Using the theory of hyperkähler manifolds, we generalize the notion of Enriques surfaces to higher dimensions, explore their properties, and construct several examples using group actions on Hilbert schemes of points or moduli spaces of stable sheaves. Brought to you by | Tokyo Daigaku Authenticated Download Date | 5/27/15 6:14 AM In some sense, there are not too many known examples of hyperkähler manifolds. Beauville [3] constructed two infinite series, namely the Hilbert scheme of points for K3 surfaces, and his generalized Kummer variety Km n ðAÞ, defined as a Bogomolov factor in Hilb nþ1 ðAÞ for abelian surfaces A. Furthermore, there are two sporadic examples of O'Grady (see [27], [28]). The first idea to construct Enriques manifolds is to look at Hilbert schemes for Enriques surfaces or bielliptic surfaces, but this does not work out. Rather, it leads to an interesting new construction of Calabi-Yau manifolds:Theorem. Let S be an Enriques surface or a bielliptic surface, and n f 2. Then Hilb n ðSÞ has a finite étale covering that is a Calabi-Yau manifold or is the product of a Calabi-Yau manifold with an elliptic curve, respectively.However, if one starts with an Enriques surface S 0 , say with universal covering S, and an odd number n f 1, then the induced action of G ¼ p 1 ðS 0 Þ on X ¼ Hilb n ðSÞ is free, and the corresponding quotient is an Enriques manifold Y of dimension dimðY Þ ¼ 2n and index d ¼ 2. There is a variant with generalized Kummer varieties, and the preceding construction can be extended from Hilbert schemes of points to moduli spaces of sheaves:Theorem. Suppose S 0 is an Enriques surface whose corresponding K3 surface has Picard number rðSÞ ¼ 10. Let v ¼ ðr; l; w À rÞ A H ev ðS; ZÞ be a primitive Mukai vector with v 2 f 0 and w A Z odd. Then for very general polarizations H A NSðSÞ R , the moduli space X ¼ M H ðvÞ is a hyperkähler manifold endowed with a free action of G ¼ p 1 ðS 0 Þ, and Y ¼ X =G is an Enriques manifold of dimension v 2 þ 2 and index d ¼ 2.Recall that a bielliptic surface S has, by definition, a finite étale covering A that is an abelian surface. To construct examples of Enriques of higher index, we use the classification of bielliptic surfaces due to Bagnera and de Franchis and study the induced action of G ¼ p 1 ðSÞ on Hilb n ðAÞ. This yields:Theorem. There are Enriques manifolds with index d ¼ 2; 3; 4.The paper is organized as follows: In the first section, we recall several results about the Bogomolov decomposition of manifolds with trivial first Chern class and the theory of hyperkähler manifolds. In the second section, we introduce the notion of Enriques manifolds and collect their basic properties. In the third section, we examine Hilbert schemes of points for Enriques surfaces and bielliptic surfaces. The first examples of Enriques manifolds appear in Section 4 as quotients of Hilbert schemes of points for the K3 covering of an Enriques surface. We extend this construction to moduli spaces of stable sheaves in Section 5. In Section 6 we use the classification of b...
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