The Weil-Petersson current for moduli of vector bundles and applications to orbifolds
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 4, pp. 895-917.

Nous étudions les fibrés vectoriels holomorphes stables sur une varieté kählérienne compacte ou plus généralement sur une orbifold possédant une structure kählérienne. Dans ce contexte, nous utilisons l’existence d’une connexion Hermite-Einstein et construisons une forme de Weil-Petersson généralisée sur l’espace des modules des fibrés holomorphes stables à fibré déterminant fixé. Nous montrons que la forme de Weil-Petersson s’étend en un courant (semi-)positif fermé pour des dégénerescences de familles qui sont des restrictions de faisceaux cohérents. Ce courant sera appelé un courant de Weil-Petersson . Dans le cas d’une orbifold de type Hodge, un fibré en droite déterminant existe sur l’espace des modules. Ce fibré en droites est muni d’une métrique de Quillen dont la courbure coincide avec la forme de Weil-Petersson généralisée. En application, nous montrons que le fibré en droites déterminant s’étend à une compactification de l’espace des modules.

We investigate stable holomorphic vector bundles on a compact complex Kähler manifold and more generally on an orbifold that is equipped with a Kähler structure. We use the existence of Hermite-Einstein connections in this set-up and construct a generalized Weil-Petersson form on the moduli space of stable vector bundles with fixed determinant bundle. We show that the Weil-Petersson form extends as a (semi-)positive closed current for degenerating families that are restrictions of coherent sheaves. Such an extension will be called a Weil-Petersson current. When the orbifold is of Hodge type, there exists a certain determinant line bundle on the moduli space; this line bundle carries a Quillen metric, whose curvature coincides with the generalized Weil-Petersson form. As an application we show that the determinant line bundle extends to a suitable compactification of the moduli space.

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     title = {The {Weil-Petersson} current for moduli of vector bundles and applications to orbifolds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Biswas, Indranil; Schumacher, Georg. The Weil-Petersson current for moduli of vector bundles and applications to orbifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 4, pp. 895-917. doi : 10.5802/afst.1514. http://archive.numdam.org/articles/10.5802/afst.1514/

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