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.
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.
DOI: 10.5802/afst.1514
Indranil Biswas 1; Georg Schumacher 2
@article{AFST_2016_6_25_4_895_0,
author = {Indranil Biswas and Georg Schumacher},
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},
pages = {895--917},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 25},
number = {4},
year = {2016},
doi = {10.5802/afst.1514},
zbl = {1356.32008},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1514/}
}
TY - JOUR AU - Indranil Biswas AU - Georg Schumacher TI - The Weil-Petersson current for moduli of vector bundles and applications to orbifolds JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2016 SP - 895 EP - 917 VL - 25 IS - 4 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1514/ DO - 10.5802/afst.1514 LA - en ID - AFST_2016_6_25_4_895_0 ER -
%0 Journal Article %A Indranil Biswas %A Georg Schumacher %T The Weil-Petersson current for moduli of vector bundles and applications to orbifolds %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2016 %P 895-917 %V 25 %N 4 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1514/ %R 10.5802/afst.1514 %G en %F AFST_2016_6_25_4_895_0
Indranil Biswas; Georg Schumacher. The Weil-Petersson current for moduli of vector bundles and applications to orbifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 4, pp. 895-917. doi : 10.5802/afst.1514. https://afst.centre-mersenne.org/articles/10.5802/afst.1514/
[1] Atiyah (M.), Singer (I.M.).— The index of elliptic operators: III. Ann. Math. 87, p. 546-604 (1968). | DOI | MR | Zbl
[2] Baily (W. L.).— On the imbedding of V-manifolds in projective space. Amer. Jour. Math. 79, p. 403-430 (1957). | DOI | MR | Zbl
[3] Bismut (J.-M.), Gillet (H.) and Soulé (C.).— Analytic torsion and holomorphic determinant bundles. I: Bott-Chern forms and analytic torsion. II: Direct images and Bott-Chern forms. III: Quillen metrics on holomorphic determinants. Commun. Math. Phys. 115, 49-78, 79-126, p. 301-351 (1988). | DOI | Zbl
[4] Bismut (J.-M.), Lebeau (G.).— Complex immersions and Quillen metrics. Publ. math. IHES 74, p. 1-298 (1991). | DOI
[5] Bismut (J.-M.), Ma (X.).— Holomorphic immersions and equivariant torsion forms. J. reine angew. Math. 575, p. 189-235 (2004). | DOI | MR | Zbl
[6] Demailly (J.P.).— Complex Analytic and Differential Geometry, Grenoble (1997).
[7] Donaldson (S. K.).— Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. 50, p. 1-26 (1985). | DOI | MR | Zbl
[8] Forster (O.), Knorr (K.).— Ein Beweis des Grauertschen Bildgarbensatzes nach ldeen von B. Malgrange. Manuscripta Math. 5, p. 19-44 (1971). | DOI | MR | Zbl
[9] Forster (O.), Knorr (K.).— Über die Deformationen von Vektorraumbündeln auf kompakten komplexen Räumen. Math. Ann. 209, p. 291-346 (1974). | DOI | Zbl
[10] Fujiki (A.) and Schumacher (G.).— The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics. Publ. Res. Inst. Math. Sci. 26, p. 101-183 (1990). | DOI | Zbl
[11] Grauert (H.).— Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, p. 331-368 (1962). | DOI | Zbl
[12] Kawasaki (T.).— The Riemann-Roch theorem for complex V-manifolds. Osaka Jour. Math. 16, p. 151-159 (1979). | MR | Zbl
[13] Kawasaki (T.).— The signature theorem for V-manifolds. Topology 17, p. 75-83 (1978). | DOI | MR | Zbl
[14] Lieblich (M.).— Moduli of twisted sheaves. Duke Math. Jour. 138, p. 23-118 (2007). | DOI | MR | Zbl
[15] Ma (X.).— Submersions and equivariant Quillen metrics. Ann. Inst. Fourier, Grenoble 50, p. 1539-1588 (2000). | DOI | MR | Zbl
[16] Ma (X.).— Orbifolds and analytic torsion. Preprint, Mathematical Section ICTP, Trieste, 59 p. | DOI | MR
[17] Ma (X.).— Orbifolds and analytic torsion. Trans. Am. Math. Soc. 357, p. 2205-2233 (2005). | DOI | MR | Zbl
[18] Satake (I.).— On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. 42, p. 359-363 (1956). | DOI | MR | Zbl
[19] Schumacher (G.).— Positivity of relative canonical bundles and applications. Invent. Math. 190 (2012), 1-56, and G. Schumacher: Erratum to: Positivity of relative canonical bundles and applications. Invent. Math. 192, p. 253-255 (2013). | DOI | MR | Zbl
[20] Schumacher (G.).— An extension theorem for hermitian line bundles. arXiv:1507.06195 | DOI | MR | Zbl
[21] Schumacher (G.) and Toma (M.).— On the Petersson-Weil metric for the moduli space of Hermite-Einstein bundles and its curvature. Math. Ann. 293, p. 101-107 (1992). | DOI | MR | Zbl
[22] Siu (Y.-T.).— Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, p. 53-156 (1974). | DOI | MR | Zbl
[23] Siu (Y.-T.).— Lectures on Hermitian Einstein metrics for stable bundles and Kähler-Einstein metrics. DMV Seminar, Bd. 8. Basel-Boston: Birkhäuser Verlag (1987). | DOI | Zbl
[24] Teleman (A.).— Families of holomorphic bundles. Comm. Contemp. Math. 10, p. 523-551 (2008). | DOI | MR | Zbl
[25] Uhlenbeck (K.) and Yau (S.-T.).— On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Frontiers of the mathematical sciences: Comm. Pure Appl. Math. 39, p. 257-293 (1986). | DOI | MR
[26] Varouchas (J.).— Stabilité de la classe des variétés Kähleriennes par certains morphismes propres. Invent. Math. 77, p. 117-127 (1984). | DOI | MR | Zbl
Cited by Sources: