Exponential sheaf sequence
# 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 $H^0$ 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 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 $H^1(O_M*)$ 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.

