— Publication: In Groupes de Galois arithmetiques et differentiels (Luminy 2004; eds. D. Bertrand and P. Dèbes), Sem. et Congres, Vol. 13 (2006), 165–230. The genus of projective curves discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli Mg of genus g curves. Yet, modern applications require a data variable (function) on such curve...

A category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits. 1. Recently, Toms in [26] used the refinement of the invariant K0 introduced by Cuntz almost thirty years ago in [4] to show that certain C*-algebras are not isomorphic. Anticipating the possible use of this invariant to establish isomorphism, we take the liberty of reporting ...

Projective clustering is a problem with both theoretical and practical importance and has received a great deal of attentions in recent years. Given a set of points P in R space, projective clustering is to find a set F of k lower dimensional j-flats so that the average distance (or squared distance) from points in P to their closest flats is minimized. Existing approaches for this problem are ...

Let k be a field of characteristic zero. For every smooth, projective k-variety Y of dimension n which admits a connected, proper morphism f : Y → S of relative dimension one, we construct idempotent correspondences (projectors) πij(Y ) ∈ CH (Y × Y,Q) generalizing a construction of Murre. If n = 3 and the transcendental cohomology group H tr(Y ) has the property that H tr(Y,C) = f H tr(S,C) + I...

The notion of a multiplier of a group X is generalized to that of a C-multiplier by allowing it to have values in an arbitrary C-algebra A. On the other hand, the notion of the action of X in A is generalized to that of a projective action of X as linear transformations of the space of continuous functions with compact support in X and with values in A. It is shown that there exists a one-to-on...

All spaces below are Tychonov. We define the pro-jective π-character p πχ(X) of a space X as the supremum of the values πχ(Y) where Y ranges over all continuous images of X. Our main result says that every space X has a π-base whose order is ≤ p πχ(X), that is every point in X is contained in at most p πχ(X)-many members of the π-base. Since p πχ(X) ≤ t(X) for compact X, this provides a signifi...

1. Introduction. Let S be an oriented closed surface of genus g > 1. A projec-tive structure on S is a maximal system of local coordinates modeled on the Riemann sphere C, whose transition functions are Möbius transformations. For a given pro-jective structure on S, we have a pair (f, ρ) of a local homeomorphism f from the universal cover S of S to C, called a developing map, and a group homomo...

Let P be the n-dimensional projective space over C. Denote by G(1,P) the grassmannian of the all projective lines in P. By the Plücker embedding G(1,P) →֒ P( n+1 2 )−1 the grassmannian is an algebraic subset of P( n+1 2 ). For any projective line L ⊂ P we will denote by [L] the corresponding point of G(1,P) and for any P,Q ∈ P, P 6= Q, we will denote by PQ the unique projective line in P spanned...

1.1. A fake projective plane is a smooth compact complex surface which is not the complex projective plane but has the same Betti numbers as the complex projective plane. Such a surface is known to be projective algebraic and it is the quotient of the (open) unit ball B in C (B is the symmetric space of PU(2, 1)) by a torsion-free cocompact discrete subgroup of PU(2, 1) whose Euler-Poincaré cha...