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Relative mirror symmetry for non-Fano varieties

Fenglong You

TL;DR

This work extends mirror symmetry to log Calabi–Yau pairs $(X,D)$ with non-nef anticanonical divisors by employing the Gross–Siebert intrinsic mirror to form a proper LG model $(reve X,W)$. It clarifies how the classical period of $W$ and the regularized quantum period of $X$ differ, attributing the discrepancy to curve classes meeting $D$ and encoding it via the mirror map in $D$, along with an explicit formula for the proper potential $W$ in terms of relative invariants. The paper proves that, after incorporating the $D$-mirror map, quantum periods and the proper potential carry the same information and demonstrates this equivalence through degeneration techniques and Lagrange inversion with Bell polynomials. It also extends the relative mirror theorem to non-Fano situations, derives identities for relative Gromov–Witten invariants in this setting, and provides a concrete example involving a non-Fano blow-up to illustrate the framework. Overall, the results deepen the understanding of mirror symmetry beyond nef settings and reveal the enumerative meaning of the $D$-mirror map in shaping the LG potential and its periods.

Abstract

Given a smooth projective variety $X$ with a smooth anticanonical divisor $D$, we study mirror symmetry for the log Calabi--Yau pair $(X,D)$ without assuming that $D$ is nef. We consider the mirror proper Landau--Ginzburg model $(\check X,W)$ from the intrinsic mirror construction of Gross--Siebert. We examine the relationship between the regularized quantum period of $X$ and the classical period of $W$, and identify the discrepancy between them as originating from curve counts in $D$, governed by the mirror map associated with $D$. We also obtain an explicit formula for the proper potential $W$ that encodes this discrepancy. In the end, we show that the quantum period, together with the mirror map, gives exactly the same information as the proper potential.

Relative mirror symmetry for non-Fano varieties

TL;DR

This work extends mirror symmetry to log Calabi–Yau pairs with non-nef anticanonical divisors by employing the Gross–Siebert intrinsic mirror to form a proper LG model . It clarifies how the classical period of and the regularized quantum period of differ, attributing the discrepancy to curve classes meeting and encoding it via the mirror map in , along with an explicit formula for the proper potential in terms of relative invariants. The paper proves that, after incorporating the -mirror map, quantum periods and the proper potential carry the same information and demonstrates this equivalence through degeneration techniques and Lagrange inversion with Bell polynomials. It also extends the relative mirror theorem to non-Fano situations, derives identities for relative Gromov–Witten invariants in this setting, and provides a concrete example involving a non-Fano blow-up to illustrate the framework. Overall, the results deepen the understanding of mirror symmetry beyond nef settings and reveal the enumerative meaning of the -mirror map in shaping the LG potential and its periods.

Abstract

Given a smooth projective variety with a smooth anticanonical divisor , we study mirror symmetry for the log Calabi--Yau pair without assuming that is nef. We consider the mirror proper Landau--Ginzburg model from the intrinsic mirror construction of Gross--Siebert. We examine the relationship between the regularized quantum period of and the classical period of , and identify the discrepancy between them as originating from curve counts in , governed by the mirror map associated with . We also obtain an explicit formula for the proper potential that encodes this discrepancy. In the end, we show that the quantum period, together with the mirror map, gives exactly the same information as the proper potential.
Paper Structure (15 sections, 22 theorems, 162 equations)

This paper contains 15 sections, 22 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminary in relative Gromov--Witten theory
  3. Relative mirror theorem
  4. The mirror map in $D$
  5. The relative mirror map
  6. The extended $I$-function
  7. An example
  8. Quantum periods and relative mirror maps
  9. Quantum periods from degeneration
  10. The nef anticanonical case
  11. The non-Fano case
  12. The proper potential for non-Fano varieties
  13. Some identities of relative Gromov--Witten invariants
  14. The proper potential from the relative mirror theorem
  15. The Lagrange inversion and Bell polynomial identities

Key Result

Theorem 1.1

Let $X$ be a smooth projective variety and $D$ be a smooth anticanonical divisor of $X$. Suppose $D$ is nef, and the classical period defined by the proper potential $W$ coincides with the regularized quantum period of $X$.

Theorems & Definitions (44)

  • Theorem 1.1: =Theorem \ref{['thm-class-quantum']}
  • Theorem 1.2: =Theorem \ref{['thm-classical-quantum']}
  • Theorem 1.4: =Theorem \ref{['thm-proper-potential']}
  • Remark 1.5
  • Proposition 2.1: =FWY, Proposition 3.4
  • Definition 2.2: FWY, Definition 5.7
  • Theorem 3.1: =You25*Corollary 3.6
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 34 more