Solving equations with Hodge theory

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 $L^2$-Hodge theory on Poincaré disk, we present a simple and unified method to solve the Beltrami equations with measurable coefficients for quasi-conformal maps.

37 pages. All comments are welcome

References in corpus (1)

• Power series proofs for local stabilities of Kähler and balanced structures with mild $\partial \bar{\partial }$-lemma

Cited by in corpus (4)

• Deformations of Dolbeault cohomology classes
• Deformations of Dolbeault cohomology classes for Lie algebra with complex structures
• A logarithmic $\bar{\partial }$-equation on a compact Kähler manifold associated to a smooth divisor
• Extension of holomorphic canonical forms on complete d-bounded Kahler manifolds