By Shigeyuki Kondō (auth.), Radu Laza, Matthias Schütt, Noriko Yui (eds.)

In fresh years, learn in K3 surfaces and Calabi–Yau types has visible mind-blowing growth from either mathematics and geometric issues of view, which in flip keeps to have an incredible impression and effect in theoretical physics—in specific, in string idea. The workshop on mathematics and Geometry of K3 surfaces and Calabi–Yau threefolds, held on the Fields Institute (August 16-25, 2011), aimed to provide a cutting-edge survey of those new advancements. This lawsuits quantity incorporates a consultant sampling of the large variety of issues coated through the workshop. whereas the themes diversity from mathematics geometry via algebraic geometry and differential geometry to mathematical physics, the papers are obviously similar by way of the typical subject of Calabi–Yau forms. With the wide variety of branches of arithmetic and mathematical physics touched upon, this region unearths many deep connections among topics formerly thought of unrelated.

Unlike such a lot different meetings, the 2011 Calabi–Yau workshop began with three days of introductory lectures. a range of four of those lectures is integrated during this quantity. those lectures can be utilized as a kick off point for the graduate scholars and different junior researchers, or as a advisor to the topic.

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**Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds**

In recent times, study in K3 surfaces and Calabi–Yau types has noticeable stunning development from either mathematics and geometric issues of view, which in flip keeps to have a tremendous impact and impression in theoretical physics—in specific, in string thought. The workshop on mathematics and Geometry of K3 surfaces and Calabi–Yau threefolds, held on the Fields Institute (August 16-25, 2011), aimed to provide a state of the art survey of those new advancements.

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**Additional resources for Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds**

**Example text**

2 that ϕ˜ is represented by an automorphism g of X. Obviously g commutes with σ and hence it induces an automorphism of Y. Enriques surfaces with a finite group of automorphisms are very rare. Such Enriques surfaces were classified by Nikulin [32] and Kond¯o [20]. There are seven classes of such Enriques surfaces. Two of them consist of one-dimensional irreducible families and the others are unique. Moreover Nikulin [32] introduced the notion of the root invariant of an Enriques surface, which describes the group of automorphisms of the Enriques surface up to finite groups.

Table 3: The number of fixed points of finite symplectic automorphisms of K3 surfaces m2345678 fm 8 6 4 4 2 3 2 Recall that the Mathieu group M24 acts on the set Ω = {1, . . , 24} of 24 letters. Let M23 be the stabilizer subgroup of the letter 1. Then M23 is also a finite sporadic simple group, called the Mathieu group of degree 23. The conjugacy classes of M23 are determined by their orders and are given in the following Table 4. Table 4: Conjugacy classes of M23 |σ| 2 3 σ (2)8 (3)6 4 (4)4 (2)2 5 6 7 8 (5)4 (6)2 (3)2 (2)2 (7)3 (8)2 (4)(2) |σ| 11 14 15 23 σ (11)2 (14)(7)(2) (15)(5)(3) (23) Denote by (|σ|) the number of fixed points of σ ∈ M23 on Ω.

Nikulin, Integral symmetric bilinear forms and its applications. Math. USSR Izv. 14, 103–167 (1980) 31. V. Nikulin, Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. J. Sov. Math. 22, 1401–1475 (1983) 32. V. Nikulin, On a description of the automorphism groups of Enriques surfaces. Sov. Math. Dokl. 30, 282–285 (1984) 33. V. Nikulin, Surfaces of type K3 with a finite automorphism group and a Picard number three. Proc. Steklov Inst.