Strongly homotopy Lie algebras and deformations of calibrated submanifolds

For an element $\Psi$ in the graded vector space $\Omega^\ast (M,TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined as a submanifold $N$ of $M$ such that $\Psi_{\lvert N} \in \Omega^\ast (N,TN)$. The class of $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $\Omega^\ast (M,TM)$ carries a natural graded Lie algebra structure, given by the Frölicher–Nijenhuis bracket $[-,-]^{FN}$. When $\Psi$ is an odd degree element with $[\Psi,\Psi]^{FN}=0$, we associate to a $\Psi$-submanifold $N$ a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of $N$ as a $\Psi$-submanifold, and we show that under certain assumptions these deformations form an analytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of $\varphi$-calibrated closed submanifolds, where $\varphi$ is a parallel form in a real analytic Riemannian manifold.

Keywords

$\Psi$-submanifold, calibrated submanifold, Frölicher–Nijenhuis bracket, strongly homotopy Lie algebra, derived bracket, formal deformation, smooth deformation, complex submanifold

2010 Mathematics Subject Classification

Primary 32G10, 53C38. Secondary 17B55, 53C29, 58D27.

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The research of H.V.L. was supported by RVO:67985840, and by the GAČR-project 18-00496S.

L.V. acknowledges partial support by grant SCHW893/5-1 of the Deutsche Forschungsgemeinschaft.

Received 6 March 2019

Accepted 2 September 2020

Published 14 March 2022