# How can I compute the cotangent sheaf for a projective variety?

Given a projective variety $X \subset \mathbb{P}^n_\mathbb{C}$, how can I compute the cotangent sheaf? Is it just the sheafification of the kahler differentials? For example, if I take the K3 surface $$X = \textbf{Proj}(R) = \textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(x^4 + y^4 + z^4 + w^4)} \right)$$ then the kahler differentials of the underlying graded ring is the graded module $$\frac{Rdx \oplus Rdy \oplus Rdz \oplus Rdw}{x^3dx + y^3dy + z^3dz + w^3dw}$$ Given a finite presentation of a smooth projective variety $S/(f_1,\ldots, f_k)$, the diffrerential of the $f_i$'s should always be homogeneous.

• Does your description work locally? – Mariano Suárez-Álvarez Dec 23 '16 at 23:29
• Do you mean in an affine chart? – 54321user Dec 23 '16 at 23:38
• For example. ${}{}$ – Mariano Suárez-Álvarez Dec 23 '16 at 23:56

The answer is yes. The sheaf $\Omega_X^1$ is the sheafification of the module $\Omega_{S/\mathbb C}^1$, where $S$ is the homogeneous coordinate ring of $X$.
If you want to compute the cohomology of $\mathbb \Omega_X^1$, you can use standard exact sequences:
The Euler sequence: $$0 \to \Omega_{\mathbb P}^1|X \to \mathscr O_X(-1)^{N+1} \to \mathscr O_X \to 0.$$ The cotangent sequence: $$0 \to \mathscr I/\mathscr I^2 \to \Omega_{\mathbb P}^1|X \to \Omega_X^1 \to 0.$$ Note that if $X$ is a hypersurface of degree $d$, then $\mathscr I/\mathscr I^2 = \mathscr O_X(-d)$.
Tensoring the ideal sequence we also get the sequence (in the case of a hypersurface; the general case is similar): $$0 \to \Omega_{\mathbb P}^1(-d) \to \Omega_{\mathbb P}^1 \to \Omega_{\mathbb P}^1|X \to 0.$$
Now playing around with long exact sequences should give you all the information needed to compute the cohomology of $X$.
• Oh, so if I evaluate $\Omega^1_X$ on one of the affine opens $U_i$ by inverting one of the coordinates, this will be isomorphism to the localization of $\Omega^1_{S/\mathbb{C}}$ – 54321user Dec 24 '16 at 20:03