Mixed Hodge structures and Sullivan’s minimal models of Sasakian manifolds

Kasuya, Hisashi

doi:10.5802/aif.3142

Cited by 5 publications

(7 citation statements)

References 13 publications

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“…This result is optimal; indeed, for each n ě 1, the p2n `1q-dimensional Heisenberg compact nilmanifold H n is a Sasakian manifold, yet it is not n-formal. Theorem 1.6 strengthens a statement of Kasuya [30], who claimed that, for n ě 2, a Sasakian manifold as above is 1-formal. The proof of that claim, though, has a gap, which we avoid by giving a proof based on a very different approach.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 78%

“…It turns out that the proof from [30] has a gap, which we now proceed to explain. Given a cdga A, the (degree 2) decomposable part is the subspace DH 2 pAq Ď H 2 pAq defined as the image of the product map in homology, H 1 pAq ^H1 pAq Ñ H 2 pAq.…”

Section: Sasakian Manifoldsmentioning

confidence: 96%

“…What Kasuya actually shows is that (17) DH 2 pM 1 pMqq " H 2 pM 1 pMqq, for a compact Sasakian manifold M 2n`1 with n ą 1, where M 1 pMq is the Sullivan 1minimal model of M, over R. Equality ( 17) is an easy consequence of 1-formality. Kasuya deduces the 1-formality of M from (17), by invoking in [30,Proposition 4.1] as a crucial tool Lemma 3.17 from [2]. Unfortunately, though, this lemma is false, as shown by Mȃcinic in Example 4.5 and Remark 4.6 from [33].…”

Section: Sasakian Manifoldsmentioning

confidence: 99%

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The Topology of Compact Lie Group Actions Through the Lens of Finite Models

Papadima¹,

Suciu

2018

International Mathematics Research Notices

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Abstract. Given a compact, connected Lie group K, we use principal K-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let M be a compact, connected, smooth manifold which supports an almost free K-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for M with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of M and of the representation varieties of π 1 pMq. As an application, we show that compact Sasakian manifolds of dimension 2n`1 are pn´1q-formal, and that their fundamental groups are filtered-formal. Further applications to the study of weighted-homogeneous isolated surface singularities are also given.

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Section: Introduction and Statement Of Resultssupporting

confidence: 78%

Section: Sasakian Manifoldsmentioning

confidence: 96%

Section: Sasakian Manifoldsmentioning

confidence: 99%

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The Topology of Compact Lie Group Actions Through the Lens of Finite Models

Papadima¹,

Suciu

2018

International Mathematics Research Notices

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“…We notice that Vaisman metrics are closely related to Sasakian structures. We can also obtain nice de Rham models of Sasakian manifolds like the above DGA (see [26]) and we can develop Morgan's mixed Hodge theory on Sasakian manifolds (see [15]).…”

Section: Mixed Hodge Diagrams For Transverse Kähler Structures On Cenmentioning

confidence: 99%

Transverse Kähler structures on central foliations of complex manifolds

Ishida

Kasuya

2018

Annali di Matematica

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For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. If there exists a transverse Kähler structure on such a foliation, then we obtain a nice differential graded algebra which is quasi-isomorphic to the de Rham complex and a nice differential bi-graded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kähler manifolds. Moreover, under certain additional condition, we can develop Morgan's theory of mixed Hodge structures as similar to the study on smooth algebraic varieties.

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Section: Simple Examplesmentioning

confidence: 99%

Transverse Kähler structures on central foliations of complex manifolds

Ishida¹,

Kasuya²

2016

Preprint

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For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. If there exists a transverse Kähler structure on such a foliation, then we obtain a nice differential graded algebra which is quasi-isomorphic to the de Rham complex and a nice differential bigraded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kähler manifolds. Moreover, under certain additional condition, we can develop Morgan's theory of mixed Hodge structures as similar to the study on smooth algebraic varieties.

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Mixed Hodge structures and Sullivan’s minimal models of Sasakian manifolds

Cited by 5 publications

References 13 publications

The Topology of Compact Lie Group Actions Through the Lens of Finite Models

The Topology of Compact Lie Group Actions Through the Lens of Finite Models

Transverse Kähler structures on central foliations of complex manifolds

Transverse Kähler structures on central foliations of complex manifolds

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