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Let $A$ be a principally polarized abelian variety. Let $X\subset A$ be an abelian subvariety. Is $X$ also principally polarized? Here's what I think should be a proof. Is it correct?

We may and do assume $A= X\times Y$, where $Y$ is also an abelian variety. By assumption, there is an isomorphism $A\to A^t$, where $A^t$ is the dual of $A$. Now, the composed morphism $X\to X\times Y \to (X\times Y)^t = X^t\times Y^t \to X^t$ is an isomorphism. In fact, it is easily seen to be injective. For surjectivity, you dualize this construction and obtain a morphism $X^t \to X$.

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up vote 4 down vote accepted

No, a subvariety of a principally polarized abelian variety need not be principally polarizable. This follows from:

1) Not every abelian variety is principally polarizable. (E.g. there are abelian varieties which are not isomorphic to their dual abelian variety.)

2) By the Zarhin trick, for any abelian variety $A$, $(A \times A^{\vee})^4$ is principally polarizable.

With regard to your argument: it is not true that any subvariety of an abelian variety is necessarily a direct factor. This is only true up to isogeny (and, indeed, every abelian variety over an algebraically closed field is isogenous to a principally polarized abelian variety).

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