Search results for "Projective variety"

showing 6 items of 6 documents

Non-archimedean hyperbolicity and applications

2018

Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field $K$ of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is $K$-analytically Brody hyperbolic in equal characteristic zero. These two results are predicted by the Green-Griffiths-Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use …

Abelian varietyPure mathematicsConjectureMathematics - Number TheoryApplied MathematicsGeneral Mathematics010102 general mathematics[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Field (mathematics)01 natural sciencesModuli spaceMathematics - Algebraic GeometryMorphism0103 physical sciencesUniformization theoremFOS: MathematicsNumber Theory (math.NT)[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]010307 mathematical physics0101 mathematicsAbelian groupAlgebraic Geometry (math.AG)Projective varietyMathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
researchProduct

Quasi-Projective Varieties

2000

We have developed the theory of affine and projective varieties separately. We now introduce the concept of a quasi-projective variety, a term that encompasses both cases. More than just a convenience, the notion of a quasi-projective variety will eventually allow us to think of an algebraic variety as an intrinsically defined geometric object, free from any particular embedding in affine or projective space.

AlgebraComputer scienceAffine spaceEmbeddingProjective spaceAlgebraic varietyAffine transformationVariety (universal algebra)Projective testProjective variety
researchProduct

Albanese Maps and Fundamental Groups of Varieties With Many Rational Points Over Function Fields

2020

We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field, and prove that every linear quotient of the fundamental group of such a variety is virtually abelian, as well as that its Albanese map is surjective, has connected fibres, and has no multiple fibres in codimension one.

Fundamental groupPure mathematicsGeneral Mathematics01 natural sciencesSurjective functionMathematics - Algebraic GeometryMathematics::Algebraic Geometry0103 physical sciencesFOS: MathematicsNumber Theory (math.NT)0101 mathematicsAbelian groupAlgebraic Geometry (math.AG)Projective varietyQuotientFunction fieldMathematicsMathematics - Number Theory010102 general mathematics[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG][MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV]Codimension[MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV][MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]010307 mathematical physicsVariety (universal algebra)International Mathematics Research Notices
researchProduct

Generically split projective homogeneous varieties. II

2012

AbstractThis article gives a complete classification of generically split projective homogeneous varieties. This project was begun in our previous article [PS10], but here we remove all restrictions on the characteristic of the base field, give a new uniform proof that works in all cases and in particular includes the case PGO2n+ which was missing in [PS10].

Pure mathematicsAlgebra and Number TheoryCollineationComplex projective spaceProjective lineProjective spaceGeometry and TopologyRational normal curveQuaternionic projective spaceProjective varietyMathematicsTwisted cubicJournal of K-Theory
researchProduct

Homological Projective Duality for Determinantal Varieties

2016

In this paper we prove Homological Projective Duality for crepant categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a n x m matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we discuss the relation between rationality and categorical representability in codimension two for determinantal varieties.

Pure mathematicsGeneral MathematicsHomological projective dualitySemi-orthogonal decompositionsDeterminantal varieties01 natural sciencesDerived categoryMathematics - Algebraic GeometryMathematics::Algebraic GeometryMathematics::Category Theory0103 physical sciencesFOS: MathematicsProjective spaceCategory Theory (math.CT)0101 mathematicsAlgebraic Geometry (math.AG)Categorical variableMathematics::Symplectic GeometryPencil (mathematics)Projective varietyComputingMilieux_MISCELLANEOUSMathematicsDiscrete mathematicsDerived category010308 nuclear & particles physicsProjective varietiesComplex projective space010102 general mathematicsFano varietyMathematics - Category TheoryCodimension[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG]Rationality questions[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]
researchProduct

Hodge Theory and Algebraic Cycles

2006

Algebraic cycles and Hodge theory, in particular Chow groups, Deligne cohomology and the study of cycle class maps were some of the themes of the Schwerpunkt ”Globale Methoden in der Komplexen Geometrie”. In this survey we report about several projects around the structure of (higher) Chow groups CH(X,n) [3] which the author has studied with his coauthors during this time by using different methods. In my opinion there are two interesting view points: first the internal structure of higher Chow groups, i.e., the existence of interesting elements and nontriviality of parts of their Bloch-Beilinson filtrations. This case has arithmetic and geometric features, and the groups in question show d…

Pure mathematicsIntersection theorymedicine.medical_specialtyHodge theoryAlgebraic cycleHodge conjectureDeligne cohomologyMathematics::Algebraic GeometryMathematics::K-Theory and HomologyAlgebraic surfacemedicineProjective varietyHodge structureMathematics
researchProduct