Deformation of pairs and semiregularity

TL;DR

This paper investigates the deformation theory of maps into Kähler manifolds with divisorial images, establishing conditions under which deformations are possible based on Hodge classes and linking semiregularity to classical geometric notions.

Contribution

It refines the variational Hodge conjecture by connecting semiregularity of maps with the Hodge property of cycle classes in families.

Findings

Deformation of maps is possible iff the cycle class remains Hodge.

Semiregularity relates to Cayley-Bacharach conditions.

Results provide new insights into classical stability notions.

Abstract

In this paper, we study relative deformations of maps into a family of K\"ahler manifolds whose images are divisors. We show that if the map satisfies a condition called semiregularity, then it allows relative deformations if and only if the cycle class of the image remains Hodge in the family. This gives a refinement of the so-called variational Hodge conjecture. We also show that the semiregularity of maps is related to classical notions such as Cayley-Bacharach conditions and d-semistability.