# Computation of cohomology of $End(TX)$ for a ruled surface $X$.

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?

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.
• Thanks for the answer. Using this so far I coud prove that $h^2(End(TX)) = 3g(C)-3$, But I'm stuck with $H^0$ and $H^1$. Any idea? – Alan Muniz Oct 9 '18 at 21:33
• Tensorizing $0 \to T_{X/C} \to T_X \to p^*T_C \to 0$ with other loc free sheaves it boils down to computing the cohomology of $T_{X/C}\otimes p^*T_C^\vee$ and its dual $T_{X/C}^\vee\otimes p^*T_C$. – Alan Muniz Oct 10 '18 at 11:10
• Supposing that $e< 2g(C)-2$ and $g(C)>1$ we have $T_{X/C}\otimes p^*T_C^\vee$ ample and we can use Kodaira vanishing to its dual. Riemann-Roch completes the calculation for $T_{X/C}^\vee\otimes p^*T_C$. – Alan Muniz Oct 10 '18 at 11:12
• However I could not figure out the cohomology of $T_{X/C}\otimes p^*T_C^\vee$. – Alan Muniz Oct 10 '18 at 11:14