Let $X$ be a complex manifold, and let $A^{0,k}(X,T_X)$ denote the smooth $(0,k)$-forms with values in the holomorphic tangent bundle $T_X$, equipped with the Dolbeault operator $\bar\partial$ and the Kodaira-Spencer bracket $[\cdot,\cdot]$. Let $\varphi_1 \in A^{0,1}(X,T_X)$ satisfy $\bar\partial\varphi_1=0$, and let $[\varphi_1]\in H^1(X,T_X)$ be its Dolbeault cohomology class. Suppose that $[\varphi_1]$ extends to a second-order deformation in the Maurer-Cartan sense: there exists $\varphi_2\in A^{0,1}(X,T_X)$ such that, for the formal Kodaira-Spencer element

\begin{align*} \varphi(t)=t\varphi_1+t^2\varphi_2, \end{align*}

one has

\begin{align*} \bar\partial\varphi(t)+\frac{1}{2}[\varphi(t),\varphi(t)]\equiv 0 \pmod{t^3} \end{align*}

in $A^{0,*}(X,T_X)[[t]]$. Then

\begin{align*} \left[\frac{1}{2}[\varphi_1,\varphi_1]\right]=0 \quad \text{in } H^2(X,T_X). \end{align*}