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I'm studying an example in book Yuji Shimizu and Kenji Ueno. Advances in Moduli Theory. Translations of Mathematical Monographs, vol. 206, that shows the importance of isomorphism as principally polarized abelian varieties in Torelli's theorem.

I'm studying the following example:

For a compact Riemann surface $R$ of genus $2$, $W^1=\varphi(R)$ is isomorphic to $R$, where $\varphi:R \longrightarrow J(R)$ is the Abel map. Hence, in this case the theta divisor $\Theta$ is also a compact Riemann surface of genus $2$. Hence the Jacobian variety $J(R)$ contains a compact Riemann surface $C:=\Theta- [k]$ of genus 2, and $(J(R), [C])$ gives a principal polarization. Let $E$ be a elliptic curve. Suppose that the self product $E \times E$ contains a compact Riemann surface of genus 2 and that $E \times E$ is isomorphic to a two-dimensional abelian variety $A$. In this case $(J(C), [C])\cong (A, [C])\cong (E\times E, [C])$; isomorphisms as principally abelian varieties. On the other hand, for points $a, b \in E$ a divisor $D = a \times E+E \times b$ is ample and $(E \times E, [D])$ is also a principally polarized abelian variety. But $(E\times E, [C])$ and $(E\times E, [D])$ are not isomorphic as principally polarized abelian varieties.

A first question is: For $(E \times E, [D])$ to be a principally polarized abelian variety, it should not be that $h^0(D)=1$? If so, how do I conclude that $h^0(D)=1$?

And another question is: Why $(E\times E, [C])$ and $(E\times E, [D])$ are not isomorphic as principally polarized abelian varieties? How was this seen so quickly?

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    $\begingroup$ forgive me if not precise, but just off the top of my head, first question: use riemann roch (i.e.compute intersection numbers); second question: the theta divisors are not isomorphic. $\endgroup$
    – roy smith
    Commented Nov 12, 2018 at 2:06
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    $\begingroup$ For the first question, it is more direct to use the Künneth formula. $\endgroup$
    – abx
    Commented Nov 12, 2018 at 5:40
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    $\begingroup$ As abx nicely observes, that h^0(D) = 1 (and that h^j(D) = 0 for j>0), are seen most clearly and directly here by the Kunneth formula. But if you are curious how someone knew immediately that D (and C) defines a principal polarization, they may also have mentally calculated an intersection number, since an effective divisor on an abelian variety of dimension g is a principal polarization iff its self intersection number is g! This can be seen immediately for D, (and for C by the adjunction formula). The cohomology calculations follow in general, but for D Kunneth just lays them out. $\endgroup$
    – roy smith
    Commented Nov 13, 2018 at 0:28
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    $\begingroup$ About the second question, may be it is possible to use that the Riemann period matrix of the product of two elliptic curves is diagonal, while for a genus 2 curves is not (but I am not sure how to show this "elementarily"). $\endgroup$
    – Xarles
    Commented Nov 13, 2018 at 8:21

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