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

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

2005-06-27

arxiv.org/abs/math/0506550v3

J. Math. Phys. Vol.52, 102305 (2011)

Mathematics

Algebraic Geometry

10 pages - The Siegel induced volume form on M_g is expressed in terms of the Riemann period matrix only, it is simply related

Scientific paper

10.1063/1.3653550

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.

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