Abstract. The semi-topological K-theory K ∗ (X) of a quasi-projective complex algebraic variety X is based on the notion of algebraic vector bundles modulo algebraic equivalence. This theory is given as the homotopy groups of an infinite loop space K(X) which is equipped with maps K(X) → K(X), K(X) → Ktop(Xan) whose composition is the natural map from the algebraic K-theory of X to the topologi...

— Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Given a semistable vector bundle E over X, we show that its direct image F∗E under the Frobenius map F of X is again semistable. We deduce a numerical characterization of the stable rank-p vector bundles F∗L, where L is a line bundle over X. Résumé (Semi-stabilité des images...

A new technique for the Geometry of Numbers is exhibited. This technique provides sharp estimates on the number of bounded height rational rational points in subsets of projective space whose “projective cone” is semi–algebraic. This technique improves existing techniques as the one introduced by H. Davenport in [15]. As main outcome, we conclude that systems of rational polynomial equations of...

Consider a semi-algebraic set A in R constructed from the sets which are determined by inequalities pi(x) > 0, pi(x) ≥ 0, or pi(x) = 0 for a given list of polynomials p1, . . . , pm. We prove several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f . Then f is a factor of a...

The paper deals with a semi-algebraic set A in R constructed by the inequalities pi(x) > 0, pi(x) ≥ 0, and pi(x) = 0 for a given list of polynomials p1, . . . , pm, and presents several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f . Then f is a factor of a certain multi...

let $mathcal{x}$ be a class of $r$-modules‎. ‎in this paper‎, ‎we investigate ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective (projective) complexes which are generalizations of injective (projective) and dg-injecti‎‎ve (projective) complexes‎. ‎we prove that some known results can be extended to the class of ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective ...

We consider those projective bundles (or Brauer-Severi varieties) over an abelian variety that are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative group schemes; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homo...

We establish structure theorems for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. first prove that is rationally connected if has no truly flat tangent vectors at some point (which satisfied when the curvature quasi-positive). This result solves Yau's conjecture on positive in strong form. Moreover, we admits locally trivial morphism $\phi:X\to Y$ such fibe...

Introduction 1 1. Heights on the projective space 3 1.1. Basic height function 3 1.2. Height function on the projective space 5 1.3. Behavior under maps 7 2. Heights on varieties 9 2.1. Divisors 9 2.2. Heights 13 3. Conjectures 19 3.1. Zeta functions and counting 19 3.2. Height zeta function 20 3.3. Results and methods 22 3.4. Examples 24 4. Compactifications of Semi-Simple Groups 26 4.1. A Con...