Lifting cycles to deformations of two-dimensional pseudoconvex manifolds

Abstract:

Let $M$ be a strictly pseudoconvex manifold with exceptional set $A$. Let $D \geqslant 0$ be a cycle on $A$. Let $\omega :\mathfrak {M} \to Q$ be a deformation of $M$. Kodaira’s theory for deforming submanifolds of $\mathfrak {M}$ is extended to the subspace $D$. Let $\mathfrak {J}$ be the sheaf of germs of infinitesimal deformations of $D$. Suppose that ${H^1}(D,\mathfrak {J}) = 0$. If $\omega$ is the versal deformation, then $D$ lifts to above a submanifold of $Q$. This lifting is a complete deformation of $D$ with a smooth generic fiber. If all of the fibers of $\mathfrak {M}$ are isomorphic, then $\omega$ is the trivial deformation. If $M$ has no exceptional curves of the first kind, then there exists $\omega$ such that only any given irreducible component of $A$ disappears as part of the exceptional set.