scholarly journals Equivariant Chow Ring and Chern Classes of Wonderful Symmetric Varieties of Minimal Rank

2008 ◽  
Vol 13 (3-4) ◽  
pp. 471-493 ◽  
Author(s):  
M. Brion ◽  
R. Joshua
Keyword(s):  
Chern Classes ◽  
Chow Ring ◽  
Minimal Rank ◽  
Symmetric Varieties

scholarly journals On the Chow Ring of Cynk–Hulek Calabi–Yau Varieties and Schreieder Varieties

2019 ◽  
Vol 72 (2) ◽  
pp. 505-536 ◽  
Author(s):  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Positive Characteristic ◽  
K3 Surfaces ◽  
The Self ◽  
Dimensional Case ◽  
Chern Classes ◽  
Chow Ring ◽  
Cycle Class

AbstractThis note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

scholarly journals Zero-cycles on double EPW sextics

2020 ◽  
pp. 2050040
Author(s):  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Type Theorem ◽  
Chow Group ◽  
K3 Surface ◽  
Chern Classes ◽  
Chow Ring ◽  
Fourier Decomposition ◽  
Complete Family ◽  
Chow Rings ◽  
Rank One ◽  
Hyperkähler Varieties

The Chow rings of hyperKähler varieties are conjectured to have a particularly rich structure. In this paper, we focus on the locally complete family of double EPW sextics and establish some properties of their Chow rings. First, we prove a Beauville–Voisin type theorem for zero-cycles on double EPW sextics; precisely, we show that the codimension-4 part of the subring of the Chow ring of a double EPW sextic generated by divisors, the Chern classes and codimension-2 cycles invariant under the anti-symplectic covering involution has rank one. Second, for double EPW sextics birational to the Hilbert square of a K3 surface, we show that the action of the anti-symplectic involution on the Chow group of zero-cycles commutes with the Fourier decomposition of Shen–Vial.

scholarly journals Equivariant vector bundles on complete symmetric varieties of minimal rank

2014 ◽  
Vol 25 (14) ◽  
pp. 1450120 ◽  
Author(s):  
Indranil Biswas ◽  
S. Senthamarai Kannan ◽  
D. S. Nagaraj
Keyword(s):  
Vector Bundle ◽  
Symmetric Space ◽  
Vector Bundles ◽  
Borel Subgroup ◽  
Universal Cover ◽  
Minimal Rank ◽  
Symmetric Varieties ◽  
Complex Symmetric

Let X be the wonderful compactification of a complex symmetric space G/H of minimal rank. For a point x ∈ G, denote by Z the closure of BxH/H in X, where B is a Borel subgroup of G. The universal cover of G is denoted by [Formula: see text]. Given a [Formula: see text] equivariant vector bundle E on X, we prove that E is nef (respectively, ample) if and only if its restriction to Z is nef (respectively, ample). Similarly, E is trivial if and only if its restriction to Z is so.

Smooth Compactifications of Locally Symmetric Varieties

2009 ◽  
Author(s):  
Avner Ash ◽  
David Mumford ◽  
Michael Rapoport ◽  
Yung-sheng Tai
Keyword(s):  
Symmetric Varieties

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.

scholarly journals Tensors with eigenvectors in a given subspace

2021 ◽  
Author(s):  
Giorgio Ottaviani ◽  
Zahra Shahidi
Keyword(s):  
Algebraic Geometry ◽  
Linear Subspace ◽  
Chern Classes ◽  
Symmetric Tensors

AbstractThe first author with B. Sturmfels studied in [16] the variety of matrices with eigenvectors in a given linear subspace, called the Kalman variety. We extend that study from matrices to symmetric tensors, proving in the tensor setting the irreducibility of the Kalman variety and computing its codimension and degree. Furthermore, we consider the Kalman variety of tensors having singular t-tuples with the first component in a given linear subspace and we prove analogous results, which are new even in the case of matrices. Main techniques come from Algebraic Geometry, using Chern classes for enumerative computations.

scholarly journals ON THE CHOW RING OF CERTAIN LEHN–LEHN–SORGER–VAN STRATEN EIGHTFOLDS

2021 ◽  
pp. 1-24
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ROBERT LATERVEER
Keyword(s):  
Chow Group ◽  
Chow Ring ◽  
Similar Statement ◽  
Intersection Product ◽  
Finite Dimensional

Abstract We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

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