By Shigeyuki Kondō (auth.), Radu Laza, Matthias Schütt, Noriko Yui (eds.)
In fresh years, study in K3 surfaces and Calabi–Yau types has visible incredible growth from either mathematics and geometric issues of view, which in flip maintains to have an immense impact and effect 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 complaints quantity incorporates a consultant sampling of the extensive variety of themes coated through the workshop. whereas the themes diversity from mathematics geometry via algebraic geometry and differential geometry to mathematical physics, the papers are clearly comparable through the typical subject of Calabi–Yau types. 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 begun 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 info for Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds
In the following we apply this to the case of a generic Jacobian Kummer surface. Consider the root lattice R = A3 ⊕ A61 . We can embed R into L such that R is generated by Leech roots and S = R⊥ is isomorphic to the Picard lattice S X of the Kummer surface X associated to a generic smooth curve of genus 2. The faces of the finite polyhedron D(S X ) consist of 316 (= 32 + 32 + 60 + 192) hyperplanes perpendicular to (−2)-, (−4)-, (−4)- or (−12)-vectors in S X respectively. These 32 (−2)-vectors correspond to 32 smooth rational curves on X forming the Kummer (16)6 -configuration.
These 32 (−2)-vectors correspond to 32 smooth rational curves on X forming the Kummer (16)6 -configuration. The ample class w defines an embedding of X into P5 whose image is the intersection of three quadrics. The group of symmetries of the finite polyhedron D(S X ) is isomorphic to (Z/2Z)5 · S6 where (Z/2Z)5 acts on X as automorphisms (16 translations and 16 switches) and S6 is the symmetry of the Weierstrass points on the curve of genus 2. Finally the remaining 32, 60, 192 hyperplanes of D(S X ) correspond to classical automorphisms of X, that is, 16 projections and 16 correlations, 60 Cremona transformations and 192 Cremona transformations respectively.
Both actions have the same character for m ≤ 8. In fact, Mukai proved a stronger assertion. 11 Theorem () Let G be a finite group. Then the followings are equivalent. (1) G acts on a K3 surface as symplectic automorphisms. (2) G is a subgroup of M23 which has at least five orbits on Ω. K3 and Enriques Surfaces 21 The condition that G has at least five orbits is necessary because G fixes H 0 (X, Q), H 4 (X, Q), Re(ωX ), Im(ωX ) and a K¨ahler class. Mukai  determined maximal groups of symplectic automorphisms (11 types) and gave their explicit examples by equations.