Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $X^t \to X$.

share|improve this question

1 Answer 1

up vote 3 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).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.