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Let $A$ be an abelian variety (at least over $\mathbb{C}$). Suppose we have two theta divisors $\Theta_1$ and $\Theta_2$ on $A$, which give two principal polarizations on $A$.

In general, are those two divisors the same, up to something? In other words, is a principal polarization unique, if exists?

Edit: the question was edited following comments.

Thanks for help!

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    $\begingroup$ Indeed, the theta divisor is unique up to translation, essentially by definition. See any book on abelian varieties, for instance Birkenhake-Lange. $\endgroup$
    – abx
    Commented Mar 4, 2020 at 15:54
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    $\begingroup$ The line bundle satisfies $h^0(\mathscr{O}_A(\Theta ))=1$, hence it corresponds to a unique effective divisor. And a polarization defines the corresponding line bundle up to translation. $\endgroup$
    – abx
    Commented Mar 4, 2020 at 16:29
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    $\begingroup$ Are you asking whether an abelian variety has a unique principal polarization, if one exists? The answer is no. You'll find examples by searching for non-isomorphic curves with isomorphic jacobians, for instance. $\endgroup$
    – Felipe Voloch
    Commented Mar 4, 2020 at 18:44
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    $\begingroup$ I think maybe the terminology is confusing. Are you using theta divisor as a synonym for a divisor that gives a principal polarization on an arbitrary abelian variety? Then you'll find lots of counterexamples that are not algebraically equivalent. Or are you using it to be a translate of the image of $C^{g-1}$ in its Jacobian variety? $\endgroup$
    – Joe Silverman
    Commented Mar 5, 2020 at 15:08
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    $\begingroup$ By M.S. NARASIMHAN AND M. V. NORI, Polarizations on an abelian variety, Proc. Indian Accad. Sci. Math. Sci. 90 (1981) No. 2, 125-128. any abelian variety admits only a finite number of principal polarizations up to isomorphism. To compute the number of these isomorphism classe, see "Abelian varieties with several principal polarizations" by Herbert Lange available at projecteuclid.org/euclid.dmj/1077306167 $\endgroup$
    – Hacon
    Commented Mar 6, 2020 at 20:03

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