Global sections for divisors on products of varieties

Question

Let $X$ and $Y$ be normal varieties with $D$ and $E$ Cartier divisors on $X$ and $Y$, respectively. Let $(D,E)$ denote the divisor $\pi_X^*(D)+\pi_Y^*(E)$ on the product $X\times Y$, where $\pi_X$ and $\pi_Y$ are the projections from $X\times Y$.

There is a natural inclusion $H^0(X,\mathcal{O}(D))\otimes H^0(Y,\mathcal{O}(E))\to H^0(X\times Y,\mathcal{O}((D,E)))$ induced by the inclusion $K(X)\otimes K(Y)\to K(X\times Y)$.

Under what conditions is this inclusion an isomorphism? This holds for example if $X$ and $Y$ are toric.

Answer 1

This is always true, it follows from the Künneth formula for coherent sheaves (see Kunneth formula for sheaf cohomology of varieties).