The EGH Conjecture and the Sperner property of complete intersections

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.

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EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s147474802100027x ◽

2021 ◽

pp. 1-66

Author(s):

Tom Bachmann ◽

Kirsten Wickelgren

Keyword(s):

Commutative Algebra ◽

Vector Bundles ◽

Euler Class ◽

Complete Intersections ◽

Bilinear Forms ◽

Euler Numbers ◽

Coherent Sheaves ◽

Linear Subspaces ◽

Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

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On complete intersections with trivial canonical class

Advances in Mathematics ◽

10.1016/j.aim.2014.08.013 ◽

2015 ◽

Vol 268 ◽

pp. 339-349

Author(s):

Lev A. Borisov ◽

Zhan Li

Keyword(s):

Complete Intersections ◽

Canonical Class

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Branched Coverings and Minimal Free Resolution for Infinite-Dimensional Complex Spaces

Georgian Mathematical Journal ◽

10.1515/gmj.2003.37 ◽

2003 ◽

Vol 10 (1) ◽

pp. 37-43

Author(s):

E. Ballico

Keyword(s):

Projective Space ◽

Projective Spaces ◽

Free Resolution ◽

Complete Intersections ◽

Vanishing Theorem ◽

Sheaf Cohomology ◽

Minimal Free Resolution ◽

Branched Coverings ◽

Infinite Dimensional ◽

Complex Spaces

Abstract We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P(V ).

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Birationally rigid Fano complete intersections. II

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2012-0059 ◽

2014 ◽

Vol 2014 (688) ◽

Cited By ~ 5

Author(s):

Aleksandr V. Pukhlikov

Keyword(s):

Complete Intersections

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LAURENT POLYNOMIAL LANDAU–GINZBURG MODELS FOR COMINUSCULE HOMOGENEOUS SPACES

Transformation Groups ◽

10.1007/s00031-020-09636-7 ◽

2021 ◽

Author(s):

PETER SPACEK

Keyword(s):

Homogeneous Space ◽

Homogeneous Spaces ◽

Laurent Polynomial ◽

Complete Intersections ◽

Laurent Polynomials ◽

Young Diagrams ◽

Similar Shape ◽

Algebraic Torus ◽

The One ◽

Polynomial Potentials

AbstractIn this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

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