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

In fresh years, examine in K3 surfaces and Calabi–Yau kinds has visible wonderful development from either mathematics and geometric issues of view, which in flip keeps to have an important impact 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 complaints quantity encompasses a consultant sampling of the extensive variety of themes lined through the workshop. whereas the topics variety from mathematics geometry via algebraic geometry and differential geometry to mathematical physics, the papers are evidently similar via the typical subject of Calabi–Yau types. With the wide variety of branches of arithmetic and mathematical physics touched upon, this sector finds many deep connections among matters 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 incorporated 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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**Extra resources for Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds**

**Example text**

Thus C(X) is called the ample cone. The following is the Torelli type theorem for algebraic K3 surfaces due to Piatetskii-Shapiro and Shafarevich. 16 S. 2 Theorem ([35]) Let X, X be algebraic K3 surfaces. Let ϕ : H 2 (X, Z) → H 2 (X , Z) be an isometry satisfying (1) ϕ(ωX ) ∈ C · ωX , (2) ϕ(C(X)) ⊂ C(X ). Then there exists an isomorphism g : X → X with g∗ = ϕ. Since any automorphism preserves ample classes, we have a natural map ψ : Aut(X) → Aut(C(X)) = {ϕ ∈ O(S X ) : g(C(X)) = C(X)} where Aut(X) is the group of automorphisms of X.

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.

Lewis where E X are the C ∞ (p, q)-forms which in local holomorphic coordinates z = (z1 , . . , zn ) ∈ X, are of the form: p,q fIJ dzI ∧ dz J , fIJ are C − valued C ∞ f unctions, |I|=p,|J|=q I = 1 ≤ i1 < · · · < i p ≤ d, dzI = dzi1 ∧ · · · ∧ dzi p , J = 1 ≤ j1 < · · · < jq ≤ d, dz J = dz j1 ∧ · · · ∧ dz jq . One has the diﬀerential d : E Xk → E Xk+1 , and we define k (X, C) = HDR ker d : E Xk → E Xk+1 dE Xk−1 . p,q p+1,q The operator d decomposes into d = ∂ + ∂, where ∂ : E X → E X p,q+1 EX 2 p,q and ∂ : E X → .