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.
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.
Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.
Smooth Schubert varieties are rigid in rational homogeneous manifolds.
problem Classifying and understanding rigidity of Schubert varieties.
method Classification and analysis of homology classes.
result Smooth Schubert varieties are rigid unless they are linear.
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no referenc…
The paper studies foliations on homogeneous spaces and identifies specific foliations.
problem Understanding foliations on homogeneous varieties.
method Equivariant techniques applied to rational homogeneous spaces.
result Identified specific foliations on various homogeneous spaces.
The paper describes spectra of operators on rational homogeneous varieties.
problem Spectrum of invariant (1,1)-forms and Weitzenböck remainder. method Detailed analysis using index theory and Lie-theoretic data.
result Explicit formula for smallest eigenvalue and new lower bounds.
Geometric structures on special manifolds are studied for rigidity.
problem Characterize and prove rigidity of geometric structures on rational homogeneous manifolds.
method Use Cartan geometry to study horospherical varieties and geometric structures modeled on them.
result Geometric structures on smooth projective horospherical varieties of Picard number one are locally equivalent to the standard structure.
Study Kähler-Ricci flow on rational homogeneous varieties using algebraic geometry and representation theory.
problem Analyzing the Kähler-Ricci flow on rational homogeneous varieties.
method Combining projective algebraic geometry and representation theory of semisimple Lie groups and Lie algebras.
result Explicit description and computation of solutions and geometric quantities along the flow.
Study shows no hyperkähler fourfolds in specified conditions.
problem Identifying hyperkähler fourfolds in specific geometric settings.
method Classification and computation of Hodge numbers for fourfolds over rational homogeneous varieties.
result No hyperkähler fourfolds found in the specified conditions.
Strong formal properties for toric and homogeneous Kähler manifolds.
problem Understanding formal properties of Kähler manifolds.
method Analyzing rationally and strongly formal properties of toric and homogeneous Kähler manifolds.
result Toric and homogeneous Kähler manifolds are both rationally and strongly formal.
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.
The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.
problem Solving the deformed Hermitian Yang-Mills equation on rational homogeneous varieties.
method Using Lie theory to describe the Lagrangian phase and characterize solutions.
result Characterization of all supercritical and hypercritical homogeneous solutions of the dHYM equation.
Estimates quantum cohomology complexity for Fano varieties and homogeneous spaces.
problem Quantum cohomology complexity estimation for compact symplectic manifolds.
method Estimates the number of states with finite approximate complexity for Fano complete intersections and (co)minuscule homogeneous varieties.
result Sharp upper bound for the dimension of the space spanned by states with finite complexity for Gr(2, n).
This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with deg…
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.
Schubert varieties are irreducible subvarieties of homogeneous manifold, which are important to understand the geometry of homogeneous manifold G/P and the action of the semisimple Lie group G. Consider the space of effective cycles in G/P with homology class equal to an integral multiple of the homology class of a Sch…
Researchers study metrics with maximal Ricci curvature on homogeneous spaces.
problem Maximizing Ricci curvature in G-invariant metrics on homogeneous spaces.
method Used a formula for the Lichnerowicz Laplacian in terms of the moment map for the variety of algebras.
result Such metrics are generic in the compact case.
A moduli space is constructed for affine homogeneous spaces using their local geometry.
problem Describing and classifying affine homogeneous spaces locally and globally.
method Local description via curvature, torsion, and connection; algebraic variety construction; Spencer cohomology for infinitesimal deformations.
result A moduli space M(glV) is constructed for affine homogeneous spaces of dimension dimV. The paper examines bi-Lipschitz triviality of function germs on singular varieties.
problem Analyzing the bi-Lipschitz triviality of deformations of function germs on singular varieties.
method Introducing strongly rational RX-bi-Lipschitz trivial families and providing an infinitesimal criterion for bi-Lipschitz triviality. result Bi-Lipschitz triviality of deformations of f on (X,0) when X and f are homogeneous of the same degree. These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (jo…
Proves LeBrun-Salamon Conjecture for low-dimensional contact manifolds.
problem Proving the LeBrun-Salamon Conjecture in low dimensions for contact manifolds.
method Study of algebraic torus actions and use of Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems.
result Contact Fano manifolds of dimension at most 9 with reductive automorphism group are homogeneous.
We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.
We study in this paper three natural notions of convergence of homogeneous manifolds, namely infinitesimal, local and pointed, and their relationship with a fourth one, which only takes into account the underlying algebraic structure of the homogeneous manifold and is indeed much more tractable. Along the way, we intro…
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.
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.
Solves equivalence problem for curves in G(2) flag varieties.
problem Equivalence problem for unparametrized curves in G(2)/P.
method Computes algebra of differential invariants for integral and generic curves.
result Provides a solution to the equivalence problem for curves in G(2) flag varieties.
Study geometric criteria for cone structures on flag varieties.
problem Geometric criteria for isotrivial cone structures on flag varieties.
method Complete classification of flag varieties satisfying a projective-geometric criterion.
result Characterization of flag varieties with an injective Gaussian map.
New dHYM connections found on complex vector bundles.
problem Existence of dHYM connections on higher rank vector bundles.
method Constructing explicit non-trivial examples and providing algebraic conditions.
result First explicit non-trivial dHYM connections on higher rank holomorphic vector bundles.
We prove that the compact Kaehler manifolds with first Chern class nonnegative that admit holomorphic parabolic geometries are the flat bundles of rational homogeneous varieties over complex tori. We also prove that the compact Kaehler manifolds with negative first Chern class that admit holomorphic cominiscule geometr…
We consider a class of compact homogeneous CR manifolds, that we call n-reductive, which includes the orbits of minimal dimension of a compact Lie group K0 in an algebraic homogeneous variety of its complexification K. For these manifolds we define canonical equivariant fibrations onto complex flag man…
Researchers discover all affinely homogeneous models for surfaces in 4D space.
problem Identifying all affinely homogeneous models for surfaces in 4D space.
method Improved power series method of equivalence, capturing invariants at the origin, creating branches, and infinitesimalizing calculations.
result Find several inequivalent terminal branches yielding each to some nonempty moduli space of homogeneous models.
We give an elementary introduction to our papers relating the geometry of rational homogeneous varieties to representation theory. We also describe related work and recent progress.
The paper studies K-stability of spherical varieties and their degenerations.
problem Understanding K-stability and degenerations of polarized spherical varieties.
method Reduction to a variational problem on the moment polytope, convexity constraint, and solving the HMA equation.
result Determines strict semistability and polystable degenerations for Fano spherical varieties of rank two.
Let X=G/P be cominuscule rational homogeneous variety. (Equivalently, X admits the structure of a compact Hermitian symmetric space.) We say a Schubert class [S] is Schur rigid if the only irreducible subvarieties Y of X with homology class [Y] = r [S], for an integer r, are Schubert varieties. Robles and The identifie…
A new framework for shape analysis on homogeneous spaces.
problem Efficiently comparing shapes on homogeneous manifolds.
method Generalized Square Root Velocity Transform framework for shapes on homogeneous manifolds.
result Variety of possibilities based on different Lie group actions and metrics.
Let X⊂PN be a variety (respectively a patch of an analytic submanifold) and let x∈X be a general point. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre Pn×Pm, n,m≥2, a Grassmaniann G(2,n+2), $n\geq 4…
Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open GR--orbits in flag varieties G/P. We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, w…
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.
The paper classifies equivariant test configurations for spherical varieties.
problem Classifying equivariant test configurations for spherical varieties.
method Combinatorial data classification of equivariant normal R-test configurations.
result Finiteness theorem of central fibers of G-equivariant special R-test configurations.
New method integrates Poisson homogeneous spaces to symplectic groupoids.
problem Integrating Poisson homogeneous spaces to symplectic structures.
method Using Dirac geometry and explicit constructions, integrates Poisson homogeneous spaces to symplectic groupoids.
result Every Poisson homogeneous space of a Poisson Lie group integrates to a symplectic groupoid.
We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivarian…
Compact pseudo-Hermitian spaces have rigid holomorphic isometries.
problem Characterizing the rigidity of pseudo-Hermitian homogeneous spaces.
method Analysis of Tits fibration and automorphism groups of compact spaces.
result Holomorphic isometries of compact pseudo-Hermitian spaces are compact.
The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.
problem Finding Kähler-Einstein metrics on specific Fano varieties.
method Using combinatorial criteria for K-polystability and properties of Fano varieties.
result Proves existence of Kähler-Einstein metrics on Xm for m≥4 and on Ym for m=4,5. Sigma models linked to Gross-Neveu models via quiver varieties.
problem Understanding the relationship between sigma models and Gross-Neveu models.
method Exploring the mathematical correspondence between sigma models and Gross-Neveu models, including their geometric and trigonometric/elliptic deformations.
result Sigma models are mathematically equivalent to Gross-Neveu models under certain conditions.
Study classifies special Hessian rank 2 hypersurfaces in 4D space.
problem Classifying hypersurfaces with constant Hessian rank 2.
method Power series method of equivalence, Lie's classification spirit.
result 34 inequivalent terminal branches, each with a nonempty moduli space.
We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure Π. We determine the irreducible components of $\Lagr$, and we show that each irreducible component is a smo…
Study finds lowest-degree invariant PDEs over rational contact manifolds.
problem Finding the simplest invariant PDEs over rational contact manifolds.
method Analyzes simple Lie algebras to find lowest-degree invariant second-order PDEs.
result Provides explicit formulas and proves uniqueness for specific types of Lie algebras.