Almost Hermitian Identities

Joana Cirici,Scott O Wilson

doi:10.3390/math8081357

Abstract

We study the local commutation relation between the Lefschetz operator and the exterior differential on an almost complex manifold with a compatible metric. The identity that we obtain generalizes the backbone of the local Kähler identities to the setting of almost Hermitian manifolds, allowing for new global results for such manifolds.

Highlights

On a Kähler manifold ( M, J, ω ), the most fundamental local identity is perhaps the commutation relation between the exterior differential d and the adjoint Λ to the Lefschetz operator,[Λ, d] = ? I−1 d I ?, (1)where ? denotes the Hodge star operator and I denotes the extension of J to all forms.This identity, due to A
Weil [1], strongly depends on the Kähler condition, dω = 0, and is true when removing the integrability condition NJ ≡ 0. It is valid for almost Kähler and symplectic manifolds as well [2,3,4]
The local identities of [5,6] for complex non-Kähler manifolds include other algebra terms which lead to further Laplacian operators, leading to various inequalities relating the geometry with the topology of the manifold

Summary

Introduction

On a Kähler manifold ( M, J, ω ), the most fundamental local identity is perhaps the commutation relation between the exterior differential d and the adjoint Λ to the Lefschetz operator,. There the fundamental relation of Proposition 1 is used to show that the moduli spaces of unitary anti-self-dual connections over any almost Hermitian 4-manifold is almost Hermitian, whenever the Nijenhuis tensor has sufficiently small C0 -norm This generalizes a well known result for Kähler manifolds that was exploited in Donaldson’s work in the 1980s, and is expected to have consequences for the topology of almost complex 4-manifolds which are of so-called. In the integrable Kähler case both inequalities are true and so one recovers the well-known consequence of the Hodge decomposition. The local identities of [5,6] for complex non-Kähler manifolds include other algebra terms which lead to further Laplacian operators, leading to various inequalities relating the geometry with the topology of the manifold. We aim to further understand the origin of these inequalities by means of the correct version of (1) for almost Hermitian manifolds for which, a priori, the only geometric-topological inequality in the compact case is given by dim Ker (∆μ + ∆∂ ̄ + ∆∂ + ∆μ )|( p,q) ≤ bk

Preliminaries

Almost Hermitian Identities

Applications