I am studying deformations of complex structures on compact complex manifolds and at the moment I am reading the paper ''New proof for the existence of locally complete families of Complex Structures'' by Kuranishi (1965) and I want to understand equation $(8)$: \begin{align} \omega\circ e(\xi)=\omega+\overline{\partial}\xi+R(\omega,\xi), \end{align} for $\xi\in A^0:=\Gamma(T^{1,0}M)$. I will give more context below, but essentially it is a proof of the fact that isomorphic deformations of the complex structure are parametrized by $H^1(M,T^{1,0}M)$.
Let me further elaborate. The setting in the paper is as follows. We have a smooth manifold $\mathbf{M}$ equipped with a fixed complex structure. The resulting complex manifold is denoted by $M$. An almost complex structure is defined as a decomposition $T_\mathbb{C}M=T^{'}_1\oplus T^{'}_2$ such that $\overline{T_1^{'}}=T^{'}_2$ and is denoted by $M'$ (which I think is confusing notation). Then, he introduces the concept of deforming the structure while keeping it almost complex. He shows this is equivalent to (at least for small deformations) a bundle map $\epsilon: T^{0,1}\to T^{1,0}$, by identifying \begin{align} T^{'}_2=T_\xi:=\{-\xi(X)+X\mid X\in T^{0,1}M\}. \end{align} Here $T^{1,0}$ and $T^{0,1}$ refer to the $+i$- and $-i$-eigenspace of the fixed complex structure, respectively. Now, by requiring the deformation to be integrable one arrives at the Maurer-Cartan formula $\overline{\partial}\epsilon +\frac{1}{2}[\epsilon,\epsilon]=0$. From this we see that infinitesimal deformations must be $\overline{\partial}$-closed. This is all clear.
Now, we are interested in deformations of $M$ (the fixed complex structure on $\mathbf{M}$) up to isomorphism. So, he picks a diffeomorphism $f:\mathbf{M}\to \mathbf{M}$ and $\omega\in A^1:=\Omega^{0,1}(M,T^{1,0}M)$. He then claims there is a unique almost complex structure $M'$ such that $f$ induces an isomorphism from $M'$ to $M_\omega$, where $M_\omega$ is the almost complex structure corresponding to $T_\omega$ as defined above. I think there is only one thing one can write down, namely \begin{align} T_2^{'}=(df)^{-1}T_\omega. \end{align} Is this correct?
Furthermore, he supposes $M'$ has `finite distance' to $M$, which is defined as $M'$ being equal to $M_\theta$ for some $\theta\in A^1$. He then writes $\theta=\omega\circ f$ and claims this is only well-defined for $f$ close enough (in the appropriate Whitney topology) to the identity. Moreover, he claims that in local coordinates $\theta=\omega\circ f$ can be written as \begin{align} \frac{\partial f^\alpha}{\partial \overline{z}^\beta}+w_\gamma^\alpha(f(z))\frac{\partial \overline{f}^\gamma}{\partial \overline{z}^\beta}=\left(\frac{\partial f^\alpha}{\partial z^\delta}+w^\alpha_\gamma(f(z))\frac{\partial \overline{f}^\alpha}{\partial z^\beta}\right)\theta_\beta^\delta(z), \end{align} where $\omega=X_\alpha d\overline{z}^\alpha$, $X_\alpha=w_\alpha^\gamma\frac{\partial}{\partial z^\gamma}$ and similarly for $\theta$. There might be some terms that should or shouldn't be barred; I have an old copy of the paper and it is difficult to see. So, please correct me if the expression is not correct.
At this point, I have three questions:
- What is meant with with the expression $\theta=\omega\circ f$? It feels like $\theta$ and $\omega$ should be related through some pullback, i.e. $\theta=f^*\omega$. Is this true? I tried to express $f^*\omega$ in local coordinates in the hope to show the equation above, but was not successful.
- I am not sure about the well-definedness of $\omega\circ f$ and how it relates to being sufficiently close to the identity. The only thing I can imagine is that you want $\mathrm{pr}_{T^{0,1}}:(df)^{-1}T_\omega\to T^{1,0}$ to be invertible (let $\varphi$ denote the inverse), where $\mathrm{pr}_{T^{0,1}}$ is the projection onto $T^{0,1}$. This is true when $f$ is close to the identity. Using this I think I was able to show that (I can share the steps as well) \begin{align} T_2^{'}=\{-\left((df)^{-1}\circ\omega\circ df\circ\varphi-\mathrm{pr}_{T^{1,0}}\circ \varphi\right)(Z)+Z\mid Z\in T^{0,1}\}, \end{align} so that $\theta=(df)^{-1}\circ\omega\circ df\circ\varphi-\mathrm{pr}_{T^{1,0}}\circ \varphi$. Note, this does not quite match up with a pullback, because of the second term. Is this the correct expression for $\theta$?
- How do you show the local expression above. Using either point 1 or 2, I was only able to obtain some of the terms, but not the full expression.
After this, Kuranishi basically applies this formula to a diffeomorphism induced by the exponential map (in the Riemannian geometry setting) to arrive at the first equation at the top.
There is a lot of atypical notation in the original paper. So, please let me know if there is anything unclear in the questions above.