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.

  • $\begingroup$ 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? $\endgroup$ – Alan Muniz Oct 9 '18 at 21:33
  • $\begingroup$ @AlanMuniz: And where precisely is your problem? $\endgroup$ – Sasha Oct 10 '18 at 4:20
  • $\begingroup$ 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 $T_{X/C}^\vee\otimes p^*T_C$. $\endgroup$ – Alan Muniz Oct 10 '18 at 11:10
  • $\begingroup$ 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 $T_{X/C}^\vee\otimes p^*T_C$. $\endgroup$ – Alan Muniz Oct 10 '18 at 11:12
  • $\begingroup$ However I could not figure out the cohomology of $T_{X/C}\otimes p^*T_C^\vee$. $\endgroup$ – Alan Muniz Oct 10 '18 at 11:14

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