Applications to the topology of Berkovich spaces

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Closed Field ◽  
Projective Varieties ◽  
Homotopy Types ◽  
Berkovich Spaces ◽  
Galois Orbits ◽  
Finite Simplicial Complex ◽  
Upper Level ◽  
Definable Functions ◽  
Strong Deformation Retraction ◽  
Deformation Retraction

This chapter presents various applications to the topology of classical Berkovich spaces. It deduces from the main theorem several new results on the topology of V(superscript an) which were not known previously in such a level of generality. In particular, it shows that V(superscript an) admits a strong deformation retraction to a subspace homeomorphic to a finite simplicial complex and that V(superscript an) is locally contractible. The chapter also proves the existence of strong retractions to skeleta for analytifications of definable subsets of quasi-projective varieties and goes on to prove finiteness of homotopy types in families in a strong sense and a result on homotopy equivalence of upper level sets of definable functions. Finally, it describes an injection in the opposite direction (over an algebraically closed field) which in general provides an identification between points of Berkovich analytifications and Galois orbits of stably dominated points.

scholarly journals Some properties of complete intersections in “good” projective varieties

1976 ◽  
Vol 61 ◽  
pp. 103-111 ◽  
Author(s):  
Lorenzo Robbiano
Keyword(s):  
Complete Intersection ◽  
Singular Surface ◽  
Complete Intersections ◽  
Closed Field ◽  
Irreducible Curve ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Torsion Free

In [10] it was proved that, if X denotes a non singular surface which is a complete intersection in (k an algebraically closed field of characteristic 0) and C an irreducible curve on X, which is a set-theoretic complete intersection in X, then C is actually a complete intersection in X; the key point was to show that Pic (X) modulo the subgroup generated by the class of is torsion-free.

scholarly journals On ample divisors

1982 ◽  
Vol 86 ◽  
pp. 155-171 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Cartier Divisor ◽  
Closed Field ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Smooth Projective Varieties ◽  
Ample Divisors

In this paper we are dealing with the following problem: determine all normal (or smooth) projective varieties X over an algebraically closed field k supporting a given variety Y as an ample Cartier divisor.

scholarly journals Augmented base loci and restricted volumes on normal varieties, II: The case of real divisors

2015 ◽  
Vol 159 (3) ◽  
pp. 517-527
Author(s):  
ANGELO FELICE LOPEZ
Keyword(s):  
Open Subset ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Smooth Complex ◽  
Normal Varieties ◽  
Restricted Volume ◽  
Complex Projective ◽  
Base Loci

AbstractLet X be a normal projective variety defined over an algebraically closed field and let Z be a subvariety. Let D be an ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on X. Given an expression (*) D$\sim_{\mathbb R}$t1H1 +. . .+ tsHs with ti ∈ ${\mathbb R}$ and Hi very ample, we define the (*)-restricted volume of D to Z and we show that it coincides with the usual restricted volume when Z$\not\subseteq$B+(D). Then, using some recent results of Birkar [Bir], we generalise to ${\mathbb R}$-divisors the two main results of [BCL]: The first, proved for smooth complex projective varieties by Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa, is the characterisation of B+(D) as the union of subvarieties on which the (*)-restricted volume vanishes; the second is that X − B+(D) is the largest open subset on which the Kodaira map defined by large and divisible (*)-multiples of D is an isomorphism.

scholarly journals Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic — A remark to a paper of Dinh-Oguiso-Zhang

2020 ◽  
Vol 31 (08) ◽  
pp. 2050059
Author(s):  
Sichen Li
Keyword(s):  
Projective Variety ◽  
Type Theorem ◽  
Closed Field ◽  
Unipotent Group ◽  
Projective Varieties ◽  
Arbitrary Characteristic ◽  
Derived Length ◽  
Dimension Formula ◽  
Zero Entropy ◽  
Finite Index Subgroup

Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.

scholarly journals The topology of Baumslag–Solitar representations

2019 ◽  
pp. 1-13
Author(s):  
Maxime Bergeron ◽  
Lior Silberman
Keyword(s):  
Algebraic Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Reductive Algebraic Group ◽  
Strong Deformation ◽  
Strong Deformation Retraction ◽  
Deformation Retraction

Let [Formula: see text] be a Baumslag–Solitar group and let [Formula: see text] be a complex reductive algebraic group with maximal compact subgroup [Formula: see text]. We show that, when [Formula: see text] and [Formula: see text] are relatively prime with distinct absolute values, there is a strong deformation retraction of Hom([Formula: see text]) onto Hom([Formula: see text]).

scholarly journals Rational connectivity and analytic contractibility

2019 ◽  
Vol 2019 (747) ◽  
pp. 45-62
Author(s):  
Morgan Brown ◽  
Tyler Foster
Keyword(s):  
Laurent Series ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Homotopy Equivalence ◽  
Formal Laurent Series ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Rationally Connected ◽  
Smooth Projective Varieties

Abstract Let {{k}} be an algebraically closed field of characteristic 0, and let {f:X\to Y} be a morphism of smooth projective varieties over the ring {k((t))} of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the induced map {f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}} between the Berkovich analytifications of X and Y is a homotopy equivalence. Two important consequences of this result are that the Berkovich analytification of any {\mathbb{P}^{n}} -bundle over a smooth projective {k((t))} -variety is homotopy equivalent to the Berkovich analytification of the base, and that the Berkovich analytification of a rationally connected smooth projective variety over {k((t))} is contractible.

Normal bundles in flag bundles

1981 ◽  
Vol 90 (3) ◽  
pp. 395-402
Author(s):  
Samuel A. Ilori
Keyword(s):  
Vector Bundle ◽  
Normal Bundle ◽  
Closed Field ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Algebraic Vector Bundle ◽  
Algebraic Vector ◽  
Normal Bundles ◽  
Tangent Bundles

AbstractLet i: Y ↪ X be an inclusion map of non-singular irreducible algebraic quasi-projective varieties defined over an algebraically closed field. Let E be an algebraic vector bundle over X and H be a sub-bundle of the induced bundle, i*E. If j:F(H) ↪ F(E) is the corresponding inclusion map of (incomplete) flag bundles, then we derive the normal bundle N(F(H), F(E)) in terms of the bundles H and E, the tangent bundles of Y and X as well as the tautological bundles over F(H).

scholarly journals Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1645-1672 ◽  
Author(s):  
Ariyan Javanpeykar ◽  
Ljudmila Kamenova
Keyword(s):  
Moduli Spaces ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Complex Projective Variety ◽  
Hyperbolic Complex ◽  
Closed Subvariety ◽  
Algebraically Closed Fields ◽  
Complex Projective ◽  
Smooth Projective Varieties

Abstract Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that $${{\,\mathrm{Aut}\,}}(X)$$ Aut ( X ) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties.

A Relative Version of Gieseker’s Problem on Stratifications in Characteristic $\boldsymbol{p>0}$

2017 ◽  
Vol 2019 (18) ◽  
pp. 5635-5648 ◽  
Author(s):  
Hélène Esnault ◽  
Vasudevan Srinivas
Keyword(s):  
Fundamental Group ◽  
Closed Field ◽  
Fundamental Groups ◽  
Characteristic P ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Relative Version ◽  
Smooth Projective Varieties

AbstractWe prove that the vanishing of the functoriality morphism for the étale fundamental group between smooth projective varieties over an algebraically closed field of characteristic $p>0$ forces the same property for the fundamental groups of stratifications.

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