# Exponential sheaf sequence

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In [mathematics](/source/mathematics), the '''exponential sheaf sequence''' is a fundamental [short exact sequence](/source/short_exact_sequence) of [sheaves](/source/sheaf_(mathematics)) used in [complex geometry](/source/complex_geometry).

Let ''M'' be a [complex manifold](/source/complex_manifold), and write ''O''<sub>''M''</sub> for the sheaf of [holomorphic function](/source/holomorphic_function)s on ''M''. Let ''O''<sub>''M''</sub>* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of [abelian group](/source/abelian_group)s. The [exponential function](/source/exponential_function) gives a sheaf homomorphism

:<math>\exp : \mathcal O_M \to \mathcal O_M^*,</math>

because for a holomorphic function ''f'', exp(''f'') is a non-vanishing holomorphic function, and exp(''f''&nbsp;+&nbsp;''g'') =&nbsp;exp(''f'')exp(''g'').  Its [kernel](/source/Kernel_(algebra)) is the sheaf 2π''i'''''Z''' of [locally constant function](/source/locally_constant_function)s on ''M'' taking the values 2π''in'', with ''n'' an [integer](/source/integer). The '''exponential sheaf sequence''' is therefore

:<math>0\to 2\pi i\,\mathbb Z \to \mathcal O_M\to\mathcal O_M^*\to 0.</math>

The exponential mapping here is not always a surjective map on sections; this can be seen for example when ''M'' is a [punctured disk](/source/punctured_disk) in the complex plane. The exponential map ''is'' surjective on the [stalks](/source/stalk_of_a_sheaf): Given a [germ](/source/germ_of_a_function) ''g'' of an holomorphic function at a point ''P'' such that ''g''(''P'')&nbsp;≠&nbsp;0, one can take the [logarithm](/source/logarithm) of ''g'' in a neighborhood of ''P''. The [long exact sequence](/source/long_exact_sequence) of [sheaf cohomology](/source/sheaf_cohomology) shows that we have an exact sequence

:<math>\cdots\to H^0(\mathcal O_U) \to H^0(\mathcal O_U^*)\to H^1(2\pi i\,\mathbb Z|_U) \to \cdots</math>

for any open set ''U'' of ''M''. Here ''H''<sup>0</sup> means simply the sections over ''U'', and the sheaf cohomology ''H''<sup>1</sup>(2π''i'''''Z'''|<sub>''U''</sub>) is the [singular cohomology](/source/singular_cohomology) of ''U''. 

One can think of ''H''<sup>1</sup>(2π''i'''''Z'''|<sub>''U''</sub>) as associating an integer to each loop in ''U''. For each section of ''O''<sub>''M''</sub>*, the connecting homomorphism to ''H''<sup>1</sup>(2π''i'''''Z'''|<sub>''U''</sub>) gives the [winding number](/source/winding_number) for each loop. So this homomorphism is therefore a generalized [winding number](/source/winding_number) and measures the failure of ''U'' to be [contractible](/source/contractible). 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

:<math>\cdots\to H^1(\mathcal O_M)\to H^1(\mathcal O_M^*)\to H^2(2\pi i\,\mathbb Z)\to \cdots.</math>

Here ''H''<sup>1</sup>(''O''<sub>''M''</sub>*) can be identified with the [Picard group](/source/Picard_group) of [holomorphic line bundle](/source/holomorphic_line_bundle)s on ''M''. The connecting homomorphism sends a line bundle to its first [Chern class](/source/Chern_class).

==References==
* {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[John Wiley & Sons](/source/John_Wiley_%26_Sons) | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523 | year=1994}}, see especially p.&nbsp;37 and p.&nbsp;139

{{DEFAULTSORT:Exponential Sheaf Sequence}}
Category:Complex manifolds
Category:Sheaf theory

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Adapted from the Wikipedia article [Exponential sheaf sequence](https://en.wikipedia.org/wiki/Exponential_sheaf_sequence) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Exponential_sheaf_sequence?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.