- Exponential sheaf sequence
In

mathematics , the**exponential sheaf sequence**is a fundamentalshort exact sequence of sheaves used incomplex geometry .Let "M" be a

complex manifold , and write "O"_{"M"}for the sheaf ofholomorphic function s on "M". Let "O"_{"M"}* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves ofabelian group s. Theexponential 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" oflocally constant function s taking values which are 2π"in", with "n" aninteger . 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 thelogarithm of "g" close enough to "P". Thelong exact sequence ofsheaf 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 thesingular cohomology of "U". Therefore there is a kind ofwinding number invariant: if "U" is notcontractible , 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 thePicard group ofholomorphic line bundle s on "M". The homomorphism to the "H"^{2}group is essentially the firstChern class .

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