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Degenerations, fibrations and higher rank Landau-Ginzburg models

Charles F. Doran, Jordan Kostiuk, Fenglong You

TL;DR

The paper develops a higher-rank Landau–Ginzburg (LG) mirror framework for Tyurin degenerations of quasi-Fano varieties, introducing rank-$n$ LG models and a gluing paradigm that recovers the ordinary (rank-1) LG mirror of $(X,D)$ by gluing rank-2 models attached to the Tyurin pieces. It proves a topological Euler-characteristic gluing formula and derives Hadamard-product relations for periods and $I$-functions, connecting generalized functional invariants across glued pieces. Through explicit toric and non-toric examples, the authors illuminate product relations among generalized invariants and residue-based period computations, and they extend the gluing philosophy to degenerations to the normal cone and iterative Tyurin degenerations, predicting Calabi–Yau fibrations on the mirror. They also situate the discussion in a broader Gromov–Witten framework, relating relative and formal invariants of simple normal crossing pairs to periods via mirror theorems. Overall, the work offers a cohesive geometric and algebro–analytic mechanism for constructing and understanding mirrors under Tyurin-type degenerations and provides a roadmap for higher-codimension Calabi–Yau fibrations in the mirror correspondence.

Abstract

We study semi-stable degenerations of quasi-Fano varieties to unions of two pieces. We conjecture that the higher rank Landau-Ginzburg models mirror to these two pieces can be glued together to lower rank Landau-Ginzburg models which are mirror to the original quasi-Fano varieties. We prove this conjecture by relating their Euler characteristics, generalized functional invariants as well as periods. We also use it to conjecture a relation between the degenerations to the normal cones and the fibrewise compactifications of higher rank Landau-Ginzburg models. Furthermore, we use it to iterate the Doran-Harder-Thompson conjecture and obtain higher codimension Calabi-Yau fibrations.

Degenerations, fibrations and higher rank Landau-Ginzburg models

TL;DR

The paper develops a higher-rank Landau–Ginzburg (LG) mirror framework for Tyurin degenerations of quasi-Fano varieties, introducing rank- LG models and a gluing paradigm that recovers the ordinary (rank-1) LG mirror of by gluing rank-2 models attached to the Tyurin pieces. It proves a topological Euler-characteristic gluing formula and derives Hadamard-product relations for periods and -functions, connecting generalized functional invariants across glued pieces. Through explicit toric and non-toric examples, the authors illuminate product relations among generalized invariants and residue-based period computations, and they extend the gluing philosophy to degenerations to the normal cone and iterative Tyurin degenerations, predicting Calabi–Yau fibrations on the mirror. They also situate the discussion in a broader Gromov–Witten framework, relating relative and formal invariants of simple normal crossing pairs to periods via mirror theorems. Overall, the work offers a cohesive geometric and algebro–analytic mechanism for constructing and understanding mirrors under Tyurin-type degenerations and provides a roadmap for higher-codimension Calabi–Yau fibrations in the mirror correspondence.

Abstract

We study semi-stable degenerations of quasi-Fano varieties to unions of two pieces. We conjecture that the higher rank Landau-Ginzburg models mirror to these two pieces can be glued together to lower rank Landau-Ginzburg models which are mirror to the original quasi-Fano varieties. We prove this conjecture by relating their Euler characteristics, generalized functional invariants as well as periods. We also use it to conjecture a relation between the degenerations to the normal cones and the fibrewise compactifications of higher rank Landau-Ginzburg models. Furthermore, we use it to iterate the Doran-Harder-Thompson conjecture and obtain higher codimension Calabi-Yau fibrations.
Paper Structure (23 sections, 16 theorems, 216 equations)

This paper contains 23 sections, 16 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Gluing rank $2$ LG models
  3. Set-up
  4. Topological gluing
  5. Periods
  6. Example 1: a complete intersection in $\mathbb P^4\times \mathbb P^1$
  7. Functional invariants
  8. Periods
  9. Example 2: quartic threefold
  10. Functional invariants
  11. Periods
  12. Toric complete intersections
  13. Set-up
  14. Generalized functional invariants
  15. Periods
  16. ...and 8 more sections

Key Result

Theorem 1.2

where $\mathrm d=\dim_{\mathbb C}X$ and $t$ is a regular value of $W$.

Theorems & Definitions (48)

  • Conjecture 1.1
  • Theorem 1.2: see Theorem \ref{['thm:topo-gluing']}
  • Theorem 1.3: =Theorem \ref{['thm:gluing-period-general']}
  • Theorem 1.4: Theorem \ref{['thm:general-case-I-function']}
  • Conjecture 1.5: =Conjecture \ref{['conj-compactification']}
  • Conjecture 1.6: =Conjecture \ref{['conj-higher-dim-base']}
  • Conjecture 1.7
  • Remark 1.8
  • Definition 2.1
  • Definition 2.2: DHT, Definition 2.1
  • ...and 38 more