$W^{1,2}$ Bott-Chern and Dolbeault decompositions on K\"ahler manifolds

TL;DR

This paper establishes weak $W^{1,2}$ decompositions for Bott-Chern and Dolbeault forms on K"ahler manifolds, extending classical results to non-compact cases with bounded curvature.

Contribution

It introduces $W^{1,2}$ weak decompositions for Bott-Chern and Dolbeault forms on K"ahler manifolds, including non-compact cases with curvature bounds.

Findings

Proves $W^{1,2}$ weak Bott-Chern and Dolbeault decompositions.

Shows relation between Bott-Chern decomposition and harmonic forms under curvature conditions.

Extends compact K"ahler properties to non-compact manifolds with bounded curvature.

Abstract

Let $(M,J,g,\omega)$ be a K\"ahler manifold. We prove a $W^{1,2}$ weak Bott-Chern decomposition and a $W^{1,2}$ weak Dolbeault decomposition, following the $L^2$ weak Kodaira decomposition on Riemannian manifolds. Moreover, if the K\"ahler metric is complete and the sectional curvature is bounded, the $W^{1,2}$ Bott-Chern decomposition is strictly related to the space of $W^{1,2}$ Bott-Chern harmonic forms, i.e., $W^{1,2}$ smooth differential forms which are in the kernel of an elliptic differential operator of order $4$, called Bott-Chern Laplacian. We also generalize to the non compact case the well known property that on compact K\"ahler manifolds the kernel of the Dolbeault Laplacian and the kernel of the Bott-Chern Laplacian coincide.