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

${\displaystyle \exp$ :{\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

${\displaystyle 0\to 2\pi i\,\mathbb {Z} \to {\mathcal {O}}_{M}\to {\mathcal {O}}_{M}^{*}\to 0.}$

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

${\displaystyle \cdots \to H^{0}({\mathcal {O}}_{U})\to H^{0}({\mathcal {O}}_{U}^{*})\to H^{1}(2\pi i\,\mathbb {Z} |_{U})\to \cdots }$

for any open set U of M. Here 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

${\displaystyle \cdots \to H^{1}({\mathcal {O}}_{M})\to H^{1}({\mathcal {O}}_{M}^{*})\to H^{2}(2\pi i\,\mathbb {Z} )\to \cdots .}$

Here H¹(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