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 $\pi_!(F)$?
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1$\begingroup$ Where do your vector bundles live? $\endgroup$ – abx Apr 5 '17 at 12:17

$\begingroup$ @abx Thanks for asking. They live on $M\times\Sigma$. I have just made the correction. $\endgroup$ – No_way Apr 5 '17 at 12:18

2$\begingroup$ I'm afraid the answer to your question is your username... $\endgroup$ – Piotr Achinger Apr 5 '17 at 12:49

2$\begingroup$ 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 $\pi _*E=\pi _*F=0$. $\endgroup$ – abx Apr 5 '17 at 13:46

3$\begingroup$ ... 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. $\endgroup$ – Piotr Achinger Apr 5 '17 at 14:03