Sections of Hodge bundles II: deformation of $(p,p)$-classes and applications to K\"ahler geometry

TL;DR

This paper develops a new framework for understanding deformations of $(p,p)$-classes on K"ahler manifolds, introducing intrinsic period and Hodge maps, and providing applications to K"ahler cones, algebraic approximation, and the variational Hodge conjecture.

Contribution

It introduces an intrinsic period map and Hodge map for deformations of $(p,p)$-classes, and characterizes K"ahler cones via analytic cycles, advancing the understanding of K"ahler geometry.

Findings

Explicit positive representatives for deformations of K"ahler cones.

Complete local description of K"ahler cones in terms of analytic cycles.

Extension of Green's density criterion to algebraic approximation and Hodge loci.

Abstract

Let $(X,\omega_0)$ be a compact K\"ahler manifold and $\mathcal X\to B$ its Kuranishi family, where the base $B$ may be singular with $\dim_{\C} B \ge 1$. Using explicit sections of Hodge bundles obtained from algebraic and geometric constructions, we define an intrinsic period map and a Hodge map that parametrizes nearby $(p,p)$-classes. For deformations over irreducible analytic bases, we introduce $\nabla^{1,1}$-flat extensions of K\"ahler cones and obtain explicit positive representatives, leading to an upper semicontinuity property for these extensions. Combined with the characterization of K\"ahler cones due to Demailly--Paun, this yields a complete local description of K\"ahler cones in terms of analytic cycles. We further show that this upper semicontinuity persists on a large region of the base determined by a uniform bound on the operator norm of the Beltrami differential. As further applications, we generalize Green's density criterion to strong algebraic approximation and to the approximation of real $(p,p)$-forms, and give an intrinsic analytic description of Hodge loci, leading to a Beltrami-differential criterion for the variational Hodge conjecture.