Holomorphic principal bundles over projective toric varieties

Meersseman, Laurent; Verjovsky, Alberto

doi:10.1515/crll.2004.054

Cited by 38 publications

(80 citation statements)

References 11 publications

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“…To do that, we have to study another action whose quotient is a toric variety closely related to our LVMB manifold. This action was already studied in [18] and [9] but we need a more thorough ANNALES DE L'INSTITUT FOURIER study. Finally, in the fifth part, we make the inverse construction: starting with a rationally starshaped sphere, we construct a LVMB manifold whose associated complex is the given sphere.…”

Section: Introductionmentioning

confidence: 99%

LVMB manifolds and simplicial spheres

Tambour¹

2012

Ann. inst. Fourier

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LVM and LVMB manifolds are a large family of examples of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a very natural action of the real torus and the quotient of this action is a simple polytope. This quotient allows us to relate closely LVM manifolds to the moment-angle manifolds studied (for example) by Buchstaber and Panov. The aim of this paper is to generalize the polytopes associated to LVM manifolds to the LVMB case and study its properties. In particular, we show that it belongs to a very large class of simplicial spheres. Moreover, we show that for every sphere belonging to this class, we can construct a LMVB manifold whose associated sphere is the given sphere. We use this latter result to show that many moment-angle complexes can be endowed with a complex structure (up to product with circles).

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Section: Introductionmentioning

confidence: 99%

LVMB manifolds and simplicial spheres

Tambour¹

2012

Ann. inst. Fourier

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“…., n} is indispensable if i ∈ P for all P ∈ E m,n (cf. [3], [17] and [18]). There are 2 types of LVMB data; one is the case with an indispensable integer and the other one is the case with no indispensable integer.…”

Section: Theorem 92mentioning

confidence: 99%

“…It has been shown that certain LVM manifolds carry principal bundles over projective quasi-regular toric varieties. Conversely, starting from any quasiregular projective toric variety, a principal bundle over it whose total space is an LVM manifold can be constructed (see [18] for details). It turns out that any projective smooth toric variety can be obtained from an LVMB manifold as a quotient space.…”

Section: Equivariant Principal Bundlesmentioning

confidence: 99%

Complex manifolds with maximal torus actions

Ishida

2016

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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AbstractIn 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 {(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in {\mathfrak{g}}, complex vector subspace {\mathfrak{h}} of {\mathfrak{g}^{\mathbb{C}}} and compact torus G 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 {\mathbb{C}^{n}}-actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.

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“…In the case of rational simplicial normal fans Σ P a construction of MeerssemanVerjovsky [44] identifies the corresponding projective toric variety V P as the base of a holomorphic principal Seifert fibration, whose total space is the moment-angle manifold Z P equipped with a complex structure of an LVM-manifold, and fibre is a compact complex torus of complex dimension ℓ = m−n 2 . (Seifert fibrations are generalisations of holomorphic fibre bundles to the case when the base is an orbifold.)…”

Section: Holomorphic Principal Bundles Over Toric Varieties and Dolbementioning

confidence: 99%

Geometric structures on moment-angle manifolds

2015

Mathematical Surveys and Monographs

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Abstract. The moment-angle complex Z K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in toric topology, and currently are gaining much interest in homotopy theory, complex and symplectic geometry.The geometric aspects of the theory of moment-angle complexes are the main theme of this survey. We review constructions of non-Kähler complexanalytic structures on moment-angle manifolds corresponding to polytopes and complete simplicial fans, and describe invariants of these structures, such as the Hodge numbers and Dolbeault cohomology rings. Symplectic and Lagrangian aspects of the theory are also of considerable interest. Moment-angle manifolds appear as level sets for quadratic Hamiltonians of torus actions, and can be used to construct new families of Hamiltonian-minimal Lagrangian submanifolds in a complex space, complex projective space or toric varieties.

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Holomorphic principal bundles over projective toric varieties

Cited by 38 publications

References 11 publications

LVMB manifolds and simplicial spheres

LVMB manifolds and simplicial spheres

Complex manifolds with maximal torus actions

Geometric structures on moment-angle manifolds

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