We give an example of a sequence of positive harmonic functions on \mathbb{Z}^{d} , d\geq 2 , that converges pointwise to a non-harmonic function.
We provide a proof of the Baum–Connes conjecture (with no coefficients) for word hyperbolic groups, based on the Green metric, a metric adapted to random walks. Résumé Nous proposons une démonstration de la conjecture de Baum–Connes (sans coefficients) pour les groupes hyperboliques en utilisant la distance de Green, une distance adaptée à l’étude des marches aléatoires.
We study the dynamics of complex algebraic families of maps on \mathbb{P}^{N} , and the geometry of their preperiodic points. The goal of this article is to formulate a conjectural characterization of the subvarieties of S \times\mathbb{P}^{N} containing a Zariski-dense set of preperiodic points, where the parameter space S is a quasi-projective complex algebraic variety; the characterization is given in terms of the non-vanishing of a power of the invariant Green current associated to the family of maps. This conjectural characterization is inspired by and generalizes the relative Manin–Mumford conjecture for families of abelian varieties, recently proved by Gao and Habegger, and it includes as special cases the Manin–Mumford conjecture (theorem of Raynaud) and the dynamical Manin–Mumford conjecture (posed by Ghioca, Tucker, and Zhang). We provide examples where the equivalence is known to hold, and we show that many recent results can be viewed as special cases. Finally, we give the proof of one implication in the conjectural characterization.
We simplify and improve the main fundamental theorems of positive characteristic generic vanishing theory. As a quick corollary of the theory, we prove that a normal variety X of maximal Albanese dimension satisfies H^0(X, ω_X) ≠ 0 and that if Alb(X) is ordinary, then S^0(X, ω_X) ≠ 0.
A desmic quartic surface is a birational model of the Kummer surface of the self-product of an elliptic curve. We recall the classical geometry of these surfaces and study their analogs in arbitrary characteristic. Moreover, we discuss the cubic line complex G associated with the desmic tetrahedra introduced by G. Humbert. We prove that G is a rational Fano threefold with 34 nodes. The number 34 is the maximum number of nodes on a Fano threefold of degree 6 in P-5, and the group of projective automorphisms is isomorphic to S-4 (sic) 2 = (S(4 )x S-4)(sic) 2.
