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

Kefeng Liu, Yang Shen

Abstract

Let $(X,ω_0)$ be a compact Kähler 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 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.

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

Abstract

Let be a compact Kähler manifold and its Kuranishi family, where the base may be singular with . 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 -classes. For deformations over irreducible analytic bases, we introduce -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 -forms, and give an intrinsic analytic description of Hodge loci, leading to a Beltrami-differential criterion for the variational Hodge conjecture.
Paper Structure (8 sections, 31 theorems, 246 equations)

This paper contains 8 sections, 31 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Sections of Hodge bundles
  3. Characterization of (p,p)-classes
  4. Kähler stability and upper semicontinuity of Kähler cones
  5. Kähler stability on large scales
  6. Algebraic approximations of compact Kähler manifolds
  7. Approximation of (p,p)-classes by Hodge classes
  8. Hodge locus and variational Hodge conjecture

Key Result

Proposition 2

The classes in $\mathcal{K}^{\nabla^{1,1}}_{t_{0},t}$ admit positive definite representatives provided that there exists a Beltrami differential $\varphi(t)\in A^{0,1}(X_{t_{0}},\mathrm{T}^{1,0} X_{t_{0}})$ with supremum operator norm such that that is, there exists a diffeomorphism $d_{t}:\, X_{t}\to X_{t_{0}}$ which induces $\varphi(t)$.

Theorems & Definitions (68)

  • Definition 1
  • Proposition 2
  • Theorem 3: Upper semicontinuity of Kähler cones
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Definition 1.1
  • Theorem 1.2
  • ...and 58 more