Strongly homotopy Lie algebras and deformations of calibrated submanifolds

TL;DR

This paper introduces a framework linking strongly homotopy Lie algebras to the deformation theory of various special submanifolds, including calibrated and complex submanifolds, within a unified geometric setting.

Contribution

It establishes a novel connection between graded Lie algebra structures and the deformation theory of $ extPsi$-submanifolds, extending to calibrated and complex cases.

Findings

Strong homotopy Lie algebra governs deformations of $ extPsi$-submanifolds.

Deformations form an analytic variety under certain conditions.

Revisits deformation theory of complex and calibrated submanifolds.

Abstract

For an element $\Psi$ in the graded vector space $\Omega^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $\Psi$-submanifold is defined as a submanifold $N$ of $M$ such that $\Psi_{|N} \in \Omega^*(N, TN)$. The class of $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $\Omega^*(M, TM)$ carries a natural graded Lie algebra structure, given by the Fr\"olicher-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.