scholarly journals Strong cohomological rigidity of toric varieties

2017 ◽  
Vol 147 (5) ◽  
pp. 971-992
Author(s):  
Suyoung Choi ◽  
Seonjeong Park
Keyword(s):  
Betti Number ◽  
Cohomology Ring ◽  
Toric Varieties ◽  
Ring Isomorphism ◽  
Quasitoric Manifolds ◽  
Cohomological Rigidity

Every cohomology ring isomorphism between two non-singular complete toric varieties (respectively, two quasitoric manifolds), with second Betti number 2, is realizable by a diffeomorphism (respectively, homeomorphism).

scholarly journals Cohomology of torus manifold bundles

2019 ◽  
Vol 69 (3) ◽  
pp. 685-698 ◽  
Author(s):  
Jyoti Dasgupta ◽  
Bivas Khan ◽  
Vikraman Uma
Keyword(s):  
Toric Variety ◽  
Cohomology Ring ◽  
Toric Varieties ◽  
Orbit Space ◽  
Dimensional Torus ◽  
Generators And Relations ◽  
Torus Manifold ◽  
Singular Cohomology ◽  
Quasitoric Manifolds ◽  
Torus Manifolds

Abstract Let X be a 2n-dimensional torus manifold with a locally standard T ≅ (S1)n action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T-bundle p : E → B and let π : E(X) → B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as a H*(B)-algebra and the topological K-ring of E(X) as a K*(B)-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt. These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety.

scholarly journals Projective bundles over toric surfaces

2016 ◽  
Vol 27 (04) ◽  
pp. 1650032 ◽  
Author(s):  
Suyoung Choi ◽  
Seonjeong Park
Keyword(s):  
Toric Variety ◽  
Cohomology Ring ◽  
Line Bundles ◽  
Projective Bundle ◽  
Cohomology Rings ◽  
Toric Surfaces ◽  
Projective Bundles ◽  
Higher Dimensional ◽  
Quasitoric Manifolds

Let [Formula: see text] be the Whitney sum of complex line bundles over a topological space [Formula: see text]. Then, the projectivization [Formula: see text] of [Formula: see text] is called a projective bundle over [Formula: see text]. If [Formula: see text] is a nonsingular complete toric variety, then so is [Formula: see text]. In this paper, we show that the cohomology ring of a nonsingular projective toric variety [Formula: see text] determines whether it admits a projective bundle structure over a nonsingular complete toric surface. In addition, we show that two [Formula: see text]-dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic as graded rings. Furthermore, we study the smooth classification of higher dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds.

scholarly journals On the integral cohomology ring of toric orbifolds and singular toric varieties

2017 ◽  
Vol 17 (6) ◽  
pp. 3779-3810 ◽  
Author(s):  
Anthony Bahri ◽  
Soumen Sarkar ◽  
Jongbaek Song
Keyword(s):  
Cohomology Ring ◽  
Toric Varieties ◽  
Integral Cohomology

scholarly journals The vanishing of the chow cohomology ring for affine toric varieties and additional results

2019 ◽  
Author(s):  
◽  
Ryan Matthew Richey
Keyword(s):  
Recent Work ◽  
Moduli Space ◽  
Toric Variety ◽  
Cohomology Ring ◽  
Toric Varieties ◽  
Chow Ring

From the recent work of Edidin and Satriano, given a good moduli space morphism between a smooth Artin stack and its good moduli space X, they prove that the Chow cohomology ring of X embeds into the Chow ring of the stack. In the context of toric varieties, this implies that the Chow cohomology ring of any toric variety embeds into the Chow ring of its canonical toric stack. Furthermore, the authors give a conjectural description of the image of this embedding in terms of strong cycles. One consequence of their conjectural description, and an additional conjecture, is that the Chow cohomology ring of any affine toric variety ought to vanish. We prove this result without any assumption on smoothness. Afterwards, we present a series of results related to their conjectural description, and finally, we provide a conjectural toric description of the image of this embedding for complete toric varieties by utilizing Minkowski weights.

scholarly journals Topological classification of quasitoric manifolds with second Betti number 2

2012 ◽  
Vol 256 (1) ◽  
pp. 19-49 ◽  
Author(s):  
Suyoung Choi ◽  
Seonjeong Park ◽  
Dong Youp Suh
Keyword(s):  
Betti Number ◽  
Topological Classification ◽  
Quasitoric Manifolds

scholarly journals On the Equivalence of B-Rigidity and C-Rigidity for Quasitoric Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Jin Hong Kim
Keyword(s):  
Convex Polytopes ◽  
Cohomology Rings ◽  
Toric Topology ◽  
Quasitoric Manifolds ◽  
Simple Convex ◽  
Cohomological Rigidity

For quasitoric manifolds and moment-angle complexes which are central objects recently much studied in toric topology, there are several important notions of rigidity formulated in terms of cohomology rings. The aim of this paper is to show that, among other things, Buchstaber-rigidity (or B-rigidity) is equivalent to cohomological-rigidity (or C-rigidity) for simple convex polytopes supporting quasitoric manifolds.

scholarly journals Infinite families of equivariantly formal toric orbifolds

2019 ◽  
Vol 31 (2) ◽  
pp. 283-301
Author(s):  
Anthony Bahri ◽  
Soumen Sarkar ◽  
Jongbaek Song
Keyword(s):  
Cohomology Ring ◽  
Toric Varieties ◽  
Simplicial Complexes ◽  
Simple Polytopes ◽  
Toric Manifolds ◽  
Integral Cohomology

AbstractThe simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed integrally equivariantly formal. In all cases, it is possible to give a description of the cohomology ring and to relate it to the cohomology of the original orbifold.

scholarly journals On the graph labellings arising from phylogenetics

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
Weronika Buczyńska ◽  
Jarosław Buczyński ◽  
Kaie Kubjas ◽  
Mateusz Michałek
Keyword(s):  
Betti Number ◽  
Toric Varieties

AbstractWe study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.

Classification of Bott Manifolds up to Dimension 8

2014 ◽  
Vol 58 (3) ◽  
pp. 653-659 ◽  
Author(s):  
Suyoung Choi
Keyword(s):  
Cohomology Ring ◽  
Cohomology Rings ◽  
Integral Cohomology ◽  
Ring Isomorphism

AbstractWe show that three- and four-stage Bott manifolds are classified up to diffeomorphism by their integral cohomology rings. In addition, any cohomology ring isomorphism between two three-stage Bott manifolds can be realized by a diffeomorphism between the Bott manifolds.

Sign in / Sign up

Export Citation Format

Share Document