[Paper Review] The Brauer-Manin obstruction for zero-cycles on K3 surfaces

Evis Ieronymou|arXiv (Cornell University)|Apr 11, 2018

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes that the Brauer-Manin obstruction is the only obstruction to the existence of zero-cycles of degree $d$ on $K3$ surfaces over number fields, under the assumption that the Brauer-Manin obstruction is the only one for rational points over all finite extensions. By using a fibration over $\mathbb{P}^1$ and leveraging boundedness results on Brauer groups from Skorobogatov–Zarhin and Orr–Skorobogatov, the authors extend Liang’s results from rationally connected varieties to $K3$ surfaces, circumventing the failure of $H^2(X_{\overline{k}}, \mathcal{O}) = 0$.

ABSTRACT

We study local-global principles for zero-cycles on K3 surfaces defined over number fields. We follow an idea of Liang to use the trivial fibration over the projective line.

Motivation & Objective

• To extend Liang’s results on the Brauer-Manin obstruction for zero-cycles to $K3$ surfaces, which do not satisfy $H^2(X_{\overline{k}}, \mathcal{O}) = 0$.
• To establish that the Brauer-Manin obstruction is the only obstruction to the existence of zero-cycles of degree $d$ on $K3$ surfaces over number fields.
• To provide evidence for the conjecture that the Brauer-Manin obstruction controls rational points and zero-cycles on $K3$ surfaces.

Proposed method

• Use of the fibration method via a trivial fibration over $\mathbb{P}^1$ to reduce the problem to local conditions on fibers.
• Leverage boundedness results on Brauer groups of $K3$ surfaces over number fields from Skorobogatov–Zarhin and Orr–Skorobogatov to bypass the $H^2 = 0$ assumption.
• Apply results from Harpaz–Wittenberg on the fibration method to control local-to-global principles for zero-cycles.
• Use Galois cohomology and the Hochschild–Serre spectral sequence to relate Brauer groups over base fields and their extensions.
• Apply weak approximation techniques to construct zero-cycles matching local data modulo $n$.
• Use the fact that $CH_0(X_{k_v})/n$ is controlled by divisibility and local geometry to match global and local zero-cycle classes.

Experimental results

Research questions

• RQ1 Does the Brauer-Manin obstruction control the existence of zero-cycles of degree $d$ on $K3$ surfaces over number fields, assuming it controls rational points over all finite extensions?
• RQ2 Can the fibration method be adapted to $K3$ surfaces despite the failure of $H^2(X_{\overline{k}}, \mathcal{O}) = 0$?
• RQ3 Is the Brauer-Manin obstruction the only obstruction to weak approximation for zero-cycles on $K3$ surfaces, under the same rational point assumption?
• RQ4 To what extent does the Brauer group of a $K3$ surface control local-to-global principles for zero-cycles?
• RQ5 Can the conjectural exactness of the Brauer–Chow complex for $K3$ surfaces be supported by such local-global results?

Key findings

• The Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of degree $d$ on a $K3$ surface $X$ over a number field $k$, provided the same holds for rational points over all finite extensions of $k$.
• The result extends Liang’s theorem from rationally connected varieties to $K3$ surfaces by using fibration techniques and boundedness of Brauer groups.
• For any $n \geq 1$, a zero-cycle $b$ exists that matches a given family of local zero-cycles $\{z_v\}$ modulo $n$ in $CH_0(X_{k_v})$ for all places $v$, under the weak approximation assumption.
• The proof relies on the fact that $CH_0(X_{k_v})$ is divisible modulo $n$ for almost all $v$, under suitable conditions on the residue field characteristic.
• The map $\mathrm{Br}(X)/\mathrm{Br}_0(X) \to \mathrm{Br}(X_{k'})/\mathrm{Br}_0(X_{k'})$ is an isomorphism for suitable extensions $k'/k$, ensuring control over the Brauer group over the base field.
• The authors establish that the Brauer–Chow exact sequence conjecture (E) would follow from stronger divisibility properties of $A_0(X_{k_v})$, which remain open for $K3$ surfaces.