# Obstructions to deformations of complex manifolds

Roughly, a deformation of a compact complex manifold $M$ (in the sense of Kodaira-Spencer) is a triple $(\mathcal{M},w,B)$ where $w:\mathcal{M}\to B$ is a holomorphic map over domain $0\in B\subset \mathbb{C}^n$ satisfying some basic constraints, so that $\mathcal{M}$ can be viewed as a family $\lbrace M_t\rbrace_{t\in B}$ with $M_0=M$. It turns out that all $M_t$ are diffeomorphic to each other, so the interesting feature is the complex structure. We can make sense of what it means to take the derivative of the complex structure of $M_t$, and it is called the infinitesimal deformation $\theta(t)=\frac{d M_t}{dt}\in H^1(M_t,\Theta_t)$ where $\Theta_t$ is the sheaf of germs of holomorphic vector fields over $M_t$. We may ask the question of existence:

Given $M$ and a class $\theta\in H^1(M,\Theta)$, does there exist a deformation of $M$ such that $\theta(0)=\theta$ ?

There are obstructions. One can write down what it means for $\theta(t)=\lbrace \theta_{jk}(t)\rbrace$ to be a 1-cocycle, and differentiate that equation $N\ge 1$ times and set $t=0$. Taking $N=1$ gives the primary obstruction, the Lie bracket $[\theta,\theta]\in H^2(M,\Theta)$ which is essentially the cup product. That is, $[\theta,\theta]=0$ is necessary for the existence of the deformation (and there are examples where this is nonzero). Though not relevant, it turns out that the condition $H^2(M,\Theta)=0$ is sufficient!

Does there exist a positive integer $N$ such that if the first $N$ obstructions vanish, then there exists a deformation (in the direction $\theta$)? We need at least $N>1$.

Are these infinite number of obstructions sufficient? Misha's suggestion might work. If I can represent these "solutions of obstruction equations" as elements of an appropriate $N$-jet space (and let $N\to\infty$) then I would have a formal Taylor series, which should subsequently converge to an honest solution (of the deformation problem) by an appropriate use of Artin's approximation theorem.

Have people studied the "secondary obstructions"? Answered affirmatively, thanks to the appendix of the first paper linked in the comments. Douady writes down the secondary obstruction via the triple Massey product, and gives an explicit example where it doesn't vanish.

• I am quite sure that: (1) $N$ does not exist, (2) If all obstruction vanish then there exists a curve of deformations. Part (2) should follow from Artin's theorem. Examples for (1) likely come in the form of complex manifolds $SL(2,C)/\Gamma$, where $\Gamma$'s are uniform torsion-free lattices (moreover, these deformation should obey Vakil's "Murphy's Law"). You are in Berkeley, why do not you just drive to Stanford to talk to Ravi Vakil? He probably will give you simpler examples than mine. – Misha Feb 9 '15 at 3:13
• I don't know who is the appropriate person to ask, but I will email him since the commute is pretty long :-) – Chris Gerig Feb 9 '15 at 18:27
• Are you aware of this lecture by Douady in the Cartan seminar? I think it answers some (but not all) of your questions. – abx Feb 9 '15 at 21:04
• For examples of obstructed deformations of quotients of $SL(2,\mathbb{C})$ see also this paper of Ghys perso.ens-lyon.fr/ghys/articles/deformationsstructures.pdf. This paper of Rollenske math.uni-bielefeld.de/~rollenske/papers/… seems also quite relevant. – YangMills Feb 10 '15 at 20:46