# Pushforward of tensor product of holomorphic vector bundles

Let $M$ be a complex manifold, $\Sigma$ a Riemann surface of genus $g$, and $E, F$ holomorphic vector bundles on $M\times\Sigma$. If $\pi: M\times \Sigma\to M$ is the projection map, is it possible to express the pushforward $\pi_!(E\otimes F)$ in terms $\pi_!(E)$ and $\pi_!(F)$?

• Where do your vector bundles live? – abx Apr 5 '17 at 12:17
• @abx Thanks for asking. They live on $M\times\Sigma$. I have just made the correction. – No_way Apr 5 '17 at 12:18
• Take for $M$ the Jacobian of $\Sigma$. Let $E$ be the Poincaré line bundle (such that $E_{|\{\alpha \}\times \Sigma }=\alpha$), and let $F:=E^{-1}$. Then $\pi _*E=\pi _*F=0$, but $\pi _*(E\otimes F)=\mathcal{O}_M$. Variations on this theme give all sorts of vector bundles for $\pi _*(E\otimes F)$, still with $\pi _*E=\pi _*F=0$. – abx Apr 5 '17 at 13:46
• ... or take $M$ to be a point, $E=$ a line bundle of degree one with no sections. Some tensor power $F=E^{\otimes k}$ will have sections. – Piotr Achinger Apr 5 '17 at 14:03