# Cohomology of $Sym^m Q \otimes Sym^k Q \otimes L^p$

Let $V$ be a complex vector space. Let $L=\mathcal{O}(-1)$ and $Q=V/L$ be the quotient bundle over $\mathbb{P}V$.

I'm trying to compute the cohomologies with coefficients in $Sym^m Q \otimes Sym^k Q \otimes L^p$ and the basic thechniques lead to something I can not resolve (see Maps between products of symmetric powers).

Is there any standard way to compute this?

• Since the characteristic is zero, you can use the theory of irreducible representations of $\textbf{SL}_n$ to rewrite $\text{Sym}^m(Q)\otimes \text{Sym}^k(Q)$ as a direct sum of other Schur functors applied to $Q$. Then you can use Bott vanishing / Borel-Weil-Bott to compute the global sections of these summands. – Jason Starr Oct 25 '15 at 22:36