Goto's deformation theory of geometric structures, a Lie-theoretical description

TL;DR

This paper provides a Lie-theoretical perspective on Goto's deformation theory of geometric structures, demonstrating that the deformation space can be described via an $L_{}$-algebra, offering a new proof of its smoothness in key cases.

Contribution

It shows that Goto's deformation space arises from a specific $L_{}$-algebra and proves its homotopy abelian property for important geometric structures, unifying classical unobstructedness results.

Findings

Deformation space corresponds to a certain $L_{}$-algebra.

For Calabi-Yau, $G_2$, and $Spin(7)$ structures, this algebra is homotopy abelian.

Provides a new proof of Goto's smoothness theorem.

Abstract

In \cite{Goto}, Ryushi Goto has constructed the deformation space for a manifold equipped with a collection of closed differential forms and showed that in some important cases (Calabi-Yau, $G_2$- and $Spin(7)$-structures) this deformation space is smooth. This result unifies the classical Bogomolov-Tian-Todorov and Joyce theorems about unobstructedness of deformations. Using the work of Fiorenza and Manetti, we show that this deformation space could be obtained as the deformation space associated to a certain $L_{\infty}$-algebra. We also show that for Calabi-Yau, $G_2$- and $Spin(7)$-structures this $L_{\infty}$-algebra is homotopy abelian. This gives a new proof of Goto's theorem.