Related papers: Higher dimensional geometry of $p$-jets

Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We describe several mild geometric conditions ensuring that the group $A(K^{\rm perf})$ is finitely generated and that the $p$-primary torsion…

We develop a new approach to construction of numerical invariants for ramified coverings of algebraic surfaces of prime characteristic. Let A be a two-dimensional regular local ring of prime characteristic p with algebraically closed…

We study the iterated blow-up X of projective space along an arbitrary collection of linear subspaces. By replacing the universal torsor with an $\mathbb{A}^1$-homotopy equivalent model, built from $\mathbb{A}^1$-fiber bundles not just…

We study the generic behavior of Hamiltonian trajectories on a regular level set in the cotangent bundle, after projection to the base. We prove that for a generic submersive level set, projected trajectories have discrete…

Let $(X,d,p)$ be a pointed metric space. A pretangent space to $X$ at $p$ is a metric space consisting of some equivalence classes of convergent to $p$ sequences $(x_n), x_n \in X,$ whose degree of convergence is comparable with a given…

In the setting of a variety $X$ admitting a tilting bundle $T$ we consider the problem of constructing $X$ as a quiver GIT quotient of the endomorphism algebra $A=\textrm{End}_X(T)^{\textrm{op}}$ corresponding to the tilting bundle. We…

We prove Vojta's abc conjecture for projective space ${\Bbb P}^n({\Bbb C})$, assuming that the entire curves in ${\Bbb P}^n({\Bbb C})$ are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the…

We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order l in its Jacobian.…

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which…

Let $f: X \to S$ be a unipotent degeneration of projective complex manifolds over a disc such that the reduction of the central fibre $Y=f^{-1}(0)$ is simple normal crossings, and let $X_\infty$ be the canonical nearby fibre. Building on…

It is proved that given any prime ideal $\mathfrak{p}$ of height at least 2 in a countable commutative noetherian ring $A$, there are uncountably many more dualizable objects in the $\mathfrak{p}$-local $\mathfrak{p}$-torsion stratum of the…

Let $k$ be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve $X_1$ defined over $k$, there exists a lifting $X$ of the…

Let $X$ be a curve of genus $g\geq 2$ over a number field $F$ of degree $d = [F:Q]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic…

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old…

Let M be a (bounded or not) domain of C^n which is complete with respect to a K\"ahler metric, or more generally, a complete K\"ahler manifold with trivial canonical bundle. Let f be a linearly nondegenerate meromorphic map from M to the…

Let $f : X \rightarrow B$ be a proper flat dominant morphism between two smooth quasi-projective complex varieties $X$ and $B$. Assume that there exists an integer $l$ such that all closed fibres $X_b$ of $f$ satisfy $CH_j(X_b) = \Q$ for…

We consider surfaces with a double elliptic fibration, with two sections. We study the orbits under the induced translation automorphisms proving that, under natural conditions, the finite orbits are confined to a curve. This goes in a…

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the…

We give a partial positive answer to a conjecture of Tyurin (\cite {Tyu}). Indeed we prove that on a general quintic hypersurface of $\Pj^4$ every arithmetically Cohen--Macaulay rank 2 vector bundle is infinitesimally rigid.

This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle…