Higher dimensional Calabi-Yau manifolds of Kummer type. (English) Zbl 1525.14050
Higher dimensional Calabi-Yau manifolds of Kummer type were first introduced and constructed in [S. Cynk and K. Hulek, Can. Math. Bull. 50, No. 4, 486–503 (2007; Zbl 1141.14009)]: these manifolds are given as crepant resolution of the quotient of product of lower dimensional Calabi-Yau manifolds by the action of \((\mathbb{Z}/d\mathbb{Z})^k\) for certain \(d, k\in \mathbb{N}\); in fact they were born as generalization of the so-called Calabi-Yau \(3\)-folds of Borcea-Voisin type. More in details, the examples of Cynk and Hulek are constructed as follows: they considered the product of \(n\)-elliptic curves \(E_d\) with an extra automorphism of order \(d=2,3,4\) and they defined an action of \((\mathbb{Z}/d\mathbb{Z})^{n-1}\) on it such that the quotient admits a desingularization which is a Calabi-Yau \(n\)-folds.
The goal of the article under review is to fill the gap providing a construction for Calabi-Yau manifolds of Kummer type using elliptic curve \(E_6\). The main point is to construct the crepant resolution for such singular quotient of \((E_6)^{\times n}\). Indeed, as the author explains, the method of Cynk and Hulek does not work and as well a factorization of an action of order \(6\) into an action of order \(2\) and \(3\). The construction of such resolution in provided in Section \(3\) ( see Proposition \(3.1\)) and the core idea is to used a suitable toric resolution. After that, the author gives the formulas for Hodge numbers and the Euler characteristic of Calabi-Yau manifolds of Kummer type constructed using \(E_d\) with \(d=2,3,4,6\).
The author underlines that the interest in such Calabi-Yau manifolds is also related to the fact that they provide the first higher dimensional examples satisfying a conjecture of Voisin that concerns about a condition that zero-cycles on certain varieties should satisfy, see [C. Voisin, Lect. Notes Pure Appl. Math. 179, 265–285 (1996; Zbl 0912.14003)].
Finally, Kummer surfaces of supersingular elliptic curves were used in [T. Katsura and M. Schütt, “Zariski \(K3\) surfaces”, Preprint, arXiv:1710.08661] to constructed the first example of Zariski \(K3\) surface in positive characteristic. Using the examples of Calabi-Yau manifolds mentioned above and the method of Katsura and Schütt, the author of the present paper provides arbitrarily dimensional Calabi-Yau manifolds which are Zariski varieties in any characteristic \(p\not= 1 \pmod{12}\), which also provided the first examples of higher dimensional unirational Calabi-Yau manifolds.
The goal of the article under review is to fill the gap providing a construction for Calabi-Yau manifolds of Kummer type using elliptic curve \(E_6\). The main point is to construct the crepant resolution for such singular quotient of \((E_6)^{\times n}\). Indeed, as the author explains, the method of Cynk and Hulek does not work and as well a factorization of an action of order \(6\) into an action of order \(2\) and \(3\). The construction of such resolution in provided in Section \(3\) ( see Proposition \(3.1\)) and the core idea is to used a suitable toric resolution. After that, the author gives the formulas for Hodge numbers and the Euler characteristic of Calabi-Yau manifolds of Kummer type constructed using \(E_d\) with \(d=2,3,4,6\).
The author underlines that the interest in such Calabi-Yau manifolds is also related to the fact that they provide the first higher dimensional examples satisfying a conjecture of Voisin that concerns about a condition that zero-cycles on certain varieties should satisfy, see [C. Voisin, Lect. Notes Pure Appl. Math. 179, 265–285 (1996; Zbl 0912.14003)].
Finally, Kummer surfaces of supersingular elliptic curves were used in [T. Katsura and M. Schütt, “Zariski \(K3\) surfaces”, Preprint, arXiv:1710.08661] to constructed the first example of Zariski \(K3\) surface in positive characteristic. Using the examples of Calabi-Yau manifolds mentioned above and the method of Katsura and Schütt, the author of the present paper provides arbitrarily dimensional Calabi-Yau manifolds which are Zariski varieties in any characteristic \(p\not= 1 \pmod{12}\), which also provided the first examples of higher dimensional unirational Calabi-Yau manifolds.
Reviewer: Martina Monti (Milano)
MSC:
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14J50 | Automorphisms of surfaces and higher-dimensional varieties |
| 14M20 | Rational and unirational varieties |
Keywords:
Calabi-Yau manifold; generalized Kummer construction; unirational manifolds; Zariski varietiesReferences:
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