A natural hermitian metric associated with local universal families of compact Kähler manifolds with zero first Chern class

Gunnar Magnússon

doi:10.1016/j.crma.2011.11.013

Algebraic Geometry/Analytic Geometry

A natural hermitian metric associated with local universal families of compact Kähler manifolds with zero first Chern class
[Une forme hermitienne naturelle associée aux familles de variétés compactes Kähleriennes dont la première classe de Chern est nulle]

Présenté par : Jean-Pierre Demailly

Gunnar Magnússon ¹

¹ Institut Fourier, 100, rue des Maths, BP 74, 38402 St Martin dʼHères, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 63-66.

Résumé
Abstract

Soit $π : X \to S$ une famille universelle locale de variétés compactes Kähleriennes dont la première classe de Chern est nulle sur une base lisse S. Les cônes de Kähler complexifiés de chaque variété de la famille forment une fibration holomorphe $K \to S$ . Nous montrons quʼil existe une métrique hermitienne naturelle sur le produit fibré $X \times_{S} K$ . Pour finir, nous discutons lʼexemple des courbes elliptiques.

Let $π : X \to S$ be a local universal family of compact Kähler manifolds with zero first Chern class over a smooth base S. The complexified Kähler cones of each manifold of the family form a holomorphic fiber bundle $K \to S$ . We show that there exists a natural hermitian metric ω on the fiber product $X \times_{S} K$ . We then discuss the example of elliptic curves in some detail.

Reçu le : 2011-10-21
Accepté le : 2011-11-17
Publié le : 2011-11-30

DOI : 10.1016/j.crma.2011.11.013

Affiliations des auteurs :

Gunnar Magnússon ¹

¹ Institut Fourier, 100, rue des Maths, BP 74, 38402 St Martin dʼHères, France

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     author = {Gunnar Magn\'usson},
     title = {A natural hermitian metric associated with local universal families of compact {K\"ahler} manifolds with zero first {Chern} class},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {63--66},
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     doi = {10.1016/j.crma.2011.11.013},
     language = {en},
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Gunnar Magnússon. A natural hermitian metric associated with local universal families of compact Kähler manifolds with zero first Chern class. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 63-66. doi : 10.1016/j.crma.2011.11.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.11.013/

Bibliographie
Cité par

[1] K. Kodaira; D.C. Spencer On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2), Volume 71 (1960), pp. 43-76 MR 0115189 (22 #5991)

[2] Antonella Nannicini Weil–Petersson metric in the moduli space of compact polarized Kähler–Einstein manifolds of zero first Chern class, Manuscripta Math., Volume 54 (1986) no. 4, pp. 405-438 MR 829406 (87k:32038)

[3] Georg Schumacher On the geometry of moduli spaces, Manuscripta Math., Volume 50 (1985), pp. 229-267 MR 784145 (86j:32049)

[4] Gang Tian Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric, San Diego, Calif., 1986 (Adv. Ser. Math. Phys.), Volume vol. 1, World Sci. Publishing, Singapore (1987), pp. 629-646 (MR 915841)

[5] Shing Tung Yau On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I, Comm. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 MR 480350 (81d:53045)

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