Linear relations among holomorphic quadratic differentials and induced Siegel's metric on M_g

Abstract

We derive the explicit form of the (g-2)(g-3)/2 linearly independent relations among the products of pairs in a basis of holomorphic abelian differentials in the case of canonical curves of genus g greater than 3. It turns out that Petri's relations remarkably match in determinantal conditions. We explicitly express the volume form on the moduli space M_g of canonical curves induced by the Siegel metric, in terms of the period Riemann matrix only. By the Kodaira-Spencer map, the relations lead to an expression of the induced Siegel metric on M_g, that corresponds to the square of the Bergman reproducing kernel. A key role is played by distinguished bases for holomorphic differentials whose properties also lead to an immediate derivation of Fay's trisecant identity.

10 pages - The Siegel induced volume form on M_g is expressed in terms of the Riemann period matrix only, it is simply related to the Bergman reproducing kernel. Bases of holomorphic n-differentials are introduced, whose distinguished properties also lead to an immediate derivation of Fay's trisecant identity