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Fundamental groups of log Calabi-Yau surfaces

Cécile Gachet, Zhining Liu, Joaquín Moraga

TL;DR

This work investigates the orbifold fundamental groups $\pi_1^{\rm orb}(X,\Delta)$ of log canonical Calabi–Yau pairs, proving that in dimension two these groups are extensions of a nilpotent kernel of length at most $2$ and rank at most $4$ by a finite quotient of order at most $7200$, with sharpness demonstrated by explicit constructions. It establishes residual finiteness and a Jordan-type structure for broader settings, including dlt Fano pairs, and provides a thorough classification of orbifold groups arising from curves, Fano surfaces, and Mori fiber spaces. The authors develop a robust framework combining MMP reductions, a Galois correspondence for compatible finite covers, and toric/complement techniques to control group structure and yield effective bounds. They also supply necessary criteria for when the orbifold group is infinite or not virtually abelian and furnish several sharp examples, including Heisenberg-type quotients and large finite groups, to illustrate the scope and limitations of the results. Overall, the paper links positivity properties of canonical bundles to the global topology of log Calabi–Yau and Fano pairs via a detailed suite of birational-geometric and group-theoretic tools with implications for finite group actions on log Calabi–Yau surfaces.

Abstract

In this article, we study the orbifold fundamental group $π_1^{\rm orb}(X,Δ)$ of a Calabi--Yau pair $(X,Δ)$ with log canonical singularities. We conjecture that the orbifold fundamental group $π_1^{\rm orb}(X,Δ)$ of a $n$-dimensional log Calabi--Yau pair admits a normal solvable subgroup of rank at most $2n$ and index at most $c(n)$. We prove this conjecture in the case that $n=2$. More precisely, for a log Calabi--Yau surface pair $(X,Δ)$ we show that $π_1^{\rm orb}(X,Δ)$ is the extension of a nilpotent group of length at most $2$ and rank at most $4$ by a finite group of order at most $7200$. We also show that the bounds on the nilpotency length, rank, and order of the finite group quotient in this result are sharp. Finally, we provide some necessary criteria for a log Calabi--Yau surface $(X,Δ)$ to have an infinite, or a non virtually abelian orbifold fundamental group.

Fundamental groups of log Calabi-Yau surfaces

TL;DR

This work investigates the orbifold fundamental groups of log canonical Calabi–Yau pairs, proving that in dimension two these groups are extensions of a nilpotent kernel of length at most and rank at most by a finite quotient of order at most , with sharpness demonstrated by explicit constructions. It establishes residual finiteness and a Jordan-type structure for broader settings, including dlt Fano pairs, and provides a thorough classification of orbifold groups arising from curves, Fano surfaces, and Mori fiber spaces. The authors develop a robust framework combining MMP reductions, a Galois correspondence for compatible finite covers, and toric/complement techniques to control group structure and yield effective bounds. They also supply necessary criteria for when the orbifold group is infinite or not virtually abelian and furnish several sharp examples, including Heisenberg-type quotients and large finite groups, to illustrate the scope and limitations of the results. Overall, the paper links positivity properties of canonical bundles to the global topology of log Calabi–Yau and Fano pairs via a detailed suite of birational-geometric and group-theoretic tools with implications for finite group actions on log Calabi–Yau surfaces.

Abstract

In this article, we study the orbifold fundamental group of a Calabi--Yau pair with log canonical singularities. We conjecture that the orbifold fundamental group of a -dimensional log Calabi--Yau pair admits a normal solvable subgroup of rank at most and index at most . We prove this conjecture in the case that . More precisely, for a log Calabi--Yau surface pair we show that is the extension of a nilpotent group of length at most and rank at most by a finite group of order at most . We also show that the bounds on the nilpotency length, rank, and order of the finite group quotient in this result are sharp. Finally, we provide some necessary criteria for a log Calabi--Yau surface to have an infinite, or a non virtually abelian orbifold fundamental group.
Paper Structure (32 sections, 63 theorems, 105 equations, 1 figure)

This paper contains 32 sections, 63 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Jordan property and residual finiteness for fundamental groups
  4. Some conjectures
  5. Sketch of the proof of Theorem \ref{['introthm:fun-group-lcy']}
  6. Preliminaries
  7. The orbifold fundamental group
  8. Basic facts on residually finite groups
  9. Adjunction, coregularity, complements, and complexity as a characterization of toric pairs
  10. Pot-pourri of useful lemmas
  11. Proof of Theorem \ref{['introthm:fun-group-klt']}
  12. Proof of Theorem \ref{['introthm:Jordan-dlt-Fano']} and more
  13. Classifying Fano and Calabi--Yau curve pairs
  14. Log Calabi--Yau surfaces admitting Mori fiber spaces to curves
  15. Hereditarily uniruled pairs
  16. ...and 17 more sections

Key Result

Theorem 1

Let $(X,\Delta)$ be a log canonical Calabi--Yau pair with $\dim X =2$. Then the orbifold fundamental group $\pi_1^{\rm orb}(X,\Delta)$ admits a normal subgroup of index at most $7200$, that is abelian of rank at most $4$, or nilpotent of length at most $2$ and a quotient, for some $k\ge 1$, of the H

Figures (1)

  • Figure 1: Diagram of dependences between our main results

Theorems & Definitions (155)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Corollary 8
  • Conjecture 1
  • Conjecture 2
  • ...and 145 more