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## Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties (1994)

Venue: | J. Alg. Geom |

Citations: | 464 - 20 self |

### Citations

1535 |
Superstring Theory
- Green, Schwarz, et al.
- 1987
(Show Context)
Citation Context ...ng to Calabi-Yau varieties from two families F(∆) and F(∆ ∗ ). 1 Introduction Calabi-Yau 3-folds caught much attention from theoretical physics because of their connection with the superstring theory =-=[21]-=-. Physicists discovered a duality for Calabi-Yau 3-folds which is called Mirror Symmetry [1, 2, 3, 12, 17, 22, 35, 37]. This duality defines a correspondence between two topologically different Calabi... |

587 | Théorie de Hodge - Deligne - 1974 |

538 |
A pair of Calabi–Yau manifolds as an exactly soluble superconformal field theory
- Candelas, Ossa, et al.
(Show Context)
Citation Context ...ve spaces P(ω0, . . .,ω4). The property (1) for the construction of B. Greene and R. Plesser [22] was proved by S.-S. Roan [44]. In the paper of P. Candelas, X.C. de la Ossa, P.S. Green and L. Parkes =-=[13]-=- the Mirror Symmetry was applied to give predictions for the number of rational curves of various degrees on general quintic 3-folds in P4. For degrees ≤ 3 these predictions were confirmed by algebrai... |

293 |
Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties
- Oda
- 1988
(Show Context)
Citation Context ...Esnault and E. Viehweg, for providing ideal conditions for my work. 2 The geometry of toric varieties We follow notations of V. Danilov in [14]. Almost all statements of this section are contained in =-=[41]-=- and [42]. 2.1 Two definitions and basic notations First we fix notations used in the contravariant definition of toric varieties P∆ associated to a lattice polyhedron ∆. M abelian group of rank n; M ... |

132 |
Polyhedres de Newton et nombre de
- Kouchnirenko
- 1976
(Show Context)
Citation Context ...where H2k c (W) is assumed to be a 1dimensional C-space of the Hodge type (k, k). If we take U = T, then V = Zf. The Euler characteristic of Zf was calculated by Bernstein, Khovanskiî and Kushnirenko =-=[33]-=-, [29]. Theorem 3.3.4 e(Zf) = ∑ i≥0(−1) i dim H i (Zf) = (−1) n−1 dM(∆). In particular, we obtain: Corollary 3.3.5 The dimension of H n−1 (Zf) is equal to dM(∆) + n − 1. Definition 3.3.6 Let P be a co... |

112 |
Newton polyhedra and an algorithm for computing Hodge-Deligne numbers
- Danilov, Khovanskii
- 1987
(Show Context)
Citation Context ...th compact supports which we denote by H i c(∗). First we note that there exists the following analog of the Lefschetz theorem for ∆- regular hypersurfaces proved by Bernstein, Danilov and Khovanskiî =-=[15]-=-. Theorem 3.3.1 For any open toric subvariety U ⊂ P∆, the Gysin homomorphism H i c (Zf ∩ U) → H i+2 c (U) is bijective for i > n − 1 and injective for i = n − 1. Using this theorem, one can often redu... |

112 |
M.R.Plesser, Duality in Calabi-Yau moduli space, Nucl.Phys. B338
- Greene
- 1990
(Show Context)
Citation Context ...lds caught much attention from theoretical physics because of their connection with the superstring theory [21]. Physicists discovered a duality for Calabi-Yau 3-folds which is called Mirror Symmetry =-=[1, 2, 3, 12, 17, 22, 35, 37]-=-. This duality defines a correspondence between two topologically different Calabi-Yau 3-folds V and V ′ such that the Hodge numbers of V and V ′ satisfy the relations h 1,1 (V ) = h 2,1 (V ′ ), h 1,1... |

107 | Mirror symmetry and rational curves on quintic threefolds: A guide for the mathematicians
- Morrison
- 1993
(Show Context)
Citation Context ...applied to give predictions for the number of rational curves of various degrees on general quintic 3-folds in P4. For degrees ≤ 3 these predictions were confirmed by algebraic geometers [18, 26]. In =-=[39]-=- Morrison has presented a mathematical review of the calculation of P. Candelas et al. [13]. Applying analogous method based on a consideration of the Picard-Fuchs equation, he has found in [40] simil... |

100 | Picard–Fuchs equations and mirror maps for hypersurfaces
- Morrison
- 1992
(Show Context)
Citation Context ...]. In [39] Morrison has presented a mathematical review of the calculation of P. Candelas et al. [13]. Applying analogous method based on a consideration of the Picard-Fuchs equation, he has found in =-=[40]-=- similar predictions for the number of rational curves on general members of another families of Calabi-Yau 3-folds with h 1,1 = 1 constructed as hypersurfaces in weighted projective spaces. Some veri... |

99 |
V.: “Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori
- Batyrev
- 1993
(Show Context)
Citation Context ...lar the constructions of mirrors and the computations of predictions for numbers rational curves, can be extended to the case of Calabi-Yau hypersurfaces and complete intersections in toric varieties =-=[6, 7, 8, 9]-=-. We remark that the toric technique for resolving singularities and computing Hodge numbers of hypersurfaces and complete intersections was first developed by A.G. Khovansky [29, 30]. For the case of... |

98 | Straten, “Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties
- Batyrev, van
- 1995
(Show Context)
Citation Context ...lar the constructions of mirrors and the computations of predictions for numbers rational curves, can be extended to the case of Calabi-Yau hypersurfaces and complete intersections in toric varieties =-=[6, 7, 8, 9]-=-. We remark that the toric technique for resolving singularities and computing Hodge numbers of hypersurfaces and complete intersections was first developed by A.G. Khovansky [29, 30]. For the case of... |

54 |
On the finiteness of rational curves on quintic threefolds, Compositio Mathematica 60
- Katz
- 1986
(Show Context)
Citation Context ...Symmetry was applied to give predictions for the number of rational curves of various degrees on general quintic 3-folds in P4. For degrees ≤ 3 these predictions were confirmed by algebraic geometers =-=[18, 26]-=-. In [39] Morrison has presented a mathematical review of the calculation of P. Candelas et al. [13]. Applying analogous method based on a consideration of the Picard-Fuchs equation, he has found in [... |

39 |
A Generalized Construction of Mirror Manifolds
- Berglund, Hübsch
- 1993
(Show Context)
Citation Context ...ng h 1,1 and h 2,1 [12, 34, 45]. This fact gives an empirical evidence in favor of the conjectural duality. On the other hand, physicists have proposed some explicit constructions of mirror manifolds =-=[10, 22]-=- for several classes of Calabi-Yau 3-folds obtained from 3-dimensional hypersurfaces in 4-dimensional weighted projective spaces P(ω0, . . .,ω4). The property (1) for the construction of B. Greene and... |

39 |
on Calabi-Yau complete intersections, mirror symmetry and Picard-Fuchs equations
- Libgober, Teitelbaum, et al.
- 1993
(Show Context)
Citation Context ... these these predictions were obtained by S. Katz in [28]. The method of P. Candelas et al. was also applied to Calabi-Yau complete intersections in projective spaces by A. Libgober and J. Teitelbaum =-=[36]-=- whose calculation gave correct predictions for the number of lines and conics. Analogous results were obtained by physicists A. Font [19], A. Klemm and S.Theisen [31, 32]. In this paper we consider f... |

38 |
Considerations of one modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kahler potentials and mirror maps,” Nucl.Phys. B389
- Klemm, Theisen
- 1993
(Show Context)
Citation Context ...by A. Libgober and J. Teitelbaum [36] whose calculation gave correct predictions for the number of lines and conics. Analogous results were obtained by physicists A. Font [19], A. Klemm and S.Theisen =-=[31, 32]-=-. In this paper we consider families F(∆) of Calabi-Yau hypersurfaces which are compactifications in n-dimensional projective toric varieties P∆ of smooth affine hypersurfaces whose equations have a f... |

33 |
Lattice vertex polytopes with interior lattice points
- Hensley
- 1983
(Show Context)
Citation Context ...exist up to an unimodular transformation of the lattice M only finitely many reflexive pairs (∆, M) of fixed dimension n. This statement follows from the finiteness theorem in [5], or from results in =-=[11, 25]-=-. 4.2 Singularities and morphisms of Calabi-Yau hypersurfaces Let ∆ be a reflexive polyhedron, ∆∗ the dual reflexive polyhedron. Take a maximal projective triangulation T of ∆. It follows from the pro... |

20 |
The number of twisted cubic curves on the generic quintic threefold
- Ellingsrud, Strømme
- 1995
(Show Context)
Citation Context ...Symmetry was applied to give predictions for the number of rational curves of various degrees on general quintic 3-folds in P4. For degrees ≤ 3 these predictions were confirmed by algebraic geometers =-=[18, 26]-=-. In [39] Morrison has presented a mathematical review of the calculation of P. Candelas et al. [13]. Applying analogous method based on a consideration of the Picard-Fuchs equation, he has found in [... |

18 |
Periods and duality symmetries in Calabi-Yau compactifications,” Nucl.Phys. B391
- Font
- 1993
(Show Context)
Citation Context ...ections in projective spaces by A. Libgober and J. Teitelbaum [36] whose calculation gave correct predictions for the number of lines and conics. Analogous results were obtained by physicists A. Font =-=[19]-=-, A. Klemm and S.Theisen [31, 32]. In this paper we consider families F(∆) of Calabi-Yau hypersurfaces which are compactifications in n-dimensional projective toric varieties P∆ of smooth affine hyper... |

18 | Algebraic Geometry - Hartshorn - 1977 |

15 |
Equations of hypergeometric type and toric varieties
- Gel’fand, Zelevinski, et al.
- 1989
(Show Context)
Citation Context ... a triangulation T of ∆ is projective if the cone C(T ) has a nonempty interior. In other words, T is projective if and only if there exists a strictly upper convex function α(T ). Proposition 2.2.19 =-=[20]-=- Let ∆ be an integral polyhedron. Take an admissible subset A in ∆ ∩ Z n . Then ∆ admits at least one projective A-triangulation, in particular, there exists at least one maximal projective triangulat... |

14 |
Quantum algebraic geometry of superstring compactifications
- Aspinwall, Lütken
- 1991
(Show Context)
Citation Context ...lds caught much attention from theoretical physics because of their connection with the superstring theory [21]. Physicists discovered a duality for Calabi-Yau 3-folds which is called Mirror Symmetry =-=[1, 2, 3, 12, 17, 22, 35, 37]-=-. This duality defines a correspondence between two topologically different Calabi-Yau 3-folds V and V ′ such that the Hodge numbers of V and V ′ satisfy the relations h 1,1 (V ) = h 2,1 (V ′ ), h 1,1... |

14 | Rational curves on Calabi–Yau manifolds: verifying predictions of mirror symmetry,” in Projective Geometry with Applications
- Katz
- 1994
(Show Context)
Citation Context ... members of another families of Calabi-Yau 3-folds with h 1,1 = 1 constructed as hypersurfaces in weighted projective spaces. Some verifications of these these predictions were obtained by S. Katz in =-=[28]-=-. The method of P. Candelas et al. was also applied to Calabi-Yau complete intersections in projective spaces by A. Libgober and J. Teitelbaum [36] whose calculation gave correct predictions for the n... |

13 |
On Ricci flat 3-fold
- Roan, Yau
- 1987
(Show Context)
Citation Context ...[29, 30]. For the case of 3-dimensional varieties with trivial canonical class, toric methods for resolving quotient singularities were first applied by D. Markushevich [38], S.-S. Roan and S.-T. Yau =-=[43]-=-. Let us give an outline of the paper. Section 2 is devoted to basic terminology and well-known results on toric varieties. In this section we fix our notation for the rest of the paper. We use two de... |

11 |
in Superstrings, unified theories, and cosmology
- Dixon
- 1987
(Show Context)
Citation Context ...lds caught much attention from theoretical physics because of their connection with the superstring theory [21]. Physicists discovered a duality for Calabi-Yau 3-folds which is called Mirror Symmetry =-=[1, 2, 3, 12, 17, 22, 35, 37]-=-. This duality defines a correspondence between two topologically different Calabi-Yau 3-folds V and V ′ such that the Hodge numbers of V and V ′ satisfy the relations h 1,1 (V ) = h 2,1 (V ′ ), h 1,1... |

10 | Constructing mirror manifolds - Greene - 1997 |

5 |
Landau-Ginzburg theories as orbifolds, Phys
- Lynker, Schimmrigk
- 1990
(Show Context)
Citation Context |

4 |
On the Hodge Structure
- Batyrev, Cox
(Show Context)
Citation Context ...lar the constructions of mirrors and the computations of predictions for numbers rational curves, can be extended to the case of Calabi-Yau hypersurfaces and complete intersections in toric varieties =-=[6, 7, 8, 9]-=-. We remark that the toric technique for resolving singularities and computing Hodge numbers of hypersurfaces and complete intersections was first developed by A.G. Khovansky [29, 30]. For the case of... |

4 |
Resolution of singularities (toric method), appendix to
- Markushevich
- 1987
(Show Context)
Citation Context ...st developed by A.G. Khovansky [29, 30]. For the case of 3-dimensional varieties with trivial canonical class, toric methods for resolving quotient singularities were first applied by D. Markushevich =-=[38]-=-, S.-S. Roan and S.-T. Yau [43]. Let us give an outline of the paper. Section 2 is devoted to basic terminology and well-known results on toric varieties. In this section we fix our notation for the r... |

4 |
The construction of mirror symmetry
- Schimmrigk
(Show Context)
Citation Context ... obtained from hypersurfaces in weighted projective spaces have shown a a striking symmetry for possible pairs of integers (h 1,1 , h 2,1 ) relative to the transposition interchanging h 1,1 and h 2,1 =-=[12, 34, 45]-=-. This fact gives an empirical evidence in favor of the conjectural duality. On the other hand, physicists have proposed some explicit constructions of mirror manifolds [10, 22] for several classes of... |

3 |
The Mirror of Calabi-Yau Orbifold, Int
- Roan
- 1991
(Show Context)
Citation Context ... obtained from 3-dimensional hypersurfaces in 4-dimensional weighted projective spaces P(ω0, . . .,ω4). The property (1) for the construction of B. Greene and R. Plesser [22] was proved by S.-S. Roan =-=[44]-=-. In the paper of P. Candelas, X.C. de la Ossa, P.S. Green and L. Parkes [13] the Mirror Symmetry was applied to give predictions for the number of rational curves of various degrees on general quinti... |

2 |
Construction and couplings of mirror manifolds
- Aspinwall, Lütken, et al.
- 1990
(Show Context)
Citation Context |

2 | Geometry of mirror manifolds - Aspinwall, Lütken - 1991 |

2 |
Boundness of the degree of higher-dimensional toric Fano varieties
- Batyrev
- 1982
(Show Context)
Citation Context ...a. Theorem 4.1.12 There exist up to an unimodular transformation of the lattice M only finitely many reflexive pairs (∆, M) of fixed dimension n. This statement follows from the finiteness theorem in =-=[5]-=-, or from results in [11, 25]. 4.2 Singularities and morphisms of Calabi-Yau hypersurfaces Let ∆ be a reflexive polyhedron, ∆∗ the dual reflexive polyhedron. Take a maximal projective triangulation T ... |

2 |
The geometry of toric varieities
- Danilov
- 1978
(Show Context)
Citation Context ... for the support, and the University of Essen, especially H. Esnault and E. Viehweg, for providing ideal conditions for my work. 2 The geometry of toric varieties We follow notations of V. Danilov in =-=[14]-=-. Almost all statements of this section are contained in [41] and [42]. 2.1 Two definitions and basic notations First we fix notations used in the contravariant definition of toric varieties P∆ associ... |

2 |
polyhedra and the genus of full intersections
- Khovanskiî, Newton
- 1978
(Show Context)
Citation Context ...ic varieties [6, 7, 8, 9]. We remark that the toric technique for resolving singularities and computing Hodge numbers of hypersurfaces and complete intersections was first developed by A.G. Khovansky =-=[29, 30]-=-. For the case of 3-dimensional varieties with trivial canonical class, toric methods for resolving quotient singularities were first applied by D. Markushevich [38], S.-S. Roan and S.-T. Yau [43]. Le... |

2 |
Abelian Landau-Ginzburg Orbifolds and Mirror
- Kreuzer, Schimmrigk, et al.
- 1992
(Show Context)
Citation Context ... obtained from hypersurfaces in weighted projective spaces have shown a a striking symmetry for possible pairs of integers (h 1,1 , h 2,1 ) relative to the transposition interchanging h 1,1 and h 2,1 =-=[12, 34, 45]-=-. This fact gives an empirical evidence in favor of the conjectural duality. On the other hand, physicists have proposed some explicit constructions of mirror manifolds [10, 22] for several classes of... |

1 |
Multiple Mirror Manifolds and Topology
- Aspinwall
(Show Context)
Citation Context ...ouis, R. Schimmrigk, S. Theisen. I am very grateful to D. Morrison, B. Greene and P. Aspinwall who found a serious error in my earlier formulas for the Hodge number h 1,1 while working on their paper =-=[4]-=-. Correcting these errors, I found an easy proof of the formula for h n−2,1 . I would like to thank the DFG for the support, and the University of Essen, especially H. Esnault and E. Viehweg, for prov... |

1 |
polyhedra (resolution of singularities), Sovremen. Probl
- Khovanskiî, Newton
- 1983
(Show Context)
Citation Context ...ic varieties [6, 7, 8, 9]. We remark that the toric technique for resolving singularities and computing Hodge numbers of hypersurfaces and complete intersections was first developed by A.G. Khovansky =-=[29, 30]-=-. For the case of 3-dimensional varieties with trivial canonical class, toric methods for resolving quotient singularities were first applied by D. Markushevich [38], S.-S. Roan and S.-T. Yau [43]. Le... |

1 |
Mirror Maps and Instanton Sums for Complete
- Theisen
- 1993
(Show Context)
Citation Context ...by A. Libgober and J. Teitelbaum [36] whose calculation gave correct predictions for the number of lines and conics. Analogous results were obtained by physicists A. Font [19], A. Klemm and S.Theisen =-=[31, 32]-=-. In this paper we consider families F(∆) of Calabi-Yau hypersurfaces which are compactifications in n-dimensional projective toric varieties P∆ of smooth affine hypersurfaces whose equations have a f... |