Related papers: Topological reconstruction theorems for varieties

Using a Zariski topology associated to a finite field extensions, we give new proofs and generalize the primitive and normal basis theorems.

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the…

We formulate and prove a generalization of Zariski-van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz-Zariski-van Kampen type…

Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal…

In algebraic geometry specialisations and valuations play and important role. In this paper we start investigating analogous structures for Zariski structures. Specifically, we look into the existence and uniqueness properties of extensions…

The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring…

A reduced divisor on a nonsingular variety defines the sheaf of logarithmic 1-forms. We introduce a certain coherent sheaf whose double dual coincides with this sheaf. It has some nice properties, for example, the residue exact sequence…

We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.

Given a normal complete variety $Y$ over an algebraically closed field $\mathbb K$, distinct effective Weil divisors $D_1,... D_n$ of $Y$ and positive integers $d_1,... d_n$, we spell out the conditions for the existence of an abelian cover…

We establish the geometric Bogomolov conjecture for semiabelian varieties over function fields. We show a closed subvariety contains Zariski dense sets of small points, if and only if, after modulo its stabilizer, it is a torsion translate…

Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in…

The purpose of this paper is to show how the generic vanishing theorems of M. Green and the second author can be used to settle several questions and conjectures concerning the geometry of irregular complex projective varieties. First, we…

We establish a unified group-theoretic framework bridging the arithmetic homotopy exact sequence of a variety and the Birman exact sequence of a surface. Within this framework, we reinterpret classical arithmetic notions - such as the…

We study the tropicalizations of analytic subvarieties of normal toric varieties over complete non-archimedean valuation fields. We show that a Zariski closed analytic subvariety of a normal toric variety is algebraic if its tropicalization…

In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $-K_S.C+p_g(C)-1$, where $C$…

There is a close relationship between the embedded topology of complex plane curves and the (group-theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves which include a special…

We study a special kind of local invariant sets of singular holomorphic foliations called nodal separators. We define notions of equisingularity and topological equivalence for nodal separators as intrinsic objects and, in analogy with the…

In this note, we give a short proof of the Torelli theorem for cubic fourfolds that relies on the global Torelli theorem for irreducible holomorphic symplectic varieties proved by Verbitsky.

On Zariski Main Theorem in Algebraic Geometry and Analytic Geometry. We fill a surprising gap of Complex Analytic Geometry by proving the analogue of Zariski Main Theorem in this geometry, i.e. proving that an holomorphic map from an…

In very rough terms, the main theorem is that the set, which consists of semistable vector bundles with trivial rational Chern classes and nontrivial kth cohomology on a smooth complex projective variety, is a degeneration of a union of…