  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.
