[Paper Review] Severi-Brauer varieties; a geometric treatment

Janós Kollár|arXiv (Cornell University)|Jun 14, 2016

Algebraic Geometry and Number Theory6 references21 citations

TL;DR

This paper presents a geometric treatment of Severi-Brauer varieties by constructing a canonical non-split vector bundle $ F(P) $ on a Severi-Brauer variety $ P $, defined as the unique non-split extension $ 0 \to \mathcal{O}_P \to F(P) \to T_P \to 0 $. The key contribution is that $ \operatorname{End}(F(P))^{\text{op}} $ recovers the associated central simple algebra, establishing a purely geometric construction without relying on Galois cohomology or central simple algebra theory.

ABSTRACT

These notes present a geometric treatment of Severi-Brauer varieties, without using any results from the theory of central simple algebras or from Galois cohomology. 2026 version: major revisions

Motivation & Objective

• To develop a geometric theory of Severi-Brauer varieties independent of central simple algebras or Galois cohomology.
• To show that the geometry of a Severi-Brauer variety $ P $ is best understood through a canonical vector bundle $ F(P) $, defined as a non-split extension of $ \mathcal{O}_P $ by the tangent bundle $ T_P $.
• To establish that the endomorphism algebra $ \operatorname{End}(F(P))^{\text{op}} $ recovers the central simple algebra associated to $ P $, thus inverting the classical Châtelet construction.

Proposed method

• Define a twisted line bundle on a $ k $-scheme $ X $ as a line bundle on $ X_{\bar{k}} $ such that the category of vector bundles on $ X $ that become sums of its pullback over $ \bar{k} $ contains a nonzero bundle.
• Introduce the vector bundle $ F(P) $ as the unique non-split extension $ 0 \to \mathcal{O}_P \to F(P) \to T_P \to 0 $, which exists and is unique up to isomorphism on a Severi-Brauer variety $ P $.
• Use the relative de Rham complex and residue maps to construct a canonical extension $ 0 \to p^*\mathcal{L} \to F(P) \to T_{P/S} \to 0 $, with $ \mathcal{L} = (R^1p_*\Omega^1_{P/S})^* $, and show that $ R^1p_*\Omega^1_{P/S} \cong \mathcal{O}_S $.
• Prove that the formation of $ F(P) $ commutes with arbitrary base change, using the canonical isomorphism $ \delta_f: \mathcal{O}_S \cong R^1\pi_*\Omega^1_{P/S} $ via residue maps on divisors.
• Establish functoriality of $ R^1\pi_*\Omega^1_{P/S} $, showing it is isomorphic to $ \mathcal{O}_S $ via base change and pullback arguments on projective bundles.
• Show that $ \operatorname{End}(F(P))^{\text{op}} $ is a sheaf of algebras whose fiberwise structure recovers the central simple algebra associated to $ P $, thus realizing the inverse of the Châtelet construction.

Experimental results

Research questions

• RQ1 Can the theory of Severi-Brauer varieties be developed purely geometrically, without appealing to central simple algebras or Galois cohomology?
• RQ2 What is the canonical vector bundle on a Severi-Brauer variety that encodes its geometric and arithmetic structure?
• RQ3 How can the endomorphism algebra of a geometrically constructed vector bundle $ F(P) $ recover the associated central simple algebra?
• RQ4 Is the construction of $ F(P) $ as a non-split extension $ 0 \to \mathcal{O}_P \to F(P) \to T_P \to 0 $ canonical and compatible with base change?
• RQ5 What is the role of the relative canonical bundle and residue maps in constructing the extension class of $ F(P) $?

Key findings

• The vector bundle $ F(P) $, defined as the unique non-split extension $ 0 \to \mathcal{O}_P \to F(P) \to T_P \to 0 $, exists and is unique up to isomorphism on any Severi-Brauer variety $ P $.
• The formation of $ F(P) $ commutes with arbitrary base change, ensuring its functoriality and geometric naturality.
• The sheaf $ R^1\pi_*\Omega^1_{P/S} $ is isomorphic to $ \mathcal{O}_S $, which is essential for constructing the extension class of $ F(P) $.
• The endomorphism algebra $ \operatorname{End}(F(P))^{\text{op}} $ is a sheaf of algebras whose restriction to each geometric fiber is a central simple algebra, thus recovering the algebra associated to $ P $.
• The construction provides a geometric inverse to the classical Châtelet map from central simple algebras to Severi-Brauer varieties.
• The twisted Picard group $ \operatorname{Pic}^{\text{tw}}(X) $ captures the quotient $ \operatorname{Pic}^\text{tw}(X)/\operatorname{Pic}(X) $, and the bundle $ F(P) $ lies in the category $ T(\mathcal{L}) $ for $ \mathcal{L} = \mathcal{O}_{P_{\bar{k}}}(1) $, showing its geometric origin.