A note on $\tmd$-operator

Dexie Lin; Hengyu Zhou

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arXiv · 2503.15939

A note on $\tmd$-operator

Dexie Lin·Hengyu Zhou

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Abstract

In almost K\"ahler 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 K\"ahler 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.

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Dexie Lin, Hengyu Zhou. 2025-03-20. A note on $\tmd$-operator. https://arxiv.org/abs/2503.15939

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