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Irregular Hodge numbers of stacky Clarke mirror pairs

Andrew Harder, Sukjoo Lee

TL;DR

This work establishes a duality for the irregular Hodge filtration on twisted cohomology for Clarke mirror pairs of stacky Landau–Ginzburg models, providing strong evidence that Clarke duals are mirror in the Hodge-theoretic sense. It unifies and extends classical results (Batyrev–Borisov, Krawitz, Gross–Katzarkov–Ruddat) and confirms a Katzarkov–Kontsevich–Pantev-type duality for toric complete intersections via nef partitions and Cayley tricks, including orbifold settings. A key methodological backbone combines tropical geometry, tropical Jacobian sheaves, and quasi-stable degenerations to connect irregular Hodge numbers with tropical invariants, enabling combinatorial proofs of dualities. The paper also treats non-convex Clarke dual pairs through stacky enhancements and demonstrates tropical realizations of irregular Hodge numbers in degenerations, broadening the scope of mirror symmetry in toric and orbifold contexts with practical implications for Hodge theory and LG-model dualities.

Abstract

We prove a duality between the graded pieces of the irregular Hodge filtration on the twisted cohomology for a large class of Clarke mirror pairs of stacky Landau-Ginzburg models. We use this to recover results of Batyrev--Borisov, generalize results of Ebeling-Gusein-Zade-Takahashi and Krawitz, and prove results similar to those of Gross-Katzarkov-Ruddat. We apply our results to prove a generalized version of a conjecture of Katzarkov-Kontsevich-Pantev for orbifold toric complete intersections with nef anticanonical divisors and orbifold Fano stacks, and we prove the Hodge number duality result for orbifold log Calabi-Yau complete intersections. Along the way, we study the behaviour of twisted cohomology under degeneration and prove that for certain degenerations of toric Landau--Ginzburg models, irregular Hodge numbers admit a tropical realization.

Irregular Hodge numbers of stacky Clarke mirror pairs

TL;DR

This work establishes a duality for the irregular Hodge filtration on twisted cohomology for Clarke mirror pairs of stacky Landau–Ginzburg models, providing strong evidence that Clarke duals are mirror in the Hodge-theoretic sense. It unifies and extends classical results (Batyrev–Borisov, Krawitz, Gross–Katzarkov–Ruddat) and confirms a Katzarkov–Kontsevich–Pantev-type duality for toric complete intersections via nef partitions and Cayley tricks, including orbifold settings. A key methodological backbone combines tropical geometry, tropical Jacobian sheaves, and quasi-stable degenerations to connect irregular Hodge numbers with tropical invariants, enabling combinatorial proofs of dualities. The paper also treats non-convex Clarke dual pairs through stacky enhancements and demonstrates tropical realizations of irregular Hodge numbers in degenerations, broadening the scope of mirror symmetry in toric and orbifold contexts with practical implications for Hodge theory and LG-model dualities.

Abstract

We prove a duality between the graded pieces of the irregular Hodge filtration on the twisted cohomology for a large class of Clarke mirror pairs of stacky Landau-Ginzburg models. We use this to recover results of Batyrev--Borisov, generalize results of Ebeling-Gusein-Zade-Takahashi and Krawitz, and prove results similar to those of Gross-Katzarkov-Ruddat. We apply our results to prove a generalized version of a conjecture of Katzarkov-Kontsevich-Pantev for orbifold toric complete intersections with nef anticanonical divisors and orbifold Fano stacks, and we prove the Hodge number duality result for orbifold log Calabi-Yau complete intersections. Along the way, we study the behaviour of twisted cohomology under degeneration and prove that for certain degenerations of toric Landau--Ginzburg models, irregular Hodge numbers admit a tropical realization.
Paper Structure (53 sections, 66 theorems, 264 equations, 4 figures)

This paper contains 53 sections, 66 theorems, 264 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Irregular Hodge numbers of a Landau--Ginzburg model
  4. Irregular Hodge number duality for Clarke mirror pairs
  5. Katzarkov--Kontsevich--Pantev conjecture for toric complete intersections
  6. Non-convex Clarke dual pairs
  7. Outline of the proof of Theorem \ref{['thm:intro-clarkedual']}
  8. Comparison results
  9. Combinatorial results
  10. Structure of the article
  11. Acknowledgements
  12. Background
  13. Toric varieties
  14. Toric Deligne--Mumford stacks
  15. Orbifolds and their cohomology
  16. ...and 38 more sections

Key Result

Theorem 1.1

Let $({\bf\Sigma}, {\bf \check{\Sigma}})$ be a Clarke dual pair. For $\lambda,\mu \in \mathbb{Q}$, we have the identification of the orbifold irregular Hodge numbers

Figures (4)

  • Figure 1: $\mathrm{SD}(w)$ (Left) and $\mathrm{PC}(w)$ (Right) in Example \ref{['eg:trop']}.
  • Figure 2: $\mathrm{nf}(\Delta(w))$ (Left) and $\Sigma$ (Right) in Example \ref{['eg:trop2']}.
  • Figure 3: $\mathrm{Trop}(\Sigma, w)$ in Example \ref{['eg:trop']}.
  • Figure 4:

Theorems & Definitions (167)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3: Theorem \ref{['t:kkp']}
  • Theorem 1.4: Theorem \ref{['t:gkr']}
  • Remark 1.5
  • Theorem 1.6: Theorem \ref{['t:nearbyfibcoh']}, Proposition \ref{['p:hodgeorb']}
  • Definition 2.3
  • Remark 2.4
  • Definition 2.6
  • Definition 2.7
  • ...and 157 more