Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T\_X),H^1(X,T\_X),H^2(X,T\_X)$? Here $T\_X$ is the tangent sheaf of $X$.
For instance $h^0(X,T\_X)$ gives the dimension of the automorphism group of $X$.



Cohomology of tangent sheaf of a hypersurface
Use the normal sequence
$$
0 \to T\_X \to T\_{\mathbb{P}^n}\vert\_X \to N\_{X/\mathbb{P}^n} \to 0,
$$
exact sequences
$$
0 \to \mathcal{O}\_{\mathbb{P}^n} \to \mathcal{O}\_{\mathbb{P}^n}(d) \to i\_\*N\_{X/\mathbb{P}^n} \to 0
$$
(we identify here $N\_{X/\mathbb{P}^n}$ with $\mathcal{O}\_X(d)$ and denote by $i$ the embedding $X \to \mathbb{P}^n$) and
$$
0 \to T\_{\mathbb{P}^n}(-d) \to T\_{\mathbb{P}^n} \to i\_\*(T\_{\mathbb{P}^n}\vert\_X) \to 0,
$$
and Borel-Bott-Weil Theorem to compute cohomology on $\mathbb{P}^n$.


