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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 | Abstract: | Given 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 | Services: | Forum | Review | PDF | Favorites |
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