Asian Journal of Mathematics

Volume 25 (2021)

Number 3

Strongly homotopy Lie algebras and deformations of calibrated submanifolds

Pages: 341 – 368



Domenico Fiorenza (Dipartimento di Matematica, Università di Roma, Italy)

Hông Vân Lê (Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic)

Lorenz Schwachhöfer (Faculty for Mathematics, Technische Universität Dortmund, Germany)

Luca Vitagliano (Dipartimento di Matematica, Università degli Studi di Salerno, Italy; and Istituto Nazionale di Fisica Nucleare, Italy)


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.


$\Psi$-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.

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