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Forum Math. 24 (2012), 609–625 DOI 10.1515/FORM.2011.076 Forum Mathematicum © de Gruyter 2012 Weyl and Zariski chambers on K3 surfaces Thomas Bauer and Michael Funke Communicated by Karl Strambach Abstract. The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl cham- bers. In the present paper we compare these two decompositions and we study their mutual relationship: First, we give a numerical criterion for the two decompositions to coincide. Secondly, we study

The canonical zero cycle on a K3 surface S is defined in [ 1 ] as the rational equivalence class of any point lying on a rational curve C ⊂ S . The paper [ 1 ] shows that the intersection of any two divisors in S is proportional to the canonical cycle in CH 0 ( S ). It is also shown that the second Chern class c 2 ( S ) is proportional to this canonical zero cycle o . Both results can be obtained as consequences of the following theorem. Theorem (1)([1, Proposition 4.2]) Let S be a K3 surface . In CH 2 ( S 3 ) Q there is a decomposition Δ 123 = Δ 12

[1] Artin M., Supersingular K3 surfaces, Ann. Sci. École Norm. Sup., Sér. 4, 1974, 7, 543–567 [2] Artin M., Mazur B., Formal groups arising from algebraic varieties, Ann. Sci. École Norm. Sup., Sér. 4, 1977, 10, 87–131 [3] Berthelot P., Ogus A., Notes on crystalline cohomology, Princeton University Press, Princeton, 1978 [4] Bogomolov F.A., Sur l’algébricité des représentations ℓ-adiques, C. R. Acad. Sci. Paris Sér. A-B, 1980, 290, A701–A703 (in French) [5] Bogomolov F.A., Points of finite order on abelian varieties, Izv. Akad. Nauk SSSR Ser. Mat., 1980, 44


The aim of this paper is to derive explicitly a connection between the Zagier elliptic trilogarithm and Mahler measures of certain families of three-variable polynomials defining K3 surfaces. In addition, we prove some linear relations satisfied by the elliptic trilogarithm evaluated at torsion points on elliptic curves.

Introduction Statement of the main results A K3 surface X over a perfect field k of characteristic p is called ordinary if it satisfies the following equivalent conditions: i the Hodge and Newton polygons of H crys 2 ⁢ ( X / W ⁢ ( k ) ) {{\rm H}^{2}_{\rm crys}(X/W(k))} coincide, ii the Frobenius endomorphism of H 2 ⁢ ( X , 𝒪 X ) {{\rm H}^{2}(X,{\mathcal{O}}_{X})} is a bijection, iii the formal Brauer group of X (see [ 1 ]) has height 1. If k is finite, then these are also equivalent with | X ⁢ ( k ) | ≢ 1 mod p {|X(k)|\not\equiv 1\bmod p

Adv. Geom. 8 (2008), 413–440 Advances in Geometry DOI 10.1515 / ADVGEOM.2008.027 c© de Gruyter 2008 Projective models of K3 surfaces with an even set Alice Garbagnati and Alessandra Sarti∗ (Communicated by R. Miranda) Abstract. The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a careful analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient

Potier J., Systèmes Cohérents et Structures de Niveau, Astérisque, 214, Société Mathématique de France, Paris, 1993 [16] Leyenson M., On the Brill-Noether theory for K3 surfaces II, preprint available at [17] Maruyama M., Construction of moduli spaces of stable sheaves via Simpson’s idea, In: Moduli of Vector Bundles, Sanda, Kyoto, 1994, Lecture Notes in Pure and Appl. Math., 179, Dekker, New York, 1996, 147–187 [18] Mukai S., Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math., 1984, 77

On Homotopy K 3 Surfaces KVNIHIKO KODAIRA By a surface we shall mean a compact complex manifold of complex dimension 2. A surface is said to be regular if its first Betti number van- ishes. A K 3 surface is defined to be a regular surface of which the first Chern class vanishes. Every K 3 surface is diffeomorphic to a non- singular quartic surface in a complex projective 3-space (see [1], Theorem 13). Thus there is a unique diffeomorphic type of K 3 surface. By a homotopy K 3 sllrface we mean a surface of the oriented homotopy type of K 3 surface. The


This note deals with Lagrangian fibrations of elliptic K3 surfaces and the associated Hamiltonian monodromy. The fibration is constructed through the Weierstraß normal form of elliptic surfaces. There is given an example of K3 dynamical models with the identity monodromy matrix around 12 elementary singular loci.

J. reine angew. Math. 648 (2010), 13—67 DOI 10.1515/CRELLE.2010.078 Journal für die reine und angewandte Mathematik ( Walter de Gruyter Berlin New York 2010 Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic By Jordan Rizov at London Abstract. Kuga and Satake associate with every polarized complex K3 surface ðX ;LÞ a complex abelian variety called the Kuga-Satake abelian variety of ðX ;LÞ. We use this construction to define morphisms between moduli spaces of polarized K3 surfaces with certain level structures and moduli spaces of polarized