A New Proof of Torelli's Theorem

Abstract

gave a modern proof, valid in the abstract case, based on an idea of Andreotti. Other abstract proofs were later given by Matsusaka [3] and Andreotti [1]. The proof to be given here is based on a modification of two of Weil's lemmas which enables us to recover Torelli's theorem as a combinatorial consequence of the Riemann-Roch theorem and Abel's theorem. We begin by proving four preliminary lemmas of which the second and fourth may be characterized as modifications of Weil's Hilfssatze 3 and 1, respectively. Lemmas 2, 3, and 4 admit generalizations which, however, are not needed for our purposes. We denote, as usual, by Wr the translate of Wr by an element a e J(X). Following Weil [5], we denote by (Wr)* the image of Wr under the map u -u + (p(Z) where Z is a canonical divisor on X. We recall [2] that the sets Wr and (Wr)* are subvarieties of J(X). Our first lemma is a known result which we prove for convenience: