[Paper Review] Kazhdan's Theorem on Arithmetic Varieties

J. S. Milne|ArXiv.org|Jun 23, 2001

Geometry and complex manifolds19 references3 citations

TL;DR

This paper provides a uniform, classification-free proof of Kazhdan's theorem on arithmetic varieties, showing that the conjugate of an arithmetic variety under any automorphism of ℂ remains arithmetic. By leveraging the Bergmann metric, group-theoretic structures, and properties of Hermitian symmetric domains, the author establishes the result under codimension and splitting conditions, avoiding case-by-case analysis and relying on Clozel's work on limit multiplicities.

ABSTRACT

Define an arithmetic variety to be the quotient of a bounded symmetric domain by an arithmetic group. An arithmetic variety is algebraic, and the theorem in question states that when one applies an automorphism of the field of complex numbers to the coefficients of an arithmetic variety the resulting variety is again arithmetic. This article simplifies Kazhdan's proof. In particular, it avoids recourse to the classification theorems. It was originally completed on March 28, 1984, and distributed in handwritten form. July 23, 2001: Fixed about 30 misprints.

Motivation & Objective

• To provide a uniform, classification-free proof of Kazhdan's theorem on arithmetic varieties.
• To show that the conjugate of an arithmetic variety under any field automorphism σ ∈ Aut(ℂ) remains arithmetic.
• To avoid case-by-case analysis by using geometric and group-theoretic techniques instead of classification theorems.
• To establish the result under codimension ≥3 and maximal torus splitting conditions over imaginary quadratic extensions.
• To rely on Clozel’s result on limit multiplicities and Yau’s Kähler-Einstein theorem as foundational tools.

Proposed method

• Define an arithmetic variety as the quotient of a bounded symmetric domain by an arithmetic group, with the group arising from a Q-simple, simply connected algebraic group G.
• Use the Bergmann metric on the universal cover to characterize the geometry of conjugated varieties σX.
• Construct a projective system of arithmetic subvarieties Xα associated with reductive subgroups Hα of G, using Piatetsky-Shapiro and Borovoi’s embedding techniques.
• Apply the criterion that a variety is arithmetic if its tangent bundle is generated by subbundles from arithmetic subvarieties of lower dimension.
• Use the fact that σXα is arithmetic (by induction and the A1 case) to deduce that the tangent bundle of σX is generated by such subbundles.
• Derive a contradiction from the assumption that the index set I of non-compact components is neither empty nor full, using Lie algebra decomposition and the maximality of H_I'.

Experimental results

Research questions

• RQ1 Does the conjugate of an arithmetic variety under any σ ∈ Aut(ℂ) remain arithmetic, without relying on classification theorems?
• RQ2 Can Kazhdan’s theorem be proven uniformly for all arithmetic varieties satisfying codimension ≥3 and torus splitting conditions?
• RQ3 To what extent can the proof be made independent of case-by-case analysis, especially for non-compact types like E6, E7, and mixed Dℝ/Dℍ?
• RQ4 How can the role of the Bergmann metric and Kähler-Einstein metrics be unified in the proof?
• RQ5 Can the group-theoretic structure of the fundamental group and its congruence subgroups be used to establish arithmeticity of conjugated varieties?

Key findings

• The conjugate variety σX is arithmetic for all σ ∈ Aut(ℂ), provided X satisfies the codimension ≥3 and maximal torus splitting conditions.
• The proof avoids the classification of symmetric domains by using geometric and group-theoretic arguments on tangent bundles and subvarieties.
• The tangent space of σX is generated by the tangent spaces of arithmetic subvarieties Xασ, ensuring the variety is arithmetic.
• A contradiction arises if the index set I of non-compact components is neither empty nor full, proving that such a configuration cannot occur.
• The construction of H_I' as a subgroup of G' with H_I' = G' leads to a contradiction, confirming the theorem.
• The result holds uniformly across all types, including E6, E7, and mixed Dℝ/Dℍ, under the stated conditions, relying on Clozel’s work on limit multiplicities.