Quot schemes and Ricci semipositivity

Indranil Biswas; Harish Seshadri

doi:10.1016/j.crma.2017.03.012

Analytic geometry

Quot schemes and Ricci semipositivity
[Schéma quot et semi-positivité de Ricci]

Présenté par : Jean-Pierre Demailly

Indranil Biswas ¹ ; Harish Seshadri ²

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
² Indian Institute of Science, Department of Mathematics, Bangalore 560003, India

Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 577-581.

Résumé
Abstract

Soit X une surface de Riemann compacte et connexe de genre au moins deux, et soit $Q_{X} (r, d)$ le schéma quot qui paramétrise tous les quotients torsion cohérents de $O_{X}^{\oplus r}$ de degré d. L'espace $Q_{X} (r, d)$ est aussi un espace de modules de vortex sur X. Nous démontrons que le fibré anticanonique de X n'a pas la propriété nef. De façon équivalente, $Q_{X} (r, d)$ n'admet aucune métrique kählérienne dont la courbure de Ricci soit semi-positive.

Let X be a compact connected Riemann surface of genus at least two, and let $Q_{X} (r, d)$ be the quot scheme that parameterizes all the torsion coherent quotients of $O_{X}^{\oplus r}$ of degree d. This $Q_{X} (r, d)$ is also a moduli space of vortices on X. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of $Q_{X} (r, d)$ is not nef. Equivalently, $Q_{X} (r, d)$ does not admit any Kähler metric whose Ricci curvature is semipositive.

Reçu le : 2016-10-19
Accepté le : 2017-03-22
Publié le : 2017-03-29

DOI : 10.1016/j.crma.2017.03.012

Affiliations des auteurs :

Indranil Biswas ¹ ; Harish Seshadri ²

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
² Indian Institute of Science, Department of Mathematics, Bangalore 560003, India

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     title = {Quot schemes and {Ricci} semipositivity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {577--581},
     publisher = {Elsevier},
     volume = {355},
     number = {5},
     year = {2017},
     doi = {10.1016/j.crma.2017.03.012},
     language = {en},
}

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Indranil Biswas; Harish Seshadri. Quot schemes and Ricci semipositivity. Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 577-581. doi : 10.1016/j.crma.2017.03.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.03.012/

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