How can I calculate $H^i(Gr(2,7),\mathcal{O}_{Gr(2,7)})$ over the base field $\mathbb{C}$?
Or in general $H^i(Gr(k,n),\mathcal{O}_{Gr(k,n)})$?
The first idea came to my mind is using Plücker embedding $i:Gr(2,7)\hookrightarrow\mathbb{P}^{20}$ and the long exact sequence associated to the ideal sheaf exact sequence $$0\rightarrow\mathcal{I}_{Gr(2,7)}\rightarrow\mathcal{O}_{\mathbb{P}^{20}}\rightarrow{i_*}\mathcal{O}_{Gr(2,7)}\rightarrow{0}\,.$$
But I don't know how to calculate $H^i(Gr(2,7),\mathcal{I}_{Gr(2,7)})$ either.
Best Answer
Question: "How can I calculate $H^i(Gr(2,7),\mathcal{O}_{Gr(2,7)})$ over the base field $\mathbb{C}$?"
Answer: Let $K$ be a field of characteristic zero and let $V:=K^n$ be an $n$-dimensional vector space over $K$ and $W \subseteq V$ a $k$-dimensional subspace. You may let $P \subseteq SL(V)$ be the (parabolic) subgroup of elements fixing $W$. There is a "canonical" isomorphism $SL(V)/P \cong G(k,V)$ where $G(k,V)$ is the grassmannian variety of $k$-dimensional subspaces of $V$. If you look in Fulton-Harris book "Representation theory: A first course" in Section 15.4 you will find a relation between global sections of invertible sheaves $\mathcal{L}$ on $G(k,V)$ and $SL(V)$-modules: For any $\mathcal{L} \in Pic(G(k,V))$ it follows $H^i(G(k,V), \mathcal{L})$ is an irreducible $SL(V)$-module. The book also gives explicit formulas for the dimension $h^i(G(k,V), \mathcal{L})$. If $\mathcal{L}:=\mathcal{O}$ is the structur sheaf, the book gives explicit formulas for $h^i(G(k,V), \mathcal{O})$.
Example 1.If $K$ is a field of characteristic zero and of $dim_K(V)=n$, let $G(k,V)$ be the grassmannian of $k$-dimensional subspaces in $V$. There is the Plucker embedding
$$ \rho: G(k,V) \rightarrow \mathbb{P}:=\mathbb{P}(\wedge^k V^*)$$
sending a subspace $W \subseteq V$ to the line $\wedge^k W \subseteq \wedge^k V$. The space of homogeneous polynomials of degree $m$ on $\mathbb{P}$ is the $m$'th symmetric power $Sym^m(\wedge^k V^*)$. Let $I(G)_m$ be the polynomials of degree $m$ vanishing on $G(k,V)$. We get an exact sequence of $SL(V)$-modules
$$0 \rightarrow I(G)_m \rightarrow Sym^m(\wedge^k V^*) \rightarrow W_m \rightarrow 0.$$
In Exercise 15.43 they calculate the $SL(V)$-module $I(G)_2$: $SL(V)$ is a semi simple linear algebraic group and any finite dimensional $SL(V)$-module decompose as a direct sum of irreducible modules. Such a decomposition is given in the exercise. In the book they give an elementary introduction to the representation theory of $SL(V), GL(V),..$ and give explicit and elementary examples from geometry.