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Compact homogeneous locally conformally Kähler manifolds. (English) Zbl 1361.32027

Let \((M,J)\) be a manifold with a complex structure \(J\). A Hermitian structure \(h\) (i.e., a Riemannian metric such that \(\Omega :=h\circ J = h(J \cdot, \cdot)\) is a 2-form (the Kähler form)) is called locally conformally Kähler (or shortly l.c.K.) if \( d \Omega = \theta \wedge \Omega\) for some closed 1-form \(\theta\), called the Lee form. The manifold \((M,J)\) is called an l.c.K. manifold if it admits a l.c.K. structure. It is called a Vaisman manifold if it admits a l.c.K. structure \(h\) such that the Lee form is parallel with respect to the Levi-Civita connection, and \((M,J,h)\) is called a homogeneous l.c.K. manifold if a Lie group \(G\) of authomorpisms of \((M, J, h)\) acts transitively on \(M\). The first theorem describes the holomorphic structure of compact homogeneous l.c.K. manifolds. It states that a compact homogeneous l.c.K. manifold \(M \) is biholomorphic to a holomorphic principal bundle over a flag manifold, with fibre a 1-dimensional complex torus \(T^1_{\mathbb{C}}\). More precisely, \(M\) can be identified with \( G/H\), where \(G \) is a compact connected Lie group of holomorphic automorphisms of \(M \), which is of the form \(G = S^1 \times S\), where \(S\) is a compact simply connected semi-simple Lie group, which contains the connected component \(H_0\) of the stability subgroup \( H\). Moreover, \( S/H_0\) is a compact simply connected homogeneous Sasaki manifold, which is a principal fiber bundle over a flag manifold \(S/Q\) with circle fibre \( S^1 = Q/H_0\), where \(Q \supset H_0\) is a parabolic subgroup of \( S\). The manifold \( M \) can be expressed as \(M = T^1 \times_{\Gamma}S/H_0\), where \(\Gamma = H/H_0 \) is a finite abelian group acting holomorphically on the fiber \(T^1_{\mathbb{C}}\) of the fibration \( G/H_0 \to G/Q \) from the right.
The second theorem claims that a compact homogeneous l.c.K. manifold \( (M= G/H,h)\) is a Vaisman manifold, that is, the Lee form \( \theta \) is parallel, hence the dual vector field \(\xi = h^{-1}\circ \theta \) is a parallel Killing vector field.

MSC:

32M10 Homogeneous complex manifolds
32M05 Complex Lie groups, group actions on complex spaces
53A30 Conformal differential geometry (MSC2010)
53B35 Local differential geometry of Hermitian and Kählerian structures
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Full Text: arXiv Euclid