Holomorphic torus actions on compact locally conformal Kähler manifolds
The author studies compact non-Kähler Hermitian manifolds which admit holomorphic complex torus actions \(T^1_\mathbb{C}\times M\to M\), \((t,x)\mapsto tx\) on complex manifolds \(M\). For each \(x\in M\), the orbit map \(T^1_\mathbb{C}\to M\), \(t\mapsto tx\) induces a homomorphism \(\mathbb{Z}^2\to H_1(M; \mathbb{Z})\), and the action is called a rank-\(k\) torus action if the rank of the image of the homomorphism has rank \(k\) \((=0,1\), or 2). By \textit{J. B. Carrell} [Lect. Notes Math. 299, 205-236 (1972; Zbl 0252.57017)], every holomorphically isometric complex torus action on a compact Kähler manifold is homologically injective (i.e., rank-2). The main theorem of the article asserts that a holomorphic complex torus action on \((M,g)\) is rank-1 assuming that \((M,g)\) is a compact locally conformal non-Kähler manifold of real dimension at least 4 and the torus acts as a group of conformal transformations with respect to the Hermitian metric \(g\) on \(M\). The author also notes that some elliptic surfaces admit rank-1 torus actions.
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