We show that the Hodge conjecture is false in general for products of surfaces. We construct a K3 surface whose transcendental lattice has a self-isomorphism which is not a linear combination of self-isomorphisms which preserve cup products over Q up to nonzero multiples. We then find another surface mapping into it in which the transcendental lattice is generated by H 1 cup products according ...

For a smooth projective surface X the finite dimensionality of the Chow motive h(X), as conjectured by S.I Kimura, has several geometric consequences. For a complex surface of general type with pg = 0 it is equivalent to Bloch’s conjecture. The conjecture is still open for a K3 surface X which is not a Kummer surface. In this paper we give some evidence to Kimura’s conjecture for a K3 surface :...

In a recent paper Külshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general H-analogue of the Nakayama conjecture where H is a set of positive integers.

H is the theory extending β-conversion by identifying all closed unsolvables. Hω is the closure of this theory under the ω-rule (and β-conversion). A long-standing conjecture of H. Barendregt states that the provable equations of Hω form a Π11-complete set. Here we prove that conjecture.

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular we give a partial answer to a conjecture of Kalai by proving that h-vectors of flag Cohen-Macaulay simplicial complexes are h-vectors of Cohen-Macaulay balanced simplicial complexes.

H is the theory extending β-conversion by identifying all closed unsolvables. Hω is the closure of this theory under the ω-rule (and β-conversion). A long-standing conjecture of H. Barendregt states that the provable equations of Hω form a Π11-complete set. Here we prove that conjecture.

Vizing’s conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this note we use a new, transparent approach to prove Vizing’s conjecture for graphs with domination number 3; that is, we prove that for any graph G with γ(G) = 3 and an arbitrary graph H, γ(G H) > 3γ(H).