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Galois covers of Calabi-Yau manifolds

Matteo Verni

TL;DR

This work develops a framework for studying Galois rational maps between Calabi–Yau and hyper‑Kähler manifolds, defining Calabi–Yau Galois covers and Galois‑like factorization concepts and analyzing monodromy and birational automorphisms. It proves strong obstructions to nontrivial CY Galois covers in key HK4‑fold settings (notably for very general cubic fourfolds) and shows Voisin maps fail to be CY Galois‑like or abelian CY Galois‑like in studied cases, supported by cohomological and BB‑form arguments. The paper also establishes bounds on automorphisms in abelian Galois covers, provides elliptic K3 examples illustrating both CY Galois‑like and non‑CY Galois‑like behavior, and uses Beauville–Bogomolov theory to deduce negativity properties of branch divisors and lower bounds on b_2 for HK manifolds admitting abelian CY covers. Collectively, these results clarify when CY maps can lift to Galois covers and reveal structural constraints from monodromy, cohomology, and the BB form in CY/HK geometry.

Abstract

We study Galois rational maps between smooth projective varieties with trivial canonical bundle, with a particular interest in the case where the codomain is Hyper-Kähler. We obtain results about the birational geometry and the Galois closure of rational maps between Calabi-Yau manifolds. We apply these general results to a number of well-known examples.

Galois covers of Calabi-Yau manifolds

TL;DR

This work develops a framework for studying Galois rational maps between Calabi–Yau and hyper‑Kähler manifolds, defining Calabi–Yau Galois covers and Galois‑like factorization concepts and analyzing monodromy and birational automorphisms. It proves strong obstructions to nontrivial CY Galois covers in key HK4‑fold settings (notably for very general cubic fourfolds) and shows Voisin maps fail to be CY Galois‑like or abelian CY Galois‑like in studied cases, supported by cohomological and BB‑form arguments. The paper also establishes bounds on automorphisms in abelian Galois covers, provides elliptic K3 examples illustrating both CY Galois‑like and non‑CY Galois‑like behavior, and uses Beauville–Bogomolov theory to deduce negativity properties of branch divisors and lower bounds on b_2 for HK manifolds admitting abelian CY covers. Collectively, these results clarify when CY maps can lift to Galois covers and reveal structural constraints from monodromy, cohomology, and the BB form in CY/HK geometry.

Abstract

We study Galois rational maps between smooth projective varieties with trivial canonical bundle, with a particular interest in the case where the codomain is Hyper-Kähler. We obtain results about the birational geometry and the Galois closure of rational maps between Calabi-Yau manifolds. We apply these general results to a number of well-known examples.

Paper Structure

This paper contains 12 sections, 22 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Overview
  5. Conventions
  6. Proof of Theorem \ref{['thm_HK2_rho=1']}
  7. Topological bounds on CY rational covers of HK manifolds
  8. CY covers of HK fourfolds
  9. Bounds on automorphisms of abelian Galois covers
  10. Some elliptic K3 examples
  11. Uniruledness above the branch divisor
  12. Beauville-Bogomolov negativity of branch divisors

Key Result

Theorem 1.14

Let $X$ be a hyper-Kähler manifold with $b_2(X)=23$ and $\rho_X=1$. Let $\phi \colon Y \dashrightarrow X$ be a Calabi-Yau rational cover. Then $Y$ admits no nontrivial birational automorphism commuting with $\phi$, i.e., In particular, if $\phi$ is Galois, then it is an isomorphism.

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.6
  • Remark 1.9
  • Remark 1.10
  • Example 1.11: Voisin map on $F_1(W)$
  • Example 1.12: Voisin map to LLSvS
  • Example 1.13: CY fibered in abelian varieties
  • Theorem 1.14
  • Corollary 1.15
  • Corollary 1.16
  • ...and 41 more