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Let X a projective variety over an algebraically closed field on which an abelian variety $A$ acts freely. Then $X/A$ is a projective variety. Let $f:X\rightarrow X/A$

Let $K\in D_{c}^{\leq 0}(X,\bar{\mathbb{Q}}_{l})$ a complex of sheaves not necessarily pure.

We assume that $K$ is equivariant with respect to $A$.

Under which conditions, can we descend $K$ to a complex $K'$ on $X/A$ such that $f^{*}K'=K$.

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What exactly do you mean by "$K$ is equivariant with respect to $A$" if $K$ is just an object in the derived category? Is $a^* K$ required to be isomorphic, or just quasi-isomorphic to $K$? – Piotr Achinger Dec 11 '12 at 22:22
They are isomorphic. – prochet Dec 12 '12 at 0:54
Then I guess this is just a descent question for $\bar\mathbb{Q}_l$-sheaves and morphisms between them, and the derived category doesn't play a role here. I think this should be true and that you should just push $K$ forward to $X/A$ and take invariants. – Piotr Achinger Dec 12 '12 at 11:22
but with your argument I have the impression that everyhing descend without any hypotheses, it's a bit suspect. – prochet Dec 12 '12 at 18:09

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