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 o 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"). Its kernel can be identified as the sheaf denoted by 2π"i"Z, meaning the sheaf on "M" of locally constant functions taking values which are 2π"in", with "n" an integer. The exponential sheaf sequence is therefore

:0 o 2pi i,mathbb Z o mathcal O_M omathcal O_M^* o 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 o H^0(mathcal O_U) o H^0(mathcal O_U^*) o H^1(2pi i,mathbb Z) o cdots

for any open set "U" of "M". Here "H"0 means simply the sections over "U"; while the sheaf cohomology "H"1 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 o H^1(mathcal O_M) o H^1(mathcal O_M^*) o H^2(2pi i,mathbb Z) o cdots.

Here "H"1("O""M"*) can be identified with the Picard group of holomorphic line bundles on "M". The homomorphism to the "H"2 group is essentially the first Chern class.


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