##
**Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori.**
*(English)*
Zbl 0812.14035

The polynomial representation of cohomology classes in the middle cohomology group of hypersurfaces allows one to explicitly calculate the Picard-Fuchs equations, or in fact the Gauss-Manin connection for families of hypersurfaces. These differential equations seem to contain sufficient information to describe some superconformal theories. We remark that there exists a general philosophy that \(N = 2\) supergravity and superconformal models can be identified with special variations of abstract mixed Hodge structures. – Projective spaces and weighted projective spaces are special cases of so-called toric varieties. It is natural to ask whether the method of Jacobian rings for the calculation of the Hodge structure can be extended to the case of hypersurfaces in projective toric varieties.

There is one more strong reason why the calculation of the variation of the Hodge structure of hypersurfaces in toric varieties is especially important for physicists. The most interesting class of variations of Hodge structure in physics is connected with families of Calabi-Yau varieties. First, it is possible to construct many examples of families of Calabi-Yau hypersurfaces in toric varieties \(\mathbb{P}_ \Delta\) associated with reflexive polyhedra \(\Delta\). Second, for Calabi-Yau hypersurfaces in toric varieties we can use a precise mathematical definition of mirrors via the notion of dual polyhedron. The main idea can be also explained for families of Calabi-Yau 3-folds as follows:

There are two kinds of Yukawa couplings on the Hodge spaces \(H^{2,1} (W)\) and \(H^{1,1} (W)\) considered by physicists for Calabi-Yau 3-folds \(W\). It is known that the Yukawa coupling on \(H^{2,1} (W)\) is defined by the deformation of the complex structure, but the definition of the Yukawa coupling on \(H^{1,1} (W)\) is outside the classical algebraic geometry. Physicists have discovered a trick, the so-called mirror symmetry, which allows one to replace the calculation of the “mysterious Yukawa coupling” on \(H^{1,1} (W)\) by the calculation of the Yukawa coupling on the space \(H^{2,1} (W')\) corresponding to the deformation of the complex structure on another Calabi-Yau 3-fold \(W'\). The Calabi- Yau variety \(W'\) is called a mirror of \(W\). Since the mirror mapping for Calabi-Yau hypersurfaces in toric varieties admits a precise mathematical definition, we can compute both of the Yukawa couplings using only deformations of the classical Hodge structures. One should remark that in the general case the construction of mirrors for Calabi-Yau manifolds is a rather difficult problem, and Calabi-Yau hypersurfaces in toric varieties are now the only known cases in which this problem is solved by mathematical methods.

Denote by \(\mathbb{T}\) the \(n\)-dimensional algebraic torus \((\mathbb{C}^*)^ n\). The main purpose of this paper is to give the explicit description of the mixed Hodge structure of the affine \(\Delta\)-regular hypersurface \(Z_ f \subset \mathbb{T}\) defined by a Laurent polynomial \(f\) with the Newton polyhedron \(\Delta\). As an application of our calculation of the infinitesimal variations of mixed Hodge structures, we show under some restrictions the following property of the mirror mapping for Calabi-Yau varieties:

The topological cup product on the cohomology ring \(\bigoplus^{n- 1}_{i = 0} H^{i,i}(W)\) of smooth Calabi-Yau hypersurfaces in toric Fano \(n\)-folds can be obtained as a “limit” of the multiplicative structure on the chiral ring \(\bigoplus^{n-1}_{i = 0} H^{n-1-i,i} (W')\) of the mirrors \(W'\), where the “limit” corresponds to some degeneration of \(W'\) at a boundary point of the moduli space of complex structures on \(W'\).

This property of the mirror mapping agrees with predictions of physicists.

There is one more strong reason why the calculation of the variation of the Hodge structure of hypersurfaces in toric varieties is especially important for physicists. The most interesting class of variations of Hodge structure in physics is connected with families of Calabi-Yau varieties. First, it is possible to construct many examples of families of Calabi-Yau hypersurfaces in toric varieties \(\mathbb{P}_ \Delta\) associated with reflexive polyhedra \(\Delta\). Second, for Calabi-Yau hypersurfaces in toric varieties we can use a precise mathematical definition of mirrors via the notion of dual polyhedron. The main idea can be also explained for families of Calabi-Yau 3-folds as follows:

There are two kinds of Yukawa couplings on the Hodge spaces \(H^{2,1} (W)\) and \(H^{1,1} (W)\) considered by physicists for Calabi-Yau 3-folds \(W\). It is known that the Yukawa coupling on \(H^{2,1} (W)\) is defined by the deformation of the complex structure, but the definition of the Yukawa coupling on \(H^{1,1} (W)\) is outside the classical algebraic geometry. Physicists have discovered a trick, the so-called mirror symmetry, which allows one to replace the calculation of the “mysterious Yukawa coupling” on \(H^{1,1} (W)\) by the calculation of the Yukawa coupling on the space \(H^{2,1} (W')\) corresponding to the deformation of the complex structure on another Calabi-Yau 3-fold \(W'\). The Calabi- Yau variety \(W'\) is called a mirror of \(W\). Since the mirror mapping for Calabi-Yau hypersurfaces in toric varieties admits a precise mathematical definition, we can compute both of the Yukawa couplings using only deformations of the classical Hodge structures. One should remark that in the general case the construction of mirrors for Calabi-Yau manifolds is a rather difficult problem, and Calabi-Yau hypersurfaces in toric varieties are now the only known cases in which this problem is solved by mathematical methods.

Denote by \(\mathbb{T}\) the \(n\)-dimensional algebraic torus \((\mathbb{C}^*)^ n\). The main purpose of this paper is to give the explicit description of the mixed Hodge structure of the affine \(\Delta\)-regular hypersurface \(Z_ f \subset \mathbb{T}\) defined by a Laurent polynomial \(f\) with the Newton polyhedron \(\Delta\). As an application of our calculation of the infinitesimal variations of mixed Hodge structures, we show under some restrictions the following property of the mirror mapping for Calabi-Yau varieties:

The topological cup product on the cohomology ring \(\bigoplus^{n- 1}_{i = 0} H^{i,i}(W)\) of smooth Calabi-Yau hypersurfaces in toric Fano \(n\)-folds can be obtained as a “limit” of the multiplicative structure on the chiral ring \(\bigoplus^{n-1}_{i = 0} H^{n-1-i,i} (W')\) of the mirrors \(W'\), where the “limit” corresponds to some degeneration of \(W'\) at a boundary point of the moduli space of complex structures on \(W'\).

This property of the mirror mapping agrees with predictions of physicists.

### MSC:

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

14J30 | \(3\)-folds |

32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |

### Keywords:

toric varieties; Calabi-Yau hypersurfaces; Calabi-Yau 3-folds; mirror symmetry; mixed Hodge structures
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\textit{V. V. Batyrev}, Duke Math. J. 69, No. 2, 349--409 (1993; Zbl 0812.14035)

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### References:

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