Exponential sheaf sequence

In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex manifold theory. Let M be a complex manifold, and write

O_M

for the sheaf of analytic functions on M. Let

O_M*

the subsheaf consisting of the non-vanishing analytic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphism

exp:O_M → O_M*

because for an analytic function f, exp(f) is a non-vanishing analytic function, and

exp(f+g) = exp(f)exp(g).

It can be shown that this homomorphism is a surjection. Its kernel can be identified as the sheaf denoted by

2πiZ,

meaning the sheaf on M of locally constant functions taking values which are 2πin, with n an integer. The exponential sheaf sequence is therefore

0 → 2πiZ → O_M → O_M* → 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; because given a germ g of an analytic function at a point P, such that g(P) ≠ 0, one can take the logarithm of g close enough to P. The long exact sequence of sheaf cohomology shows that we have an exact sequence

... → H⁰(O_U) → H⁰(O_U*) → H¹(2πiZ) → ...

for any open set U of M. Here H⁰ means simply the sections over U; while the sheaf cohomology H¹ in this case is essentially the singular cohomology of U. Therefore there is a kind of winding number invariant: if U is not contractible, the exponential map on sections may not be surjective. In other words, there is a potential topological obstruction to taking a global logarithm of a non-vanishing analytic function, something that is always locally possible.

A further consequence of the sequence is the exactness of

... → H¹(O_M) → H¹(O_M*) → H²(2πiZ) → ... .

Here H¹(O_M*) can be identified with the Picard group of holomorphic line bundles on M. The homomorphism to the H² group is essentially the first Chern class.