An Obstruction Theory for the Existence of Maurer-Cartan Elements in curved $L_{\infty }$-algebras and an Application in Intrinsic Formality of $P_{\infty }$-Algebras

Abstract

Let $\mathfrak{g}$ be a curved $L_{\infty }$-algebra endowed with a complete filtration $\mathfrak{Fg}$. Suppose there exists an integer $r\in {\mathbb{N}}_0$ for which the curvature ${\mu }_0$ satisfies ${\mu }_0\in {\mathfrak{F}}_{2r+1}\mathfrak{g}$ and the spectral sequence yields $E_{r+1}^{p,q}=0$ for $p,q$ with $p+q=2$. We prove that then a Maurer-Cartan element exists. In addition, we show, as a typical application, that for $P$ a possibly inhomogeneous Koszul operad with generating set in arities 1,2 (e.g. $P$=Com,As,BV,Lie,Ger), a $P_{\infty }$-algebra $A$ is intrinsically formal if its twisted deformation complex $\mathrm{Def}(H(A)\stackrel{\mathrm{id}}{\to }H(A))$ is acyclic in total degree 1.