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Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the intersection form (or cup-product) on $H^1(X,\mathbb{Z})$. My question is: Is this the only principal polarization on $J(X)$? In other words, is there another unimodular, alternating, non-degenerate bilinear form on $H^1(X,\mathbb{Z})$ different from the cup-product (this will induce a different principal polarization on $J(X)$)?

I would think this is very basic, but I am not able find a good literature for this question. Any hint/reference will be most welcome.

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3 Answers 3

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It is possible for the Jacobian's of non-isomorphic curves to be isomorphic as abelian varieties, but obviously, not as principally polarized abelian varieties. This paper https://arxiv.org/pdf/math/0304471.pdf by Howe gives examples. Also from the same paper - "it has been known since the late 1800s that distinct curves can have isomorphic unpolarized Jacobians. I"

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  • $\begingroup$ @agniesky: Are you saying that the Jacobian of a fixed curve can have more than one principal polarization? That is my question. $\endgroup$
    – Jana
    Commented Aug 6, 2018 at 19:37
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    $\begingroup$ Yes, that is exactly what the answer says. To take another example, it is well-known that the Jacobian $J$ of the Klein quartic is isomorphic to $E^3$, where $E$ is the elliptic curve with complex multiplication by $i$; thus $J$ admits another principal polarization, which is reducible. $\endgroup$
    – abx
    Commented Aug 6, 2018 at 19:50
  • $\begingroup$ @abx: Is there any condition under which an abelian variety has an unique principal polarization? $\endgroup$
    – Jana
    Commented Aug 6, 2018 at 20:40
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    $\begingroup$ @Jana In general it is difficult to compute the number of ppal polarizations on an ab. variety. Generically, if it has one, it is the only one. But also, for any $N$ there exists $N$ genus 2 curves non isomorphic with all jacobians isomorphic. $\endgroup$
    – Xarles
    Commented Aug 6, 2018 at 22:20
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    $\begingroup$ There may be more than one curve, but there are only finitely many as proved by Narasimhan and Nori. $\endgroup$
    – Kapil
    Commented Dec 9, 2021 at 3:14
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If the Neron-Severi group of an abelian variety is $\mathbb{Z}$, then a principal polarisation, if it exists, is unique. This is the generic case, as well as for generic jacobians.

To find counter-examples, one can look for abelian varieties with an automorphism which acts non-trivially on the Neron-Severi group. The example suggested by abx has this form.

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The answer is no. Your question is equivalent to the question that is there only one curve on an abelian variety? I mean that let $C$ be a smooth, genus 2 curve on an abelian surface $A$. Then $J_C\simeq A$. Let $P_A$ be the set of isomorphism classes of smooth genus 2 curves on $A$. Then your question is that $|P_A|=1$? The answer is no. Moreover, it is unbounded. Hayashida and Nishi(1965,1968) raised this question and gave some partial results, i.e., formulae for $|P_A|$. Recently, E. Kani(2014,2016) gave some answer for this question. Check their related papers.

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