Every compact Riemann surface X admits a natural projective structure pu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_u$$\end{document} as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on X, namely the Hodge projective structure ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_h$$\end{document}, related to the second fundamental form of the period map. We then describe how projective structures correspond to (1, 1)-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that pu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_u$$\end{document} and ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_h$$\end{document} are not the same structure.

Projective structures and Hodge theory / Causin, A.; Pirola, G. P.. - In: BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA. - ISSN 2198-2759. - (2024). [10.1007/s40574-024-00424-9]

### Projective structures and Hodge theory

#### Abstract

Every compact Riemann surface X admits a natural projective structure pu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_u$$\end{document} as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on X, namely the Hodge projective structure ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_h$$\end{document}, related to the second fundamental form of the period map. We then describe how projective structures correspond to (1, 1)-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that pu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_u$$\end{document} and ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_h$$\end{document} are not the same structure.
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2024
Projective structures and Hodge theory / Causin, A.; Pirola, G. P.. - In: BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA. - ISSN 2198-2759. - (2024). [10.1007/s40574-024-00424-9]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11388/340530
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