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Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Victor V. Batyrev

TL;DR

The paper develops a comprehensive toric-geometric framework for Calabi-Yau hypersurfaces based on reflexive polyhedra, establishing a precise duality between the families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$ that mirrors physicists' Mirror Symmetry. It provides explicit combinatorial formulas for Hodge numbers and Euler characteristics, ties these to lattice-point counts, and shows that dual pairs have opposite Euler characteristics with exchanged Hodge numbers in Calabi-Yau 3-folds. A robust Galois correspondence and category of reflexive pairs organize mirror constructions, including mirror candidates in hypersurfaces of degree $n+1$ in $\mathbf{P}_n$ and quotients in weighted projective spaces. The work generalizes known constructions (Greene–Plesser, Roan) to a broad toric context, enabling systematic generation and analysis of Calabi-Yau mirrors via reflexive polyhedra and MPCP-desingularizations.

Abstract

We consider families ${\cal F}(Δ)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $Δ$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $Δ$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(Δ)$ defined by a Newton polyhedron $Δ$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron $Δ^*$ in the dual space defines another family ${\cal F}(Δ^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(Δ)$ and ${\cal F}(Δ^*)$.

Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

TL;DR

The paper develops a comprehensive toric-geometric framework for Calabi-Yau hypersurfaces based on reflexive polyhedra, establishing a precise duality between the families and that mirrors physicists' Mirror Symmetry. It provides explicit combinatorial formulas for Hodge numbers and Euler characteristics, ties these to lattice-point counts, and shows that dual pairs have opposite Euler characteristics with exchanged Hodge numbers in Calabi-Yau 3-folds. A robust Galois correspondence and category of reflexive pairs organize mirror constructions, including mirror candidates in hypersurfaces of degree in and quotients in weighted projective spaces. The work generalizes known constructions (Greene–Plesser, Roan) to a broad toric context, enabling systematic generation and analysis of Calabi-Yau mirrors via reflexive polyhedra and MPCP-desingularizations.

Abstract

We consider families consisting of complex -dimensional projective algebraic compactifications of -regular affine hypersurfaces defined by Laurent polynomials with a fixed -dimensional Newton polyhedron in -dimensional algebraic torus . If the family defined by a Newton polyhedron consists of -dimensional Calabi-Yau varieties, then the dual, or polar, polyhedron in the dual space defines another family of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau -folds. Our method allows to construct many new examples of Calabi-Yau -folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families and .

Paper Structure

This paper contains 20 sections, 76 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. The geometry of toric varieties
  3. Two definitions and basic notations
  4. Singularities and morphisms of toric varieties
  5. Hypersurfaces in toric varieties
  6. Regularity conditions for hypersurfaces
  7. Birational and finite morphisms of hypersurfaces
  8. The Hodge structure of hypersurfaces
  9. Calabi-Yau hypersurfaces in toric varieties
  10. Reflexive polyhedra and reflexive pairs
  11. Singularities and morphisms of Calabi-Yau hypersurfaces
  12. The Hodge number $h^{n-2,1}(\hat{Z}_f)$
  13. The Hodge number $h^{1,1}(\hat{Z}_f)$
  14. Calabi-Yau $3$-folds
  15. Mirror symmetry
  16. ...and 5 more sections

Key Result

Proposition 2.1.1

For every $l$-dimensional face $\Theta \subset \Delta$, define the convex $n$-dimensional cone ${\check {\sigma}}(\Theta) \subset M_{\bf Q}$ consisting of all vectors $\lambda (p - p')$, where $\lambda \in{\bf Q}_{\geq 0}$, $p \in \Delta$, $p' \in \Theta$. Let $\sigma( \Theta) \subset N_{\bf Q}$ be

Theorems & Definitions (118)

  • Proposition 2.1.1
  • Proposition 2.1.2
  • Definition 2.1.3
  • Remark 2.1.4
  • Proposition 2.1.5
  • Definition 2.1.6
  • Definition 2.2.1
  • Proposition 2.2.2
  • Remark 2.2.3
  • Proposition 2.2.4
  • ...and 108 more