Sections of Hodge bundles II: deformation of $(p,p)$-classes and applications to Kähler geometry

Let $(X,{\omega }_0)$ be a compact Kähler manifold and $\mathcal{X}\to B$ its Kuranishi family, where the base $B$ may be singular with ${\dim }_{\text{C}}B\ge 1$. 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 ${\mathrm{\nabla }}^{1,1}$-flat extensions of Kähler cones and obtain explicit positive representatives, leading to an upper semicontinuity property for these extensions. Combined with the characterization of Kähler cones due to Demailly–Paun, this yields a complete local description of Kähler 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.