Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Universe of Deligne-Mostow Varieties

Klaus Hulek, Yota Maeda

TL;DR

This work extends Deligne–Mostow period maps to the broad setting of partly unordered point configurations, identifying a concrete, checkable criterion $( ext{T})$ that exactly characterizes when the DM map lifts to an isomorphism between the Kirwan blow-up $M^{ m K}_{w,S}$ and the toroidal compactification $X^{ m T}_{w,S}$. By introducing a partial order on the Deligne–Mostow universe $A_{ m DM}$, the authors reduce the problem to extremal (minimal or maximal) cases, enabling a tractable analysis of 85 DM varieties and their birational relationships. They prove a sharp dichotomy: when $( ext{T})$ holds, the lift exists and is an isomorphism (with LMMP implications); when it fails, the Kirwan and toroidal compactifications are not naturally isomorphic, and the Kirwan model is not a log minimal model. The results unify and generalize prior work on ancestral Gaussian/Eisenstein cases, offer reductions that streamline proofs, and illuminate the birational geometry of DM varieties with potential applications to related moduli problems and semi-tToroidal compactifications.

Abstract

Deligne and Mostow investigated period maps on the configuration spaces $M_{0,n}$ of $n$ ordered points on $\mathbb{P}^1$. The images of these maps are open subsets of certain ball quotients. Moreover, they extend to isomorphisms between GIT-quotients and the Baily-Borel compactifications. Building on a theorem of Gallardo, Kerr and Schaffler, the period maps lift to isomorphisms between two natural compactifications, namely the Kirwan blow-up and the toroidal compactification. In this paper, we look at the more general situation where we also allow unordered or partially ordered $n$-tuples. Our main result is an easily verifiable criterion that, in this broader setting, determines when the Deligne-Mostow period maps still lift to isomorphisms between the Kirwan blow-up and the toroidal compactification. We further investigate a partial ordering among Deligne-Mostow varieties, which reduces this problem to considering minimal or maximal Deligne-Mostow varieties with respect to this partial ordering. As a byproduct, we prove that, in general, Kirwan's resolution pair is not a log canonical log minimal model and not log $K$-equivalent to the unique toroidal compactification.

The Universe of Deligne-Mostow Varieties

TL;DR

This work extends Deligne–Mostow period maps to the broad setting of partly unordered point configurations, identifying a concrete, checkable criterion that exactly characterizes when the DM map lifts to an isomorphism between the Kirwan blow-up and the toroidal compactification . By introducing a partial order on the Deligne–Mostow universe , the authors reduce the problem to extremal (minimal or maximal) cases, enabling a tractable analysis of 85 DM varieties and their birational relationships. They prove a sharp dichotomy: when holds, the lift exists and is an isomorphism (with LMMP implications); when it fails, the Kirwan and toroidal compactifications are not naturally isomorphic, and the Kirwan model is not a log minimal model. The results unify and generalize prior work on ancestral Gaussian/Eisenstein cases, offer reductions that streamline proofs, and illuminate the birational geometry of DM varieties with potential applications to related moduli problems and semi-tToroidal compactifications.

Abstract

Deligne and Mostow investigated period maps on the configuration spaces of ordered points on . The images of these maps are open subsets of certain ball quotients. Moreover, they extend to isomorphisms between GIT-quotients and the Baily-Borel compactifications. Building on a theorem of Gallardo, Kerr and Schaffler, the period maps lift to isomorphisms between two natural compactifications, namely the Kirwan blow-up and the toroidal compactification. In this paper, we look at the more general situation where we also allow unordered or partially ordered -tuples. Our main result is an easily verifiable criterion that, in this broader setting, determines when the Deligne-Mostow period maps still lift to isomorphisms between the Kirwan blow-up and the toroidal compactification. We further investigate a partial ordering among Deligne-Mostow varieties, which reduces this problem to considering minimal or maximal Deligne-Mostow varieties with respect to this partial ordering. As a byproduct, we prove that, in general, Kirwan's resolution pair is not a log canonical log minimal model and not log -equivalent to the unique toroidal compactification.

Paper Structure

This paper contains 12 sections, 17 theorems, 38 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background of period maps
  3. Main results
  4. Organisation of the paper
  5. Proof of main results
  6. Reduction methods to minimal or maximal varieties
  7. (Non)Functoriality of the Kirwan blow-up
  8. Reduction method
  9. Applications to birational geometry
  10. On the structure of the universe of Deligne-Mostow varieties
  11. The ancestral Deligne-Mostow varieties
  12. 85 elements in $A_{\mathrm{DM}}$

Key Result

Theorem 1.2

Let $(w, S)\in A_{\mathrm{DM}}$ be a Deligne-Mostow pair. Then the open immersion mor:unordered_period_map, the complement of whose image is the discriminant divisor, can be extended to the GIT and Baily-Borel compactifications:

Figures (2)

  • Figure 1: Reduction method
  • Figure 2: Inclusions of $X_{w_i}^{\mathrm{B}}$ in the Gaussian case

Theorems & Definitions (37)

  • Theorem 1.2: DM86Mos86
  • Theorem 1.3: GKS21
  • Theorem 1.4: Theorem \ref{['thm:main_extendability']}
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7: Reduction Method \ref{['redmethodsub']}
  • Theorem 1.8: HKM24
  • Corollary 1.9: Section \ref{['section:applications']}
  • Remark 1.10
  • Lemma 2.1
  • ...and 27 more