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A note on $\tmd$-operator

Dexie Lin, Hengyu Zhou

TL;DR

This work tackles the lack of a direct symplectic analogue to the $\partial\bar{\partial}$-lemma in almost Kähler 4-manifolds by revisiting the $\tilde{D}^+_J$ operator and related structures. It develops a local-to-global analytic framework using $L^2$ methods and the Atiyah-Hitchin-Singer operator to prove that every $d$-exact real $(1,1)$-form is $\tilde{D}^+_J$-exact when the form is exact on a compact taming symplectic 4-manifold, under the condition $d^-_J a=0$ for the primitive 1-form $a$. The paper also analyzes local exactness obstacles, computes Chern-connection aspects, and demonstrates the necessity of a global elliptic approach to achieve the main result. These results contribute to a broader elliptic-analytic program toward Donaldson’s taming conjecture and underscore the interplay between symplectic taming, almost complex geometry, and global differential operators on 4-manifolds.

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.

A note on $\tmd$-operator

TL;DR

This work tackles the lack of a direct symplectic analogue to the -lemma in almost Kähler 4-manifolds by revisiting the operator and related structures. It develops a local-to-global analytic framework using methods and the Atiyah-Hitchin-Singer operator to prove that every -exact real -form is -exact when the form is exact on a compact taming symplectic 4-manifold, under the condition for the primitive 1-form . The paper also analyzes local exactness obstacles, computes Chern-connection aspects, and demonstrates the necessity of a global elliptic approach to achieve the main result. These results contribute to a broader elliptic-analytic program toward Donaldson’s taming conjecture and underscore the interplay between symplectic taming, almost complex geometry, and global differential operators on 4-manifolds.

Abstract

In almost Kähler manifolds, one of the challenges is to construct an elliptic operator on functions that plays a role analogous to the operator in complex or Kähler manifolds. One of the aims of this paper is to revisit the -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 -exact -form is globally -exact for any compact taming symplectic -manifold.

Paper Structure

This paper contains 6 sections, 8 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminary on $\tilde{D}^+_J$-operator on tamed almost complex $4$ manifold
  3. Local behavior on $\mathcal{\tilde{D}}^+_J$-operator
  4. Local exactness
  5. Formula on Chern connection
  6. The proof

Key Result

Theorem 1.1

Let $(M, J, g, F)$ be a compact almost Hermitian $4$-manifold, where $J$ is tamed by a symplectic form $\omega$. If a real-valued $(1,1)$-form $\psi$ can be expressed as $\psi = da$ for some real-valued $1$-form $a$, then $\psi$ is $\mathcal{\tilde{D}}^+_J$-exact.

Theorems & Definitions (18)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Draghici Li and Zhang DLiZh10
  • Theorem 2.4: Draghici Li and Zhang DLiZh10
  • Theorem 2.5: Lejmi Lejmi06
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8: Tan Wang Zhou and Zhu TWZZ
  • Lemma 2.9: Tan Wamg Zhou and Zhu TWZZ
  • ...and 8 more