Moduli spaces of untwisted wild Riemann surfaces

TL;DR

The paper develops a comprehensive framework for untwisted wild Riemann surfaces by constructing analytic moduli stacks $ ext{WM}_{g,m}^{oldsymbol{t}, ext{≤}oldsymbol{p}}$ and substacks with fixed root data, and defines the global wild mapping class groups as fundamental groups of these stacks. It introduces irregular types via the tail sheaf $oldsymbol{ extscr{T}}^{ ext{≤}k}$, establishes a root stratification, and proves that the stacks are analytic with fibers over $ ext{M}_{g,m}$ given by products of irregular-type spaces, yielding a Serre fibration structure for the associated fundamental groups. The work provides explicit genus-zero and genus-one examples, analyzes Deligne–Mumford properties for algebraic analogues, and develops a parallel algebraic theory of wild curves with admissible strata and Levi-filtration parametrizations. These results lay the groundwork for studying wild character varieties via wild mapping class groups, with potential applications to irregular isomonodromy and moduli of connections.

Abstract

We construct moduli stacks of wild Riemann surfaces in the (pure) untwisted case, for any complex reductive structure group, and we define the corresponding (pure) wild mapping class groups.

Theorems & Definitions (40)

• Definition 2.1.1
• Example 2.2.2
• Lemma 2.2.4
• proof
• Lemma 2.2.5
• proof
• Lemma 2.2.7
• proof
• Lemma 2.2.8
• proof