# 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

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 2π*i***Z** of locally constant functions on *M* taking the values 2π*in*, with *n* an integer. The **exponential sheaf sequence** is therefore

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

for any open set *U* of *M*. Here *H*^{0} means simply the sections over *U*, and the sheaf cohomology *H*^{1}(2π*i***Z**|_{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

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.

## 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