arXiv research

Study identifies subvarieties of projective varieties mapping to models.

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…

Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.

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.

The paper describes spectra of operators on rational homogeneous varieties.

Study Kähler-Ricci flow on rational homogeneous varieties using algebraic geometry and representation theory.

Study shows no hyperkähler fourfolds in specified conditions.

Strong formal properties for toric and homogeneous Kähler manifolds.

Study characteristic classes of a specific type of determinantal varieties.

The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…

Estimates quantum cohomology complexity for Fano varieties and homogeneous spaces.

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…

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…

We show that every flag variety contains a naturally defined homogeneous cominuscule subvariety. From the Dynkin diagram of the flag variety, we compute the Dynkin diagram of that subvariety.

Researchers study metrics with maximal Ricci curvature on homogeneous spaces.

We classify smooth Schubert varieties S_0 in a rational homogeneous manifold S associated to a short root, and show that they are rigid in the sense that any subvariety of S having the same homology class as S_0 is induced by the action of Aut_0(S), unless S_0 is linear.

The paper examines bi-Lipschitz triviality of function germs on singular varieties.

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…

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.

New connections on symmetric spaces with invariant properties.

Solves equivalence problem for curves in G(2) flag varieties.

New dHYM connections found on complex 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 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.

We consider a class of compact homogeneous CR manifolds, that we call $\mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag man…

We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily …

Researchers discover all affinely homogeneous models for surfaces in 4D space.

The paper studies K-stability of spherical varieties and their degenerations.

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…

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…

Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--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…

Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local geometry of an affine homogeneous space we construct an algebraic variety $\mathfrak{…

The paper classifies equivariant test configurations for spherical varieties.

Let $X\subset P^N$ be a variety (respectively a patch of an analytic submanifold) and let $x\in X$ 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 $P^n\times P^m$, $n,m\geq 2$, a Grassmaniann $G(2,n+2)$, $n\geq 4…

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.

The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.

Sigma models linked to Gross-Neveu models via quiver varieties.

Study classifies special Hessian rank 2 hypersurfaces in 4D space.

Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle $M\to M/H$, integrations of a Dirac structure o…

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…

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$. We prove that if $E$ admits a $ω$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of $ω$-balanced metrics of certain dir…

We develop the connection between equivariant completions of algebraic homogeneous spaces of reductive groups and lower bounds for the Mabuchi energy of a polarized manifold over the space of Bergman metrics. We provide a new definition of Tian's CM Polarization and discuss its properties.

Study positive characteristic Fano 4-folds with nef tangent bundles.