Re: Understanding Serre duality [Riemann surface context]

On Apr 1, 11:47 am, jane <jane1...@xxxxxxxxxx> wrote:
I would be grateful if someone could explain me the
following:

Let X be a Riemann surface, Q(X) = the space of
mremorphic quadratic differentials on X with only
simple poles.
Theta - sheaf of holomorphic vector fields on X.

How exactly one can understand that

H^1(X, Theta)* isomorphic Q(x)

I know this should be a consequence of the Serre
duality, but i don't see how exactly. Let me state
the Serre duality theorem i know:

H^1(X, O)* isomorphic to H^0(X, Omega),

where O is the sheaf of holomorphic functions on X
and Omega is the sheaf of meromorphic 1-forms on X.

Thanks a lot in advance,

There is a slightly more general version, which
states that

H^1(X, F)* = H^0(X, F* tensor Omega)

Thanks a lot for the answer. So you suggest take F to be the sheaf of holomorphic vector fields on X.
What does the notation F* tensor mean, what you denote by F* (pullback under what ?)

and could you please explain how do we get as F* tensor Omega = Q(x), the space of integrable quadratic forms ?

Thanks.





for a locally free sheave F on X, and for example
X projective and smooth.

-- m
.