X, assigns the set of isomorphism classes of complex vector bundles of rank n over X and, to a map, its pullback. By definition, a <a href="Characteristic_class" title="Characteristic class">characteristic class</a> is a natural transformation from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }=[-,G_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }=[-,G_{n}]}</annotation>

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</math><img src="./a335e563799383694e3073de608043b248286367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.411ex; height:3.176ex;" alt="{\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }=[-,G_{n}]}" loading="lazy"> to the cohomology functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{*}(-,\mathbb {Z} ).}">

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<annotation encoding="application/x-tex">{\displaystyle H^{*}(-,\mathbb {Z} ).}</annotation>

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</math><img src="./5016f08eabfaa682baa7f73204d6b6dfc5cc2b01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.006ex; height:2.843ex;" alt="{\displaystyle H^{*}(-,\mathbb {Z} ).}" loading="lazy"> Characteristic classes form a ring because of the ring structure of cohomology ring. <a href="Yoneda's_lemma" class="mw-redirect" title="Yoneda's lemma">Yoneda's lemma</a> says this ring of characteristic classes is exactly the cohomology ring of Gn:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Nat} ([-,G_{n}],H^{*}(-,\mathbb {Z} ))=H^{*}(G_{n},\mathbb {Z} )=\mathbb {Z} [\sigma _{1},\ldots ,\sigma _{n}].}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Computation_formulae">Computation formulae</h2></div>

Let E be a vector bundle of rank r and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{t}(E)=\sum _{i=0}^{r}c_{i}(E)t^{i}}">

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</math><img src="./553f891a242752bd16d73e17c7156d26a3df32c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.289ex; height:6.843ex;" alt="{\displaystyle c_{t}(E)=\sum _{i=0}^{r}c_{i}(E)t^{i}}" loading="lazy"> the <a href="#Chern_polynomial">Chern polynomial</a> of it.

<ul><li>For the <a href="Dual_bundle" title="Dual bundle">dual bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle E^{*}}</annotation>

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</math><img src="./a91d2119445b3b65384ba491c4b95f1557571ecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.848ex; height:2.343ex;" alt="{\displaystyle E^{*}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle c_{i}(E^{*})=(-1)^{i}c_{i}(E)}</annotation>

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</math><img src="./b68edb676e554cc0901f039f159290a90c417b54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.533ex; height:3.176ex;" alt="{\displaystyle c_{i}(E^{*})=(-1)^{i}c_{i}(E)}" loading="lazy">.<a href="#cite_note-7">[7]</a></li>

<li>If L is a line bundle, then<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{t}(E\otimes L)=\sum _{i=0}^{r}c_{i}(E)c_{t}(L)^{r-i}t^{i}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E\otimes L)=\sum _{i=0}^{r}c_{i}(E)c_{t}(L)^{r-i}t^{i}}</annotation>

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</math> and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}(E\otimes L),i=1,2,\dots ,r}">

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<annotation encoding="application/x-tex">{\displaystyle c_{i}(E\otimes L),i=1,2,\dots ,r}</annotation>

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</math><img src="./40ce31cf71039d0db354268a52173345b160af80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.335ex; height:2.843ex;" alt="{\displaystyle c_{i}(E\otimes L),i=1,2,\dots ,r}" loading="lazy"> are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(E)+rc_{1}(L),\dots ,\sum _{j=0}^{i}{\binom {r-i+j}{j}}c_{i-j}(E)c_{1}(L)^{j},\dots ,\sum _{j=0}^{r}c_{r-j}(E)c_{1}(L)^{j}.}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(E)+rc_{1}(L),\dots ,\sum _{j=0}^{i}{\binom {r-i+j}{j}}c_{i-j}(E)c_{1}(L)^{j},\dots ,\sum _{j=0}^{r}c_{r-j}(E)c_{1}(L)^{j}.}</annotation>

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</math></li>

<li>For the Chern roots <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1},\dots ,\alpha _{r}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{1},\dots ,\alpha _{r}}</annotation>

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</math><img src="./9567ba60d2f51b93da450746ffc3356ef223bc3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.181ex; height:2.009ex;" alt="{\displaystyle \alpha _{1},\dots ,\alpha _{r}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">,<a href="#cite_note-10">[10]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}c_{t}(\operatorname {Sym} ^{p}E)&=\prod _{i_{1}\leq \cdots \leq i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t),\\c_{t}(\wedge ^{p}E)&=\prod _{i_{1}<\cdots <i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t).\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c_{t}(\operatorname {Sym} ^{p}E)&=\prod _{i_{1}\leq \cdots \leq i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t),\\c_{t}(\wedge ^{p}E)&=\prod _{i_{1}<\cdots <i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t).\end{aligned}}}</annotation>

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</math> In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(\wedge ^{r}E)=c_{1}(E).}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(\wedge ^{r}E)=c_{1}(E).}</annotation>

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</math><img src="./82be79197760832393bb491d7641fd56e2968ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.561ex; height:2.843ex;" alt="{\displaystyle c_{1}(\wedge ^{r}E)=c_{1}(E).}" loading="lazy"></li>

<li>For example,<a href="#cite_note-11">[11]</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}=c_{i}(E)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{i}=c_{i}(E)}</annotation>

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</math><img src="./dfd562e839962cbc1a9082b7ea72373aceb91b7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.296ex; height:2.843ex;" alt="{\displaystyle c_{i}=c_{i}(E)}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle r=2}</annotation>

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</math><img src="./c19b110d7eb52a69381b88554e63c8a2aef376c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=2}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\operatorname {Sym} ^{2}E)=1+3c_{1}+2c_{1}^{2}+4c_{2}+4c_{1}c_{2},}">

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<annotation encoding="application/x-tex">{\displaystyle c(\operatorname {Sym} ^{2}E)=1+3c_{1}+2c_{1}^{2}+4c_{2}+4c_{1}c_{2},}</annotation>

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</math><img src="./b2a92fe7738fdafa6efcbc65c626b7fa65ab9bb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.714ex; height:3.343ex;" alt="{\displaystyle c(\operatorname {Sym} ^{2}E)=1+3c_{1}+2c_{1}^{2}+4c_{2}+4c_{1}c_{2},}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle r=3}</annotation>

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</math><img src="./2452d01ccc3bc930e3e2bff9fb18d2a425272ec9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=3}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\operatorname {Sym} ^{2}E)=1+4c_{1}+5c_{1}^{2}+5c_{2}+2c_{1}^{3}+11c_{1}c_{2}+7c_{3}.}">

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<annotation encoding="application/x-tex">{\displaystyle c(\operatorname {Sym} ^{2}E)=1+4c_{1}+5c_{1}^{2}+5c_{2}+2c_{1}^{3}+11c_{1}c_{2}+7c_{3}.}</annotation>

</semantics>

</math><img src="./e16ae4e5cdcafbc09d7b48b448f9e47c090225de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.004ex; height:3.343ex;" alt="{\displaystyle c(\operatorname {Sym} ^{2}E)=1+4c_{1}+5c_{1}^{2}+5c_{2}+2c_{1}^{3}+11c_{1}c_{2}+7c_{3}.}" loading="lazy"></dd></dl></li></ul>

<dl><dd>(cf. <a href="Segre_class#Example_2" title="Segre class">Segre class#Example 2</a>.)</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Applications_of_formulae">Applications of formulae</h3></div>

We can use these abstract properties to compute the rest of the chern classes of line bundles on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {CP} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {CP} ^{1}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(-1)^{*}\cong {\mathcal {O}}(1)}</annotation>

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</math><img src="./06c86c750d7c1744739109430947d1963f4ad87b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.604ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(-1)^{*}\cong {\mathcal {O}}(1)}" loading="lazy"> showing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}({\mathcal {O}}(1))=1\in H^{2}(\mathbb {CP} ^{1};\mathbb {Z} )}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}({\mathcal {O}}(1))=1\in H^{2}(\mathbb {CP} ^{1};\mathbb {Z} )}</annotation>

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</math><img src="./f0ce203e68e2429ff26eba1c5af4a9f4f9e28d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.497ex; height:3.176ex;" alt="{\displaystyle c_{1}({\mathcal {O}}(1))=1\in H^{2}(\mathbb {CP} ^{1};\mathbb {Z} )}" loading="lazy">. Then using tensor powers, we can relate them to the chern classes of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}({\mathcal {O}}(n))=n}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{1}({\mathcal {O}}(n))=n}</annotation>

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</math><img src="./99187af0993fab4dc06257ef8e1e06276843081e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.418ex; height:2.843ex;" alt="{\displaystyle c_{1}({\mathcal {O}}(n))=n}" loading="lazy"> for any integer.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Given a <a href="Complex_vector_bundle" title="Complex vector bundle">complex vector bundle</a> E over a <a href="Topological_space" title="Topological space">topological space</a> X, the Chern classes of E are a sequence of elements of the <a href="Cohomology" title="Cohomology">cohomology</a> of X. The k-th Chern class of E, which is usually denoted ck(E), is an element of

<msup>

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<annotation encoding="application/x-tex">{\displaystyle H^{2k}(X;\mathbb {Z} ),}</annotation>

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</math>

the cohomology of X with <a href="Integer" title="Integer">integer</a> coefficients. One can also define the total Chern class

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<annotation encoding="application/x-tex">{\displaystyle c(E)=c_{0}(E)+c_{1}(E)+c_{2}(E)+\cdots .}</annotation>

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</math>

Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example.

<div class="mw-heading mw-heading3"><h3 id="Classical_axiomatic_definition">Classical axiomatic definition</h3></div>

The Chern classes satisfy the following four axioms:

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{0}(E)=1}</annotation>

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</math><img src="./c9e488c17aa2454b75bf20ceb18bcf9ad0a4f32b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.907ex; height:2.843ex;" alt="{\displaystyle c_{0}(E)=1}" loading="lazy"> for all E.</li>

<li>Naturality: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:Y\to X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f:Y\to X}</annotation>

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</math><img src="./654e8fe3d0dc6757c546d482a59b9c5220eaff7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:Y\to X}" loading="lazy"> is <a href="Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> and f*E is the <a href="Pullback_bundle" title="Pullback bundle">vector bundle pullback</a> of E, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}(f^{*}E)=f^{*}c_{k}(E)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{k}(f^{*}E)=f^{*}c_{k}(E)}</annotation>

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</math><img src="./56a8cf5711b3a8f78f2d3eaf0aeb736c0791ce3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.209ex; height:2.843ex;" alt="{\displaystyle c_{k}(f^{*}E)=f^{*}c_{k}(E)}" loading="lazy">.</li>

<li><a href="Hassler_Whitney" title="Hassler Whitney">Whitney</a> sum formula: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\to X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle F\to X}</annotation>

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</math><img src="./ea4cab1d3733e066f66be3c686c66eff597b7a8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.335ex; height:2.176ex;" alt="{\displaystyle F\to X}" loading="lazy"> is another complex vector bundle, then the Chern classes of the <a href="Direct_sum_of_vector_bundles" class="mw-redirect" title="Direct sum of vector bundles">direct sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oplus F}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle E\oplus F}</annotation>

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</math><img src="./4548df9d548d438b02c47efd6eabca20aa46aabf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.357ex; height:2.343ex;" alt="{\displaystyle E\oplus F}" loading="lazy"> are given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(E\oplus F)=c(E)\smile c(F);}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle c(E\oplus F)=c(E)\smile c(F);}</annotation>

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</math> that is, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}(E\oplus F)=\sum _{i=0}^{k}c_{i}(E)\smile c_{k-i}(F).}">

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<annotation encoding="application/x-tex">{\displaystyle c_{k}(E\oplus F)=\sum _{i=0}^{k}c_{i}(E)\smile c_{k-i}(F).}</annotation>

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</math></li>

<li>Normalization: The total Chern class of the <a href="Tautological_line_bundle" class="mw-redirect" title="Tautological line bundle">tautological line bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {CP} ^{k}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {CP} ^{k}}</annotation>

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</math><img src="./6755b322b39820388afaf5ecce735f5bc13a6845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.187ex; height:2.676ex;" alt="{\displaystyle \mathbb {CP} ^{k}}" loading="lazy"> is 1−H, where H is <a href="Poincar%C3%A9_duality" title="Poincaré duality">Poincaré dual</a> to the <a href="Hyperplane" title="Hyperplane">hyperplane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {CP} ^{k-1}\subseteq \mathbb {CP} ^{k}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {CP} ^{k-1}\subseteq \mathbb {CP} ^{k}}</annotation>

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</math><img src="./0bd7ba04e26eedf94ff6b66c687a558080d8fa6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.573ex; height:2.843ex;" alt="{\displaystyle \mathbb {CP} ^{k-1}\subseteq \mathbb {CP} ^{k}}" loading="lazy">.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Grothendieck_axiomatic_approach">Grothendieck axiomatic approach</h3></div>

Alternatively, <a href="Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> (<a href="#CITEREFGrothendieck1958">1958</a>) replaced these with a slightly smaller set of axioms:

<ul><li>Naturality: (Same as above)</li>

<li>Additivity: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to E'\to E\to E''\to 0}">

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle 0\to E'\to E\to E''\to 0}</annotation>

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</math><img src="./0f74c322bd7af75969b647b935b05b305dbde850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.966ex; height:2.509ex;" alt="{\displaystyle 0\to E'\to E\to E''\to 0}" loading="lazy"> is an <a href="Exact_sequence" title="Exact sequence">exact sequence</a> of vector bundles, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(E)=c(E')\smile c(E'')}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle c(E)=c(E')\smile c(E'')}</annotation>

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</math><img src="./bd738789b0b8d3e5975d27f57dd8a0666c43c565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.346ex; height:3.009ex;" alt="{\displaystyle c(E)=c(E')\smile c(E'')}" loading="lazy">.</li>

<li>Normalization: If E is a <a href="Line_bundle" title="Line bundle">line bundle</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(E)=1+e(E_{\mathbb {R} })}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c(E)=1+e(E_{\mathbb {R} })}</annotation>

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</math><img src="./92f080aad36ab6beda42bb79cda2fc9e82e4a1c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.72ex; height:2.843ex;" alt="{\displaystyle c(E)=1+e(E_{\mathbb {R} })}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(E_{\mathbb {R} })}">

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<annotation encoding="application/x-tex">{\displaystyle e(E_{\mathbb {R} })}</annotation>

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</math><img src="./4a764e4c1c6b634fa165bdd0e7f48b370371880d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.027ex; height:2.843ex;" alt="{\displaystyle e(E_{\mathbb {R} })}" loading="lazy"> is the <a href="Euler_class" title="Euler class">Euler class</a> of the underlying real vector bundle.</li></ul>

He shows using the <a href="Leray%E2%80%93Hirsch_theorem" title="Leray–Hirsch theorem">Leray–Hirsch theorem</a> that the total Chern class of an arbitrary finite rank complex vector bundle can be defined in terms of the first Chern class of a tautologically-defined line bundle.

Namely, introducing the projectivization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (E)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (E)}</annotation>

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</math><img src="./2cbec9c733c19049794c8e97dbc6a456b63ae466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (E)}" loading="lazy"> of the rank n complex vector bundle E → B as the fiber bundle on B whose fiber at any point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in B}">

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<annotation encoding="application/x-tex">{\displaystyle b\in B}</annotation>

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</math><img src="./61dbfba9ff608c8700a30596649d98dcc6147d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle b\in B}" loading="lazy"> is the projective space of the fiber Eb. The total space of this bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (E)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (E)}</annotation>

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</math><img src="./2cbec9c733c19049794c8e97dbc6a456b63ae466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (E)}" loading="lazy"> is equipped with its tautological complex line bundle, that we denote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">

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<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>

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</math><img src="./38a7dcde9730ef0853809fefc18d88771f95206c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" loading="lazy">, and the first Chern class

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(\tau )=:-a}</annotation>

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restricts on each fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (E_{b})}">

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (E_{b})}</annotation>

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</math><img src="./0b2e690f0de4d3d509ed7f0f4fbc5ff3e063e3b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.882ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (E_{b})}" loading="lazy"> to minus the (Poincaré-dual) class of the hyperplane, that spans the cohomology of the fiber, in view of the cohomology of <a href="Complex_projective_space" title="Complex projective space">complex projective spaces</a>.

The classes

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<annotation encoding="application/x-tex">{\displaystyle 1,a,a^{2},\ldots ,a^{n-1}\in H^{*}(\mathbb {P} (E))}</annotation>

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therefore form a family of ambient cohomology classes restricting to a basis of the cohomology of the fiber. The <a href="Leray%E2%80%93Hirsch_theorem" title="Leray–Hirsch theorem">Leray–Hirsch theorem</a> then states that any class in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{*}(\mathbb {P} (E))}">

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<annotation encoding="application/x-tex">{\displaystyle H^{*}(\mathbb {P} (E))}</annotation>

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</math><img src="./9485ef5716376ddcf91ea08719d7c4bb4e86832f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.972ex; height:2.843ex;" alt="{\displaystyle H^{*}(\mathbb {P} (E))}" loading="lazy"> can be written uniquely as a linear combination of the 1, a, a2, ..., an−1 with classes on the base as coefficients.

In particular, one may define the Chern classes of E in the sense of Grothendieck, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(E),\ldots c_{n}(E)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(E),\ldots c_{n}(E)}</annotation>

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</math><img src="./0119c7287f5485e3681cdc8ea44e890cdca0c3aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.601ex; height:2.843ex;" alt="{\displaystyle c_{1}(E),\ldots c_{n}(E)}" loading="lazy"> by expanding this way the class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle -a^{n}}</annotation>

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</math><img src="./328c5b6638bb4e0898e6d4bec43cd0e2455c3ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.256ex; height:2.509ex;" alt="{\displaystyle -a^{n}}" loading="lazy">, with the relation:

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<annotation encoding="application/x-tex">{\displaystyle -a^{n}=c_{1}(E)\cdot a^{n-1}+\cdots +c_{n-1}(E)\cdot a+c_{n}(E).}</annotation>

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</math>

One then may check that this alternative definition coincides with whatever other definition one may favor, or use the previous axiomatic characterization.

<div class="mw-heading mw-heading3"><h3 id="The_top_Chern_class">The top Chern class</h3></div>

In fact, these properties uniquely characterize the Chern classes. They imply, among other things:

<ul><li>If n is the complex rank of V, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}(V)=0}">

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<annotation encoding="application/x-tex">{\displaystyle c_{k}(V)=0}</annotation>

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</math><img src="./2cc8b5c520d3717cc1957f805e692fe1e7a2e458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.953ex; height:2.843ex;" alt="{\displaystyle c_{k}(V)=0}" loading="lazy"> for all k > n. Thus the total Chern class terminates.</li>

<li>The top Chern class of V (meaning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}(V)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{n}(V)}</annotation>

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</math><img src="./057d4dcd31c3f357c16d7dbb09d5cb97bdd81e34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.822ex; height:2.843ex;" alt="{\displaystyle c_{n}(V)}" loading="lazy">, where n is the rank of V) is always equal to the <a href="Euler_class" title="Euler class">Euler class</a> of the underlying real vector bundle.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="In_algebraic_geometry">In algebraic geometry</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Axiomatic_description">Axiomatic description</h3></div>

There is another construction of Chern classes which take values in the algebrogeometric analogue of the cohomology ring, the <a href="Chow_ring" class="mw-redirect" title="Chow ring">Chow ring</a>.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a nonsingular quasi-projective variety of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle E\to X}</annotation>

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</math><img src="./c6e385963fc9a729e22866cd3c05d5002c361bbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.37ex; height:2.176ex;" alt="{\displaystyle E\to X}" loading="lazy"> to elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}(E)\in A^{i}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{i}(E)\in A^{i}(X)}</annotation>

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</math><img src="./0e184b3374b893b52f3964da89adc10ca8946694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.564ex; height:3.176ex;" alt="{\displaystyle c_{i}(E)\in A^{i}(X)}" loading="lazy"> called Chern classes, with Chern polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{t}(E)=c_{0}(E)+c_{1}(E)t+\cdots +c_{n}(E)t^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E)=c_{0}(E)+c_{1}(E)t+\cdots +c_{n}(E)t^{n}}</annotation>

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</math><img src="./665717e424b05761d73d2260d5ba331f68f63c84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.761ex; height:2.843ex;" alt="{\displaystyle c_{t}(E)=c_{0}(E)+c_{1}(E)t+\cdots +c_{n}(E)t^{n}}" loading="lazy">, satisfying the following (similar to <a class="mw-selflink-fragment" href="#Grothendieck_axiomatic_approach">Grothendieck's axiomatic approach</a>). <a href="#cite_note-12">[12]</a>

<ol><li>If for a Cartier divisor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\cong {\mathcal {O}}_{X}(D)}">

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<annotation encoding="application/x-tex">{\displaystyle E\cong {\mathcal {O}}_{X}(D)}</annotation>

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</math><img src="./9df2ef555fb1700aa30a484b1e924327deff64d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.09ex; height:2.843ex;" alt="{\displaystyle E\cong {\mathcal {O}}_{X}(D)}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{t}(E)=1+Dt}">

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E)=1+Dt}</annotation>

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</math><img src="./c4cdd4a1b7cc44e00bf7369e672b640a0513f251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.283ex; height:2.843ex;" alt="{\displaystyle c_{t}(E)=1+Dt}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle f:X'\to X}</annotation>

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</math><img src="./9454ea66877cf385ab67de95634a63d7588f9ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.491ex; height:2.843ex;" alt="{\displaystyle f:X'\to X}" loading="lazy"> is a morphism, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}(f^{*}E)=f^{*}c_{i}(E)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{i}(f^{*}E)=f^{*}c_{i}(E)}</annotation>

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</math><img src="./83bf99c25eaff271332def61d251ca4ec019b04d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.631ex; height:2.843ex;" alt="{\displaystyle c_{i}(f^{*}E)=f^{*}c_{i}(E)}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle 0\to E'\to E\to E''\to 0}</annotation>

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</math><img src="./0f74c322bd7af75969b647b935b05b305dbde850.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.966ex; height:2.509ex;" alt="{\displaystyle 0\to E'\to E\to E''\to 0}" loading="lazy"> is an exact sequence of vector bundles on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, the Whitney sum formula holds: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{t}(E)=c_{t}(E')c_{t}(E'')}">

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<annotation encoding="application/x-tex">{\displaystyle c_{t}(E)=c_{t}(E')c_{t}(E'')}</annotation>

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</math><img src="./b07f6751ee4b1c3eb2137e92735ba471c58203ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.21ex; height:3.009ex;" alt="{\displaystyle c_{t}(E)=c_{t}(E')c_{t}(E'')}" loading="lazy">.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Normal_sequence">Normal sequence</h3></div>

Computing the characteristic classes for projective space forms the basis for many characteristic class computations since for any smooth projective subvariety <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subset \mathbb {P} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle X\subset \mathbb {P} ^{n}}</annotation>

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</math><img src="./59ec74d93cc805a1c25dcee28747a3c4aa8fec7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.717ex; height:2.343ex;" alt="{\displaystyle X\subset \mathbb {P} ^{n}}" loading="lazy"> there is the short exact sequence

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<annotation encoding="application/x-tex">{\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{n}}|_{X}\to {\mathcal {N}}_{X/\mathbb {P} ^{n}}\to 0}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Quintic_threefold">Quintic threefold</h4></div>

For example, consider a nonsingular <a href="Quintic_threefold" title="Quintic threefold">quintic threefold</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{4}}">

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</math><img src="./0ef0f42e9d791c201ab1f11cfeaef6372d876364.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{4}}" loading="lazy">. Its normal bundle is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}(5)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}(5)}</annotation>

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</math><img src="./bc9c218b6f470e30e44eda339594d51ea2062299.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.454ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{X}(5)}" loading="lazy"> and we have the short exact sequence

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<annotation encoding="application/x-tex">{\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{4}}|_{X}\to {\mathcal {O}}_{X}(5)\to 0}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> denote the hyperplane class in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\bullet }(X)}">

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<annotation encoding="application/x-tex">{\displaystyle A^{\bullet }(X)}</annotation>

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</math><img src="./d0df82026948b51a3f21c2453c6b950c46b8a740.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.587ex; height:2.843ex;" alt="{\displaystyle A^{\bullet }(X)}" loading="lazy">. Then the Whitney sum formula gives us that

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<annotation encoding="application/x-tex">{\displaystyle c({\mathcal {T}}_{X})c({\mathcal {O}}_{X}(5))=(1+h)^{5}=1+5h+10h^{2}+10h^{3}}</annotation>

</semantics>

</math>

Since the Chow ring of a hypersurface is difficult to compute, we will consider this sequence as a sequence of coherent sheaves in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{4}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{4}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {1+5h+10h^{2}+10h^{3}}{1+5h}}\\&=\left(1+5h+10h^{2}+10h^{3}\right)\left(1-5h+25h^{2}-125h^{3}\right)\\&=1+10h^{2}-40h^{3}\end{aligned}}}</annotation>

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</math>

Using the Gauss-Bonnet theorem we can integrate the class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{3}({\mathcal {T}}_{X})}">

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{3}({\mathcal {T}}_{X})}</annotation>

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</math><img src="./b95bee730c24fabb949458a45d37d5e46d608768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.77ex; height:2.843ex;" alt="{\displaystyle c_{3}({\mathcal {T}}_{X})}" loading="lazy"> to compute the Euler characteristic. Traditionally this is called the <a href="Euler_class" title="Euler class">Euler class</a>. This is

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<annotation encoding="application/x-tex">{\displaystyle \int _{[X]}c_{3}({\mathcal {T}}_{X})=\int _{[X]}-40h^{3}=-200}</annotation>

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since the class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{3}}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{3}}</annotation>

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</math><img src="./89571cb3cd52b5c1a859ab7035c7b94e85b37149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.393ex; height:2.676ex;" alt="{\displaystyle h^{3}}" loading="lazy"> can be represented by five points (by <a href="B%C3%A9zout's_theorem" title="Bézout's theorem">Bézout's theorem</a>). The Euler characteristic can then be used to compute the Betti numbers for the cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

<div class="mw-heading mw-heading4"><h4 id="Degree_d_hypersurfaces">Degree d hypersurfaces</h4></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subset \mathbb {P} ^{3}}">

<msup>

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</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X\subset \mathbb {P} ^{3}}</annotation>

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</math><img src="./10e28a73a64a505fb07c108838230318d7ce2728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.553ex; height:2.676ex;" alt="{\displaystyle X\subset \mathbb {P} ^{3}}" loading="lazy"> is a degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> smooth hypersurface, we have the short exact sequence <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{3}}|_{X}\to {\mathcal {O}}_{X}(d)\to 0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{3}}|_{X}\to {\mathcal {O}}_{X}(d)\to 0}</annotation>

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</math> giving the relation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c({\mathcal {T}}_{X})={\frac {c({\mathcal {T}}_{\mathbb {P} ^{3}|_{X}})}{c({\mathcal {O}}_{X}(d))}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c({\mathcal {T}}_{X})={\frac {c({\mathcal {T}}_{\mathbb {P} ^{3}|_{X}})}{c({\mathcal {O}}_{X}(d))}}}</annotation>

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</math> we can then calculate this as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {(1+[H])^{4}}{(1+d[H])}}\\&=(1+4[H]+6[H]^{2})(1-d[H]+d^{2}[H]^{2})\\&=1+(4-d)[H]+(6-4d+d^{2})[H]^{2}\end{aligned}}}</annotation>

</semantics>

</math>

Giving the total chern class. In particular, we can find <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a spin 4-manifold if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4-d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4-d}</annotation>

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</math><img src="./eeac7622c484b63f3c9da41be28f71461cdcd4f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.219ex; height:2.343ex;" alt="{\displaystyle 4-d}" loading="lazy"> is even, so every smooth hypersurface of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2k}</annotation>

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</math><img src="./ab358eb7defb4d2b0fc1f9e8a4e2d189fe600eb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.374ex; height:2.176ex;" alt="{\displaystyle 2k}" loading="lazy"> is a <a href="Spin_manifold" class="mw-redirect" title="Spin manifold">spin manifold</a>.

<div class="mw-heading mw-heading2"><h2 id="Proximate_notions">Proximate notions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="The_Chern_character">The Chern character</h3></div>

Chern classes can be used to construct a homomorphism of rings from the <a href="Topological_K-theory" title="Topological K-theory">topological K-theory</a> of a space to (the completion of) its rational cohomology. For a line bundle L, the Chern character ch is defined by

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ch} (L)=\exp(c_{1}(L)):=\sum _{m=0}^{\infty }{\frac {c_{1}(L)^{m}}{m!}}.}</annotation>

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</math>

More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=L_{1}\oplus \cdots \oplus L_{n}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V=L_{1}\oplus \cdots \oplus L_{n}}</annotation>

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</math><img src="./8f265a18c97d0cc7eb39dc717c521e86e1338754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.728ex; height:2.509ex;" alt="{\displaystyle V=L_{1}\oplus \cdots \oplus L_{n}}" loading="lazy"> is a direct sum of line bundles, with first Chern classes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=c_{1}(L_{i}),}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}=c_{1}(L_{i}),}</annotation>

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</math><img src="./216382799ae2cec12ece617a6c3ad7e0209798f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.128ex; height:2.843ex;" alt="{\displaystyle x_{i}=c_{1}(L_{i}),}" loading="lazy"> the Chern character is defined additively

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</munderover>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ch} (V)=e^{x_{1}}+\cdots +e^{x_{n}}:=\sum _{m=0}^{\infty }{\frac {1}{m!}}(x_{1}^{m}+\cdots +x_{n}^{m}).}</annotation>

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</math>

This can be rewritten as:<a href="#cite_note-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ch} (V)=\operatorname {rk} (V)+c_{1}(V)+{\frac {1}{2}}(c_{1}(V)^{2}-2c_{2}(V))+{\frac {1}{6}}(c_{1}(V)^{3}-3c_{1}(V)c_{2}(V)+3c_{3}(V))+\cdots .}</annotation>

</semantics>

</math>

This last expression, justified by invoking the <a href="Splitting_principle" title="Splitting principle">splitting principle</a>, is taken as the definition ch(V) for arbitrary vector bundles V.

If a connection is used to define the Chern classes when the base is a manifold (i.e., the <a href="Chern%E2%80%93Weil_theory" class="mw-redirect" title="Chern–Weil theory">Chern–Weil theory</a>), then the explicit form of the Chern character is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ch} (V)=\left[\operatorname {tr} \left(\exp \left({\frac {i\Omega }{2\pi }}\right)\right)\right]}</annotation>

</semantics>

</math>

where Ω is the <a href="Curvature_form" title="Curvature form">curvature</a> of the connection.

The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. Specifically, it obeys the following identities:

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ch} (V\oplus W)=\operatorname {ch} (V)+\operatorname {ch} (W)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ch} (V\otimes W)=\operatorname {ch} (V)\operatorname {ch} (W).}</annotation>

</semantics>

</math>

As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ch is a <a href="Homomorphism" title="Homomorphism">homomorphism</a> of <a href="Abelian_group" title="Abelian group">abelian groups</a> from the <a href="K-theory" title="K-theory">K-theory</a> K(X) into the rational cohomology of X. The second identity establishes the fact that this homomorphism also respects products in K(X), and so ch is a homomorphism of rings.

The Chern character is used in the <a href="Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem" title="Hirzebruch–Riemann–Roch theorem">Hirzebruch–Riemann–Roch theorem</a>.

<div class="mw-heading mw-heading3"><h3 id="Chern_numbers">Chern numbers</h3></div>

If we work on an <a href="Orientable_manifold" class="mw-redirect" title="Orientable manifold">oriented manifold</a> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 2n}</annotation>

</semantics>

</math><img src="./134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy">, then any product of Chern classes of total degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2n}</annotation>

</semantics>

</math><img src="./134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy"> (i.e., the sum of indices of the Chern classes in the product should be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">) can be paired with the <a href="Orientation_homology_class" class="mw-redirect" title="Orientation homology class">orientation homology class</a> (or "integrated over the manifold") to give an integer, a Chern number of the vector bundle. For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}^{3}}">

</mrow>

</mrow>

</msubsup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1}^{3}}</annotation>

</semantics>

</math><img src="./40b996d342d64c449a5cdb6d2f2df0a63f2b1242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.061ex; height:3.176ex;" alt="{\displaystyle c_{1}^{3}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}c_{2}}">

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{1}c_{2}}</annotation>

</semantics>

</math><img src="./5d7e22249be4a8242faf83c46d8a04ccb52d0308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.122ex; height:2.009ex;" alt="{\displaystyle c_{1}c_{2}}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{3}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{3}}</annotation>

</semantics>

</math><img src="./b1dc52bfbaf6e577fbed72a716068f4533700bd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{3}}" loading="lazy">. In general, if the manifold has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2n}</annotation>

</semantics>

</math><img src="./134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy">, the number of possible independent Chern numbers is the number of <a href="Integer_partition" title="Integer partition">partitions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

The Chern numbers of the tangent bundle of a complex (or almost complex) manifold are called the Chern numbers of the manifold, and are important invariants.

<div class="mw-heading mw-heading3"><h3 id="Generalized_cohomology_theories">Generalized cohomology theories</h3></div>

There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a <a href="Generalized_cohomology_theory" class="mw-redirect" title="Generalized cohomology theory">generalized cohomology theory</a>. The theories for which such generalization is possible are called <a href="Complex_cobordism#Formal_group_laws" title="Complex cobordism">complex orientable</a>. The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a <a href="Formal_group_law" title="Formal group law">formal group law</a>.

<div class="mw-heading mw-heading3"><h3 id="Algebraic_geometry">Algebraic geometry</h3></div>

In algebraic geometry there is a similar theory of Chern classes of vector bundles. There are several variations depending on what groups the Chern classes lie in:

<ul><li>For complex varieties the Chern classes can take values in ordinary cohomology, as above.</li>

<li>For varieties over general fields, the Chern classes can take values in cohomology theories such as <a href="Etale_cohomology" class="mw-redirect" title="Etale cohomology">etale cohomology</a> or <a href="L-adic_cohomology" class="mw-redirect" title="L-adic cohomology">l-adic cohomology</a>.</li>

<li>For varieties V over general fields the Chern classes can also take values in homomorphisms of <a href="Chow_group" title="Chow group">Chow groups</a> CH(V): for example, the first Chern class of a line bundle over a variety V is a homomorphism from CH(V) to CH(V) reducing degrees by 1. This corresponds to the fact that the Chow groups are a sort of analog of homology groups, and elements of cohomology groups can be thought of as homomorphisms of homology groups using the <a href="Cap_product" title="Cap product">cap product</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Manifolds_with_structure">Manifolds with structure</h3></div>

The theory of Chern classes gives rise to <a href="Cobordism" title="Cobordism">cobordism</a> invariants for <a href="Almost_complex_manifold" title="Almost complex manifold">almost complex manifolds</a>.

If M is an almost complex manifold, then its <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> is a complex vector bundle. The Chern classes of M are thus defined to be the Chern classes of its tangent bundle. If M is also <a href="Compact_space" title="Compact space">compact</a> and of dimension 2d, then each <a href="Monomial" title="Monomial">monomial</a> of total degree 2d in the Chern classes can be paired with the <a href="Fundamental_class" title="Fundamental class">fundamental class</a> of M, giving an integer, a Chern number of M. If M′ is another almost complex manifold of the same dimension, then it is cobordant to M if and only if the Chern numbers of M′ coincide with those of M.

The theory also extends to real <a href="Symplectic_geometry" title="Symplectic geometry">symplectic</a> vector bundles, by the intermediation of compatible almost complex structures. In particular, <a href="Symplectic_manifold" title="Symplectic manifold">symplectic manifolds</a> have a well-defined Chern class.

<div class="mw-heading mw-heading3"><h3 id="Arithmetic_schemes_and_Diophantine_equations">Arithmetic schemes and Diophantine equations</h3></div>

(See <a href="Arakelov_geometry" class="mw-redirect" title="Arakelov geometry">Arakelov geometry</a>)

<ul><li><a href="Pontryagin_class" title="Pontryagin class">Pontryagin class</a></li>

<li><a href="Stiefel%E2%80%93Whitney_class" title="Stiefel–Whitney class">Stiefel–Whitney class</a></li>

<li><a href="Euler_class" title="Euler class">Euler class</a></li>

<li><a href="Segre_class" title="Segre class">Segre class</a></li>

<li><a href="Schubert_calculus" title="Schubert calculus">Schubert calculus</a></li>

<li><a href="Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li>

<li><a href="Localized_Chern_class" title="Localized Chern class">Localized Chern class</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFBottTu1995" class="citation book cs1"><a href="Raoul_Bott" title="Raoul Bott">Bott, Raoul</a>; Tu, Loring (1995). Differential forms in algebraic topology (Corr. 3. print. ed.). New York [u.a.]: Springer. p. 267ff. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-90613-4</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHatcher" class="citation web cs1"><a href="Allen_Hatcher" title="Allen Hatcher">Hatcher, Allen</a>. <a rel="nofollow" class="external text" href="https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf">"Vector Bundles and K-theory"</a> (PDF). Proposition 3.10.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Editorial note: Our notation differs from Milnor−Stasheff, but seems more natural.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> The sequence is sometimes called the <a href="Euler_sequence" title="Euler sequence">Euler sequence</a>.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFHartshorne">Hartshorne</a>, Ch. II. Theorem 8.13.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> In a ring-theoretic term, there is an isomorphism of graded rings:

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{2*}(M,\mathbb {Z} )\to \oplus _{k}^{\infty }\eta (H^{2*}(M,\mathbb {Z} ))[t],x\mapsto xt^{|x|/2}}</annotation>

</semantics>

</math>

where the left is the cohomology ring of even terms, η is a ring homomorphism that disregards grading and x is homogeneous and has degree |x|.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFFulton">Fulton</a>, Remark 3.2.3. (a)

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFFulton">Fulton</a>, Remark 3.2.3. (b)

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFFulton">Fulton</a>, Example 3.2.2.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFFulton">Fulton</a>, Remark 3.2.3. (c)

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> Use, for example, WolframAlpha to expand the polynomial and then use the fact <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation>

</semantics>

</math><img src="./01acb7953ba52c2aa44264b5d0f8fd223aa178a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}" loading="lazy"> are elementary symmetric polynomials in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{i}}</annotation>

</semantics>

</math><img src="./3b1fb627423abe4988b7ed88d4920bf1ec074790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.287ex; height:2.009ex;" alt="{\displaystyle \alpha _{i}}" loading="lazy">'s.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFHartshorne">Hartshorne</a>, Appendix A. 3 Chern Classes.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> (See also <a href="#Chern_polynomial">§ Chern polynomial</a>.) Observe that when V is a sum of line bundles, the Chern classes of V can be expressed as <a href="Elementary_symmetric_polynomials" class="mw-redirect" title="Elementary symmetric polynomials">elementary symmetric polynomials</a> in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}(V)=e_{i}(x_{1},\ldots ,x_{n}).}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{i}(V)=e_{i}(x_{1},\ldots ,x_{n}).}</annotation>

</semantics>

</math><img src="./465f7772b037c65ac28d1a93d7eb6d2cb2a01911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.951ex; height:2.843ex;" alt="{\displaystyle c_{i}(V)=e_{i}(x_{1},\ldots ,x_{n}).}" loading="lazy">

In particular, on the one hand

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c(V):=\sum _{i=0}^{n}c_{i}(V),}</annotation>

</semantics>

</math>

while on the other hand

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}c(V)&=c(L_{1}\oplus \cdots \oplus L_{n})\\&=\prod _{i=1}^{n}c(L_{i})\\&=\prod _{i=1}^{n}(1+x_{i})\\&=\sum _{i=0}^{n}e_{i}(x_{1},\ldots ,x_{n})\end{aligned}}}</annotation>

</semantics>

</math>

Consequently, <a href="Newton's_identities#Expressing_power_sums_in_terms_of_elementary_symmetric_polynomials" title="Newton's identities">Newton's identities</a> may be used to re-express the power sums in ch(V) above solely in terms of the Chern classes of V, giving the claimed formula.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFChern1946" class="citation cs2"><a href="Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern, Shiing-Shen</a> (1946), "Characteristic classes of Hermitian Manifolds", <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>, Second Series, 47 (1): 85–121, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969037">10.2307/1969037</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969037">1969037</a></cite></li>

<li><cite id="CITEREFFulton" class="citation book cs1">Fulton, W. (29 June 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gCXsCAAAQBAJ">Intersection Theory</a>. Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-662-02421-8</bdi>.</cite></li>

<li><cite id="CITEREFGrothendieck1958" class="citation cs2"><a href="Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexander</a> (1958), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=BSMF_1958__86__137_0">"La théorie des classes de Chern"</a>, Bulletin de la Société Mathématique de France, 86: 137–154, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fbsmf.1501">10.24033/bsmf.1501</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0037-9484">0037-9484</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0116023">0116023</a></cite></li>

<li><cite id="CITEREFHartshorne" class="citation book cs1">Hartshorne, Robin (29 June 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7z4mBQAAQBAJ">Algebraic Geometry</a>. Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4757-3849-0</bdi>.</cite></li>

<li><cite id="CITEREFJost2005" class="citation cs2"><a href="J%C3%BCrgen_Jost" title="Jürgen Jost">Jost, Jürgen</a> (2005), Riemannian Geometry and Geometric Analysis (4th ed.), <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-25907-7</bdi></cite> (Provides a very short, introductory review of Chern classes).</li>

<li><cite id="CITEREFMay1999" class="citation cs2"><a href="J._Peter_May" title="J. Peter May">May, J. Peter</a> (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g8SG03R1bpgC&q=%22Chern+class%22&pg=PA1">A Concise Course in Algebraic Topology</a>, University of Chicago Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780226511832</bdi></cite></li>

<li><cite id="CITEREFMilnorStasheff1974" class="citation cs2"><a href="John_Milnor" title="John Milnor">Milnor, John Willard</a>; <a href="Jim_Stasheff" title="Jim Stasheff">Stasheff, James D.</a> (1974), Characteristic classes, Annals of Mathematics Studies, vol. 76, Princeton University Press; University of Tokyo Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-08122-9</bdi></cite></li>

<li><cite id="CITEREFRubei2014" class="citation cs2">Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Walter De Gruyter, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-11-031622-3</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html">Vector Bundles & K-Theory</a> – A downloadable book-in-progress by <a href="Allen_Hatcher" title="Allen Hatcher">Allen Hatcher</a>. Contains a chapter about characteristic classes.</li>

<li><a href="Dieter_Kotschick" title="Dieter Kotschick">Dieter Kotschick</a>, <a rel="nofollow" class="external text" href="http://www.physorg.com/news163858041.html">Chern numbers of algebraic varieties</a></li></ul>

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<li><a href="Geometric_topology" title="Geometric topology">Geometric</a>

<ul><li><a href="Low-dimensional_topology" title="Low-dimensional topology">low-dimensional</a></li></ul></li>

<li><a href="Homology_(mathematics)" title="Homology (mathematics)">Homology</a>

<ul><li><a href="Cohomology" title="Cohomology">cohomology</a></li></ul></li>

<li><a href="Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li>

<li><a href="Digital_topology" title="Digital topology">Digital</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><a href="Klein_bottle" title="Klein bottle"></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Open_set" title="Open set">Open set</a> / <a href="Closed_set" title="Closed set">Closed set</a></li>

<li><a href="Interior_(topology)" title="Interior (topology)">Interior</a></li>

<li><a href="Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">Continuity</a></li>

<li><a href="Topological_space" title="Topological space">Space</a>

<ul><li><a href="Compact_space" title="Compact space">compact</a></li>

<li><a href="Connected_space" title="Connected space">connected</a></li>

<li><a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a></li>

<li><a href="Metric_space" title="Metric space">metric</a></li>

<li><a href="Uniform_space" title="Uniform space">uniform</a></li></ul></li>

<li><a href="Homotopy" title="Homotopy">Homotopy</a>

<ul><li><a href="Homotopy_group" title="Homotopy group">homotopy group</a></li>

<li><a href="Fundamental_group" title="Fundamental group">fundamental group</a></li></ul></li>

<li><a href="Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li>

<li><a href="CW_complex" title="CW complex">CW complex</a></li>

<li><a href="Polyhedral_complex" title="Polyhedral complex">Polyhedral complex</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Bundle_(mathematics)" title="Bundle (mathematics)">Bundle (mathematics)</a></li>

<li><a href="Second-countable_space" title="Second-countable space">Second-countable space</a></li>

<li><a href="Cobordism" title="Cobordism">Cobordism</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Metrics and properties</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li>

<li><a href="Betti_number" title="Betti number">Betti number</a></li>

<li><a href="Winding_number" title="Winding number">Winding number</a></li>

<li><a href="Orientability" title="Orientability">Orientability</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

<li><a href="Invariance_of_domain" title="Invariance of domain">Invariance of domain</a></li>

<li><a href="Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li>

<li><a href="Tychonoff's_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a></li>

<li><a href="Urysohn's_lemma" title="Urysohn's lemma">Urysohn's lemma</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li> Category</li>

<li> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li>

<li> <a href="https://en.wikibooks.org/wiki/Topology" class="extiw external" title="wikibooks:Topology">Wikibook</a></li>

<li> <a href="https://en.wikiversity.org/wiki/Topology" class="extiw external" title="wikiversity:Topology">Wikiversity</a></li>

<li> <a href="List_of_topology_topics" title="List of topology topics">Topics</a>

<ul><li><a href="List_of_general_topology_topics" title="List of general topology topics">general</a></li>

<li><a href="List_of_algebraic_topology_topics" title="List of algebraic topology topics">algebraic</a></li>

<li><a href="List_of_geometric_topology_topics" title="List of geometric topology topics">geometric</a></li></ul></li>

<li> <a href="List_of_important_publications_in_mathematics" class="mw-redirect" title="List of important publications in mathematics">Publications</a></li></ul>

</div></td></tr></tbody></table></div>

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