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 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.