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One knows that every Abelian variety is a quotient of a Jacobian. Does every Abelian variety have a finite resolution by Jacobians?

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I believe this is still open, although I would also be interested in the precise status. –  Donu Arapura Feb 12 '12 at 17:30
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Do you have a specific motivation for this question ? For instance, are there examples of statements about abelian varieties that could be reduced to jacobians in this way ? –  Olivier Benoist Aug 1 '12 at 17:28

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