  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.
