Abstract:A conjecture of Colliot-Thélène predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a divisible group and a finite group. In this article we show that if $\pi:X\dashrightarrow Y$ is a generically finite rational map between smooth projective surfaces and the conjecture is true for $X\otimes_k L$ for every finite extension $L/k$, then it is true for $Y$. Using work of Raskind and Spiess, this proves the conjecture for surfaces that are geometrically dominated by products of curves, under some assumptions on the reduction type of the Jacobians. The method involves studying similar questions for an open subvariety $U$ of a projective surface $X$ by replacing the Chow group of $0$-cycles with Suslin's singular homology $H_0^{\text{sus}}(U)$.
| Comments: | 35 pages |
| Subjects: | Algebraic Geometry (math.AG) |
| Cite as: | arXiv:2507.20076 [math.AG] |
| (or arXiv:2507.20076v2 [math.AG] for this version) | |
| https://doi.org/10.48550/arXiv.2507.20076 arXiv-issued DOI via DataCite |
From: Evangelia Gazaki Ms [view email]
[v1]
Sat, 26 Jul 2025 22:53:25 UTC (45 KB)
[v2]
Tue, 26 May 2026 15:22:39 UTC (39 KB)
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