What is the DGLA controlling the deformation theory of a complex submanifold?

Question

Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. A classical result, proven by Kodaira, is that if the cohomology group $H^1(Y,\mathcal{O}(T_{X/Y}))$ vanishes, then the deformation theory of $Y$ as a complex submanifold of $X$ is unobstructed. More precisely, there exists a "maximal family" of deformations of $Y$ in $X$: this consists of a complex manifold $W$ (say with a marked point $w_0\in W$), a complex submanifold $V$ of $W\times X$, such that for each point $w\in W$ the intersection of $V$ with $w\times X$ is a complex compact submanifold of $X$, which in the case $w=w_0$ is equal to $Y$. Furthermore, this family of submanifolds is "maximal" (or "universal") in an appropriate sense. By saying the deformation problem is unobstructed, I mean the following: for any family as described above (maximal or not) there is a canonical injective complex linear map from the tangent space of $W$ at $w_0$ to the space of holomorphic sections of $T_{X/Y}$, i.e. the cohomology group $H^0(Y,\mathcal{O}(T_{X/Y}))$. Given the above assumption that $H^1(Y,\mathcal{O}(T_{X/Y}))=0$, this map is an isomorphism. Roughly speaking, one can view a holomorphic section of the normal bundle of $Y$ as a first order deformation of $Y$ in $X$, and unobstructedness means that each such first order deformation can be extended to an "honest" family of deformations.

On the other hand, Kodaira does not explore what happens if one weakens the condition $H^1(Y,\mathcal{O}(T_{X/Y}))=0$. For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y})$, it seems plausible that it isn't necessary for the entire group $H^1(Y,\mathcal{O}(T_{X/Y}))=0$ to vanish in order for $v$ to extend to an honest family of deformations.

$\textbf{Question 1}$: For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y}))$, is there a way to determine when $v$ extends to an honest family of deformations? In particular, is there a map $T:H^0(Y,\mathcal{O}(T_{X/Y}))\to H^1(Y,\mathcal{O}(T_{X/Y}))$ which can be described in a reasonably explicit way such that $v$ extends if and only if $T(v)=0$?

One possible solution to this problem would be to derive a Maurer-Cartan type of equation. It's my understanding that, on general grounds, one expects the deformation theory of almost any structure to be controlled by a differential graded Lie algebra (DGLA), with (perhaps formal) deformations corresponding to solutions of the Maurer-Cartan equation. One candidate for the DGA in the above example would be the space of sections of the Dolbeault complex $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ associated to the holomorphic vector bundle $\Lambda^{\bullet} T_{X/Y}$. This comes equipped with the Dolbeault differential, but no obvious Lie bracket (at least not obvious to me).

$\textbf{Question 2}:$ Can $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ be given the structure of a DGLA (i.e. equipped with a bracket compatible with the Dolbeault differential), such that formal deformations of $Y$ as a complex submanifold of $X$ are given by solutions to the Maurer-Cartan equation? If not, is there some other DGLA controlling the deformation theory of a complex submanifold which has a "geometric" description (i.e. as the space of sections of some vector bundle)?

Answer 1

I'd too suggest the paper by Donatella Iacono as a basic reference. I'll just add here a few lines to complement Urs Schreiber's answer by explaining in which sense the $\infty$-groupoids point of view clarifies what happens here (if my memory is not playing a bad trick on my, in the paper with Elena Martinengo we do not address the problem of deformations of submanifalds, although the needed technology is there). So these few lines are to be read as a "How to read Donatella Iacono's results by an $\infty$-groupoids point of view".

What one does is moving from Set-valued deformation functors to $\infty$-groupoids valued deformation functors (these are called "formal moduli problems" in Lurie's DAG X). One recovers the classical deformation functor from the $\infty$-groupoid valued one simply by taking $\pi_0$. However (and this is the reason to prefer the $\infty$-groupoid valued version), things are behaved much more naturally in the $\infty$-groupoids setting since here one can make homotopy invariant constructions.

More precisely, if we denote by $Def_{\mathfrak{g}}$ the $\infty$-groupoids-valued deformation functor associated with a differential graded Lie algebra $\mathfrak{g}$, then the association $\mathfrak{g}\mapsto Def_{\mathfrak{g}}$ establishes an equivalence of $(\infty,1)$ categories between differential graded Lie algebras and formal moduli problems (see Lurie's DAG X or Pridham's "Unifying derived deformation theories"). This means in particular that if the deformation problem we are interested in arises as a (homotopy) limit of simpler deformation problems for which we know differential graded Lie algebras $\mathfrak{g}_i$ governing them, then the problem we are interested in will be governed by the (homotopy) limit of the $\mathfrak{g}_i$'s.

For instance, consider the problem of $Def_{Z\hookrightarrow X}$ of infinitesimal deformations of a complex submanifold $Z$ inside a complex manifold $X$. Such a deformation is equivalently the datum of a deformation of the pair $(Z,X)$ together with the datum of a trivialization of the deformation of $X$. So if we denote by $Def_{(Z,X)}$ and by $Def_X$ the deformation functors describing infinitesimal deformations of the pair and of $X$ respectively, we see that the problem $Def_{Z\hookrightarrow X}$ we are interested in is the (homotopy) fiber of the forgetful morphism $Def_{(Z,X)}\to Def_X$. For both $Def_X$ and $Def_{(Z,X)}$ it is simple to describe dglas governing them. They are the Dolbeault complex on $X$ with coefficients in the tangent sheaf of $X$ and the sub-dgla of this given by the kernel of the natural morphism to the Dolbeault complex on $Z$ with coefficients in the normal sheaf of $Z$ (i.e., by differential forms on $X$ with coefficients in the tangent sheaf of $X$ which, restricted to $Z$ are tangent to $Z$, thus inducing a deformation of $Z$). The dgla governing $Def_{Z\hookrightarrow X}$ will therefore be the homotopy fiber of this inclusion.

Since the model category structure on differential graded Lie algebras is induced by that on chain complexes, as a chain complex the homotopy fiber of the inclusion of the kernel above is quasi-isomorphic to the Dolbeault complex on $Z$ with coefficients in the normal sheaf of $Z$ shifted by one degree. And by the homotopical transfer of $L_\infty$-structures this means that there is a natural $L_\infty$-algebra structure on the shifted Dolbeault complex of $Z$ with values in $N_{X/Z}$, extending the chain complex structure, such that the associated deformation functor is $Def_{Z\hookrightarrow X}$. This answers Question 2 above.

In particular one recovers the well known fact that the tangent space to $Def_{Z\hookrightarrow X}$ is $H^0(Z, N_{X/Z})$: this is a degree 1 cohomology group for the shifted Dolbeault complex. Similarly one sees that $H^1(Z, N_{X/Z})$ is an obstruction space for $Def_{Z\hookrightarrow X}$.

Answer 2

This paper by Donatella Iacono could be of some help for you.

Answer 3

This here might be helpful:

Domenico Fiorenza, Elena Martinengo, A short note on ∞-groupoids and the period map for projective manifolds, Publications of the nLab vol. 2 no. 1 (2012) (arXiv:0911.3845)