Its kernel is the sheaf 2πiZ of locally constant functions on M taking the values 2πin, with n an integer.
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 H0 means simply the sections over U, and the sheaf cohomology H1(2πiZ|U) is the singular cohomology of U.
For each section of OM*, the connecting homomorphism to H1(2πiZ|U) gives the winding number for each loop.
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 H1(OM*) 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.