[Paper Review] Sur le groupe de Brauer transcendant

Jean-Louis Colliot-Thélène, Alexei N. Skorobogatov|arXiv (Cornell University)|Jun 30, 2011

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes the finiteness of the cokernel of the natural map from the Brauer group of a smooth projective variety $X$ over a field $k$ of characteristic zero to the Galois-invariant subgroup of the Brauer group of $X_{ar{k}}$. Using the interplay between the Brauer group, Picard group, and intersection pairings modulo numerical equivalence, the authors derive explicit bounds on the order and exponent of this finite cokernel, particularly under number field assumptions, with applications to K3 surfaces and products of curves.

ABSTRACT

For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions on k, e.g. over number fields, we give estimates for the order of this cokernel. ---- Soit X une variété projective et lisse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s'envoie dans les invariants, sous le groupe de Galois absolu de k, du groupe de Brauer de la même variété considérée sur une clôture algébrique de k. Nous montrons que le quotient est fini. Sous des hypothèses supplémentaires, par exemple sur k un corps de nombres, nous donnons des estimations sur l'ordre de ce quotient.

Motivation & Objective

• To establish the finiteness of the cokernel of the natural map $\mathrm{Br}(X) \to \mathrm{Br}(\overline{X})^\Gamma$ for smooth projective varieties over fields of characteristic zero.
• To provide effective bounds on the order and exponent of this cokernel under additional assumptions, such as when the base field is a number field.
• To clarify the role of the intersection pairing between divisors and 1-cycles modulo numerical equivalence in controlling the Brauer group structure.
• To extend the finiteness result to smooth quasi-projective varieties over finitely generated fields over $\mathbb{Q}$, affirming a question by Szamuely.
• To apply the results to specific classes of varieties, including K3 surfaces and products of curves, where the bounds simplify significantly.

Proposed method

• Construct a natural complex $\mathrm{Br}(X) \xrightarrow{\alpha} \mathrm{Br}(\overline{X})^\Gamma \xrightarrow{\beta} \mathrm{H}^2(k, \mathrm{Pic}(\overline{X}))$, which is exact under certain conditions (e.g., existence of a rational point or $k$ a number field).
• Use functoriality and Tsen's theorem to show that the image of $\beta$ vanishes when restricted to any closed curve in $X$, leading to finiteness via the first method.
• Apply a spectral sequence argument involving the divisible subgroup $\mathrm{Br}^0(\overline{X})$ and the Kummer sequence to interpret $\beta$ as a connecting homomorphism.
• Leverage Lieberman's theorem on the coincidence of numerical and homological equivalence for 1-cycles to relate the image of $\beta$ to the discriminant groups of the intersection form on $\mathrm{NS}(\overline{X})$.
• Use the exponent of the discriminant groups of the intersection pairing between $\mathrm{NS}(\overline{X})$ and $\mathrm{CH}_1(\overline{X})$ modulo numerical equivalence to bound the exponent of the cokernel.
• Combine both methods to derive comparable but non-identical estimates for the order and exponent of the finite cokernel.

Experimental results

Research questions

• RQ1 Is the cokernel of the map $\mathrm{Br}(X) \to \mathrm{Br}(\overline{X})^\Gamma$ finite for smooth projective varieties over fields of characteristic zero?
• RQ2 Can effective bounds be given for the order and exponent of this cokernel when the base field is a number field?
• RQ3 What is the role of the intersection pairing between divisors and 1-cycles modulo numerical equivalence in constraining the Brauer group?
• RQ4 Does the finiteness result extend to smooth quasi-projective varieties over finitely generated fields over $\mathbb{Q}$?
• RQ5 How do the bounds simplify for special classes of varieties such as K3 surfaces or products of curves with rational points?

Key findings

• The cokernel of $\alpha: \mathrm{Br}(X) \to \mathrm{Br}(\overline{X})^\Gamma$ is finite for any smooth, projective, geometrically integral variety $X$ over a field $k$ of characteristic zero.
• For $k$ a number field, the order of the cokernel is bounded by the product of the exponents of the discriminant groups of the intersection pairing on $\mathrm{NS}(\overline{X})$ and $\mathrm{CH}_1(\overline{X})$ modulo numerical equivalence.
• The exponent of the cokernel divides the exponent of the discriminant group of the intersection form on $\mathrm{NS}(\overline{X})$.
• For K3 surfaces with a $k$-rational point, the cokernel is annihilated by the exponent of the discriminant group of the Néron–Severi group under the intersection form.
• The finiteness result extends to smooth quasi-projective varieties over finitely generated fields over $\mathbb{Q}$, affirming a question by Szamuely.
• The two methods—functoriality with Tsen's theorem and spectral sequence analysis—yield comparable but distinct bounds, both relying on the structure of the intersection pairing.