Algebraic flat connections and o-minimality

2025/06/09 by Hélène Esnault, Esnault, Hélène, Moritz Kerz +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2506.07498

openalex publication_date 2025/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that an algebraic flat connection has definable flat sections in the analytic exponential structure if and only if it is regular singular with unitary monodromy eigenvalues at infinity, refining previous work of Bakker and Mullane. This provides an o minimal characterisation of classical properties of the Gauss-Manin connection.

Citations

• Definable structures on flat bundles
• o-minimal GAGA and a conjecture of Griffiths
• The Field of Reals with Multisummable Series and the Exponential Function
• The Hodge filtration on nonabelian cohomology

Related

• Good formal structure for meromorphic flat connections on smooth projective surfaces
• Definability of the Integrability Locus in Polynomially Bounded o-Minimal Structures
• Characteristic classes of flat bundles and determinant of the Gauss-Manin connection
• Log-Noetherian functions
• Finite Nocommutative Coverings and Flat Connections