Holomorphic Torus Actions on Compact Locally Conformal Kähler Manifolds

Abstract

Given a torus action (T², M) on a smooth manifold, the orbit map ev_x(t)=t·x for each x∈M induces a homomorphism ev_*: $\mathbb Z$²→H₁(M;$\mathbb Z$). The action is said to be Rank-k if the image of ev_* has rank k ([les ]2) for each point of M. In particular, if ev_* is a monomorphism, then the action is called homologically injective. It is known that a holomorphic complex torus action on a compact Kähler manifold is homologically injective. We study holomorphic complex torus actions on compact non-Kähler Hermitian manifolds. A Hermitian manifold is said to be a locally conformal Kähler if a lift of the metric to the universal covering space is conformal to a Kähler metric. We shall prove that a holomorphic conformal complex torus action on a compact locally conformal Kähler manifold M is Rank-1 provided that M has no Kähler structure.