The characteristic cycle and the singular support of a constructible sheaf
type: Item
instance of: scholarly article
title: The characteristic cycle and the singular support of a constructible sheaf
zbMATH Open document ID: 1437.14016
author: Takeshi Saito
published in: Inventiones Mathematicae
publication date: 2017-02-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1510.03018
review text: This paper was inspired by Kashiwara-Schapira's analogous theory over the field of complex numbers \(\mathbb{C}\) and various conjectures of Deligne. Using Belinson's singular supports, the author first constructs the characteristic cycles for constructible étale sheaves. Let \(X\) be a smooth and projective variety of dimension \(d\) over a perfect field of characteristic \(p\), and \(\mathcal F\) a constructible étale sheaf on \(X\). The singular support \(SS(\mathcal{F})\) of \(\mathcal{F}\), defined by \textit{A. Beilinson} [Sel. Math., New Ser. 22, No. 4, 1797--1819 (2016; Zbl 1375.14058)], is a closed subscheme of \(T^*X\), the cotangent bundle of \(X\). All its irreducible components \(C_1, \dots, C_k\) are of dimension \(d\). The characteristic cycle \(CC(\mathcal{F}) \) of \(\mathcal{F}\) is a cycle supported on \(SS(\mathcal{F})\): \[ CC(\mathcal {F}) = \sum_{i=1}^{n}n_iC_i \] where \(n_i\in \mathbb{Z}\). The coefficients \(n_i\) are defined using a generalization of Milnor's formula for the number of vanishing cycles. Its main interest is that it can be used to compute the Euler-Poincaré characteristic of \(\mathcal{F}\) by the following index formula: \[ \chi(X, F) = (CC(\mathcal{F}), T^*_XX)_{T^*X} \] where \(T^*_XX\) is the zero section. This provides a generalization to higher dimension of the Grothendieck-Ogg-Shafarevich formula. The main ingredients of this paper are: Radon and Legendra transforms after Brylinski, geometric theory of Lefschetz pensils, ramifications theory for imperfect residue fields and Deligne's theory of vanishing cycles over general bases.
reviewed by: Yigeng Zhao
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14F06
zbMATH DE Number: 6685347
zbMATH Keywords: characteristic cycles
zbMATH Keywords: singular supports
zbMATH Keywords: index formula
zbMATH Keywords: Euler-Poincaré characteristic
zbMATH Keywords: characteristic class
MaRDI profile type: Publication
OpenAlex ID: W3104756461
arXiv ID: 1510.03018
cites work: Ramification of local fields with imperfect residue fields. II
cites work: The characteristic class and ramification of an \(\ell\)-adic étale sheaf
cites work: Faisceaux constructibles Cohomologie d'une courbe algebrique
cites work: Theoreme de finitude pour un morphisme propre ; Dimension cohomologique des schemas algebriques affines
cites work: Morphismes acycliques
cites work: Théorème de changement de base par un morphisme lisse, et applications
cites work: Q4640173
cites work: Constructible sheaves are holonomic
cites work: Q3762606
cites work: Smoothness, semi-stability and alterations
cites work: La formule de dualite globale
cites work: Q5682091
cites work: Q4119315
cites work: Etale Cohomology Theory
cites work: Q4342000
cites work: Characteristic varieties and vanishing cycles
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Séconde partie)
cites work: Q4128903
cites work: \(\varepsilon\)-factor of a tamely ramified sheaf on a variety
cites work: Q4325441
cites work: Around the Thom-Sebastiani theorem, with an appendix by Weizhe Zheng
cites work: Q3999005
cites work: Q3807334
cites work: Ramification theory for varieties over a perfect field
cites work: Q4740028
cites work: Q3921333
cites work: Chern classes for singular algebraic varieties
cites work: Modifications et cycles proches sur une base generale
cites work: Wild ramification and the cotangent bundle
cites work: Characteristic cycle and the Euler number of a constructible sheaf on a surface
cites work: Characteristic cycle of the external product of constructible sheaves
cites work: Q5509717
cites work: Logarithmic version of the Milnor formula
DOI: 10.1007/S00222-016-0675-3
description: scientific article