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". 
One can think of "H"1(2π"iZ|"U") as associating an integer to each loop in "U". For each section of "O""M"*, the connecting homomorphism to "H"1(2π"iZ|"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
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.