Computations with modified diagonals
type: Item
instance of: scholarly article
title: Computations with modified diagonals
zbMATH Open document ID: 1305.14002
DOI: 10.4171/RLM/677
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2014-11-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.0757
reviewed by: Makiko Mase
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 6363537
zbMATH Keywords: Chow ring
zbMATH Keywords: hyperkähler varieties
zbMATH Keywords: modified diagonals
MaRDI profile type: Publication
Wikidata QID: Q60141286
arXiv ID: 1311.0757
cites work: Variétés kähleriennes dont la première classe de Chern est nulle
cites work: Q3592898
cites work: On the Chow ring of a K3 surface
cites work: An intersection number for the punctual Hilbert scheme of a surface
cites work: Q3328651
cites work: The modified diagonal cycle on the triple product of a pointed curve
cites work: Beauville-Voisin Conjecture for Generalized Kummer Varieties
cites work: Some remarks on modified diagonals
cites work: The Fourier Transform for Certain HyperKähler Fourfolds
cites work: On the Chow ring of certain algebraic hyper-Kähler manifolds
cites work: Infinitesimal invariants for cycles modulo algebraic equivalence and 1-cycles on Jacobians
OpenAlex ID: W3122068241
author: Kieran G. O'Grady
review text: The \(m\)-th modified diagonal on a variety \(X\) over a field \(\mathbb{K}\) is a cycle defined by NEWLINE\[NEWLINE \Gamma^m(X;a) := \sum_{J\not=\emptyset}(-1)^{m-|J|}\Delta^m_J(X;a), NEWLINE\]NEWLINE where for a \(\mathbb{K}\)-rational point \(a\in X(\mathbb{K})\) and a subset \(J\subset \{ 1,\ldots , m\}\) NEWLINENEWLINE\[NEWLINE\Delta^m_J(X;a) := \Big\{ (x_1,\ldots , x_m)\in X^m \, \Big| \, \begin{cases} x_i=x_j\, &\text{if \(i\) and \(j\) are in \(J\)},\\ x_k=a \, &\text{if \(k\) are not in \(J\)}\end{cases} \Big\}. NEWLINE\]NEWLINENEWLINENEWLINEBeing motivated by Beauville's conjecture of the vanishing of the modified diagonal on a hyperkähler variety in the rational Chow group, the article under review states the main results in Propositions 0.2 and 0.3.NEWLINENEWLINEIn Proposition 0.2, for smooth projective varieties \(X\) and \(Y\), if \(\Gamma^m(X;a)\) and \(\Gamma^n(Y;b)\) are both rationally equivalent to zero for some \(a\in X(\mathbb{K})\) and \(b\in Y(\mathbb{K})\), then, \(\Gamma^{m+n-1}(X\times Y;(a,\, b))\) is shown to be rationally equivalent to zero. By applying to smooth genus-2 curves, it is proved that the \(5\)-th modified diagonal of a complex abelian surface is rationally equivalent to zero at some point.NEWLINENEWLINEIn Proposition 0.3, a blow-up \(X\to Y\) centred along a codimension-\(e\) smooth subvariety \(V\) in a smooth projective variety \(Y\) of dimension \(n\) is considered. If for some \(b\in V(\mathbb{K})\), \(\Gamma^{n+1}(Y;b)\) and \(\Gamma^{n-e+1}(V;b)\) are both rationally equivalent to zero, then, it is proved that \(\Gamma^{n+1}(X;a)\) vanishes for some \(a\in f^{-1}(b)\).NEWLINENEWLINEThe Hilbert scheme \(S^{[n]}\) of a \(K3\) surface \(S\) of subvarieties of length \(n\) is constructed by blowing-up \(S^{[n+1]}\to S^{[n]}\times S\) centred along the tautological subscheme \(\mathcal{L}_n\) of \(S^{[n]}\times S\). It was shown by Beauville that \(S^{[n]}\) is hyperkähler, and by Beauville-Voisin that the \(3\)rd modified …
description: scientific article; zbMATH DE number 6363537