I want to compute few sheaf cohomology groups of sub tori of a complex tori and having some trouble doing it. Could any of you give suggestion on how to compute the cohomology groups? Those are what I want to compute
For 3 complex dimensional tori X, let D be a 2 complex dimensional tori, which is also a divisor in X. Then how could I compute $$ H^{\bullet}(D,K_X)$$ where$ K_X$ is the canonical bundle of the 3 dimensional complex tori X.
For a question I asked, I first thought that I could perhaps use a short exact sequence $$ 0\rightarrow \mathcal{O}_X(-D)\rightarrow \mathcal{O}_X\rightarrow \mathcal{O}_D\rightarrow 0 $$ by tensoring $ K_X$ to the short exact sequence. But then, I do not know the cohomology groups $$ H^\bullet (X, \mathcal{O}_X(-D))$$ and $$ H^\bullet (X, \mathcal{O}_X(-D)\otimes K_X)$$
Could you give me any suggestion regarding problem I have?
Thank you in advance