Analytic fields on compact balanced Hermitian manifolds

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 1form to be harmonic. We prove that if H is positive definite then the first Betti number b1 = 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. We prove that 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. Running title: Analytic fields on balanced manifolds