International Journal of Mathematics

We show that the algebraicity of Weil's Hodge cycles implies the usual Hodge conjecture for a general member of a PEL-family of abelian varieties of type III. We deduce the general Hodge conjecture for certain 6-dimensional abelian varieties of type III, and the usual Hodge and Tate conjectures for certain 4-dimensional abelian varieties of type III.

Let ε be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section s ∈ Γ(ε) whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X-r:= n - r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.

Let ℒ be a very ample line bundle on ℳ, a projective manifold of dimension n ≥3. Under the assumption that K_ℳ + (n-2) ℒ has Kodaira dimension n, we study the degree of the map ϕ associated to the complete linear system |2(K_M + (n-2) L)|, where (M, L) is the first reduction of (ℳ, ℒ). In particular we show that under a number of conditions, e.g. n ≥ 5 or K_ℳ + (n-3)ℒ having nonnegative Kodaira dimension, the degree of ϕ is one, i.e. ϕ is birational. We also show that under a mild condition on the linear system |K_M + (n-2) L| satisfied for all known examples, ϕ is birational unless (ℳ, ℒ) is a three dimensional variety with very restricted invariants. Moreover there is an example with these invariants such that deg ϕ= 2.

The universal C*-algebras of discrete product systems generalize the Toeplitz–Cuntz algebras and the Toeplitz algebras of discrete semigroups. We consider a semigroup P which is quasi-lattice ordered in the sense of Nica, and, for a product system p : E→ P, we study those representations of E, called covariant, which respect the lattice structure of P. We identify a class of product systems, which we call compactly aligned, for which there is a purely C*-algebraic characterization of covariance, and study the algebra which is universal for covariant representations of E. Our main theorem is a characterization of the faithful representations of when P is the positive cone of a free product of totally-ordered amenable groups.

Given a pair of simple dimension groups with isomorphic state spaces we try to express it as the pair K₀(A),K₀(A×_αR) of K₀ groups for a C*-algebra A with an action α of R, where both A and the crossed product A×_αR are supposed to be simple AT algebras of real rank zero. We solve this when the state spaces are finite-dimensional.

A theorem of Bernstein type is established saying that a complete affinely harmonic graph in R⁴ is affinely equivalent to {(x,y,xy,y² - x²)|(x,y)∈ R²}. The structures of flat affinely harmonic surfaces are determined.