In deformation theory of complex structure, the Maurer-Cartan equation takes the form $$\bar{\partial}\varphi(t)+\frac{1}{2}[\varphi(t),\varphi(t)]=0.$$ where $\varphi(t)\in\Gamma_{smooth}(X,\bigwedge^{0,1}\otimes T_X)$. If $X$ is a compact Calabi-Yau manifold, using the Tian-Todorov lemma, one can prove that such $\varphi(t)$ always exists with any given initial direction $[\varphi_1]\in H^1(X,T_X)$. Suppose now we are given an honest deformation $\varphi(t)$, and a $\bar{\partial}$-closed element $\alpha_0\in\Gamma_{smooth}(X,\bigwedge^{p,q}\otimes T_X)$, I am wondering whether the following equation can be solved under the assumption of Calabi-Yau: $$\bar{\partial}\alpha(t)+[\varphi(t),\alpha(t)]=0.$$ with $\alpha(t)\in\Gamma_{smooth}(X,\bigwedge^{p,q}\otimes T_X)$ and $\alpha(0)=\alpha_0$ and the bracket is extended as usual, i.e. $$[\omega_1\otimes v_1,\omega_2\otimes v_2]=\omega_1\wedge\omega_2\otimes[v_1,v_2].$$ By solving I mean finding $\alpha(t)$ with $\alpha(0)=\alpha_0$ and solving $$\bar{\partial}\alpha(t)+[\varphi(t),\alpha(t)]=0.$$ for any given $\varphi(t)$ satisfying the Maurer-Cartan equation.
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2$\begingroup$ Are you sure you want the factor of $\frac12$ in the equation for $\alpha$? More natural is to linearize the Maurer--Cartan equation: if you ask that $\bar\partial\phi + \frac12[\phi,\phi]=0$ for $\phi=\varphi+\epsilon\alpha$, then the $O(\epsilon)$ term is $\bar\partial\alpha+[\phi,\alpha]=0$. $\endgroup$– Theo Johnson-FreydJul 16, 2013 at 17:39
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$\begingroup$ Well, yes. You have a dgla (the Kodaira-Spencer dgla), with differential $d=\overline{\partial}$. In any dgla you have $(d+ad(\phi))^2= ad Q(\phi)$, where $Q(\phi)=d\phi+\frac{1}{2}[\phi,\phi]$. In other words, if you start with a deformation, $d+ad(\phi)$ is a differential. This is the same calculation as the one you do for flat connections... $\endgroup$– Peter DalakovAug 22, 2013 at 12:26
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