Almost Complex Structure approach to Deformation of Compact Complex Manifolds

Question

I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book Complex Geometry: An Introduction. There are two parts to this chapter. The second goes through the standard approach, that is, considering a family of compact complex manifolds as a proper holomorphic submersion between two connected complex manifolds. My question is about the approach taken in the first section, which I will briefly outline.

One can instead consider a deformation of complex structures on a fixed smooth manifold, as opposed to deformations of complex manifolds – by Ehresmann's result, a deformation over a connected base is nothing but a deformation of complex structure on a fixed smooth manifold. This point of view is difficult to work with because a complex structure is a complicated object, so we instead consider almost complex structures – by the Newlander-Niremberg Theorem, complex structures correspond to integrable almost complex structures.

Fix a smooth even-dimensional manifold $M$. Now Huybrechts considers a continuous family of almost complex structures $I(t)$. He does not say where $t$ comes from, but I have interpreted it to be an open neighbourhood of $0$ in $\mathbb{C}$. Now, let $I(0) = I$. The complexified tangent bundle to $M$ splits with respect to $I$. That is, $TM\otimes_{\mathbb{R}}\mathbb{C} = T^{1,0}M\oplus T^{0,1}M$. But this is true of each almost complex structure $I(t)$. Denote the corresponding decompositions by $TM\otimes_{\mathbb{R}}\mathbb{C} = T^{1,0}M_t\oplus T^{0,1}M_t$ – this is deliberately suggestive notation; we can consider the compact (soon-to-be) complex manifold $(M, I(t))$ as the fibre of a complex family over a point $t$ in the base.

For small $t$, we can encode the given information by a map $\phi(t) : T^{0,1}M \to T^{1,0}M$ where, for $v \in T^{0,1}M$, $v + \phi(t)v \in T^{0,1}M_t$. Huybrechts then says:

Explicitly, one has $\phi(t) = -\text{pr}_{T^{1,0}M_t}\circ j$, where $j : T^{0,1}M \subset TM\otimes_{\mathbb{R}}\mathbb{C}$ and $\text{pr}_{T^{1,0}M_t} : TM\otimes_{\mathbb{R}}\mathbb{C} \to T^{1,0}M_t$ are the natural inclusion respectively projection.

According to this, the codomain of $\phi(t)$ is $T^{1,0}M_t$, not $T^{1,0}M$. Is this a typo or am I missing something? Added later: As Peter Dalakov points out in his answer, it is a typo.

Anyway, Huybrechts continues with this approach. Enforcing the integrability condition $[T^{0,1}M_t, T^{0,1}M_t] \subset T^{0,1}M_t$ ensures that each almost complex structure is induced by a complex structure. Under the assumption that $I$ is integrable, $[T^{0,1}M_t, T^{0,1}M_t] \subset T^{0,1}M_t$ is equivalent to the Maurer-Cartan equation $\bar{\partial}\phi(t) + [\phi(t), \phi(t)] = 0$, where $\bar{\partial}$ is the natural operator on the holomorphic vector bundle $T^{1,0}M$, and $[\bullet, \bullet]$ is an extension of the Lie bracket.

I like this approach because if you take a power series $\sum_{t=0}^{\infty}\phi_it^i$ of $\phi(t)$ you can deduce:

1. $\phi_1$ defines the Kodaira-Spencer class of the deformation;
2. all the obstructions to finding the coefficients $\phi_i$ lie in $H^2(M, T^{1,0}M)$.

Does anyone know of some other places where I would be able to learn about this approach, or is there some reason why this approach is not that common?

Just for the record, I have looked at Kodaira's Complex Manifolds and Deformation of Complex Structures, but I haven't been able to find anything resembling the above.

Answer 1

This approach to deformations is taken, for instance, in all of the original papers of Kodaira-Spencer and Nirenberg. You can have a look at On the existence of deformations of complex analytic structures, Annals, Vol.68, No.2, 1958

http://www.jstor.org/discover/10.2307/1970256?uid=3737608&uid=2129&uid=2&uid=70&uid=4&sid=47699092130607

but there are many other papers by the same authors.

For a nice and compact exposition, you can look at these class notes of Christian Schnell: http://www.math.sunysb.edu/~cschnell/pdf/notes/kodaira.pdf

Of course, the Maurer-Cartan equation and deformations (of various structures) via dgla's have been used by many other people since the late 1950-ies: Goldman & Millson, Gerstenhaber, Stasheff, Deligne, Quillen, Kontsevich.

Regarding the formula: that's a typo, indeed. You have two eigen-bundle decompositions, for $I$ and $I_t$:

$$ T_{M, \mathbb{C}} = T^{1,0}\oplus T^{0,1}\simeq T^{1,0}_t\oplus T^{0,1}_t $$

and you write $T^{0,1}_{t}=\textrm{graph }\phi$, where $\phi: T^{0,1}_M\to T^{1,0}_M$. So actually

$$\phi = \textrm{pr}^{1,0}\circ \left.\left(\textrm{pr}^{0,1}\right)\right|_{T^{1,0}_t}^{-1}.$$

In local coordinates, $$ \phi = \sum_{j,k=1}^{\dim_{\mathbb{C}} M}h_{jk}(t,z)d\overline{z}_j\otimes \frac{\partial}{\partial z_k}, $$ and $T^{0,1}_t$ is generated (over the smooth functions) by

$$\frac{\partial}{\partial \overline{z_j}} + \sum_{k=1}^{\dim_{\mathbb{C}}M}h_{jk}\frac{\partial}{\partial z_k}. $$

Regarding the question "where does $t$ come from?", the answer is "From Ehresmann's Theorem": given a proper holomorphic submersion $\pi:\mathcal{X}\to \Delta$, you can choose a holomorphically transverse trivialisation $\mathcal{X}\simeq X\times \Delta$, $X=\pi^{-1}(0)= (M,I)$. In this way you get yourself two (almost) complex structures on $X\times \Delta$, which you can compare.

ADDENDUM I second YangMills' suggestion to have a look at Chapter 2 of Gross-Huybrechts-Joyce. You can also try Chapter 1 of K. Fukaya's book "Deformation Theory, Homological algebra, and Mirror Symmetry", as well as the Appendix to Homotopy invariance of the Kuranishi Space by Goldman and Millson (Illinois J. of Math, vol.34, No.2, 1990). In particular, you'll see how one uses formal Kuranishi theory to avoid dealing with the convergence of the power series for $\phi(t)$. For deformations of compact complex manifolds, the convergence was proved by Kodaira-Nirenberg-Spencer. Fukaya says a little bit about the convergence of this series in general, i.e., for other deformation problems.