[Paper Review] Periodic de Rham bundles over curves

Raju Krishnamoorthy, Mao Sheng|arXiv (Cornell University)|Nov 6, 2020

Algebraic Geometry and Number Theory24 references4 citations

TL;DR

This paper introduces periodic de Rham bundles over smooth complex curves, proving that motivic de Rham bundles—arising from Gau{ extbackslash}ss-Manin systems—are periodic. It conjectures that irreducible periodic de Rham bundles are motivic, and proves this for rank one and rigid objects, using nonabelian Hodge theory in positive characteristic and parabolic structures via spreading-out and Frobenius periodicity.

ABSTRACT

In this article, we introduce the notion of periodic de Rham bundles over smooth complex curves. We prove that motivic de Rham bundles over smooth complex curves are periodic. We conjecture that irreducible periodic de Rham bundles over smooth complex curves are motivic. We show that the conjecture holds for rank one objects and rigid objects.

Motivation & Objective

• To define and study periodic de Rham bundles over smooth complex curves using reduction modulo primes and Frobenius actions.
• To prove that motivic de Rham bundles—arising from Gau{ extbackslash}ss-Manin systems—are periodic.
• To conjecture that irreducible periodic de Rham bundles are motivic, and to verify this for rank one and rigid objects.
• To extend Ogus-Vologodsky nonabelian Hodge theory to the parabolic setting for de Rham bundles with logarithmic structures.
• To establish a dynamical characterization of Picard-Fuchs equations via periodicity of the twisted Frobenius action in positive characteristic.

Proposed method

• Spreading out the de Rham bundle and its Hodge filtration to a scheme over a finitely generated Z-algebra to enable reduction modulo primes.
• Using the parabolic inverse Cartier transform and Griffiths transversality to analyze the behavior of connections and Higgs bundles in positive characteristic.
• Applying the Ogus-Vologodsky correspondence in positive characteristic to relate de Rham and Dolbeault structures.
• Employing the twisted Frobenius action and periodicity conditions on reductions at geometric points to define periodic de Rham bundles.
• Using the Simpson correspondence and rigidity theorems to relate rigid local systems to motivic structures.
• Proving bijectivity of a self-map on moduli spaces of rigid parabolic connections via finite set arguments and inverse Cartier transforms.

Experimental results

Research questions

• RQ1 Are all motivic de Rham bundles over smooth complex curves periodic?
• RQ2 Is every irreducible periodic de Rham bundle over a smooth complex curve motivic?
• RQ3 How does the periodicity of de Rham bundles in positive characteristic relate to their motivic origin?
• RQ4 Can the parabolic inverse Cartier transform preserve rigidity and determinant structure under reduction modulo primes?
• RQ5 What is the role of the twisted Frobenius action in characterizing Picard-Fuchs equations?

Key findings

• Motivic de Rham bundles over smooth complex curves are periodic, as shown via spreading-out and reduction modulo primes.
• The conjecture that irreducible periodic de Rham bundles are motivic holds for rank one objects, proven using determinant line bundle analysis.
• The conjecture also holds for rigid de Rham bundles, established via the rigidity of parabolic connections and the bijectivity of the inverse Cartier transform on moduli spaces.
• The self-map induced by the composition of the inverse Cartier transform and grading is bijective on the moduli space of rigid parabolic connections, implying periodicity.
• Irreducible weakly physically semi-rigid (WPSR) connections are motivic, as they arise as summands of Gau{ extbackslash}ss-Manin systems via hypersurface compactifications.
• The determinant of the inverse Cartier transform of a rigid bundle is isomorphic to a tensor power of the original determinant, preserving the structure needed for motivicity.