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Abelianized Descent Obstruction for 0-Cycles

Hui Zhang

TL;DR

The paper develops an abelianized descent framework for 0-cycles by leveraging Borovoi’s abelian Galois cohomology, and proves that the algebraic Brauer–Manin obstruction coincides with the abelianized descent obstruction for 0-cycles. It extends descent theory from rational points to 0-cycles for commutative groups and then to connected linear groups, establishing a central equality: Z_{0,A}(X)^{Br} = Z_{0,A}(X)^{conn}_ab. It also provides a topology on adelic 0-cycles, showing that the abelianized descent obstructions can be closed and, in certain geometries (rationally connected, K3), coincide with the closure of BB descent. The work ties universal torsors, abelian cohomology, and Brauer obstructions together, and discusses implications for weak approximation of 0-cycles and the role of universal torsors under tori in this setting.

Abstract

Classical descent theory of Colliot-Thélène and Sansuc for rational points tells that, over a smooth variety $X$, the algebraic Brauer--Manin subset equals the descent obstruction subset defined by a universal torsor. Moreover, Harari shows that the Brauer--Manin subset equals the descent obstruction subset defined by torsors under connected linear groups. By using the abelian cohomology theory by Borovoi, we define abelianized descent obstructions for 0-cycles by torsors under connected linear groups. As an analogy, we show the equality between the Brauer--Manin obstruction and the abelianized descent obstruction for 0-cycles. We also show that the abelianized descent obstruction is the closure of the descent obstruction defined by Balestrieri and Berg when $X$ is a projective rationally connected variety or a projective K3 surface.

Abelianized Descent Obstruction for 0-Cycles

TL;DR

The paper develops an abelianized descent framework for 0-cycles by leveraging Borovoi’s abelian Galois cohomology, and proves that the algebraic Brauer–Manin obstruction coincides with the abelianized descent obstruction for 0-cycles. It extends descent theory from rational points to 0-cycles for commutative groups and then to connected linear groups, establishing a central equality: Z_{0,A}(X)^{Br} = Z_{0,A}(X)^{conn}_ab. It also provides a topology on adelic 0-cycles, showing that the abelianized descent obstructions can be closed and, in certain geometries (rationally connected, K3), coincide with the closure of BB descent. The work ties universal torsors, abelian cohomology, and Brauer obstructions together, and discusses implications for weak approximation of 0-cycles and the role of universal torsors under tori in this setting.

Abstract

Classical descent theory of Colliot-Thélène and Sansuc for rational points tells that, over a smooth variety , the algebraic Brauer--Manin subset equals the descent obstruction subset defined by a universal torsor. Moreover, Harari shows that the Brauer--Manin subset equals the descent obstruction subset defined by torsors under connected linear groups. By using the abelian cohomology theory by Borovoi, we define abelianized descent obstructions for 0-cycles by torsors under connected linear groups. As an analogy, we show the equality between the Brauer--Manin obstruction and the abelianized descent obstruction for 0-cycles. We also show that the abelianized descent obstruction is the closure of the descent obstruction defined by Balestrieri and Berg when is a projective rationally connected variety or a projective K3 surface.

Paper Structure

This paper contains 15 sections, 39 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Descent obstruction for a group of multiplicative type
  3. Notations
  4. Restricted product $\mathbf{P}^1(k,G)$
  5. Comparison of the algebraic Brauer--Manin obstruction and the descent obstruction for universal torsors
  6. abelianization of Galois cohomology of connected reductive group
  7. Abelian Galois cohomology of reductive groups
  8. Abelianization map in degree 1
  9. Base-change of the abelianization maps
  10. Abelian Galois cohomology of connected linear groups over a number field
  11. descent obstruction for a connected linear group
  12. Topology of adelic 0-cycles
  13. Topology of adelic 0-cycles defined by the weak approximation
  14. Decomposition of the abelianized descent obstruction for 0-cycles
  15. Universal torsors under tori.

Key Result

Theorem 1.1

Let $X$ be a smooth, proper and geometrically integral variety over a number field $k$ such that $\mathrm{Pic}(X_{\overline k})$ is of finite type. (1) Let $f:Y\rightarrow X$ be a universal torsor over $X$, then we have (2) The algebraic Brauer--Manin subset $X(\mathbf{A})^{\mathrm{Br}_1}\neq\emptyset$ if and only if there is a universal torsor $f:Y\rightarrow X$ such that $Y(\mathbf{A})\neq\empt

Theorems & Definitions (81)

  • Theorem 1.1: Sko01, Theorem 6.1.2 (a) and Corollary 6.1.3 (1)
  • Theorem 1.2: Theorem \ref{['cor1']}
  • Theorem 1.3: Har02, Theorem 2 and Remark 4
  • Theorem 1.4: Theorem \ref{['main3']}
  • Theorem 1.5: Corollary \ref{['main-geo']}
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • ...and 71 more