中科院数学与系统科学研究院
数学研究所
学术报告
拓扑研讨班
报告人：Taras Panov (Moscow State University)
题 目：Complex geometry of moment-angle manifolds
时 间：2023.11.15（星期三）15:30-16:30
地 点：数学院南楼N802
摘 要：Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds with a holomorphic torus action. A complex structure on a moment-angle manifold Z is defined by a marked complete simplicial fan. When the fan is rational, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In this case, the invariants of the complex structure of Z, such Dolbeault cohomology and the Hodge numbers, can be analysed using the Borel spectral sequence of the holomorphic bundle.
In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F which is equivariant with respect to the algebraic torus action. Examples of moment-angle manifolds include the Hopf manifolds, Calabi-Eckmann manifolds, and their deformations. The holomorphic foliated manifolds (Z,F) are models for irrational toric varieties.
We describe the basic de Rhama and Dolbeault cohomology algebras of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology algebra of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg-Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence and bring it to bear on the study of smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.
We also provide a DGA model for the ordinary Dolbeault cohomology algebra of Z. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kaehler (equivalently, polytopal) case using a foliated analogue of toric blow-up.
The talk is based on joint works with Hiroaki Ishida and Roman Krutowski.
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