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

Abstract

Let $(X,\omega_0)$ be a compact K\"ahler manifold and $\mathcal X\to B$ its Kuranishi family, where $B$ may be singular and $\dim_{\C}B\ge1$. Using explicit sections of Hodge bundles, we define an intrinsic period map and a Hodge map parametrizing nearby $(p,p)$-classes. For deformations over irreducible analytic bases, we introduce two flat extensions of K\"ahler cones defined by the reference and moving Hodge connections. The extension associated with the reference connection admits explicit positive representatives and yields uniform upper semicontinuity, while that associated with the moving connection identifies the K\"ahler cones away from a countable union of proper analytic subsets and admits an explicit expression in terms of the period map and the Beltrami differential. These constructions provide a description of K\"ahler cones through analytic cycles and yield both local and large-scale K\"ahler stability without assuming unobstructedness. As further applications, we generalize Green's density criterion to strong algebraic approximation and to the approximation of real $(p,p)$-forms. We also obtain an intrinsic analytic description of Hodge loci, leading to a Beltrami-differential criterion for the variational Hodge conjecture.