Exponential sheaf sequence

In mathematics, the '''exponential sheaf sequence''' is a fundamental short exact sequence of sheaves used in complex geometry.

Let ''M'' be a complex manifold, and write ''O''_''M'' for the sheaf of holomorphic functions on ''M''. Let ''O''_''M''* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The 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 is the sheaf 2π''i'''''Z''' of locally constant functions on ''M'' taking the values 2π''in'', with ''n'' an 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 in the complex plane. The exponential map ''is'' surjective on the stalks: Given a germ ''g'' of an holomorphic function at a point ''P'' such that ''g''(''P'')&nbsp;≠&nbsp;0, one can take the logarithm of ''g'' in a neighborhood of ''P''. The long exact sequence of 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''⁰ means simply the sections over ''U'', and the sheaf cohomology ''H''¹(2π''i'''''Z'''|_''U'') is the singular cohomology of ''U''.

One can think of ''H''¹(2π''i'''''Z'''|_''U'') as associating an integer to each loop in ''U''. For each section of ''O''_''M''*, the connecting homomorphism to ''H''¹(2π''i'''''Z'''|_''U'') gives the winding number for each loop. So this homomorphism is therefore a generalized winding number and measures the failure of ''U'' to be 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''¹(''O''_''M''*) can be identified with the Picard group of holomorphic line bundles on ''M''. The connecting homomorphism sends a line bundle to its first 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 | 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