Let $(A,\Theta)$ be a ppav such that $A=X\times Y$ for $X$ and $Y$ complementary abelian subvarieties. Suppose that $\mathrm{Hom}(X,Y)=\{0\}$. I'm trying to prove the following: Let $B\subset A$ be an abelian sub-variety of $A$, then $B\subset X$ or $B\subset Y$.

I know there is a bijection between the abelian subvarieties of $A$ and the abelian subvarieties of $X\times Y$. Therefore, I just need to describe all abelian sub-varieties of $X\times Y$, and I think they are:

  • $X\times Y$
  • $X\times \{0\}$
  • $\{0\}\times Y$
  • $B_X\times\{0\}$
  • $\{0\}\times B_Y$ where $B_X$ is an abelian sub-variety of $X$ and $B_Y$ is an abelian sub-variety of $Y$. am I correct? I'm not sure how to verify this.


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