A note on $\tmd$-operator

Abstract

In almost Kähler manifolds, one of the challenges is to construct an elliptic operator on functions that plays a role analogous to the $\partial \bar{\partial }$ operator in complex or Kähler manifolds. One of the aims of this paper is to revisit the $\tmd$-operator introduced in \cite{TWZZ}. We will provide some local analysis estimates and highlight several difficulties that remain to be addressed. Additionally, we use the Atiyah-Hitchin-Singer operator to demonstrate that every $d$-exact $(1,1)$-form is globally $\tmd$-exact for any compact taming symplectic $4$-manifold.

12 pages. Comments are welcom