[Paper Review] XI Solomon Lefschetz Memorial Lecture Series: Hodge structures in non-commutative geometry. (Notes by Ernesto Lupercio)

Maxim Kontsevich|ArXiv.org|Jan 30, 2008

Algebraic Geometry and Number Theory12 references12 citations

TL;DR

This paper introduces a non-commutative generalization of Hodge structures using Hochschild homology and Connes' B-operator in deformation quantization and derived algebraic geometry. It proposes that the spectral sequence degeneration in positive characteristic may be understood via the cohomology of a complex with differential ∂ + uB, suggesting a mechanism for Hodge-to-de Rham degeneration and linking non-commutative geometry to L-functions and the Weil conjectures.

ABSTRACT

Traditionally, Hodge structures are associated with complex projective varieties. In my expository lectures I discussed a non-commutative generalization of Hodge structures in deformation quantization and in derived algebraic geometry.

Motivation & Objective

• To generalize classical Hodge structures—typically defined on cohomology of smooth projective varieties—into the realm of non-commutative geometry.
• To explore how Hochschild homology and Connes' B-operator can serve as non-commutative analogues of differential forms and the de Rham complex.
• To investigate the degeneration of the Hodge-to-de Rham spectral sequence in positive characteristic using a deformation complex with differential ∂ + uB.
• To propose a non-commutative L-function associated with saturated non-commutative spaces, inspired by the Weil conjectures and Beilinson's conjectures.
• To establish a conceptual bridge between non-commutative geometry, arithmetic geometry, and topological string theory via L-functions and D-brane-like sums.

Proposed method

• Uses the Hochschild complex $ C_{\bullet}(A,A) $ of a unital associative algebra $ A $ over $ \mathbb{C} $, with differential $ \partial $, to model non-commutative differential forms.
• Introduces Connes' B-operator of degree $-1$ on the Hochschild complex, satisfying $ B^2 = 0 $, $ \partial B + B\partial = 0 $, and $ \partial^2 = 0 $, forming a mixed complex.
• Considers the reduced Hochschild complex $ C^{\mathrm{red}}_{\bullet}(A,A) $, where all but the first factor are reduced modulo scalars, preserving cohomology.
• Analyzes the complex $ (C^{\mathrm{red}}_{\bullet}(A,A)[u], \partial + uB) $, treating it as a family over $ \mathbb{A}^1 $, with $ u $ a formal parameter.
• Studies the cohomology of this complex in characteristic $ p > 0 $, showing it may be a coherent sheaf, and conjectures quasi-isomorphism to $ (C^{\mathrm{red}}_{\bullet}(A_0,A_0)[u,u^{-1}], \partial) $ for flat dga over $ \mathbb{Z}_p $.
• Proposes a non-commutative L-function via $ L(X) = \prod_p L_p(s) $, with $ L_p(s) = \det(1 - \mathrm{Fr}_p / p^s)^{-1} $, and links it to K-theory and the Riemann hypothesis.

Experimental results

Research questions

• RQ1 Can Hodge structures be generalized beyond commutative algebraic varieties to non-commutative algebras using Hochschild homology and Connes' B-operator?
• RQ2 Does the spectral sequence associated with the mixed complex $ (\partial + uB) $ on $ C^{\mathrm{red}}_{\bullet}(A,A)[u] $ degenerate in positive characteristic, and if so, why?
• RQ3 Is the cohomology of $ (C^{\mathrm{red}}_{\bullet}(A,A)[u], \partial + uB) $ a coherent sheaf in positive characteristic, and what does this imply for Hodge-to-de Rham degeneration?
• RQ4 Can a non-commutative L-function be defined for saturated non-commutative spaces, and does it satisfy the Riemann hypothesis and Beilinson's conjectures?
• RQ5 What is the role of the filtration $ \mathrm{Fil}_{\leq n} $ on the reduced Hochschild complex in understanding the behavior of $ \partial + B $ in characteristic $ p $, particularly when $ n $ is divisible by $ p $?

Key findings

• In characteristic $ p > 0 $, the complex $ (C^{\mathrm{red}}_{\bullet}(A_0,A_0)[u,u^{-1}], \partial + uB) $ is conjectured to be quasi-isomorphic to $ (C^{\mathrm{red}}_{\bullet}(A_0,A_0)[u,u^{-1}], \partial) $ as $ \mathbb{Z}/p[u,u^{-1}] $-modules, suggesting a mechanism for spectral sequence degeneration.
• For $ n $ coprime to $ p $, the graded piece $ \mathrm{gr}_n(\mathrm{Fil}) $ with differential $ \partial + B $ is acyclic in characteristic $ p $, implying vanishing cohomology in those degrees.
• When $ n = kp $, the complex $ \mathrm{gr}_n(\mathrm{Fil}) $ with $ \partial + B $ becomes isomorphic to $ V^{\otimes k} \xrightarrow{1 - \sigma} V^{\otimes k} $, where $ \sigma $ generates $ \mathbb{Z}/k\mathbb{Z} $, indicating potential non-trivial cohomology.
• In characteristic 0, the complex $ (C^{\mathrm{red}}_{\bullet}(A,A)[u,u^{-1}], \partial + uB) $ is acyclic, meaning all cohomology is concentrated at $ u = 0 $, forming an infinite Jordan block, indicating a singular, non-vector bundle structure.
• The conjecture that $ (C^{\mathrm{red}}_{\bullet}(A_0,A_0)[u,u^{-1}], \partial + uB) \simeq (C^{\mathrm{red}}_{\bullet}(A_0,A_0)[u,u^{-1}], \partial) $ in characteristic $ p $ would imply that the spectral sequence degenerates if the cohomology is finite-dimensional.
• A non-commutative $ L $-function $ L(X) = \prod_p L_p(s) $ is proposed, with $ L_p(s) = \det(1 - \mathrm{Fr}_p / p^s)^{-1} $, and it is conjectured to satisfy the Riemann hypothesis and Beilinson's conjectures via connections to $ K_{1-2s}(X) $.