Hypertoric varieties are hyperkähler analogues of toric varieties, and are constructed as abelian hyperkähler quotients of a quaternionic affine space. Just as symplectic toric orbifolds are determined by labelled polytopes, orbifold hypertoric varieties are intimately related to the combinatorics of hyperplane arrange…
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48 results for “Hypertoric varieties”
Study of Hilbert schemes and Coulomb branches of hypertoric varieties.
problem Understanding the geometry and topology of Coulomb branches of hypertoric varieties.
method Investigation of transverse equivariant Hilbert schemes and Hamiltonian reductions, proposing new metrics.
result Coulomb branches of hypertoric varieties can be constructed as Hilbert schemes or Hamiltonian reductions.
The geometry of the universal hyperKaehler implosion for SU(n) is explored. In particular, we show that the universal hyperKaehler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a gauge theoretic approach to hyperKaehler implosion.
Let M be a Hamiltonian T space with a proper moment map, bounded below in some component. In this setting, we give a combinatorial description of the T-equivariant cohomology of M, extending results of Goresky, Kottwitz and MacPherson and techniques of Tolman and Weitsman. Moreover, when M is equipped with an antisympl…
Let X be a Kahler manifold that is presented as a Kahler quotient of C^n by the linear action of a compact group G. We define the hyperkahler analogue M of X as a hyperkahler quotient of the cotangent bundle T^*C^n by the induced G-action. Special instances of this construction include hypertoric varieties and quiver v…
Strands algebras and the affine highest weight property for equivariant hypertoric categoriesmath.RT
New homological results for bordered Floer algebras derived from hypertoric categories.
problem Homological properties of bordered Floer algebras.
method Affine quasi hereditary property of equivariant hypertoric convolution algebras and computation of Ext groups.
result Existence of standard modules and isomorphism of Ext groups to bordered strands dg algebras.
New algebraic description connects Fukaya category to bordered Floer homology.
problem Connecting Fukaya category to bordered Floer homology.
method Algebraic description using toric varieties and amplituhedron combinatorics.
result Isomorphic Fukaya category to bordered Floer homology algebras.
We study the geometry of the twistor space of the universal hyperkaehler implosion Q for SU(n). Using the description of Q as a hyperkaehler quiver variety, we construct a holomorphic map from the twistor space Z_Q of Q to a complex vector bundle over P^1, and an associated map of Q to the affine space R of the bundle'…
Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.
problem Symplectic singularities and their degenerations.
method Combining volume minimization, deformation theory, and rigidity results.
result Kaledin's conjecture confirmed for symplectic singularities.
We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension with a tri-Hamiltonian action of a torus of dimension , without any assumption on the finiteness of the Betti numbers. As a result we find that the hyp…
In this thesis we study the topology and geometry of hyperkähler quotients, as well as some related non-compact Kähler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. We prove a Lojasiewicz inequality which permits the use of Morse theo…
Survey and generalization of implosion and contraction in symplectic and hyperkähler geometry.
problem Exploring implosion and contraction in symplectic and hyperkähler geometry.
method Survey and extension of implosion construction to general reductive groups, interpretation in Moore-Tachikawa category, generalization of contraction construction.
result Generalization of implosion and contraction concepts to hyperkähler and complex symplectic situations.
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.
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.
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.
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.
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.
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.
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.
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…
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.
We investigate the secant dimensions and the identifiablity of flag varieties parametrizing flag of sub vector spaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.
Character varieties get a natural Poisson structure.
problem Character varieties of surfaces.
method Natural Poisson structure.
result Character varieties have a Poisson structure.
Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…
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…
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.
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.
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…
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.
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.
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.
Solves open problems on curved projective varieties.
problem Structure theorems for curved projective varieties.
method Supplements and proposes open problems.
result Provides new insights into structure of curved projective varieties.
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.
Classifies holomorphic parabolic geometries on complex manifolds.
problem Classifying holomorphic parabolic geometries on complex manifolds.
method Bounding numerical dimension and using geometric invariants.
result Uncovering foliations and fibrations on smooth projective varieties.
Character variety of Borromean link solved, Alexander polynomial formula found.
problem Character variety of Borromean link
method Determined irreducible SL(2,C) character variety, found Alexander polynomial formula
result Formula for twisted Alexander polynomial on character variety
The article examines twisted cohomologies on algebraic and analytic varieties.
problem Understanding and comparing twisted cohomologies on algebraic and analytic varieties.
method Comparison and definition of twisting parameters in both categories, algebraic and analytic.
result Reviewed isomorphisms of twisted cohomologies for cohomologous twisting parameters.
Proves properness of K-moduli spaces for Fano varieties.
problem Proving properness of moduli spaces of K-polystable Fano varieties.
method Algebraic approach, studying test configurations, constructing stratification.
result Proves properness under specific divisorial valuation condition.
Extends Kollár's result to fibered Calabi-Yau varieties with cohomological assumption.
problem Deformations of fibered Calabi-Yau varieties under cohomological constraints.
method Hodge theoretic techniques and -lifting criterion of Kawamata-Ran.
result Small deformations of fibered Calabi-Yau varieties remain fibered.
BNSR invariants are contained in the complement of tropical varieties.
problem Understanding the relationship between algebraic and topological invariants.
method Showed containment of BNSR invariants in the complement of tropical varieties.
result BNSR invariants are contained in the complement of tropical varieties.
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
A family of new algebraic Poisson varieties will be constructed, generalising the complex character varieties of Riemann surfaces. Then the well-known (Poisson) mapping class group actions on the character varieties will be generalised.
Study characteristic classes of a specific type of determinantal varieties.
problem Understanding the geometric properties of a special class of determinantal varieties.
method Used Schubert calculus to derive explicit formulas for Chern-Schwartz-MacPherson and Chern-Mather classes.
result Explicit formulas for sectional Euler characteristics, characteristic cycles, and polar classes were obtained.
In this paper, we study the analogue of the Shafarevich conjecture for polarized Calabi-Yau varieties. We use variations of Hodge structures and Higgs bundles to establish a criterion for the {\it rigidity} of families. We then apply the criterion to obtain that some important and typical families of Calabi-Yau varieti…
We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the -Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem e.g., var…