Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
Zeta functions of SL2-character varieties match Dedekind zeta functions of trace fields.
problem Characterizing zeta functions of SL2-character varieties of hyperbolic 3-manifolds.
method Proving equality of zeta functions and expressing special values.
result Zeta functions of SL2-character varieties are equal to Dedekind zeta functions of trace fields.
Study quantized SL2-character variety of a once-punctured torus, finding three Coulomb branch isomorphisms.
problem Understanding the structure of quantized SL2-character variety of a once-punctured torus.
method Analyzing the quantized algebra and its subalgebras isomorphic to Coulomb branches.
result Three Z2-invariant subalgebras of the quantized algebra are isomorphic to Coulomb branches. Skein modules over 3-manifolds are shown to form line bundles.
problem Understanding the structure of skein modules over 3-manifolds.
method Using the Frobenius morphism, the skein module is mapped to a coherent sheaf over the SL2 character scheme.
result When the character scheme is reduced, the sheaf is a line bundle.
New invariant derived from skein algebra representations at roots of unity.
problem Understanding skein algebras of open surfaces and their invariants.
method Extended skein algebras, constructed isomorphisms, embedding and isomorphism constructions.
result Invariant associated to each skein algebra representation class.
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 CPr 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.
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.
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.
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.
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.
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…
Character varieties get a natural Poisson structure.
problem Character varieties of surfaces.
method Natural Poisson structure.
result Character varieties have a Poisson structure.
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 L2-Betti numbers for irregular varieties.
problem Bounding growth of normalized Hodge numbers and computing L2-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 L2-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.
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…
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.
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 bounds found for certain types of algebraic varieties.
problem Bounding properties of algebraic varieties with specific invariants.
method Analyzing the boundedness of Q-Fano varieties with controlled anti-canonical degrees and alpha-invariants. result Fixed-dimensional Q-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 G-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 AGLr(C)-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.
Uniformizes varieties of general type with harmonic metrics.
problem Uniformizing varieties of general type with harmonic metrics.
method Criterion for harmonic metrics on Higgs bundles on klt varieties.
result Complete numerical characterisation of singular quotients of the unit ball.
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.
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.
Geometrically connects toric varieties to normed spaces.
problem Connecting algebraic geometry with convex analysis.
method Establishes a 1-1 correspondence using topological models.
result Toric varieties correspond to horofunction compactifications of polyhedral norms.
Proves Yau-Tian-Donaldson conjecture for certain singular Fano varieties.
problem Proving the conjecture for a specific class of singular Fano varieties.
method Using log smooth resolutions and K-polystability criteria.
result Kähler-Einstein metrics exist for K-polystable singular Fano 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 T1-lifting criterion of Kawamata-Ran. result Small deformations of fibered Calabi-Yau varieties remain fibered.
New proof for K-stability of certain Fano varieties.
problem K-stability of specific Fano varieties.
method Proving K-stability and semistability of birationally superrigid Fano varieties.
result Alpha invariant and K-stability conditions for Fano varieties.