# Kodaira-Spencer maps and deformation theory

This post concerns the following question: Can we black-box the analysis of PDE's which arises in the construction of Kuranishi families for complex analytic structures?

The deformation theory of compact complex manifolds hit a benchmark with the work of Kodaira-Spencer, Nirenberg and Kuranishi which culminated in the following results. Given a compact complex manifold $X$ and a deformation $p: \mathcal{X}\rightarrow B$ of $X$ over a pointed base $(B, b_{0})$, there exists a natural map \begin{align} KS: T_{b_{0}}B\rightarrow H^{1}(X, \Theta) \end{align} where $\Theta$ is the sheaf of germs of holomorphic vector fields on $X.$ Moreover, there is a natural obstruction map \begin{align} Ob: H^{1}(X, \Theta)\rightarrow H^{2}(X, \Theta) \end{align} such that $\alpha\in H^{1}(X, \Theta)$ is tangent to an honest deformation of $X$ if and only if $Ob(\alpha)=0.$ Pushing a bit further, $Ob^{-1}(0)$ is a complex analytic space which is (locally around 0) the base of a versal deformation of $X.$

In complex analytic geometry, there are a plethora of natural refinements of such deformation problems, a few examples include deformations of a pair $(X, E)$ where $X$ is a complex manifold and $E$ is a holomorphic vector bundle, or deformations of a pair $(Y, X)$ where $X$ is a complex manifold and $Y\hookrightarrow X$ is an embedded complex sub-manifold.

In each of these aforementioned deformation problems, there is an analogous story of some sheaf (or complex of sheaves) whose (hyper)-cohomology governs the deformation problem, and an attendant Kodaira-Spencer and obstruction map connecting the infinitesimal theory to the local theory.

The infinitesimal theory has a very beautiful general framework (given by DGLA's and functors from local analytic Artin $\mathbb{C}$-algebras to sets), but I've found it a bit discouraging that there seems to be a lack of such a general framework for passing from the infinitesimal to the local picture.

Let me explain, the pioneering work of Kodaira-Spencer and Nirenberg used now standard, but non-trivial analysis (Hodge theory and the attendent analysis of elliptic PDE) to establish the existence of a versal family $\mathcal{X}\rightarrow Ob^{-1}(0)$ where we really intend an open subset around $0\in H^{1}(X, \Theta)$ when we say $Ob^{-1}(0).$

For the problem $(X, E)$ of a complex manifold and a holomorphic vector bundle, the paper of Chan-Suen https://arxiv.org/abs/1406.6753 proves the same theorem, but feel the need to reiterate the same analytic strategy to establish the existence of a Kuranishi family.

There are many other examples of this phenomenon, where the infinitesimal theory is standard, but due to lack of general theorems, the passage to the existence of a local versal deformation requires a standard application of the strategy outlined above. The result of this situation is that many papers in the literature end up spending many pages to establish the existence of Kuranishi families. My question follows:

Question: Do there exist general theorems in the literature giving infinitesimal conditions on general deformation problems which guarantee the existence of a local Kuranishi family? If not, why not? My impression is that it's just too annoying to prove these general statements, but perhaps there's something much less trivial going on.

To be a bit more precise, I am asking if there is a (finite) list of conditions, let's call then Condition A, such that the following theorem is true:

"Theorem:" Let $X$ be a compact complex manifold equipped with auxiliary holomorphic data (this is imprecise, but I'm thinking of things like holomorphic vector bundles, holomorphic connections on them, sections of them, etc.) Suppose that $\mathcal{A}^{\bullet}$ is a complex of sheaves governing the deformation problem, by which I mean that (up to degree considerations)

1. $\mathbb{H}^{0}(X, \mathcal{A}^{\bullet})$ identifies with infinitesimal automorphisms.

2. $\mathbb{H}^{1}(X, \mathcal{A}^{\bullet})$ identifies with infinitesimal deformations.

3. There exists an obstruction map $Ob: \mathbb{H}^{1}(X, \mathcal{A}^{\bullet})\rightarrow \mathbb{H}^{2}(X, \mathcal{A}^{\bullet}\otimes \mathcal{A}^{\bullet})$ such that $\alpha\in \mathbb{H}^{1}(X, \mathcal{A}^{\bullet})$ satisfies $Ob(\alpha)=0$ if and only if $\alpha$ is tangent to an honest deformation.

Then, given any family of such structures over a pointed base $(B, b_{0}),$ there exists a Kodaira-Spencer map

$KS: T_{b_{0}}B\rightarrow \mathbb{H}^{1}(X, \mathcal{A}^{\bullet}).$

Furthermore, if Condition A is satisfied, then there exists an open neighborhood $B$ of $0\in Ob^{-1}(0)$ and a versal family over $B$ giving a (germ of a) deformation of the structure on $X.$

Before closing, let me remark that while very powerful, for the scope of this question I'm not interested in algebraic techniques (say when $X$ is a projective) which circumvent this issue by constructing a global moduli space (or stack...). In particular, this problem probably can't be solved using geometric invariant theory, or any of the other standard techniques in algebraic geometry. I am insisting we stay in the category of complex manifolds, or perhaps the larger category of complex analytic spaces.

I can go into more detail about the kind of problems I'm interested in, but to save space, let me stop for now and leave the question as it stands. I appreciate any references or insights anyone may have.

• Do you know the book "Advances in Moduli Theory" by Shimizu and Ueno? In the first chapter, there are some theorems about the existence of a Kuranishi family. In any case, I don't really understand what do you mean by "infinitesimal conditions". Do you want to upgrade a formal deformation to an honest deformation or something like this? – user40276 Sep 1 '17 at 23:10
• Regarding your question: yes, I mean general conditions which allow you to upgrade a formal deformation to an honest deformation. Here, this should appear as guaranteeing that some cohomology class appears in the image of a relevant Kodaira-Spencer map for some family/deformation of the chosen object. I am not familiar with "Advances in moduli theory," thank you for the reference. – Andy Sanders Sep 2 '17 at 0:10
• Try looking at work of Bingener (he did a lot with adapting Artin's techniques to the purely complex-analytic setting) and Flenner (some joint with Bingener, some not). – nfdc23 Sep 2 '17 at 1:34