Exponential sheaf sequence

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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 an holomorphic function f, exp(f) is a non-vanishing holomorphic 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\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; because given a germ g of an holomorphic 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

\cdots\to H^0(\mathcal O_U) \to H^0(\mathcal O_U^*)\to H^1(2\pi i\,\mathbb Z) \to \cdots

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 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 homomorphism to the H2 group is essentially the first Chern class.