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



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> **Question.** How does one compute $H^1(A,\Theta\_A)$ ?
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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 
$h^1(A,\Theta\_A)=h^0(A,\mathcal{O}\_A)=1$.



complex deformations of abelian varieties
Hugo,


Although this was already discussed in the comments, perhaps I can write few more details
here. The material can be found in many books such as Mumford's Abelian Varieties or
the book on the same by Birkenhake and Lange.


Claim $\dim H^1(A,\Theta)= g^2$.


The first thing to observe is that $A$ is a group, so a basis for the tangent space
at $0$ can be translated to give a global basis. Thus the tangent bundle 
$\Theta=\mathcal{O}\_A^g$ where $g=\dim A$. Thus $H^1(A,\Theta)= H^1(A,\mathcal{O}\_A)^g$.
So this reduces the claim to checking $\dim H^1(A,\mathcal{O}\_A)=g$. For this, let me use
the Hodge theorem (alternatives can be found in the above refs.).
Write $A$ as the
quotient of $\mathbb{C}^g$ by a lattice. The Euclidean metric induces a Kähler metric
on $A$, with respect to which $H^1(A,\mathcal{O}\_A)$ can be realized as the space
of harmonic forms of type $(0,1)$. These are necessarily invariant under the group, because the metric is. $\lbrace d\bar z\_1,\ldots, d\bar z\_g\rbrace$ give a basis for the invariant $(0,1)$-forms, and they are clearly harmonic. So this proves the claim.


Finally, by Kodaira-Spencer, $H^1(A,\Theta)$ is the space of first order deformations of $A$. As noted above, the moduli space of principally polarized abelian varieties has
dimension only $g(g+1)/2$. Which means that roughly half these deformations 
are nonalgebraic!


