# Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold

Let $$X$$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $$X\rightarrow\mathrm{Gr}(2,5):=G$$. Consider the normal bundle $$\mathcal{N}_{X|G}$$, how to compute cohomology of this vector bundle and related vector bundle, i.e, what is $$H^k(X,\mathcal{N}_{X|G}\otimes\mathcal{O}_X(2))$$, $$H^k(X,\mathcal{N}_{X|G})?$$

Let $$T_X$$ be the tangent bundle of $$X$$($$X$$ is general OGM threefold or more general, Fano threefold of index 1, Picard rank 1), how to compute $$H^0(X,\bigwedge^2T_X)$$? I was trying to use normal bundle exact sequence and some other exact sequence involving tangent bundle on $$X$$, but I could not decide which one I should use?

$$\mathcal{N}_{X|G}$$ is in your case $$F|_X$$, where $$F=O_G(1)^2 \oplus O_G(2))$$. In order to compute the first two spaces, you can simply use the Koszul complex for X $$0 \to det(F^{\vee}) \to \wedge^2 F^{\vee} \to F^{\vee} \to O_G \to O_X \to 0,$$ twisted with $$F$$ (or $$F(2)$$ in the second case).
You can use Borel-Bott-Weil to compute all the terms needed above. Saving you the trouble, the cohomologies you are asking for are 46 dimensional for $$H^0(\mathcal{N}_{X|G})$$, 313 dimensional for $$H^0(\mathcal{N}_{X|G}(2))$$, 0 for all other $$k$$.
Answering your second question. $$X$$ is a Fano threefold of index 1, therefore $$\wedge^2 T_X \cong \Omega^1_X(1)$$. You can use the (twisted) cotangent sequence to compute the dimension of the latter. You have $$0 \to O_X^2 \oplus O_X(-1) \to \Omega^1_G|_X(1) \to \Omega^1_X(1) \to 0$$
Clearly $$H^1( O_X^2 \oplus O_X(-1))=0$$. On the other hand you can again use Koszul to compute the cohomology groups of the central object. If I made no mistakes, $$H^0(\Omega^1_G|_X(1))= \mathbb C^2$$. Therefore $$H^0(\wedge^2 T_X)=0$$.