Solving equations with Hodge theory

2018/03/04 by Kefeng Liu, Liu, Kefeng, Shengmao Zhu +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1803.01272

openalex publication_date 2018/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We treat two quite different problems related to changes of complex structures on Kähler manifolds by using global geometric method. First, by using operators from Hodge theory on compact Kähler manifold, we present a closed explicit extension formula for holomorphic canonical forms in different complex structures. As applications, we give a closed explicit formula for certain canonical sections of Hodge bundles on marked and polarized moduli spaces of projective manifolds, and provide a closed explicit extension formula for holomorphic pluricanonical forms under certain natural conditions. Second, by using the operators in L²-Hodge theory on Poincaré disk, we present a simple and unified method to solve the Beltrami equations with measurable coefficients for quasi-conformal maps.

Citations

• Power series proofs for local stabilities of Kähler and balanced structures with mild ∂∂-lemma
• Quasi-isometry and deformations of Calabi-Yau manifolds

Related

• Sections of Hodge bundles II: Deformation of (p,p)-classes and applications to Kähler geometry
• A Solution to the Hodge Conjecture
• The raising steps method. Applications to the L^r Hodge theory in a compact riemannian manifold
• Harmonic Forms, Hodge Theory and the Kodaira Embedding Theorem
• Quasi-isometry and deformations of Calabi-Yau manifolds