ZERO-CYCLES ON VARIETIES OVER p-ADIC FIELDS AND BRAUER GROUPS

In this paper, we study the Brauer-Manin pairing of smooth proper varieties over local fields, and determine the p-adic part of the kernel of one side. We also compute the A0 of a potentially rational surface which splits over a wildly ramified extension. CH0( X ) � Br( X ) ! Q= Z; (M) where Br( X ) denotes the Grothendieck-Brauer group H 2 ´ ( X; Gm) . When dim( X ) = 1 , us- ing the Tate duality theorem for abelian varieties over p -adic local fields, Lichtenbaum (L1) proved that (M) is non-degenerate and induces an isomorphism (L) A0( X ) � ! Hom( Br( X ) = Br( k ) ; Q= Z) : Here Br( X ) = Br( k ) denotes the cokernel of the natural map Br( k ) ! Br( X ) , and A0( X ) denotes the subgroup of CH0( X ) generated by 0 -cycles of degree 0 . An interesting question is as to whether the pairing (M) is non-degenerate when dim( X ) � 2 . See (PS) for surfaces with non-zero left kernel. See (Y2) for varieties with trivial left kernel. In this paper, we are concerned with the right kernel of (M) in the higher-dimensional case. 1.1. We assume that X has a regular model X which is proper flat of finite type over the integer ring ok of k . It is easy to see that the pairing (M) induces homomorphisms CH0( X ) ! Hom( Br( X ) = Br( X ) ; Q= Z) ;