Re: Understanding Serre duality [Riemann surface context]

On 2 abr, 05:15, jane <jane1...@xxxxxxxxxx> wrote:
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 ?)

That's just the dual sheaf.

Ah, so that F* becomes a sheaf of cotangent vectors on X, right ?, i am probably misunderstanding something, but this sheaf in general is bigger than the sheaf of 1-forms on X, and why do we have then

F* Omega = Q(x) ?

Thanks.

And how

-- m
.

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