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On a conjecture of Hosono-Lee-Lian-Yau

Andrew Harder, Sukjoo Lee

TL;DR

The paper extends the Hosono–Lee–Lian–Yau mirror framework from branched double covers to broader $(\mathbb{Z}/2)^k$-Galois Calabi–Yau covers, establishing Hodge-number duality for both original and extended mirrors via a Cayley-trick–type correspondence between branched-cover de Rham theory and irregular Hodge theory of Landau–Ginzburg models. It develops a detailed LG-model/irregular-Hodge formalism, proving a key decomposition theorem that expresses twisted cohomology of LG spaces as a direct sum over branching data, and then leverages Clarke duality in toric geometry to obtain dualities for toric CY varieties and their HLLY mirrors. The results yield Hodge-number dualities for Clarke mirror pairs, toric extremal transitions, and, crucially, a full proof of the HLLY conjecture in the toric setting by combining these dualities with combinatorial identities. Overall, the work provides a robust, largely model-agnostic framework for understanding mirror symmetry of singular Calabi–Yau varieties through LG-models and Galois covers, with potential independent interest beyond mirror symmetry.

Abstract

We extend the mirror construction of singular Calabi-Yau double covers, introduced by Hosono, Lee, Lian, and Yau, to a broader class of singular Calabi-Yau $(\mathbb{Z}/2)^k$-Galois covers, and prove Hodge number duality for both the original and extended mirror pairs. A main tool in our approach is an analogue of the Cayley trick, which relates the de Rham complex of the branched covers to the twisted de Rham complex of certain Landau-Ginzburg models. In particular, it reveals direct relations between the Hodge numbers of the covers and the irregular Hodge numbers of the associated Landau-Ginzburg models. This construction is independent of mirror symmetry and may be of independent interest.

On a conjecture of Hosono-Lee-Lian-Yau

TL;DR

The paper extends the Hosono–Lee–Lian–Yau mirror framework from branched double covers to broader -Galois Calabi–Yau covers, establishing Hodge-number duality for both original and extended mirrors via a Cayley-trick–type correspondence between branched-cover de Rham theory and irregular Hodge theory of Landau–Ginzburg models. It develops a detailed LG-model/irregular-Hodge formalism, proving a key decomposition theorem that expresses twisted cohomology of LG spaces as a direct sum over branching data, and then leverages Clarke duality in toric geometry to obtain dualities for toric CY varieties and their HLLY mirrors. The results yield Hodge-number dualities for Clarke mirror pairs, toric extremal transitions, and, crucially, a full proof of the HLLY conjecture in the toric setting by combining these dualities with combinatorial identities. Overall, the work provides a robust, largely model-agnostic framework for understanding mirror symmetry of singular Calabi–Yau varieties through LG-models and Galois covers, with potential independent interest beyond mirror symmetry.

Abstract

We extend the mirror construction of singular Calabi-Yau double covers, introduced by Hosono, Lee, Lian, and Yau, to a broader class of singular Calabi-Yau -Galois covers, and prove Hodge number duality for both the original and extended mirror pairs. A main tool in our approach is an analogue of the Cayley trick, which relates the de Rham complex of the branched covers to the twisted de Rham complex of certain Landau-Ginzburg models. In particular, it reveals direct relations between the Hodge numbers of the covers and the irregular Hodge numbers of the associated Landau-Ginzburg models. This construction is independent of mirror symmetry and may be of independent interest.

Paper Structure

This paper contains 15 sections, 29 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Double covers and Landau--Ginzburg models
  3. Applications to mirror symmetry
  4. Cyclic covers and Landau--Ginzburg models
  5. Landau--Ginzburg models and the irregular Hodge filtration
  6. Twisted de Rham cohomology
  7. Irregular Hodge filtration
  8. Affine bundles
  9. Statement of Theorem \ref{['t : multiple direct sum decomposition']} and discussion
  10. Proof of Theorem \ref{['t : multiple direct sum decomposition']}, assuming Theorem \ref{['t : single direct sum decomposition']}
  11. Proof of Theorem \ref{['t : single direct sum decomposition']}
  12. Hodge number duality for Clarke mirror pairs
  13. Mirror symmetry for Galois cover Calabi--Yau varieties
  14. Toric extremal transition
  15. A proof of HLLY conjecture

Key Result

Theorem 1.1

Let the notation be as above. There is a filtered isomorphism Here, $H^*(\widehat{X}\setminus R)^{(-)}$ and $H_B^*(X)$ carry their usual pure Hodge structures, and the notation $(1/2)$ indicates a rational shift by $1/2$ (see § sss:lg for details).

Theorems & Definitions (77)

  • Theorem 1.1: Theorem \ref{['t : single direct sum decomposition']}
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['t : multiple direct sum decomposition']}
  • Conjecture 1.4: HLLY conjecture
  • Theorem 1.5
  • Remark 1.6
  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • Remark 2.4: On irregular filtrations
  • ...and 67 more