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

In contemporary years, examine in K3 surfaces and Calabi–Yau kinds has visible fantastic development from either mathematics and geometric issues of view, which in flip maintains to have a big effect and influence in theoretical physics—in specific, in string concept. The workshop on mathematics and Geometry of K3 surfaces and Calabi–Yau threefolds, held on the Fields Institute (August 16-25, 2011), aimed to offer a cutting-edge survey of those new advancements. This lawsuits quantity incorporates a consultant sampling of the huge diversity of issues coated via the workshop. whereas the themes variety from mathematics geometry via algebraic geometry and differential geometry to mathematical physics, the papers are evidently comparable by means of the typical topic of Calabi–Yau forms. With the wide variety of branches of arithmetic and mathematical physics touched upon, this zone finds many deep connections among matters formerly thought of unrelated.

Unlike so much different meetings, the 2011 Calabi–Yau workshop begun with three days of introductory lectures. a variety 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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**Sample text**

3) Let g be of order m and let fm be the number of fixed points of g. Then fm is given in the following Table 3. 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.

3. H i (X, C) F r H i (X, C) F d−r+1 H 2d−i (X, C)∨. 5. Let A ⊂ R be a subring. An A-Hodge structure (HS) of weight N ∈ Z is given by the following datum: • A finitely generated A-module V, and either of the two equivalent statements below: •1 A decomposition VC = V p,q , V p,q = V q,p , p+q=N where − is complex conjugation induced from conjugation on the second factor C of VC := V ⊗ C. •2 A finite descending filtration VC ⊃ · · · ⊃ F r ⊃ F r−1 ⊃ · · · ⊃ {0}, satisfying VC = F r F N−r+1 , ∀ r ∈ Z.

J. Math. 12, 191–282 (1986) 21. S. Kond¯o, The rationality of the moduli space of Enriques surfaces. Compos. Math. 91, 159–173 (1994) 22. S. Kond¯o, Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K3 surfaces (with an appendix by S. Mukai). Duke Math. J. 92, 593–603 (1998) 23. S. Kond¯o, The automorphism group of a generic Jacobian Kummer surface. J. Algebr. Geom. 7, 589–609 (1998) 24. S. Kond¯o, On the Kodaira dimension of the moduli space of K3 surfaces II.