[Paper Review] Representations of quasiprojective groups, Flat connections and Transversely projective foliations

Frank Loray, Frédéric Touzet+1 more|arXiv (Cornell University)|Feb 6, 2014

Algebraic Geometry and Number Theory37 references20 citations

TL;DR

This paper establishes a structure theorem for codimension one singular transversely projective foliations on projective manifolds by extending Corlette-Simpson's classification of rank two representations of quasiprojective fundamental groups to non-quasi-unipotent cases and proving an analogue classification for rank two flat meromorphic connections. The key result is a global classification of Riccati foliations via birational reduction to standard local models, including logarithmic and irregular types, with a complete normal form after ramified covers.

ABSTRACT

The main purpose of this paper is to provide a structure theorem for codimension one singular transversely projective foliationson projective manifolds. To reach our goal, we firstly extend Corlette-Simpson's classification of rank two representationsof fundamental groups of quasiprojective manifolds by dropping the hypothesis of quasi-unipotency at infinity.Secondly we establish an analogue classification for rank two flat meromorphic connections.In particular, we prove that a rank two flat meromorphic connection with irregular singularities having non trivial Stokesprojectively factors through a connection over a curve.

Motivation & Objective

• To extend Corlette-Simpson's classification of rank two representations of quasiprojective fundamental groups by removing the quasi-unipotency hypothesis at infinity.
• To establish an analogue classification for rank two flat meromorphic connections, particularly those with irregular singularities.
• To provide a global structure theorem for transversely projective foliations on projective manifolds through birational reduction and normal form classification.
• To classify the local models of Riccati foliations after reduction, including logarithmic and irregular types, using meromorphic gauge transformations and ramified covers.

Proposed method

• Extends the classification of rank two representations of quasiprojective fundamental groups beyond the quasi-unipotent case, generalizing Corlette-Simpson's result.
• Analyzes flat meromorphic connections on trivial rank two bundles over projective manifolds via the connection matrix $ A = \begin{pmatrix} \alpha & \beta \\ \omega & -\alpha \end{pmatrix} $, with $ \omega \wedge d\omega = 0 $.
• Uses birational bundle modifications and elementary transformations to reduce the polar divisor to minimal order, enabling local normal form reduction.
• Applies Kawamata covering to eliminate ramification in irregular singularities, transforming the irregular divisor into a fixed index $ k $ across all components.
• Employs meromorphic gauge reduction to transform local models into standard forms: log-type, irregular-type, and ramified irregular-type.
• Reduces the global foliation structure to a finite set of canonical local models via global birational transformations and normal crossing divisor assumptions.

Experimental results

Research questions

• RQ1 Can the classification of rank two representations of quasiprojective fundamental groups be extended beyond the quasi-unipotent case?
• RQ2 How do flat meromorphic connections with irregular singularities factor through lower-dimensional geometric structures?
• RQ3 What is the global structure of transversely projective foliations on projective manifolds after birational modifications?
• RQ4 Which local models classify Riccati foliations after reduction to normal forms with normal crossing divisors?
• RQ5 How can ramified singularities in irregular foliations be eliminated via covering spaces to achieve uniform irregularity indices?

Key findings

• The paper proves that a rank two flat meromorphic connection with irregular singularities and nontrivial Stokes data factors through a connection over a curve.
• All transversely projective foliations on projective surfaces are birationally equivalent to a Riccati foliation with a polar divisor in normal crossing form and local models classified into five types: Log$^{\mathbb{G}_m}$, Log$^{\mathbb{G}_a}$, Irreg 0, Irreg, and Irreg$^{\text{ram}}$.
• After a suitable ramified cover, the irregular divisor acquires a uniform irregularity index $ k $, eliminating the need for the Irreg$^{\text{ram}}$ model.
• The meromorphic gauge reduction process can be performed globally along irreducible components of the polar divisor, reducing the problem to local biholomorphic trivializations.
• The classification is stable under ramified covers in variables $ x $ or $ y $, and the models are preserved under such transformations.
• The structure theorem provides a complete normal form for transversely projective foliations via birational transformations and reduction to standard Riccati forms.