For an element Ψ in the graded vector space Ω∗(M, T M ) of tangent bundle valued forms on a smooth manifold M , a Ψ-submanifold is defined as a submanifold N of M such that Ψ|N ∈ Ω∗(N, T N ). The class of Ψ-submanifolds encompasses calibrated submanifolds, complex sub- manifolds and all Lie subgroups in compact Lie groups. The graded vector space Ω∗(M, T M ) carries a natural graded Lie algebra structure, given by the Fr ̈olicher-Nijenhuis bracket [−, −]F N . When Ψ is an odd degree element with [Ψ, Ψ]F N = 0, we associate to a Ψ-submanifold N a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of N as a Ψ-submanifold, and we show that under certain assumptions these deformations form an ana- lytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of φ-calibrated closed submanifolds, where φ is a parallel form in a real analytic Riemannian manifold.
Strongly homotopy Lie algebras and deformations of calibrated submanifolds / Fiorenza, Domenico; Lê, Hông Vân; Schwachhöfer, Lorenz; Vitagliano, Luca. - In: THE ASIAN JOURNAL OF MATHEMATICS. - ISSN 1093-6106. - 25:3(2021), pp. 341-368. [10.4310/AJM.2021.v25.n3.a2]
Strongly homotopy Lie algebras and deformations of calibrated submanifolds
Fiorenza, Domenico;
2021
Abstract
For an element Ψ in the graded vector space Ω∗(M, T M ) of tangent bundle valued forms on a smooth manifold M , a Ψ-submanifold is defined as a submanifold N of M such that Ψ|N ∈ Ω∗(N, T N ). The class of Ψ-submanifolds encompasses calibrated submanifolds, complex sub- manifolds and all Lie subgroups in compact Lie groups. The graded vector space Ω∗(M, T M ) carries a natural graded Lie algebra structure, given by the Fr ̈olicher-Nijenhuis bracket [−, −]F N . When Ψ is an odd degree element with [Ψ, Ψ]F N = 0, we associate to a Ψ-submanifold N a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of N as a Ψ-submanifold, and we show that under certain assumptions these deformations form an ana- lytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of φ-calibrated closed submanifolds, where φ is a parallel form in a real analytic Riemannian manifold.File | Dimensione | Formato | |
---|---|---|---|
Fiorenza_Strongly-homotopy_2021.pdf
accesso aperto
Tipologia:
Documento in Post-print (versione successiva alla peer review e accettata per la pubblicazione)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
382.39 kB
Formato
Adobe PDF
|
382.39 kB | Adobe PDF |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.