LINEAR RELATIONS AMONG HOLOMORPHIC QUADRATIC DIFFERENTIALS AND INDUCED SIEGEL’S METRIC ON Mg MARCO MATONE AND ROBERTO VOLPATO

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 ≥ 4. It turns out that Petri’s relations remarkably match in determinantal conditions. We explicitly express the volume form on the moduli space M̂g 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 M̂g, 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.