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 $X^t \to X$.