The middle of the diagonal of a surface with p g = 0 and q = 1

Abstract

We prove that if X is a smooth projective complex surface with the invariants pg = 0 and q = 1 then the middle Murre projector π2 (see [Mu90] or [Sch94] for the definition of π2) can be generated by two natural divisors on X whose cohomology classes form a basis for the second cohomology group H 2 (X, Q). As a consequence, this provides a second, in fact, Chow-motivic, proof of the triviality of the Albanese kernel for surfaces with pg = 0 and q = 1 (the first proof was made in [BKL76]).

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