Local-global principle and integral Tate conjecture for certain varieties
Abstract
We give a geometric criterion to check the validity of the integral Tate conjecture for one-cycles on a smooth projective variety that is separably rationally connected in codimension one, and to check that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on a separably rationally connected variety defined over a global function field. We prove that the Brauer-Manin obstruction is the only obstruction to the local-global principle for zero-cycles on all geometrically rational surfaces defined over a global function field, and to the Hasse principle for rational points on del Pezzo surfaces of degree four defined over a global function field of odd characteristic. Along the way, we also prove some results about the space of one-cycles on a smooth projective variety that is separably rationally connected in codimension one, which leads to the equality of the coniveau filtration and the strong coniveau filtration on degree homology of such varieties.
Cite
@article{arxiv.2211.15915,
title = {Local-global principle and integral Tate conjecture for certain varieties},
author = {Zhiyu Tian},
journal= {arXiv preprint arXiv:2211.15915},
year = {2024}
}
Comments
V3: 79 pages, major revision. V2: 39 pages. Major revision. The part characterizing algebraic equivalence has been removed. Stronger results on algebraic equivalence of one cycles are obtained in the joint paper with J\'anos Koll\'ar in arXiv:2302.07069 . Results on coniveau filtration (previously contained in arXiv:2211.15911) are added. V1.37 pages, 2 figures