[Paper Review] A Finiteness theorem for zero-cycles over $p$-adic fields

Shuji Saito, Kanetomo Sato|University of Regensburg Publication Server (University of Regensburg)|May 6, 2006

advanced mathematical theories4 citations

TL;DR

This paper proves a finiteness theorem for zero-cycles on smooth projective varieties over $p$-adic fields, establishing that the group $A_0(V)$ of zero-cycles of degree zero is isomorphic to the direct sum of a finite group and a $p'$-divisible group under mild model-theoretic assumptions. The key result confirms a conjecture of Colliot-Thélène on the structure of $A_0(V)$, with applications to rational connectivity and $mod \ell$ divisibility for almost all $\ell$. The proof relies on resolution of singularities, blow-up formulas, and a moving lemma over discrete valuation rings.

ABSTRACT

In this paper we prove a finiteness result concerning the Chow group of zero-cycles for varieties over $p$-adic local fields. In this final version, there are several corrections concerning mathematical symbols and reference to related known results.

Motivation & Objective

• To prove a finiteness theorem for zero-cycles on smooth projective varieties over $p$-adic fields, confirming a conjecture on the structure of $A_0(V)$.
• To establish that $A_0(V)$ is isomorphic to a direct sum of a finite group and a $p'$-divisible group under a regular model assumption with simple normal crossings.
• To extend known results on $mod \ell$ divisibility of $A_0(V)$ to almost all primes $\ell$, using de Jong's alterations and the main theorem.
• To provide a geometric foundation for understanding the kernel of the Albanese map in higher dimensions over $p$-adic fields.
• To resolve singularities of curves in regular models using a moving lemma and blow-up techniques over discrete valuation rings.

Proposed method

• Utilizes a moving lemma to deform curves in regular models over discrete valuation rings to avoid singularities.
• Applies a blow-up formula to control the behavior of zero-cycles under proper transforms in successive blow-ups.
• Employs a Bertini-type theorem over discrete valuation rings to ensure the existence of smooth hyperplane sections in the model.
• Uses resolution of singularities for embedded curves via iterated blow-ups to achieve regularity of the strict transform.
• Applies a key proposition showing that the intersection multiplicity of a curve and a divisor decreases under blow-ups at points of high tangency.
• Combines the main theorem with de Jong's alterations to reduce the general case to the case of strict semistable reduction, enabling the proof of $mod \ell$ divisibility.

Experimental results

Research questions

• RQ1 Is the group $A_0(V)$ of zero-cycles of degree zero on a smooth projective variety $V$ over a $p$-adic field isomorphic to a direct sum of a finite group and a $p'$-divisible group?
• RQ2 Does $A_0(V)$ become $mod \ell$ divisible for almost all primes $\ell$ when $V$ admits a regular proper model with simple normal crossings?
• RQ3 Can the kernel of the Albanese map $\phi_V: A_0(V) \to \operatorname{Alb}_V(k)$ be described as a direct sum of a finite group and a divisible group over $p$-adic fields?
• RQ4 Does the structure of $A_0(V)$ remain stable under finite field extensions and generically finite morphisms, particularly when $V$ is rationally connected?
• RQ5 Can singularities of curves in regular models over $\mathfrak{O}_k$ be resolved via iterated blow-ups while preserving the normal crossings of the total space?

Key findings

• The group $A_0(V)$ is isomorphic to the direct sum of a finite group and a $p'$-divisible group when $V$ admits a regular projective flat model over $\mathfrak{O}_k$ with simple normal crossings on the special fiber.
• For any smooth projective variety $V$ over a $p$-adic field, $A_0(V)$ is $\ell$-divisible for almost all primes $\ell$, as a consequence of the main theorem and de Jong's alterations.
• If $V$ is rationally connected, then $A_0(V)$ is isomorphic to the direct sum of a finite group and a $p$-primary torsion group of finite exponent.
• The kernel of the Albanese map $\phi_V$ is finite if and only if $A_0(V)$ is finite, and the theorem confirms that $A_0(V)$ is not only divisible but also has finite $\ell$-primary torsion for almost all $\ell$.
• The proof establishes that after a finite sequence of blow-ups at singular points of the strict transform, the strict transform of a curve becomes regular and meets the exceptional divisor transversally.
• The intersection multiplicity of a curve and a divisor decreases under blow-ups at points of tangency of order $\geq 2$, which ensures termination of the resolution process.