Let $A$ be an abelian variety defined over $\mathbf{C}$ (of dimension $>1$) and let $\Theta_A$ be the holomorphic tangent sheaf of $A$.

Question. How does one compute $H^1(A,\Theta_A)$ ?

If $A$ has dimension $1$ then using Serre's duality one finds that $H^1(A,\Theta_A)\simeq H^0(A,\omega_A^2)$ where $\omega_A$ is the canonical line bundle of $A$. Since $\omega_A\simeq\mathcal{O}_A$ one finds that $h^1(A,\Theta_A)=h^0(A,\mathcal{O}_A)=1$.