Local-global principle and integral Tate conjecture for certain varieties

Tian, Zhiyu

doi:10.1090/jams/1054

Local-global principle and integral Tate conjecture for certain varieties
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by Zhiyu Tian;

J. Amer. Math. Soc.

DOI: https://doi.org/10.1090/jams/1054

Published electronically: February 5, 2025

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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 $3$ homology of such varieties.

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