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

Let $\mathfrak{g}$ be a curved $L_\infty$-algebra endowed with a complete filtration $\mathfrak{F}\mathfrak{g}$. 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