Perturbations of the metric in Seiberg-Witten equations

Scala, Luca

doi:10.5802/aif.2640

Perturbations of the metric in Seiberg-Witten equations
[Perturbations de la métrique dans les équations de Seiberg-Witten]

Scala, Luca ¹

¹ University of Chicago Department of Mathematics 5734 S. University Avenue 60637 Chicago IL (USA)

Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 1259-1297.

Résumé
Abstract

Soit $M$ une variété riemannienne compacte connexe orientée de dimension $4$ . On étudie l’espace $Ξ$ des structures ${Spin}^{c}$ de classe fondamentale fixée, comme fibré principal de dimension infinie sur la variété des métriques riemanniennes de $M$ . Afin d’étudier les perturbations de la métrique dans les équations de Seiberg-Witten, on étudie la transversalité des équations universelles, paramétrées par l’espace $Ξ$ de toutes les structures ${Spin}^{c}$ . On montre que, sur une surface de Kähler, pour une métrique hermitienne $h$ suffisamment proche à la métrique de Kähler de départ, l’espace de modules de monopôles de Seiberg-Witten relatif à la métrique $h$ est lisse de la dimension attendue.

Let $M$ a compact connected oriented 4-manifold. We study the space $Ξ$ of ${Spin}^{c}$ -structures of fixed fundamental class, as an infinite dimensional principal bundle on the manifold of riemannian metrics on $M$ . In order to study perturbations of the metric in Seiberg-Witten equations, we study the transversality of universal equations, parametrized with all ${Spin}^{c}$ -structures $Ξ$ . We prove that, on a complex Kähler surface, for an hermitian metric $h$ sufficiently close to the original Kähler metric, the moduli space of Seiberg-Witten monopoles relative to the metric $h$ is smooth of the expected dimension.

EuDML MR Zbl

DOI : 10.5802/aif.2640

Classification : 57R57, 58G03, 58D27, 14J80
Keywords: Seiberg-Witten theory, perturbations of the metric, Kähler surfaces, transversality
Mot clés : équations de Seiberg-Witten, perturbations de la métrique, surfaces de Kähler, transversalité

Affiliations des auteurs :

Scala, Luca ¹

¹ University of Chicago Department of Mathematics 5734 S. University Avenue 60637 Chicago IL (USA)

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     title = {Perturbations of the metric in {Seiberg-Witten} equations},
     journal = {Annales de l'Institut Fourier},
     pages = {1259--1297},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     doi = {10.5802/aif.2640},
     zbl = {1238.57029},
     mrnumber = {2918729},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2640/}
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Scala, Luca. Perturbations of the metric in Seiberg-Witten equations. Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 1259-1297. doi : 10.5802/aif.2640. https://www.numdam.org/articles/10.5802/aif.2640/

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