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?



Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold
$\mathcal{N}\_{X|G}$ 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 $H^0(\wedge^2 T\_X)=0$.


You can try to do similar computations for all other Fano 3-folds. Embedding in Grassmannians (or at most weighted projective spaces) are known. Therefore the computations might be longer, but the strategy is the same.


