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 nonvanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphism
because for a holomorphic function f, exp(f) is a nonvanishing holomorphic function, and exp(f + g) = exp(f)exp(g). Its kernel is the sheaf 2πiZ of locally constant functions on M taking the values 2πin, with n an integer. The exponential sheaf sequence is therefore
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) ≠ 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
for any open set U of M. Here H^{0} means simply the sections over U, and the sheaf cohomology H^{1}(2πiZ_{U}) is the singular cohomology of U.
One can think of H^{1}(2πiZ_{U}) as associating an integer to each loop in U. For each section of O_{M}*, the connecting homomorphism to H^{1}(2πiZ_{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 nonvanishing holomorphic function, something that is always locally possible.
A further consequence of the sequence is the exactness of
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.
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Chapter 1a: Ordinary generating functions
Transcription
References
 Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 9780471050599, MR 1288523, see especially p. 37 and p. 139