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

$\exp :{\mathcal O}_{M}\to {\mathcal O}_{M}^{*},$

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. The connecting 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(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

• 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