A Special Case of the Γ00 Conjecture

Abstract

In this paper we prove the Γ 00 conjecture of van Geemen and van der Geer [8] under the additional assumption that the matrix of coefficients of the tangent has rank at most 2 (see theorem 1 for a precise formulation). This assumption is satisfied by Jacobians (see proposition 1), and thus our result gives a characterization of the locus of Jacobians among all principally polarized abelian varieties.The proof is by reduction to the (stronger version of the) characterization of Jacobians by semidegenerate trisecants, i.e. by the existence of lines tangent to the Kummer variety at one point and intersecting it in another, proven by Krichever in [16] in his proof of Welters' [20] trisecant conjecture.

The Schottky Problem

“…However, in order to obtain a solution of the Schottky problem, proving the Γ 00 conjecture for a generic ppav is insufficient -a generic ppav is not a Jacobian anyway. At the moment there does not seem to be a promising algebro-geometric approach to the Γ 00 conjecture in general, but the following result was recently obtained by using integrable systems methods: Theorem 8.6 (Grushevsky [Gru10]). If for some A ∈ A ind g the linear dependence (17) holds with rk(c ab ) = 1, then A ∈ J g .…”