Smooth moduli spaces of associative submanifolds
| Autor: | Damien Gayet |
|---|---|
| Přispěvatelé: | Algèbre, géométrie, logique (AGL), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-08-BLAN-0291,Floer Power,Applications des courbes holomorphes en géométrie symplectique et de contact(2008) |
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
Mathematics - Differential Geometry
General Mathematics Mathematical analysis Holonomy Boundary (topology) Submanifold Curvature Manifold Moduli space Moduli of algebraic curves Differential Geometry (math.DG) [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] Metric (mathematics) FOS: Mathematics Mathematics::Differential Geometry Mathematics::Symplectic Geometry Mathematics |
| Zdroj: | Quarterly Journal of Mathematics Quarterly Journal of Mathematics, Oxford University Press (OUP), 2014, pp.10.1093/qmath/hat042. ⟨10.1093/qmath/hat042⟩ |
| ISSN: | 0033-5606 1464-3847 |
| DOI: | 10.1093/qmath/hat042. |
| Popis: | Let $M^7$ be a smooth manifold equipped with a $G_2$-structure $\phi$, and $Y^3$ be an closed compact $\phi$-associative submanifold. In \cite{McL}, R. McLean proved that the moduli space $\bm_{Y,\phi}$ of the $\phi$-associative deformations of $Y$ has vanishing virtual dimension. In this paper, we perturb $\phi$ into a $G_2$-structure $\psi$ in order to ensure the smoothness of $\bm_{Y,\psi}$ near $Y$. If $Y$ is allowed to have a boundary moving in a fixed coassociative submanifold $X$, it was proved in \cite{GaWi} that the moduli space $\bm_{Y,X}$ of the associative deformations of $Y$ with boundary in $X$ has finite virtual dimension. We show here that a generic perturbation of the boundary condition $X$ into $X'$ gives the smoothness of $\bm_{Y,X'}$. In another direction, we use the Bochner technique to prove a vanishing theorem that forces $\bm_Y$ or $\bm_{Y,X}$ to be smooth near $Y$. For every case, some explicit families of examples will be given. Comment: 27 pages |
| Databáze: | OpenAIRE |
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