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arXiv · 2211.15915

Local-global principle and integral Tate conjecture for certain varieties

Zhiyu Tian
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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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Zhiyu Tian. 2022-11-29. Local-global principle and integral Tate conjecture for certain varieties. https://arxiv.org/abs/2211.15915

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