Vol. 7, No. 2, 2022

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Unramified cohomology, integral coniveau filtration and Griffiths groups

Shouhei Ma

Vol. 7 (2022), No. 2, 223–236
Abstract

We prove that the degree k unramified cohomology Hnr k(X, ) of a smooth complex projective variety X with small CH 0(X) has a filtration of length [k2], whose first piece is the torsion part of the quotient of Hk+1(X, ) by its coniveau 2 subgroup, and whose next graded piece is controlled by the Griffiths group Griff k2+1(X) when k is even and is related to the higher Chow group CH (k+3)2(X,1) when k is odd. The first piece is a generalization of the Artin–Mumford invariant (k = 2) and the Colliot-Thélène–Voisin invariant (k = 3). We also give an analogous result for the -cohomology Hdk(X,d()), d = dim X.

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Keywords
unramified cohomology, integral coniveau filtration, Griffiths group, $\mathcal H$-cohomology, birational invariant
Mathematical Subject Classification
Primary: 14C15, 14C25, 14F43
Milestones
Received: 12 October 2020
Revised: 23 December 2021
Accepted: 12 January 2022
Published: 13 September 2022
Authors
Shouhei Ma
Department of Mathematics
Tokyo Institute of Technology
Tokyo
Japan