Analytic fields on compact balanced Hermitian manifolds

TL;DR

This paper studies the properties of harmonic and analytic 1-forms on compact balanced Hermitian manifolds, introducing a tensor H to characterize conditions for their equivalence and exploring implications for Killing vector fields.

Contribution

It constructs a symmetric tensor H from torsion and curvature, providing new criteria for harmonic and analytic forms, and establishes conditions for the existence of Killing vector fields.

Findings

If H is positive definite, then b_1=0 and h^{1,0}=0.

Non-positive definite Chern form implies all Killing vector fields are analytic.

Negative definite Chern form leads to no Killing vector fields.

Abstract

On a Hermitian manifold we construct a symmetric $(1,1)$- tensor $H$ using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor $H$ for a harmonic $1$-form to be analytic and for an analytic $1$-form to be harmonic. We prove that if $H$ is positive definite then the first Betti number $b_1 = 0$ and the Hodge number $h^{1,0} = 0$. We obtain an obstruction to the existence of Killing vector fields in terms of the Ricci tensor of the Chern connection: if the Chern form of the Chern connection on a compact balanced Hermitian manifold is non- positive definite then every Killing vector field is analytic; if moreover the Chern form is negative definite then there are no Killing vector fields. It is proved that on a compact balanced Hermitian manifold every affine with respect to the Chern connection vector field is an analytic vector field.