In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space o…
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48 results for “Severi variety”
We consider projective varieties with degenerate Gauss image whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness property.
New proof of divisibility property for certain algebraic varieties.
problem Divisibility property for LQEL varieties.
method Construction of Clifford algebra representations to Severi varieties.
result New proof of Russo's Divisibility Property for LQEL varieties.
Each of the four critical Severi varieties arises from a minimal holomorphic nilpotent orbit in a simple regular rank 3 hermitian Lie algebra and each such variety lies as singular locus in a cubic--the chordal variety--in the corresponding complex projective space; the cubic and projective space are identified in term…
The fourth Severi variety has a special geometric structure related to parallel Clifford structures.
problem Characterizing the geometric structure of the fourth Severi variety.
method Explicit construction of a sub-bundle of End(TM) and associated differential forms.
result A canonical differential 8-form on the fourth Severi variety represents a generator of its cohomology ring.
Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projec…
A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.
The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…
Continuity of Kähler-Einstein potentials at singularities proven.
problem Regularity of solutions to degenerate complex Monge-Ampère equations on singular spaces.
method Investigation of Dirichlet problem and global continuity of solutions.
result Kähler-Einstein potentials are continuous at isolated singularities.
We study the local differential geometry of varieties with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…
Uniqueness of CP^nmath.DG
Proves uniqueness of complex projective space's Kahler structure.
problem Uniqueness of complex projective space's Kahler structure.
method Exposition of Hirzebruch, Kodaira, Yau's theorem and Yau's resolution.
result Proves uniqueness of complex projective space's Kahler structure.
Let be an irreducible smooth geometrically integral projective surface over a field. In this paper we give an effective bound in terms of the Neron--Severi rank of for the number of irreducible curves on with negative self-intersection and geometric genus less than , where is t…
A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold,…
Study deformation invariants of projective surfaces using twisted sheaves and Azumaya algebras.
problem Deformation invariants of projective surfaces with specific cohomology conditions.
method Virtual intersection numbers on moduli spaces of stable twisted sheaves and Azumaya modules.
result Invariants do not depend on the choice of Brauer-Severi variety or Azumaya algebra.
We list up to Möbius equivalence all possible degrees and embedding dimensions of real surfaces that are covered by at least two pencils of circles, together with the number of such pencils. In addition, we classify incidences between the contained circles, complex lines and isolated singularities. Such geometric chara…
This research extends quasiplurisubharmonic functions on compact Kähler manifolds.
problem Extending quasiplurisubharmonic functions on compact Kähler manifolds.
method Using a cover of Zariski-open Stein sets with strictly plurisubharmonic potentials, the authors prove extension properties for plurisubharmonic functions.
result Any ω|_X-plurisubharmonic function on an analytic subvariety X of a compact Kähler manifold V extends to a ω-plurisubharmonic function on V.
This paper proves all Pfaffian varieties are area-minimizing except hypersurfaces.
problem Proving area-minimizing property of Pfaffian varieties.
method Analyzing families of minimal real matrix varieties and proving area-minimizing property.
result All Pfaffian varieties are area-minimizing except hypersurfaces.
The paper shows how certain complex projective varieties can be broken down into simpler types.
problem Understanding the structure of complex projective varieties with pseudo-effective tangent sheaves.
method Developed a theory of pseudo-effective sheaves and applied the minimal model program.
result Projective klt varieties with pseudo-effective tangent sheaves can be decomposed into Fano varieties and Q-abelian varieties.
Investigates secant dimensions and identifiability in flag varieties.
problem Secant dimensions and identifiability of flag varieties.
method Numerical conditions ensuring secant varieties have expected dimension and points are identifiable.
result Secant varieties of flag varieties have expected dimension and points are identifiable under certain conditions.
Study Lagrangian fibrations using Jacobians and Prym varieties.
problem Understanding Lagrangian fibrations in symplectic geometry.
method Survey of holomorphic symplectic varieties with Jacobians and Prym varieties as fibres.
result Characterization of Lagrangian fibrations using Jacobians and Prym varieties.
New twisted versions of wild character varieties created.
problem Constructing twisted wild character varieties.
method Creating new versions of wild character varieties.
result Twisted versions of wild character varieties constructed.
Study of braid varieties and their Legendrian isotopy.
problem Understanding the geometric properties of braid varieties.
method Examined four types of braids and their Legendrian links.
result Each open positroid stratum can be represented as an augmentation variety.
Standard embeddings of Schubert varieties in specific manifolds characterized.
problem Characterizing standard embeddings of Schubert varieties in rational homogeneous manifolds.
method Characterization through varieties of minimal rational tangents.
result Characterization of standard embeddings of smooth Schubert varieties.
We introduce the fibred toric varieties as equivariant bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these…
The study shows boundedness of certain fibered varieties in algebraic geometry.
problem Bounding fibered varieties in algebraic geometry.
method Analyzing Calabi-Yau varieties and their fibrations by abelian or symplectic varieties.
result There are only finitely many deformation classes of certain fibered varieties.
Study identifies subvarieties of projective varieties mapping to models.
problem Understanding mappings of subvarieties to models on projective varieties.
method Analyzes smooth projective varieties with holomorphic locally homogeneous structures.
result Determines all subvarieties mapping to the model.
Paper describes holomorphic polyvector fields on toric varieties.
problem No specific problem stated; general description of fields.
method Explicit description of holomorphic polyvector fields on smooth compact toric varieties.
result Generalizes Demazure's result of holomorphic vector fields on toric varieties.
The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.
problem Understanding the structure of projective klt varieties with nef tangent sheaves.
method Developing theory of positivity of coherent sheaves and proving structure theorem.
result Projective klt varieties with specific tangent sheaf properties admit rationally connected fibrations onto abelian varieties.
No complex analytic tori in complex deformations of Kummer varieties.
problem Existence of complex analytic tori in Kummer varieties.
method Proving non-existence through complex deformation analysis.
result Generic complex deformations of Kummer varieties contain no complex analytic tori.
The paper extends character varieties to marked surfaces and discovers their triangular decompositions.
problem Character varieties on marked surfaces and their properties.
method Triangulations of marked surfaces and cohomological groups.
result Stated character varieties admit triangular decompositions.
Cominuscule subvarieties found in flag varieties.
problem Identifying special subvarieties in flag varieties.
method Using Dynkin diagrams to compute subvariety structure.
result Every flag variety has a cominuscule subvariety.
Kähler-Einstein metrics found on special types of symmetric varieties.
problem Finding Kähler-Einstein metrics on smooth Fano symmetric varieties with specific properties.
method Used a combinatorial criterion for K-stability of Fano spherical varieties and computed algebraic moment polytopes and barycenters.
result Proved all smooth Fano symmetric varieties with Picard number one admit Kähler-Einstein metrics.
K{ä}hler geometry of horosymmetric varieties, and application to Mabuchi's K-energy functionalmath.DG
Develops Kähler geometry on new varieties for canonical metrics.
problem No specific problem stated; focuses on new varieties.
method Introduces new varieties, develops Kähler geometry, associates convex functions with metrics.
result Provides expression for Mabuchi functional and combinatorial sufficient condition of properness.
Study empty polar varieties' impact on singular function-germs.
problem Topology of singular function-germs with nonisolated singularities.
method Analysis of empty polar varieties.
result Topology implications of nonempty polar varieties.
Unique K-polystable degenerations for Q-Fano varieties proven.
problem Proving uniqueness of K-polystable degenerations of Q-Fano varieties.
method Analyzing K-polystability and using moduli stack properties.
result K-polystable degenerations of Q-Fano varieties are unique.
The dual variety X* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X* is a hypersurface in the dual space (P^N)*. If dim X* < N - 1, then the variety X is called dually degenerate. The authors refine these definitions for a varie…
Resolves singular cscK varieties to smooth ones.
problem Resolving conically singular cscK varieties to smooth cscK manifolds.
method Glueing result for crepant resolutions of cscK varieties with discrete automorphism groups.
result Shows a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.
Character varieties get a natural Poisson structure.
problem Character varieties of surfaces.
method Natural Poisson structure.
result Character varieties have a Poisson structure.
New Ptolemy variety detects all SL(2,C) representations, independent of triangulation.
problem Finding all SL(2,C) representations in a 3-manifold, independent of triangulation.
method Defining a new Ptolemy variety that is independent of triangulation and detects all boundary-unipotent irreducible representations.
result The new Ptolemy variety detects all boundary-unipotent irreducible SL(2,C)-representations, solving the problem of detecting all representations independent of triangulation.
The paper classifies certain singular projective varieties with specific properties.
problem Classifying projective klt pairs with nef anti-log canonical divisors.
method Establishes a structure theorem using locally trivial rationally connected fibrations.
result Projective klt pairs can be decomposed into rationally connected and Calabi-Yau varieties.
The study bounds growth of Hodge numbers and computes -Betti numbers for irregular varieties.
problem Bounding growth of normalized Hodge numbers and computing -Betti numbers for irregular varieties.
method Analysis of abelian covers, weak generic Nakano vanishing theorem, and convergence of plurigenera.
result Optimal bounds on the growth of normalized Hodge numbers and computation of -Betti numbers.
Proves minimality of tensor varieties, generalizing previous results.
problem Finding minimality conditions for tensor varieties.
method Using Lawlor's curvature criterion and deriving Lawlor's ODE.
result Proves minimality of a class of tensor varieties except for one case.
The paper examines when real matrix Schubert varieties are minimal submanifolds.
problem When are real matrix Schubert varieties minimal submanifolds?
method The authors establish minimality conditions using geometric arguments and partial permutations.
result The paper identifies specific conditions for real matrix Schubert varieties to be minimal submanifolds.
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
New varieties found without smooth curves.
problem Finding varieties without smooth rational curves.
method Constructing normal rationally connected varieties.
result Found varieties of arbitrary large dimensions without smooth rational curves.
New bounds found for certain types of algebraic varieties.
problem Bounding properties of algebraic varieties with specific invariants.
method Analyzing the boundedness of -Fano varieties with controlled anti-canonical degrees and alpha-invariants.
result Fixed-dimensional -Fano varieties with controlled invariants form a bounded family.
New connections on symmetric spaces with invariant properties.
problem Understanding invariant connections on hermitian symmetric spaces.
method Introduced a class of -invariant connections on homogeneous bundles over hermitian symmetric spaces.
result Parameter space of connections is a normal variety with a canonical anti-holomorphic involution.
Stratifies representation varieties of twisted Hopf links.
problem Stratifying representation varieties of twisted Hopf links.
method Using stratification of -representation varieties of the fundamental group of the complement of a twisted Hopf link.
result Explicit description and computation of motives for ranks 1 and 2.