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Line bundle

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a line bundle expresses the concept of a <a href="Line_(geometry)" title="Line (geometry)">line</a> that varies from point to point of a space. For example, a <a href="Curve" title="Curve">curve</a> in the plane having a <a href="Tangent" title="Tangent">tangent</a> line at each point determines a varying line: the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> is a way of organising these. More formally, in <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a> and <a href="Differential_topology" title="Differential topology">differential topology</a>, a line bundle is defined as a <a href="Vector_bundle" title="Vector bundle">vector bundle</a> of rank 1.<a href="#cite_note-1">[1]</a>

Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 <a href="Invertible" class="mw-redirect" title="Invertible">invertible</a> real matrices, which is <a href="Homotopy" title="Homotopy">homotopy</a>-equivalent to a <a href="Discrete_two-point_space" title="Discrete two-point space">discrete two-point space</a> by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle.

From the perspective of <a href="Homotopy_theory" title="Homotopy theory">homotopy theory</a>, a real line bundle therefore behaves much the same as a <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a> with a two-point fiber, that is, like a <a href="Double_cover_(topology)" class="mw-redirect" title="Double cover (topology)">double cover</a>. A special case of this is the <a href="Orientable_double_cover" class="mw-redirect" title="Orientable double cover">orientable double cover</a> of a <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>, where the corresponding line bundle is the determinant bundle of the tangent bundle (see below). The <a href="M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the <a href="Unit_interval" title="Unit interval">unit interval</a> as a fiber, or the real line.

Complex line bundles are closely related to <a href="Circle_bundle" title="Circle bundle">circle bundles</a>. There are some celebrated ones, for example the <a href="Hopf_fibration" title="Hopf fibration">Hopf fibrations</a> of <a href="Sphere" title="Sphere">spheres</a> to spheres.

In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, an <a href="Invertible_sheaf" title="Invertible sheaf">invertible sheaf</a> (i.e., <a href="Locally_free_sheaf" class="mw-redirect" title="Locally free sheaf">locally free sheaf</a> of rank one) is often called a line bundle.

Every line bundle arises from a <a href="Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisor</a> under the following conditions:

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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"> is a reduced and irreducible scheme, then every line bundle comes from a divisor.</dd>

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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"> is a <a href="Projective_scheme" class="mw-redirect" title="Projective scheme">projective scheme</a> then the same statement holds.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="The_tautological_bundle_on_projective_space">The tautological bundle on projective space</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Tautological_line_bundle" class="mw-redirect" title="Tautological line bundle">Tautological line bundle</a></div>

One of the most important line bundles in algebraic geometry is the tautological line bundle on <a href="Projective_space" title="Projective space">projective space</a>. The <a href="Projectivization" title="Projectivization">projectivization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (V)}">

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

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</math><img src="./7da05f2c51c7f0ee8631336e4d8e0ddf62591a10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (V)}" loading="lazy"> of a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is defined to be the quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\setminus \{0\}}">

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

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</math><img src="./270db11e509d13d667ca39b4cca410b08deed693.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.469ex; height:2.843ex;" alt="{\displaystyle V\setminus \{0\}}" loading="lazy"> by the action of the multiplicative group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\times }}">

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

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</math><img src="./4d5261707dc30853daeab0f2917836969aaff368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.722ex; height:2.343ex;" alt="{\displaystyle k^{\times }}" loading="lazy">. Each point of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (V)}">

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

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</math><img src="./7da05f2c51c7f0ee8631336e4d8e0ddf62591a10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (V)}" loading="lazy"> therefore corresponds to a copy of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\times }}">

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

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</math><img src="./4d5261707dc30853daeab0f2917836969aaff368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.722ex; height:2.343ex;" alt="{\displaystyle k^{\times }}" loading="lazy">, and these copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\times }}">

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

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</math><img src="./4d5261707dc30853daeab0f2917836969aaff368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.722ex; height:2.343ex;" alt="{\displaystyle k^{\times }}" loading="lazy"> can be assembled into a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\times }}">

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

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</math><img src="./4d5261707dc30853daeab0f2917836969aaff368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.722ex; height:2.343ex;" alt="{\displaystyle k^{\times }}" loading="lazy">-bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (V)}">

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

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</math><img src="./7da05f2c51c7f0ee8631336e4d8e0ddf62591a10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (V)}" loading="lazy">. But <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\times }}">

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

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</math><img src="./4d5261707dc30853daeab0f2917836969aaff368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.722ex; height:2.343ex;" alt="{\displaystyle k^{\times }}" loading="lazy"> differs from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> only by a single point, and by adjoining that point to each fiber, we get a line bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (V)}">

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

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</math><img src="./7da05f2c51c7f0ee8631336e4d8e0ddf62591a10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (V)}" loading="lazy">. This line bundle is called the tautological line bundle. This line bundle is sometimes denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(-1)}">

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

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</math><img src="./59657431499a94040fdd3c2e5a28305ba54d6777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.63ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(-1)}" loading="lazy"> since it corresponds to the dual of the <a href="Serre_twisting_sheaf" class="mw-redirect" title="Serre twisting sheaf">Serre twisting sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)}">

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

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</math><img src="./5aeb15c854068604d35a2dd82a925899fafd3690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Maps_to_projective_space">Maps to projective space</h3></div>

Suppose that <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"> is a space and that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is a line bundle 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">. A global section of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s:X\to L}">

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

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</math><img src="./56808248b7d0b1cb9c9f90ace384f6c61b50920e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.205ex; height:2.176ex;" alt="{\displaystyle s:X\to L}" loading="lazy"> such that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:L\to X}">

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

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</math><img src="./02d0746e2aadc907dd84082c64b2f62663d03a84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.373ex; height:2.509ex;" alt="{\displaystyle p:L\to X}" loading="lazy"> is the natural projection, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\circ s=\operatorname {id} _{X}}">

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<annotation encoding="application/x-tex">{\displaystyle p\circ s=\operatorname {id} _{X}}</annotation>

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</math><img src="./b1caae73dbff51dd365278b6c2ee51aa7dde4777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.214ex; height:2.509ex;" alt="{\displaystyle p\circ s=\operatorname {id} _{X}}" loading="lazy">. In a small neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> in <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"> in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is trivial, the total space of the line bundle is the product of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and the underlying field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, and the section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> restricts to a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to k}">

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

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</math><img src="./f57893fb2fc9f31258fd3fcef6f9461773ef861f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:2.176ex;" alt="{\displaystyle U\to k}" loading="lazy">. However, the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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

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</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> depend on the choice of trivialization, and so they are determined only up to multiplication by a nowhere-vanishing function.

Global sections determine maps to projective spaces in the following way: Choosing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+1}">

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

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</math><img src="./7bf1b7e1e8ed2801660c299494ed10e6f18f790d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.343ex;" alt="{\displaystyle r+1}" loading="lazy"> not all zero points in a fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> chooses a fiber of the tautological line bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{r}}">

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

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</math><img src="./4d1085636360064a909e900c4ab20cdfcf4f1113.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.8ex; height:2.343ex;" alt="{\displaystyle \mathbf {P} ^{r}}" loading="lazy">, so choosing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+1}">

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

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</math><img src="./7bf1b7e1e8ed2801660c299494ed10e6f18f790d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.052ex; height:2.343ex;" alt="{\displaystyle r+1}" loading="lazy"> non-simultaneously vanishing global sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> determines a map from <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"> into projective space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{r}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4d1085636360064a909e900c4ab20cdfcf4f1113.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.8ex; height:2.343ex;" alt="{\displaystyle \mathbf {P} ^{r}}" loading="lazy">. This map sends the fibers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> to the fibers of the dual of the tautological bundle. More specifically, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0},\dots ,s_{r}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{0},\dots ,s_{r}}</annotation>

</semantics>

</math><img src="./63dc7a9f4b9ca5750652859badda4f58ba968240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.387ex; height:2.009ex;" alt="{\displaystyle s_{0},\dots ,s_{r}}" loading="lazy"> are global sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. In a small neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> in <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">, these sections determine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-valued functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> whose values depend on the choice of trivialization. However, they are determined up to simultaneous multiplication by a non-zero function, so their ratios are well-defined. That is, over a point <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, the values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0}(x),\dots ,s_{r}(x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{0}(x),\dots ,s_{r}(x)}</annotation>

</semantics>

</math><img src="./ab4d4020fbed783f96076fd9c99a74273f52d7a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.665ex; height:2.843ex;" alt="{\displaystyle s_{0}(x),\dots ,s_{r}(x)}" loading="lazy">are not well-defined because a change in trivialization will multiply them each by a non-zero constant λ. But it will multiply them by the same constant λ, so the <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [s_{0}(x):\dots :s_{r}(x)]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [s_{0}(x):\dots :s_{r}(x)]}</annotation>

</semantics>

</math><img src="./5825ffe92dbf4d4a6c4ab3548b007232e48d66f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.378ex; height:2.843ex;" alt="{\displaystyle [s_{0}(x):\dots :s_{r}(x)]}" loading="lazy"> are well-defined as long as the sections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{0},\dots ,s_{r}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{0},\dots ,s_{r}}</annotation>

</semantics>

</math><img src="./63dc7a9f4b9ca5750652859badda4f58ba968240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.387ex; height:2.009ex;" alt="{\displaystyle s_{0},\dots ,s_{r}}" loading="lazy"> do not simultaneously vanish at <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="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. Therefore, if the sections never simultaneously vanish, they determine a form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [s_{0}:\dots :s_{r}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [s_{0}:\dots :s_{r}]}</annotation>

</semantics>

</math><img src="./0bf5c13cae00135f9652940c488586a1e5cb7206.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.1ex; height:2.843ex;" alt="{\displaystyle [s_{0}:\dots :s_{r}]}" loading="lazy"> which gives a map from <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"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{r}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4d1085636360064a909e900c4ab20cdfcf4f1113.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.8ex; height:2.343ex;" alt="{\displaystyle \mathbf {P} ^{r}}" loading="lazy">, and the pullback of the dual of the tautological bundle under this map is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. In this way, projective space acquires a <a href="Universal_property" title="Universal property">universal property</a>.

The universal way to determine a map to projective space is to map to the projectivization of the vector space of all sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. In the topological case, there is a non-vanishing section at every point which can be constructed using a bump function which vanishes outside a small neighborhood of the point. Because of this, the resulting map is defined everywhere. However, the codomain is usually far, far too big to be useful. The opposite is true in the algebraic and holomorphic settings. Here the space of global sections is often finite dimensional, but there may not be any non-vanishing global sections at a given point. (As in the case when this procedure constructs a <a href="Lefschetz_pencil" title="Lefschetz pencil">Lefschetz pencil</a>.) In fact, it is possible for a bundle to have no non-zero global sections at all; this is the case for the tautological line bundle. When the line bundle is sufficiently ample this construction verifies the <a href="Kodaira_embedding_theorem" title="Kodaira embedding theorem">Kodaira embedding theorem</a>.

<div class="mw-heading mw-heading2"><h2 id="Determinant_bundles">Determinant bundles</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Quillen_metric#Determinant_line_bundle_of_a_family_of_operators" title="Quillen metric">Quillen metric § Determinant line bundle of a family of operators</a></div>

In general if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is a vector bundle on a space <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">, with constant fibre dimension <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">, the <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">-th <a href="Exterior_power" class="mw-redirect" title="Exterior power">exterior power</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> taken fibre-by-fibre is a line bundle, called the determinant line bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. This construction is in particular applied to the <a href="Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> of a <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>. The resulting determinant bundle (more precisely, the bundle of a fixed nonegative power of the absolute values of its sections) is responsible for the phenomenon of <a href="Tensor_density" title="Tensor density">tensor densities</a>, in the sense that for an <a href="Orientable_manifold" class="mw-redirect" title="Orientable manifold">orientable manifold</a> it has a nonvanishing global section, and its tensor powers with any real exponent may be defined and used to 'twist' any vector bundle by <a href="Tensor_product" title="Tensor product">tensor product</a>.

The same construction (taking the top exterior power) applies to a <a href="Finitely_generated_module" title="Finitely generated module">finitely generated</a> <a href="Projective_module" title="Projective module">projective module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> over a Noetherian domain and the resulting invertible module is called the determinant module of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Characteristic_classes,_universal_bundles_and_classifying_spaces">Characteristic classes, universal bundles and classifying spaces</h2></div>

The first <a href="Stiefel%E2%80%93Whitney_class" title="Stiefel–Whitney class">Stiefel–Whitney class</a> classifies smooth real line bundles; in particular, the collection of (equivalence classes of) real line bundles are in correspondence with elements of the first cohomology with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2\mathbb {Z} }">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1b3bb21abe942aa9c0c63bae35a0c38905e1712c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }" loading="lazy"> coefficients; this correspondence is in fact an isomorphism of abelian groups (the group operations being tensor product of line bundles and the usual addition on cohomology). Analogously, the first <a href="Chern_class" title="Chern class">Chern class</a> classifies smooth complex line bundles on a space, and the group of line bundles is isomorphic to the second cohomology class with integer coefficients. However, bundles can have equivalent <a href="Smooth_structure" title="Smooth structure">smooth structures</a> (and thus the same first Chern class) but different holomorphic structures. The Chern class statements are easily proven using the <a href="Exponential_sequence" class="mw-redirect" title="Exponential sequence">exponential sequence</a> of <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> on the manifold.

One can more generally view the classification problem from a homotopy-theoretic point of view. There is a universal bundle for real line bundles, and a universal bundle for complex line bundles. According to general theory about <a href="Classifying_space" title="Classifying space">classifying spaces</a>, the heuristic is to look for <a href="Contractible" class="mw-redirect" title="Contractible">contractible</a> spaces on which there are <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group actions</a> of the respective groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7ec545f7870665e1028b7492746848d149878808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy">, that are free actions. Those spaces can serve as the universal <a href="Principal_bundle" title="Principal bundle">principal bundles</a>, and the quotients for the actions as the classifying spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./773ca20b2080cb3766062a5451a01d2220e9b067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}" loading="lazy">. In these cases we can find those explicitly, in the infinite-dimensional analogues of real and complex <a href="Projective_space" title="Projective space">projective space</a>.

Therefore the classifying space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BC_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./683e9105fa05653775e2e1fbd1dda2e725290a97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.48ex; height:2.509ex;" alt="{\displaystyle BC_{2}}" loading="lazy"> is of the homotopy type of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \mathbf {P} ^{\infty }}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3aae673092f1fed336c1a014764f17d319e5c551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.38ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} \mathbf {P} ^{\infty }}" loading="lazy">, the real projective space given by an infinite sequence of <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>. It carries the universal real line bundle; in terms of homotopy theory that means that any real line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> on a <a href="CW_complex" title="CW complex">CW complex</a> <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"> determines a classifying map from <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"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \mathbf {P} ^{\infty }}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3aae673092f1fed336c1a014764f17d319e5c551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.38ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} \mathbf {P} ^{\infty }}" loading="lazy">, making <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> a bundle isomorphic to the pullback of the universal bundle. This classifying map can be used to define the <a href="Stiefel-Whitney_class" class="mw-redirect" title="Stiefel-Whitney class">Stiefel-Whitney class</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">, in the first 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>

</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"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2\mathbb {Z} }">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1b3bb21abe942aa9c0c63bae35a0c38905e1712c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }" loading="lazy"> coefficients, from a standard class on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \mathbf {P} ^{\infty }}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3aae673092f1fed336c1a014764f17d319e5c551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.38ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} \mathbf {P} ^{\infty }}" loading="lazy">.

In an analogous way, the complex projective space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \mathbf {P} ^{\infty }}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d5c59f447a3c24fed0238fc01434eb65933dbce7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.38ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} \mathbf {P} ^{\infty }}" loading="lazy"> carries a universal complex line bundle. In this case classifying maps give rise to the first <a href="Chern_class" title="Chern class">Chern class</a> of <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">, in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2}(X)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d3d6447a3ab8b2d2bd770ccb81207531b48c2274.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.947ex; height:3.176ex;" alt="{\displaystyle H^{2}(X)}" loading="lazy"> (integral cohomology).

There is a further, analogous theory with <a href="Quaternion" title="Quaternion">quaternionic</a> (real dimension four) line bundles. This gives rise to one of the <a href="Pontryagin_class" title="Pontryagin class">Pontryagin classes</a>, in real four-dimensional cohomology.

In this way foundational cases for the theory of <a href="Characteristic_class" title="Characteristic class">characteristic classes</a> depend only on line bundles. According to a general <a href="Splitting_principle" title="Splitting principle">splitting principle</a> this can determine the rest of the theory (if not explicitly).

There are theories of <a href="Holomorphic_line_bundle" class="mw-redirect" title="Holomorphic line bundle">holomorphic line bundles</a> on <a href="Complex_manifold" title="Complex manifold">complex manifolds</a>, and <a href="Invertible_sheaf" title="Invertible sheaf">invertible sheaves</a> in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, that work out a line bundle theory in those areas.

<ul><li><a href="I-bundle" title="I-bundle">I-bundle</a></li>

<li><a href="Ample_line_bundle" title="Ample line bundle">Ample line bundle</a></li>

<li><a href="Line_field" title="Line field">Line field</a></li></ul>

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

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</style><cite id="CITEREFHartshorne1975" class="citation book cs1">Hartshorne (1975). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eICMfNiDdigC&pg=PA7&dq=%22line+bundle%22">Algebraic Geometry, Arcata 1974</a>. p. 7.</cite>

</li>

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

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

<ul><li>Michael Murray, <a rel="nofollow" class="external text" href="http://www.maths.adelaide.edu.au/michael.murray/line_bundles.pdf">Line Bundles</a>, 2002 (PDF web link)</li>

<li><a href="Robin_Hartshorne" title="Robin Hartshorne">Robin Hartshorne</a>. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4pieAwAAQBAJ&q=%22Line+bundle%22">Algebraic geometry</a>. AMS Bookstore, 1975. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-1429-1</bdi></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-08" href="https://en.wikipedia.org/wiki/?title=Line_bundle&oldid=1294598550">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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