Compositio Math. 142 (2006) 1522–1548 doi:10.1112/S0010437X06002363 Connectedness of classes of fields and zero-cycles on projective homogeneous varieties Vladimir Chernousov and Alexander Merkurjev Abstract We study the Chow group of zero-dimensional cycles for projective homogeneous varieties of semisimple algebraic groups. We show that in many cases this group has no torsion. 1. Introduction Let X be a proper scheme of finite type over a field F .A zero-cycle on X is the formal sum ∑ n i [x i ] where n i ∈ Z and x i are closed (zero-dimensional) points of the variety X. The factor group of the group of zero-cycles modulo rational equivalence is called the Chow group of dimension 0 and is denoted by CH 0 (X). The assignment x → deg(x) extends to the degree homomorphism deg : CH 0 (X) → Z. The image of deg coincides with n(X)Z where n(X) is the greatest common divisor of the degrees deg(x)=[F (x): F ] over all closed points x ∈ X. We denote the kernel of deg by CH 0 (X). The main purpose of this paper is to present a characteristic free uniform method of computing the group CH 0 (X) for projective homogeneous varieties of semisimple algebraic groups. The method is based on the idea of parametrization of fields over which X has a point. We illustrate the method by proving that in many cases the group CH 0 (X) is trivial and give examples of varieties when this group is not trivial. The main results of the paper can be summarized as follows. Let X be a scheme over F . We denote by A(X) the class of all field extensions L/F such that X(L) = ∅. We say that two fields L 0 ,L 1 ∈A(X) of the same degree n over F are simply X-equivalent if they are members of a continuous family of fields L t ∈ A(X), t ∈ A 1 , of degree n over F (for a precise definition see § 6). We say that L and L  are X-equivalent if they can be connected by a chain of fields L 0 = L, L 1 ,...,L r = L  such that L i and L i+1 are simply X-equivalent for i =0,...,r − 1. Furthermore we say that the class A(X) is connected if every two fields in A(X E ) of degree n(X E ) over E are X E -equivalent over any special extension E/F (see Section 6). Our first result (Theorem 6.4) asserts that if X is an arbitrary proper scheme over F such that the class A(X) is connected and CH 0 (X L ) = 0 for any field L ∈A(X), then CH 0 (X) = 0. Note that the condition CH 0 (X L ) = 0 always holds for projective homogeneous varieties X. Thus the connectedness of the class A(X) for such X implies CH 0 (X) = 0. We prove the connectedness of A(X) for various classes of projective homogeneous varieties. These include: Severi–Brauer varieties, certain generalized Severi–Brauer varieties, quadrics, involution varieties, and projective homogeneous varieties related to groups of exceptional types 3,6 D 4 , G 2 , F 4 , 1,2 E 6 , E 7 with trivial Tits algebras. As an application we get that CH 0 (X) = 0, Received 18 February 2006, accepted in final form 11 May 2006. 2000 Mathematics Subject Classification 20G15, 14M15, 14M17. Keywords: algebraic groups, projective homogeneous varieties, zero-cycles. The first author was supported in part by Canada Research Chairs Program and NSERC Canada Grant G121210944. The second author was supported in part by NSF Grant #0355166. This journal is c  Foundation Compositio Mathematica 2006. https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X06002363 Downloaded from https://www.cambridge.org/core. IP address: 54.163.42.124, on 24 May 2020 at 00:28:12, subject to the Cambridge Core terms of use, available at