scholarly journals CHOW GROUPS, CHOW COHOMOLOGY, AND LINEAR VARIETIES

2014 ◽  
Vol 2 ◽  
Author(s):  
BURT TOTARO
Keyword(s):  
Cohomology Ring ◽  
Toric Varieties ◽  
Chow Groups ◽  
Open Problems ◽  
Weight Filtration ◽  
Rational Varieties

AbstractWe compute the Chow groups and the Fulton–MacPherson operational Chow cohomology ring for a class of singular rational varieties including toric varieties. The computation is closely related to the weight filtration on the ordinary cohomology of these varieties. We use the computation to answer one of the open problems about operational Chow cohomology: it does not have a natural map to ordinary cohomology.

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 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 Toric varieties and Gröbner bases: the complete $$\mathbb {Q}$$-factorial case

2020 ◽  
Vol 31 (5-6) ◽  
pp. 461-482
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Toric Varieties ◽  
Toric Ideals ◽  
Projective Embedding ◽  
Complete Case ◽  
Open Problems ◽  
Toric Ideal ◽  
Definition Of ◽  
Nef Cone ◽  
Special Fibre

Abstract We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors V as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by V and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels’ arguments on the Gröbner fan of toric ideals to our complete case; we give a characterization of the Gröbner region and show an explicit correspondence between Gröbner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to V allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by V. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of $$\mathbb {Q}$$ Q -factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.

scholarly journals Even-Relative-Dimensional Vanishing Cycles in Bivariant Intersection Theory

2007 ◽  
Vol 187 ◽  
pp. 49-73 ◽  
Author(s):  
Hiroshi Saito
Keyword(s):  
Cohomology Ring ◽  
Ring Homomorphism ◽  
Chow Groups ◽  
Double Cover ◽  
Chow Group ◽  
Base Change ◽  
Intersection Theory ◽  
Generic Fibre ◽  
Vanishing Cycles ◽  
Special Fibre

AbstractFor a smooth variety proper over a curve having a fibre with isolated ordinary quadratic singularities, it is well-known that we have the vanishing cycles associated to the singularities in the étale cohomology of the geometric generic fibre. The base-change by a double cover of the base curve ramified at the image of the singular fibre has singularities corresponding to the singularities in the fibre. In this note, we show that in the even relative-dimensional case, there exist elements of the bivariant Chow group of the base-change with supports in the singularities and hence their images in the bivariant Chow group with supports in the special fibre and that the usual cohomological vanishing cycles are obtained as their images by a natural map, a kind of “cycle map” so that the elements in the bivariant Chow groups can be regarded as the vanishing cycles. The bivariant Chow group with supports in the special fibre has a ring structure and the natural map is a ring homomorphism to the cohomology ring of the geometric generic fibre. Also discussed is the relation of the bivariant Chow group with supports in the special fibre to the specialization map of Chow groups.

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 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 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.

Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Continuous Glucose Monitoring Time Series and Hypo/Hyperglycemia Prevention: Requirements, Methods, Open Problems

2008 ◽  
Vol 4 (3) ◽  
pp. 181-192 ◽  
Author(s):  
Giovanni Sparacino ◽  
Andrea Facchinetti ◽  
Alberto Maran ◽  
Claudio Cobelli
Keyword(s):  
Time Series ◽  
Continuous Glucose Monitoring ◽  
Glucose Monitoring ◽  
Open Problems ◽  
Monitoring Time
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