Following thinking about a question from math overflow (and answering it https://math.stackexchange.com/a/4686391/299848) I was wondering about the topic:
Given the Segre embedding $\sigma: \mathbb{P}^m \times \mathbb{P}^n \hookrightarrow \mathbb{P}^N$, where $N = mn + m + n$, I am interested in understanding the pushforward of arbitrary coherent sheaves under this embedding, especially in terms of K-theory on $\mathbb{P}^N$.
It is well-known that for line bundles, we can compute the pushforward $\sigma_* (\mathcal{O}{\mathbb{P}^m}(a) \boxtimes \mathcal{O}{\mathbb{P}^n}(b))$ and express it in terms of K-theory on $\mathbb{P}^N$. However, for arbitrary coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $\mathbb{P}^m$ and $\mathbb{P}^n$, respectively, I am wondering:
How can we compute the pushforward $\sigma_*(\mathcal{F} \boxtimes \mathcal{G})$ under the Segre embedding? What is the expression for this pushforward in terms of K-theory on $\mathbb{P}^N$? Any insights, references, or suggestions for techniques to tackle this problem would be greatly appreciated.