J ALGEBRAIC GEOMETRY S Article electronically published on January

Description

J. ALGEBRAIC GEOMETRY 13 (2004) 417–426 S 1056-3911(04)00341-8 Article electronically published on January 5, 2004 ON THE CHOW RING OF A K3 SURFACE ARNAUD BEAUVILLE AND CLAIRE VOISIN Abstract We show that the Chow group of 0-cycles on a K3 surface contains a class of degree 1 with remarkable properties: any product of divisors is proportional to this class, and so is the second Chern class c2. 1. Introduction An important algebraic invariant of a projective manifold X is the Chow ring CH(X) of algebraic cycles on X modulo rational equivalence. It is graded by the codimension of cycles; the ring structure comes from the intersection product. For a surface we have CH(X) = Z? Pic(X)? CH0(X), where the group CH0(X) parametrizes 0-cycles on X . While the structure of the Picard group Pic(X) is well understood, this is not the case for CH0(X): if X admits a nonzero holomorphic 2-form, it is a huge group, which cannot be parametrized by an algebraic variety [M]. Among the simplest examples of such surfaces are the K3 surfaces, which carry a nowhere vanishing holomorphic 2-form. In this case Pic(X) is a lattice, while CH0(X) is very large; the following result is therefore somewhat surprising: Theorem 1.

• addition map


• cx


• class


• ?x ?x


• dimensional cycles


• fibration over


• group ch0


• map µ

Informations

• Publié par: pefav
• Nombre de lectures: 7
• Langue: English

Extrait

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