Foliations modeling nonrational simplicial toric varieties
TLDR
In this article, the authors established a correspondence between simplicial fans and certain foliated compact complex manifolds called LVMB-manifolds, and showed that the basic cohomology of the foliation is generated in degree two.Abstract:
We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMB-manifolds. In the rational case, Meersseman and Verjovsky have shown that the leaf space is the usual toric variety. We compute the basic Betti numbers of the foliation for shellable fans. When the fan is in particular polytopal, we prove that the basic cohomology of the foliation is generated in degree two. We give evidence that the rich interplay between convex and algebraic geometries embodied by toric varieties carries over to our nonrational construction. In fact, our approach unifies rational and nonrational cases.read more
Citations
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Toric Topology
TL;DR: Toric topology emerged in the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra, and continues to attract experts from different fields.
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Geometric structures on moment-angle manifolds
TL;DR: A moment-angle complex ZK as discussed by the authors is a cell complex with a torus action constructed from a finite simplicial complex K. When this con- struction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold.
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Geometric structures on moment-angle manifolds
TL;DR: Recently, moment-angle manifolds and complexes are gaining much interest in homotopy theory, complex and symplectic geometry as mentioned in this paper, and the geometric aspects of the theory of momentangle complexes are the main theme of this survey.
Posted Content
Presymplectic convexity and (ir)rational polytopes
Tudor S. Ratiu,Nguyen Tien Zung +1 more
TL;DR: In this paper, the Atiyah-Guillemin-Sternberg convexity theorem and Delzant's classification of symplectic toric manifolds to presymplectic manifolds were extended to presypolymplectic manifold.
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Complex geometry of moment-angle manifolds
TL;DR: In this article, the authors constructed transversely Kaehler metrics on moment-angle manifolds, under some restriction on the combinatorial data, called complete simplicial fans.
References
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TL;DR: In this article, the authors present a rich collection of material on the modern theory of convex polytopes, with an emphasis on the methods that yield the results (Fourier-Motzkin elimination, Schlegel diagrams, shellability, Gale transforms, and oriented matroids).
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Differential Forms in Algebraic Topology
Raoul Bott,Loring W. Tu +1 more
TL;DR: This paper presents a meta-thesis on the basis of a model derived from the model developed in [Bouchut-Boyaval, M3AS (23) 2013] that states that the mode of action of the Higgs boson is determined by the modulus of the E-modulus.
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Stratified Morse theory
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TL;DR: In this paper, the fundamental problem of Morse theory is to study the topological changes in the space X ≤c as the number c varies, where X is a topological space and c is a real number.
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The geometry of toric varieties
TL;DR: Affine toric varieties have been studied in this article, where the definition of an affine Toric variety and its properties have been discussed, including cones, lattices, and semigroups.