Let $C$ be a complex projective curve curve, let $E \longrightarrow C$ be a rank two vector bundle and let $X = \mathbb{P}(E)$ be the associated ruled surface. Then define the locally free sheaf $End(TX)$ of bundle endomorphisms of the tangent bundle $TX$.
My question is whether the cohomology groups of $End(TX)$ are already described (in terms of $E$ and $C$) in the literature. If not, how can we compute them?
Best Answer
First, $$ H^i(X,End(T_X)) = Ext^i(T_X,T_X). $$ Second, there is a natural exact sequence $$ 0 \to T_{X/C} \to T_X \to p^*T_C \to 0, $$ where $p:X \to C$ is the projection. Finally, we have relative Euler sequence $$ 0 \to O_X \to p^*E \otimes O_X(1) \to T_{X/C} \to 0, $$ hence $$ T_{X/C} \cong p^*\det(E) \otimes O_X(2). $$ Combining all this and using long exact sequences of $Ext$'s one can compute the required cohomology groups.