On the monodromy map for the logarithmic differential systems

TL;DR

This paper investigates the monodromy map for logarithmic differential systems on surfaces, establishing conditions under which it is an immersion or not, thus extending previous results to the logarithmic case.

Contribution

It provides new criteria for the monodromy map to be an immersion or not in the context of logarithmic differential systems, generalizing prior work on nonsingular systems.

Findings

Monodromy map is an immersion at generic points for certain genus and dimension conditions.

Monodromy map is nowhere an immersion in specific low-genus or low-dimension cases.

Extends previous results to systems with logarithmic singularities.

Abstract

We study the monodromy map for logarithmic $\mathfrak g$-differential systems over an oriented surface $S_0$ of genus $g$, with $\mathfrak g$ being the Lie algebra of a complex reductive affine algebraic group $G$. These logarithmic $\mathfrak g$-differential systems are triples of the form $(X, D,\Phi)$, where $(X, D) \in {\mathcal T}_{g,d}$ is an element of the Teichm\"uller space of complex structures on $S_0$ with $d \geq 1$ ordered marked points $D\subset S_0= X$ and $\Phi$ is a logarithmic connection on the trivial holomorphic principal $G$-bundle $X \times G$ over $X$ whose polar part is contained in the divisor $D$. We prove that the monodromy map from the space of logarithmic $\mathfrak g$-differential systems to the character variety of $G$-representations of the fundamental group of $S_0\setminus D$ is an immersion at the generic point, in the following two cases: A) $g \geq 2$, $d \geq 1$, and $\dim_{\mathbb C}G \geq d+2$; B) $g=1$ and $\dim_{\mathbb C}G \geq d$. The above monodromy map is nowhere an immersion in the following two cases: 1) $g=0$ and $d \geq 4$; 2) $g\geq 1$ and $\dim_{\mathbb C}G < \frac{d+3g-3}{g}$. This extends to the logarithmic case the main results in \cite{CDHL}, \cite{BD} dealing with nonsingular holomorphic $\mathfrak g$-differential systems (which corresponds to the case of $d\,=\,0$).