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Index of coregularity zero log Calabi-Yau pairs

Stefano Filipazzi, Mirko Mauri, Joaquín Moraga

TL;DR

The paper proves a dimension-free bound on the index of coregularity 0 log Calabi--Yau pairs by establishing that the Weil-adjusted multiple $\lambda'(K_X+B)$ is torsion, with $\lambda' = \mathrm{lcm}(\lambda,2)$. The key mechanism links the index to the orientability of the dual complex $\mathcal{D}(B)$, and the results extend to generalized, semi-log canonical, and isolated LC singularities, yielding sharp bounds such as $2(K_X+B)\sim 0$ in the reduced boundary case. The authors further apply these findings to mirror symmetry contexts (Gross--Siebert, Kontsevich--Soibelman), group actions on Calabi--Yau and holomorphic symplectic varieties, and to degenerations, showing that degenerations in the coregularity 0 regime are highly constrained. The methods combine birational geometry, adjunction, residue techniques, and topological analysis of dual complexes to produce effective, dimension-independent index controls with clear geometric consequences. Overall, the work supplies a robust framework for understanding how topological and group-action data govern the arithmetic of Calabi--Yau pairs in the coregularity 0 setting and informs their appearance in mirror-symmetric degenerations.

Abstract

In this article, we study the index of log Calabi--Yau pairs $(X,B)$ of coregularity 0. We show that $2λ(K_X+B)\sim 0$, where $λ$ is the Weil index of $(X,B)$. This is in contrast to the case of klt Calabi--Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi--Yau pairs, semi-log canonical pairs, and isolated log canonical singularities of coregularity 0. As a consequence, we show that the index of a variety appearing in the Gross--Siebert program or in the Kontsevich--Soibelman program is at most $2$. Finally, we discuss applications to Calabi--Yau varieties endowed with a finite group action, including holomorphic symplectic varieties endowed with a purely non-symplectic automorphism.

Index of coregularity zero log Calabi-Yau pairs

TL;DR

The paper proves a dimension-free bound on the index of coregularity 0 log Calabi--Yau pairs by establishing that the Weil-adjusted multiple is torsion, with . The key mechanism links the index to the orientability of the dual complex , and the results extend to generalized, semi-log canonical, and isolated LC singularities, yielding sharp bounds such as in the reduced boundary case. The authors further apply these findings to mirror symmetry contexts (Gross--Siebert, Kontsevich--Soibelman), group actions on Calabi--Yau and holomorphic symplectic varieties, and to degenerations, showing that degenerations in the coregularity 0 regime are highly constrained. The methods combine birational geometry, adjunction, residue techniques, and topological analysis of dual complexes to produce effective, dimension-independent index controls with clear geometric consequences. Overall, the work supplies a robust framework for understanding how topological and group-action data govern the arithmetic of Calabi--Yau pairs in the coregularity 0 setting and informs their appearance in mirror-symmetric degenerations.

Abstract

In this article, we study the index of log Calabi--Yau pairs of coregularity 0. We show that , where is the Weil index of . This is in contrast to the case of klt Calabi--Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends to the context of generalized log Calabi--Yau pairs, semi-log canonical pairs, and isolated log canonical singularities of coregularity 0. As a consequence, we show that the index of a variety appearing in the Gross--Siebert program or in the Kontsevich--Soibelman program is at most . Finally, we discuss applications to Calabi--Yau varieties endowed with a finite group action, including holomorphic symplectic varieties endowed with a purely non-symplectic automorphism.
Paper Structure (24 sections, 40 theorems, 73 equations)

This paper contains 24 sections, 40 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Index conjecture
  3. Main Theorem
  4. Orientability of dual complexes
  5. Mirror symmetry
  6. Index of semi-log canonical pairs
  7. Index of log canonical singularities
  8. Calabi--Yau and holomorphic symplectic varieties with group actions
  9. Strategy of the proof of Theorem \ref{['introthm:main-thm']}
  10. Notation
  11. Preliminaries
  12. Cartier index of a numerically trivial divisor
  13. Weil index of a generalized pair
  14. Numerically trivial moduli part
  15. Coregularity and finite quotients
  16. ...and 9 more sections

Key Result

Theorem 1

Let $(X,B,\mathbf{M}_{})$ be a projective generalized log Calabi--Yau pair of coregularity 0 and Weil index $\lambda$. Then, we have that $\lambda'(K_X+B+\mathbf{M}_{X})\sim 0$, where $\lambda'=\operatorname{lcm}(\lambda,2)$.

Theorems & Definitions (99)

  • Theorem 1
  • Proposition 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Theorem 8
  • Corollary 9
  • Corollary 10
  • ...and 89 more