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Let $X$ be a compact complex manifold. One says that a Kodaira-Spencer class $c \in H^1(X, TX)$ is unobstructed if there exist $\phi_n \in \Omega_X^{0,1}(T^{1,0}X)$ for $n \ge 1$ such that $c = [\phi_1]$ and $\phi(t) := \sum_{n=1}^\infty \phi_nt^n$ converges to a solution to the Maurer-Cartan equation $$\bar{\partial}\phi(t) + [\phi(t), \phi(t)] = 0$$ for $t$ in a neighborhood of $0$ in $\mathbb{C}$. This defines a deformation of complex structures $I_t$, where the $(0, 1)$-part of $I_t$ is $(1+\phi(t))(T^{0, 1}X)$.

Question. Suppose that $c \in H^1(X, TX)$ is unobstructed. Can we solve the Maurer-Cartan equation starting with any representative $\phi_1$ of $c$?

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    $\begingroup$ Don’t any two choices differ by $\partial$ of a holomorphic section of the holomorphic tangent bundle? Since holomorphic sections of the holomorphic tangent bundle “integrate” to one-parameter families of biholomorphisms, I think you can always modify one extendable representative to any other choice of $\phi$. $\endgroup$
    – Jason Starr
    Commented Nov 4, 2021 at 20:59

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