- Exponential sheaf sequence
mathematics, the exponential sheaf sequence is a fundamental short exact sequenceof 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 functiongives a sheaf homomorphism
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
The exponential mapping here is not always a surjective map on sections; this can be seen for example when "M" is a
punctured diskin 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 logarithmof "g" close enough to "P". The long exact sequenceof sheaf cohomologyshows that we have an exact sequence
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 cohomologyof "U". Therefore there is a kind of winding numberinvariant: 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
Here "H"1("O""M"*) can be identified with the
Picard groupof holomorphic line bundles on "M". The homomorphism to the "H"2 group is essentially the first Chern class.
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