Publishing type：Research paper (scientific journal)  

DOI： 10.2969/jmsj/88718871

Publishing type：Research paper (scientific journal)  

Abstract

We give small deformations of a left-invariant complex structure on each simply connected semisimple compact Lie group of even dimension which are not biholomorphic to any left-invariant (right-invariant) complex structure by using the Kuranishi space.On such deformed complex manifolds, we prove the Borel–Weil–Bott type theorem, and we compute the cohomology of holomorphic tangent bundles.

DOI： 10.1515/forum-2021-0133

Other URL： https://www.degruyter.com/document/doi/10.1515/forum-2021-0133/pdf

Publishing type：Research paper (scientific journal)  

Abstract

We describe the basic cohomology ring 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 ring 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 as an important tool for studying 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.

DOI： 10.1093/imrn/rnaa252

Publishing type：Research paper (scientific journal)  

<title>Abstract</title>In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples <inline-formula id="j_crelle-2016-0023_ineq_9999_w2aab3b7b1b1b6b1aab1c14b1b1Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0023_eq_0410.png" /><tex-math>{(\Delta,\mathfrak{h},G)}</tex-math></alternatives></inline-formula> of nonsingular complete fan Δ in <inline-formula id="j_crelle-2016-0023_ineq_9998_w2aab3b7b1b1b6b1aab1c14b1b3Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0023_eq_1084.png" /><tex-math>{\mathfrak{g } }</tex-math></alternatives></inline-formula>, complex vector subspace <inline-formula id="j_crelle-2016-0023_ineq_9997_w2aab3b7b1b1b6b1aab1c14b1b5Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0023_eq_1090.png" /><tex-math>{\mathfrak{h } }</tex-math></alternatives></inline-formula> of <inline-formula id="j_crelle-2016-0023_ineq_9996_w2aab3b7b1b1b6b1aab1c14b1b7Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0023_eq_1080.png" /><tex-math>{\mathfrak{g}^{\mathbb{C } } }</tex-math></alternatives></inline-formula> and compact torus <italic>G</italic> satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate <inline-formula id="j_crelle-2016-0023_ineq_9995_w2aab3b7b1b1b6b1aab1c14b1c11Aa"><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2016-0023_eq_0986.png" /><tex-math>{\mathbb{C}^{n } }</tex-math></alternatives></inline-formula>-actions on compact connected complex manifolds of complex dimension <italic>n</italic>, and construction of concrete examples of non-Kähler manifolds.

DOI： 10.1515/crelle-2016-0023

Other URL： https://www.degruyter.com/document/doi/10.1515/crelle-2016-0023/pdf

Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s10231-018-0762-8

Other URL： http://link.springer.com/article/10.1007/s10231-018-0762-8/fulltext.html