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

We find the explicit form of the volume form on the moduli space of non-hyperelliptic Riemann surfaces induced by the Siegel metric, a long-standing question in string theory. This question is related to the explicit form of the (g−2)(g−3)/2 linearly independent relations among the 2-fold products of holomorphic abelian differentials, that are provided in the case of canonical curves of genus g ⩾ 4. Such relations can be completely expressed in terms of determinants of the standard normalized holomorphic abelian differentials. Remarkably, it turns out that the induced volume form is the Kodaira-Spencer map of the square of the Bergman reproducing kernel.