Perturbations of the metric in Seiberg-Witten equations

Scala, Luca

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

Soit $M$ une variété riemannienne compacte connexe orientée de dimension $4$. On étudie l’espace $\Xi$ des structures ${\mathrm{Spin}}^{\mathrm{c}}$ 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 $\Xi$ de toutes les structures ${\mathrm{Spin}}^{\mathrm{c}}$. 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 $\Xi$ of ${\mathrm{Spin}}^{\mathrm{c}}$-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 ${\mathrm{Spin}}^{\mathrm{c}}$-structures $\Xi$. 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.

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

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