Compactifications of the Eisenstein ancestral Deligne-Mostow variety

verfasst von: Klaus Hulek, Shigeyuki Kondo, Yota Maeda
Abstract: All arithmetic non-compact ball quotients by Deligne-Mostow's unitary monodromy group arise as sub-ball quotients of either of two spaces called ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms respectively. In our previous paper, we investigated the compactifications of the Gaussian Deligne-Mostow variety. Here we work on the remaining case, namely the ring of Eisenstein integers, which is related to the moduli space of unordered 12 points on P^1. In particular, we show that Kirwan's partial resolution of the moduli space is not a semi-toroidal compactification and Deligne-Mostow's period map does not lift to the unique toroidal compactification. We give two interpretations of these phenomena in terms of the log minimal model program and automorphic forms. As an application, we prove that the above two compactifications are not (stacky) derived equivalent, as the DK-conjecture predicts. Furthermore, we construct an automorphic form on the moduli space of non-hyperelliptic curves of genus 4, which is isogenous to the Eisenstein Deligne-Mostow variety, giving another intrinsic proof, independent of lattice embeddings, of a result by Casalaina-Martin, Jensen and Laza.
Organisationseinheit(en): Institut für Algebraische Geometrie
Riemann Center for Geometry and Physics
Typ: Preprint
Publikationsdatum: 27.03.2024
Publikationsstatus: Elektronisch veröffentlicht (E-Pub)

BibTeX

@techreport{f2d728e796fb4066b6103ae712d314da,
title = "Compactifications of the Eisenstein ancestral Deligne-Mostow variety",
abstract = " All arithmetic non-compact ball quotients by Deligne-Mostow's unitary monodromy group arise as sub-ball quotients of either of two spaces called ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms respectively. In our previous paper, we investigated the compactifications of the Gaussian Deligne-Mostow variety. Here we work on the remaining case, namely the ring of Eisenstein integers, which is related to the moduli space of unordered 12 points on $\mathbb P^1$. In particular, we show that Kirwan's partial resolution of the moduli space is not a semi-toroidal compactification and Deligne-Mostow's period map does not lift to the unique toroidal compactification. We give two interpretations of these phenomena in terms of the log minimal model program and automorphic forms. As an application, we prove that the above two compactifications are not (stacky) derived equivalent, as the $DK$-conjecture predicts. Furthermore, we construct an automorphic form on the moduli space of non-hyperelliptic curves of genus 4, which is isogenous to the Eisenstein Deligne-Mostow variety, giving another intrinsic proof, independent of lattice embeddings, of a result by Casalaina-Martin, Jensen and Laza. ",
keywords = "math.AG, math.NT, 14D20, 14G35, 11F03, 14E30, 14F08",
author = "Klaus Hulek and Shigeyuki Kondo and Yota Maeda",
note = "References updated, minor improvements in exposition",
year = "2024",
month = mar,
day = "27",
language = "English",
type = "WorkingPaper",
}