On monodromy of monodromy surfaces
Abstract
Different instances of the Riemann-Hilbert correspondence relate initial value spaces of Painlevé equations to affine varieties known as monodromy surfaces, built from monodromy invariants for associated linear ODEs. We show that these affine varieties admit realisations as embedded affine del Pezzo surfaces, characterised by their degree and a prescribed divisor at infinity, first in this paper for cases associated with Painlevé equations VI,IV,II,I. We prove that the monodromy groups of these monodromy surfaces form the finite parts of the affine Weyl symmetry groups of the corresponding Painlevé equations. We realise the monodromy group in each case: analytically as permutations of lines induced by continuation along loops in parameter space, Galois-theoretically in terms of the function field of the incidence variety of lines, and combinatorially via their intersection graph. This in particular yields an interpretation of the parameter spaces of Painlevé equations, modulo symmetries, as moduli spaces of categories of embedded affine varieties. Further, it shows that, despite the affine Weyl group symmetries becoming trivial when conjugated by the corresponding Riemann-Hilbert map, the finite Weyl group part survives as monodromy of the monodromy surface.
Related
- Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces
- Iterated Monodromy Groups of Quadratic Polynomials, I
- Two-step nilpotent monodromy of local systems on special varieties
- Degeneration of monodromy in the theory of Eisenstein congruences for classical groups
- Remarks on the monodromy conjecture for symmetric determinantal hypersurfaces
BibTeX
@misc{roffelsen2026monodromy,
title = {{On monodromy of monodromy surfaces}},
author = {Pieter Roffelsen and Alexander Stokes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.25594},
eprint = {2608.25594},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.25594},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On monodromy of monodromy surfaces
AU - Pieter Roffelsen
AU - Alexander Stokes
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.25594
UR - https://arxiv.org/abs/2608.25594
AB - Different instances of the Riemann-Hilbert correspondence relate initial value spaces of Painlevé equations to affine varieties known as monodromy surfaces, built from monodromy invariants for associated linear ODEs. We show that these affine varieties admit realisations as embedded affine del Pezzo surfaces, characterised by their degree and a prescribed divisor at infinity, first in this paper for cases associated with Painlevé equations VI,IV,II,I. We prove that the monodromy groups of these monodromy surfaces form the finite parts of the affine Weyl symmetry groups of the corresponding Painlevé equations. We realise the monodromy group in each case: analytically as permutations of lines induced by continuation along loops in parameter space, Galois-theoretically in terms of the function field of the incidence variety of lines, and combinatorially via their intersection graph. This in particular yields an interpretation of the parameter spaces of Painlevé equations, modulo symmetries, as moduli spaces of categories of embedded affine varieties. Further, it shows that, despite the affine Weyl group symmetries becoming trivial when conjugated by the corresponding Riemann-Hilbert map, the finite Weyl group part survives as monodromy of the monodromy surface.
ER -