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Exponential sheaf sequence

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Mathematics" title="Mathematics">mathematics</a>, the exponential sheaf sequence is a fundamental <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> of <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> used in <a href="Complex_geometry" title="Complex geometry">complex geometry</a>.

Let M be a <a href="Complex_manifold" title="Complex manifold">complex manifold</a>, and write OM for the sheaf of <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a> on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of <a href="Abelian_group" title="Abelian group">abelian groups</a>. The <a href="Exponential_function" title="Exponential function">exponential function</a> gives a sheaf homomorphism

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<annotation encoding="application/x-tex">{\displaystyle \exp :{\mathcal {O}}_{M}\to {\mathcal {O}}_{M}^{*},}</annotation>

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</math><img src="./1b64448feef81a25ee015120be3f8a259a293237.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.369ex; height:2.843ex;" alt="{\displaystyle \exp :{\mathcal {O}}_{M}\to {\mathcal {O}}_{M}^{*},}" loading="lazy"></dd></dl>

because for a holomorphic function f, exp(f) is a non-vanishing holomorphic function, and exp(f + g) = exp(f)exp(g). Its <a href="Kernel_(algebra)" title="Kernel (algebra)">kernel</a> is the sheaf 2πiZ of <a href="Locally_constant_function" title="Locally constant function">locally constant functions</a> on M taking the values 2πin, with n an <a href="Integer" title="Integer">integer</a>. The exponential sheaf sequence is therefore

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<annotation encoding="application/x-tex">{\displaystyle 0\to 2\pi i\,\mathbb {Z} \to {\mathcal {O}}_{M}\to {\mathcal {O}}_{M}^{*}\to 0.}</annotation>

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</math><img src="./14a8a94536b9ca8e689f15a2b900967a8908ec98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.281ex; height:2.843ex;" alt="{\displaystyle 0\to 2\pi i\,\mathbb {Z} \to {\mathcal {O}}_{M}\to {\mathcal {O}}_{M}^{*}\to 0.}" loading="lazy"></dd></dl>

The exponential mapping here is not always a surjective map on sections; this can be seen for example when M is a <a href="Punctured_disk" class="mw-redirect" title="Punctured disk">punctured disk</a> in the complex plane. The exponential map is surjective on the <a href="Stalk_of_a_sheaf" class="mw-redirect" title="Stalk of a sheaf">stalks</a>: Given a <a href="Germ_of_a_function" class="mw-redirect" title="Germ of a function">germ</a> g of an holomorphic function at a point P such that g(P) ≠ 0, one can take the <a href="Logarithm" title="Logarithm">logarithm</a> of g in a neighborhood of P. The <a href="Long_exact_sequence" class="mw-redirect" title="Long exact sequence">long exact sequence</a> of <a href="Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a> shows that we have an exact sequence

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<annotation encoding="application/x-tex">{\displaystyle \cdots \to H^{0}({\mathcal {O}}_{U})\to H^{0}({\mathcal {O}}_{U}^{*})\to H^{1}(2\pi i\,\mathbb {Z} |_{U})\to \cdots }</annotation>

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</math><img src="./1d41ffb295574b8829c359c365b37d61fa82b89a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:48.863ex; height:3.343ex;" alt="{\displaystyle \cdots \to H^{0}({\mathcal {O}}_{U})\to H^{0}({\mathcal {O}}_{U}^{*})\to H^{1}(2\pi i\,\mathbb {Z} |_{U})\to \cdots }" loading="lazy"></dd></dl>

for any open set U of M. Here H0 means simply the sections over U, and the sheaf cohomology H1(2πiZ|U) is the <a href="Singular_cohomology" class="mw-redirect" title="Singular cohomology">singular cohomology</a> of U.

One can think of H1(2πiZ|U) as associating an integer to each loop in U. For each section of OM*, the connecting homomorphism to H1(2πiZ|U) gives the <a href="Winding_number" title="Winding number">winding number</a> for each loop. So this homomorphism is therefore a generalized <a href="Winding_number" title="Winding number">winding number</a> and measures the failure of U to be <a href="Contractible" class="mw-redirect" title="Contractible">contractible</a>. In other words, there is a potential topological obstruction to taking a global logarithm of a non-vanishing holomorphic function, something that is always locally possible.

A further consequence of the sequence is the exactness of

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<annotation encoding="application/x-tex">{\displaystyle \cdots \to H^{1}({\mathcal {O}}_{M})\to H^{1}({\mathcal {O}}_{M}^{*})\to H^{2}(2\pi i\,\mathbb {Z} )\to \cdots .}</annotation>

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</math><img src="./13cec51c946dae9e9bb3b11462ea16b4caa80a15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.69ex; height:3.176ex;" alt="{\displaystyle \cdots \to H^{1}({\mathcal {O}}_{M})\to H^{1}({\mathcal {O}}_{M}^{*})\to H^{2}(2\pi i\,\mathbb {Z} )\to \cdots .}" loading="lazy"></dd></dl>

Here H1(OM*) can be identified with the <a href="Picard_group" title="Picard group">Picard group</a> of <a href="Holomorphic_line_bundle" class="mw-redirect" title="Holomorphic line bundle">holomorphic line bundles</a> on M. The connecting homomorphism sends a line bundle to its first <a href="Chern_class" title="Chern class">Chern class</a>.

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

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</style><cite id="CITEREFGriffithsHarris1994" class="citation cs2"><a href="Phillip_Griffiths" title="Phillip Griffiths">Griffiths, Phillip</a>; <a href="Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joseph</a> (1994), Principles of algebraic geometry, Wiley Classics Library, New York: <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-05059-9</bdi>, <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=1288523">1288523</a></cite>, see especially p. 37 and p. 139</li></ul></div><div><div class="zim-footer">

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