Exponential sheaf sequence explained

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 : l O _{M} \to l 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 Z \to l O _{M\tolO} ^{*\to} ^{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: 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

… \to H ^{0 ( l O} _{U)} \to H ^{*)\to} _{U} H ^{1(2\pi} i Z| _{U)} \to …

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 O_M*, 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

… \to H ^{1 ( l O} _{M)\to} H ^{*)\to} _{M} H ^{2(2\pi} i Z)\to … .

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

• , see especially p. 37 and p. 139