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H 1 On the Hodge-Kodaira Laplacian on the canonical bundle of a compact Hermitian complex space

Auteurs : Bei, Francesco (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Hermitian complex spaces are a large class of singular spaces that include for instance projective varieties endowed with the metric induced by the Fubini-Study metric. Many of the problems raised by Cheeger, Goresky and MacPherson in the case of complex projective varieties admit a natural extension also in this setting. The aim of this talk is to report about some recent results concerning the Hodge-Kodaira Laplacian acting on the canonical bundle of a compact Hermitian complex space. More precisely let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. Consider the Dolbeault operator $\bar{\partial}_{m,0}$ : $L^2 \Omega^{m,0}(reg(X),h) \to L^2\Omega^{m,1}(reg(X),h)$ with domain $\Omega{_c^{m,0}}(reg(X))$ and let $\bar{\mathfrak{d}}_{m,0} : L^2 \Omega^{m,0}(reg(X),h)\to L^2\Omega^{m,1}(reg(X),h)$ be any of its closed extension. Now consider the associated Hodge-Kodaira Laplacian $\bar{\mathfrak{d}^*} \circ\bar{\mathfrak{d}}_{m,0}$ : $L^2 \Omega^{m,0}(reg(X),h)\to L^2\Omega^{m,0}(reg(X),h)$. We will show that the latter operator is discrete and we will provide an estimate for the growth of its eigenvalues. Finally we will prove some discreteness results for the Hodge-Dolbeault operator in the setting of both isolated singularities and complex projective surfaces (without assumptions on the singularities in the latter case).

Codes MSC :
53C55 - Hermitian and Kählerian manifolds (global differential geometry)
58J50 - Spectral problems; spectral geometry; scattering theory

 Informations sur la Vidéo Langue : Anglais Date de publication : 01/07/16 Date de captation : 15/06/16 Collection : Research talks ; Geometry ; Topology Format : MP4 Durée : 00:21:44 Domaine : Geometry ; Topology Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2016-06-15_Bei.mp4 Informations sur la rencontre Nom de la rencontre : Analysis, geometry and topology of stratified spaces / Analyse, géométrie et topologie des espaces stratifiésOrganisateurs de la rencontre : Mazzeo, Rafe ; Leichtnam, Eric ; Piazza, PaoloDates : 13/06/2016 - 17/06/2016 Année de la rencontre : 2016 URL Congrès : http://conferences.cirm-math.fr/1422.htmlCitation Data DOI : 10.24350/CIRM.V.19002103 Cite this video as: Bei, Francesco (2016). On the Hodge-Kodaira Laplacian on the canonical bundle of a compact Hermitian complex space. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19002103 URI : http://dx.doi.org/10.24350/CIRM.V.19002103

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