Suppose I have a closed subvariety $V$ of $\mathbb P^n \times \mathbb A^m$ given by explicit equations. Is it possible in practice to compute its coherent cohomology with coefficients in a line bundle pulled back from $\mathbb P^n$ using the Cech complex for the open affine cover of $V$ by intersections of $V$ with the products of the standard affine open subsets of $\mathbb P^n$ with $\mathbb A^m$ (similarly to how cohomology of the projective space with coefficients in a line bundle is computed)? If not, how to compute this cohomology?

Suppose I have a closed subvariety $V$ of $\mathbb P^n \times \mathbb A^m$ given by explicit equations. Is it possible in practice to compute the coherent cohomology of $V$ with coefficients in a line bundle pulled back from $\mathbb P^n$ using the Cech complex for the open affine cover of $V$ by intersections of $V$ with the products of the standard affine open subsets of $\mathbb P^n$ with $\mathbb A^m$ (similarly to how cohomology of the projective space with coefficients in a line bundle is computed)? If not, how to compute this cohomology?