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 OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphism
\( \exp :{\mathcal O}_{M}\to {\mathcal O}_{M}^{*}, \)
because for a holomorphic function f, exp(f) is a non-vanishing 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
\( 0\to 2\pi i\,{\mathbb Z}\to {\mathcal O}_{M}\to {\mathcal O}_{M}^{*}\to 0. \)
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
\( \cdots \to H^{0}({\mathcal O}_{U})\to H^{0}({\mathcal O}_{U}^{*})\to H^{1}(2\pi i\,{\mathbb Z}|_{U})\to \cdots \)
for any open set U of M. Here H0 means simply the sections over U, and the sheaf cohomology H1(2πiZ|U) is the singular cohomology 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 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
\( \cdots \to H^{1}({\mathcal O}_{M})\to H^{1}({\mathcal O}_{M}^{*})\to H^{2}(2\pi i\,{\mathbb Z})\to \cdots . \)
Here H1(OM*) 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
Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523, see especially p. 37 and p. 139
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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