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29 March 2024
 
  » arxiv » 1912.8595

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A Hodge theoretic projective structure on Riemann surfaces
Indranil Biswas ; Elisabetta Colombo ; Paola Frediani ; Gian Pietro Pirola ;
Date 18 Dec 2019
AbstractGiven any compact Riemann surface $C$, there is a symmetric bidifferential $hat{eta}$ on $C imes C$, with a pole of order two on the diagonal $Deltasubset C imes C$, which is uniquely determined by the following two properties:
1. the restriction of $hateta$ to $Delta$ coincides with the constant function $1$ on $Delta$, and
2. the cohomology class in $H^2(C imes C, {mathbb C})/langle [Delta] angle$ corresponding to $hateta$ is of pure type $(1,1)$.
The restriction of $hateta$ to the nonreduced diagonal $3Delta$ defines a projective structure on $C$. Since this projective structure on $C$ is completely intrinsic, it is natural to ask whether it coincides with the one given by the uniformization of $C$. Showing that the answer to it to be negative, we actually identify $ar partial s$, where $s$ is this section of the moduli of projective structures over the moduli space of curves, to be the pullback of the Siegel form by the Torelli map.
Source arXiv, 1912.8595
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