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

In fresh years, study in K3 surfaces and Calabi–Yau kinds has obvious fabulous development from either mathematics and geometric issues of view, which in flip keeps to have an incredible effect 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 offer a state of the art survey of those new advancements. This lawsuits quantity contains a consultant sampling of the wide diversity of themes lined by way of the workshop. whereas the themes variety from mathematics geometry via algebraic geometry and differential geometry to mathematical physics, the papers are obviously similar via the typical subject matter of Calabi–Yau types. With the wide variety of branches of arithmetic and mathematical physics touched upon, this zone 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 advisor to the topic.

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**Example text**

Hence x ∈ S ⊕ T . This implies the injectivity of pS |H. Similarly pT |H is injective. Since |H|2 = |AS | · |AT |, pS |H and pT |H are isomorphic. The remaining assertions are obvious. Conversely we have the following corollary. 12 Corollary Let S , T be even lattices. Let γ : AS → AT be an isomorphism satisfying qT (γ(x)) ≡ −qS (x) mod 2 for any x ∈ AS . Then there exist an even unimodular lattice L and K3 and Enriques Surfaces 9 a primitive embedding of S into L such that T is isomorphic to the orthogonal complement of S in L.

Hence h preserves both the Hodge and complexified weight filtrations. Now use the fact that A ⊗ Q is a field to deduce that h preserves the weight filtration over A ⊗ Q. 1 Transcendental Methods in the Study of Algebraic Cycles. . 13. Let U be a compact Riemann surface, Σ ⊂ U a finite set of points, and put U := U\Σ. According to Deligne, H 1 (U, Z(1)) carries a Z-MHS. The Hodge filtration on H 1 (U, C) is defined in terms of a filtered complex of holomorphic diﬀerentials on U with logarithmic poles along Σ ([16], but also see (10) below).

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. Let R be the orthogonal complement of x in E8 . Then R⊥ = U ⊕ U ⊕ E8 ⊕ E8 ⊕ (−2d) L2d where (−2d) is the lattice generated by x. Thus we have a primitive embedding of L2d into L which induces an embedding of the period domain D2d of polarized K3 surfaces of degree 2d into D(L).