[Paper Review] The local-global exact sequence for Chow groups of zero-cycles

Yongqi Liang|arXiv (Cornell University)|Jun 7, 2012

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper establishes the exactness of the local-global sequence for Chow groups of zero-cycles on normic bundles over projective space by leveraging rational connectedness and Brauer–Manin obstruction techniques. It proves that for rationally connected varieties over number fields, the sequence is exact under a weakened linear disjointness condition on extensions, and extends this to families of normic bundles via restriction-corestriction arguments, confirming the Brauer–Manin obstruction as the sole obstruction to the local-global principle for zero-cycles.

ABSTRACT

A local-global sequence for Chow groups of zero-cycles involving Brauer groups has been conjectured to be exact for all proper smooth algebraic varieties. We apply existing methods to construct several new families of varieties verifying the exact sequence. The examples are explicit, they are normic bundles over the projective space.

Motivation & Objective

• To verify the local-global principle for zero-cycles on proper smooth varieties over number fields, with the Brauer–Manin obstruction as the sole obstruction.
• To extend known exactness results for the local-global sequence to new families of varieties, particularly normic bundles over projective space.
• To establish exactness under weaker assumptions than previous results, specifically requiring only linear disjointness from a fixed extension rather than all finite extensions.
• To apply restriction-corestriction techniques to propagate exactness from simpler varieties to more general fibrations.

Proposed method

• Utilizes a strengthened version of a recent result (Liaa) by weakening the assumption on finite extensions to only those linearly disjoint from a fixed finite extension L/k.
• Applies the restriction-corestriction argument to transfer exactness from varieties with function fields L_i to those with function field L, under coprime degree conditions.
• Employs the modified Chow group CH₀′(X_v) to handle archimedean places and ensure compatibility with the Brauer–Manin pairing.
• Reduces the problem to birational invariants: CH₀(X), A₀(X), and Br(X), which depend only on the function field k(X).
• Constructs explicit families of normic bundles over projective space whose function fields are composita of rational function fields with finite extensions of coprime degrees.
• Relies on known results on rational points (e.g., DSWb, SJ) to verify the Brauer–Manin obstruction as the only obstruction to weak approximation, enabling application of Theorem 2.1.

Experimental results

Research questions

• RQ1 Under what conditions is the local-global sequence for Chow groups of zero-cycles exact on proper smooth varieties over number fields?
• RQ2 Can the exactness of the local-global sequence be extended to normic bundles over projective space using cohomological and fibration techniques?
• RQ3 Does the Brauer–Manin obstruction alone control the failure of the local-global principle for zero-cycles on rationally connected varieties?
• RQ4 How does the restriction-corestriction argument propagate exactness from simpler to more complex varieties with function fields of coprime degrees?
• RQ5 To what extent can the exactness of the sequence be deduced from the arithmetic of rational points on base changes?

Key findings

• The local-global sequence (E) is exact for all proper smooth rationally connected varieties X over a number field k, provided that for all finite extensions K/k linearly disjoint from a fixed finite extension L/k, the Brauer–Manin obstruction is the only obstruction to weak approximation on X_K.
• The sequence (E) is exact for normic bundles over projective space defined by equations of the form N_{K/k}(x) = P(t), where K/k is a degree 4 extension and P(t) is a quadratic irreducible polynomial splitting over K.
• The sequence (E) is exact for varieties defined by N_{K/k}(x) = c t^m (1-t)^n, generalizing results of Heath-Brown–Skorobogatov and Colliot-Thélène–Harari–Skorobogatov.
• The global-to-local homomorphism on degree 0 zero-cycles is surjective for such varieties when Br(X)/Br(k) = 0, as confirmed in Example 3.9.
• The exactness of (E) is preserved under taking fibrations with function fields that are finite extensions of coprime degrees over rational function fields, via Theorem 2.3.
• The results confirm that the Brauer–Manin obstruction controls the failure of the local-global principle for zero-cycles on a broad class of normic bundles over projective space.