Abstract
As for any symmetric space the tangent space to Siegel upper-half space is endowed with an operation coming from the Lie bracket on the Lie algebra. We consider the pull-back of this operation to the moduli space of curves via the Torelli map. We characterize it in terms of the geometry of the curve, using the Bergman kernel form associated to the curve. It is known that the second fundamental form of the Torelli map outside the hyperelliptic locus can be seen as the multiplication by a certain meromorphic form. Our second result says that the Bergman kernel form is the harmonic representative—in a suitable sense—of this meromorphic form.
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Acknowledgements
The idea to study the Lie bracket restricted to moduli space arose from discussions with Gian Pietro Pirola. The idea of using Bergman kernel in the study of the map \(\mathbf {B}^*\) was suggested by Indranil Biswas. We heartily thank both of them. We also thank Paola Frediani for several interesting discussions related to the subject of this paper.
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The authors were partially supported by MIUR PRIN 2017 “Moduli spaces and Lie Theory” , by MIUR, Programma Dipartimenti di Eccellenza (2018–2022)—Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia and by INdAM (GNSAGA).
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Ghigi, A., Tamborini, C. Bergman kernel and period map for curves. Geom Dedicata 216, 5 (2022). https://doi.org/10.1007/s10711-021-00670-7
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DOI: https://doi.org/10.1007/s10711-021-00670-7