This is somewhat related to the question found at What is the DGLA controlling the deformation theory of a complex submanifold?, though not exactly the same, so I hope it's not duplicating too much. There also seem to be a few threads about deformations of pairs where the sub-manifold is a hyper-surface, but this is not the case I'm interested in.

The basic question is: what are the basic obstructions to obtaining a deformation of a pair consisting of a complex, compact manifold $X$ and a compact, complex submanifold $M\subset X.$ Morally, I know this should involve the cohomology of the tangent sheaf to $X$ and the cohomology of the normal sheaf of $M\subset X.$

My current definition of a deformation of the pair $(X,M)$ is a tuple $(\mathcal{X},\mathcal{M},B,\pi,0)$ where $\mathcal{X}$ is a complex manifold, $\mathcal{M}\subset \mathcal{X}$ is a closed, complex submanifold, $B$ is a complex manifold and, \begin{align} \pi:\mathcal{X}\rightarrow B, \end{align} is a proper, holomorphic submersion. Furthermore, $\pi^{-1}(0)=X$ and $\pi^{-1}(0)\cap \mathcal{M}=M.$

The Kodaira-Spencer deformation theory tells us that if we don't care about $M,$ then a sufficient condition to obtain such a family is that the sheaf cohomology group vanishes; $H^2(X, TX)=0,$ where $TX$ is the sheaf of holomorphic vector fields. In this case, the tangent space to the base is identified with the first sheaf cohomology group $H^{1}(X,TX)$ via the Kodaira-Spencer map. On a related note, if we wish to deform $M$ inside of a fixed $X,$ then the first order obstruction comes from the first sheaf cohomology of $M$ with coefficients in the normal bundle of $M$ inside of $X.$ Nonetheless, I can't seem to find any basic references governing the deformation theory of the pair $(X,M),$ though I imagine to the right crowd this is very basic stuff.

So, perhaps to just give one very specific question, is there a basic condition in addition to $H^2(X,TX)=0$ which guarantees that the deformation of $X$ induces a deformation of $M.$ Contrarily, when does a deformation of $M$ as a complex manifold extend to a deformation of $X?$

In the simplest case of a complex torus $X$ of dimension greater than $1,$ and some ellipctic curve $M\subset X,$ it seems likely that every deformation of $X$ gives rise to a deformation of $M$ and vice-versa. But, at the moment I'm having some trouble fishing out what the necessary objects for this problem are.

Thank you for your help.


1 Answer 1


The infinitesimal deformations of $(X,M)$ are controlled by the sheaf $T_X\langle M\rangle$ of vector fields on $X$ which are tangent to $M$: see for instance this paper, Prop. 1.1. Thus the obstruction you are looking for lies in $H^2(X,T_X\langle M\rangle)$. The exact sequence $$0\rightarrow T_X\langle M\rangle \rightarrow T_X\rightarrow N_{M/X}\rightarrow 0$$ relates it to the obstructions to deform $X$ and $M$ in $X$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.