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

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.