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

ISBN-10: 1461464021

ISBN-13: 9781461464020

ISBN-10: 146146403X

ISBN-13: 9781461464037

In contemporary years, study in K3 surfaces and Calabi–Yau kinds has noticeable magnificent growth from either mathematics and geometric issues of view, which in flip keeps to have an enormous effect and influence 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 offer a cutting-edge survey of those new advancements. This court cases quantity contains a consultant sampling of the huge diversity of issues coated through the workshop. whereas the themes variety from mathematics geometry via algebraic geometry and differential geometry to mathematical physics, the papers are certainly similar via the typical subject matter of Calabi–Yau types. With the wide variety of branches of arithmetic and mathematical physics touched upon, this sector finds 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 consultant to the topic.

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**Example text**

3) By the assertion (2), α(g) = ζm 1 is a primitive m-th root of unity. We show that g∗ has no non-zero fixed vectors in T X ⊗ Q. Let x ∈ T X ⊗ Q with g∗ (x) = x. Then ωX , x = g∗ (ωX ), g∗ (x) = ζm ωX , x . This implies that ωX , x = 0 and hence x ∈ (S X ∩ T X ) ⊗ Q = {0}. The above argument shows that if (g∗ )n 1, then (g∗ )n has no non-zero fixed vectors. This implies that g∗ |T X ⊗ Q is an irreducible representation of a cyclic group Z/mZ of degree ϕ(m) defined over Q. Hence we have the last assertion.

Freitag, R. Salvati-Manni, Modular forms for the even unimodular lattice of signature (2, 10). J. Algebr. Geom. 16, 753–791 (2007) 16. E. Freitag, R. Salvati-Manni, The modular variety of hyperelliptic curves of genus three. Trans. Am. Math. Soc. 363, 281–312 (2011) 17. V. Gritsenko, K. Hulek, G. Sankaran, The Kodaira dimension of the moduli of K3 surfaces. Invent. Math. 169, 519–567 (2007) 18. E. Horikawa, On the periods of Enriques surfaces I, II. Math. Ann. 234, 78–108 (1978); Math. Ann. 235, 217–246 (1978) 19.

2 Example ([5]) Let L be an even unimodular lattice of signature (2, 26). Then AL = {0}. We take a modular form f = 1/Δ(τ) = q−1 + 24 + · · · of weight −12, where 24 S. Kond¯o Δ(τ) = q (1 − qn )24 , q=e √ 2π −1τ . n>0 Then we have a holomorphic automorphic form Ψ12 on D(L) of weight 12 = 24/2 with zeros along λ⊥ . λ∈L, λ2 =−2 In the following we discuss some applications of Ψ12 to the moduli spaces of polarized K3 surfaces. 5). Consider the last component E8 in the above decomposition of L, and let x ∈ E8 be a primitive vector with x2 = −2d.

### Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds by Shigeyuki Kondō (auth.), Radu Laza, Matthias Schütt, Noriko Yui (eds.)

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