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Riemann reciprocity in higher dimensions

Yakov Karpishpan
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Abstract

The reciprocity law for abelian differentials of first and second kind is generalized to higher-dimensional varieties. It is shown that $H^1(V)$ of a polarized variety $V$ is encoded in the Laurent data along a curve germ in $V$, with the polarization form on $H^1(V)$ corresponding to the {\em one-dimensional} residue pairing. This associates an {\em extended abelian variety} to $V$; if $V$ is an abelian variety itself, our construction ``extends" it, even when $V$ is not a Jacobian.

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Yakov Karpishpan. 1994-09-09. Riemann reciprocity in higher dimensions. https://arxiv.org/abs/alg-geom/9409004

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