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

### Abstract

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.

© 2011 American Institute of Physics

Received 26 June 2011
Accepted 29 September 2011
Published online 25 October 2011

Article outline:

I. INTRODUCTION
II. DETERMINANTAL CHARACTERIZATION OF CANONICAL CURVES
III. DISTINGUISHED BASES OF
IV. PROOFS OF THEOREM 2.1 AND COROLLARY 2.2
V. SIEGEL's INDUCED MEASURE ON AND BERGMAN REPRODUCING KERNEL

/content/aip/journal/jmp/52/10/10.1063/1.3653550

http://aip.metastore.ingenta.com/content/aip/journal/jmp/52/10/10.1063/1.3653550

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