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On the volumes of linear subvarieties in moduli spaces of projectivized Abelian differentials

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorNGUYEN, Duc-Manh
dc.date.accessioned2024-04-04T02:35:58Z
dc.date.available2024-04-04T02:35:58Z
dc.date.created2022
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190688
dc.description.abstractEnFor $k \in \mathbb{Z}_{>0}$, let $\mathcal{H}^{(k)}_{g,n}$ denote the vector bundle over $\mathfrak{M}_{g,n}$ whose every fiber consists of meromorphic $k$-differentials with poles of order at most $k-1$ on a fixed Riemman surface of genus $g$ with $n$ marked points (all the poles must be located at the marked points). The bundle $\mathcal{H}^{(k)}_{g,n}$ and its associated projective bundle $\mathbb{P}\mathcal{H}^{(k)}_{g,n}$ admit natural extensions, denoted by $\overline{\mathcal{H}}^{(k)}_{g,n}$ and $\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}$ respectively, to the Deligne-Mumford compactification $\overline{\mathfrak{M}}_{g,n}$ of $\mathfrak{M}_{g,n}$. We prove the following statement: let $\mathcal{M}$ be a subvariety of dimension $d$ of the projective bundle $\mathbb{P}\mathcal{H}^{(k)}_{g,n}$. Denote by $\mathscr{O}(-1)_{\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}}$ the tautological line bundle over $\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}$. Then the integral of the $d$-th power of the curvature form of the Hodge norm on $\mathscr{O}(-1)_{\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}}$ over the smooth part of $\mathcal{M}$ is equal to the intersection number of the $d$-th power of the divisor representing $\mathscr{O}(-1)_{\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}}$ and the closure of $\mathcal{M}$ in $\mathbb{P}\overline{\mathcal{H}}^{(k)}_{g,n}$. As a consequence, if $\mathcal{M}$ is a linear subvariety of the projectivized Hodge bundle $\mathbb{P}\mathcal{H}_{g,n}(=\mathbb{P}\mathcal{H}^{(1)}_{g,n})$ whose local coordinates do not involve relative periods, then the volume of $\mathcal{M}$ can be computed by the self-intersection number of the tautological line bundle on the closure of $\mathcal{M}$ in $\mathbb{P}\overline{\mathcal{H}}_{g,n}(=\mathbb{P}\overline{\mathcal{H}}^{(1)}_{g,n})$.
dc.description.sponsorshipEspaces de modules de différentielles: surfaces plates et interactions - ANR-19-CE40-0003
dc.language.isoen
dc.title.enOn the volumes of linear subvarieties in moduli spaces of projectivized Abelian differentials
dc.typeDocument de travail - Pré-publication
dc.subject.halMathématiques [math]
dc.identifier.arxiv2209.12815
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-03943472
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-03943472v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=NGUYEN,%20Duc-Manh&rft.genre=preprint


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