arXiv research

Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are…

The study classifies transverse spheres in flag manifolds and finds new examples.

Study finds conditions for Kähler-Einstein metrics on flag manifolds.

The aim of this paper is to classify all invariant generalized complex structure on a partial flag manifold $\mathbb{F}_Θ$ with at most four isotropy summands. To classify them all we proved that an invariant generalized almost complex structure on $\mathbb{F}_Θ$ is `constant' in each component of the isotropy represen…

This paper constructs Brownian motion on complex flag manifolds and finds joint distribution of stochastic areas.

We give an algorithm to compute the integer cohomology groups of any real partial flag manifold, by computing the incidence coefficients of the Schubert cells. For even flag manifolds we determine the integer cohomology groups, by proving that any torsion class has order 2 (generalizing a result of Ehresmann). We conje…

Study on constant curvature immersions of surfaces into flag manifolds.

We study equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these typ…

For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.

For a Kähler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special atte…

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F_n=SU(n+2)/S(U(n)\times U(1)\times U(1)). For all n>1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles …

We prove a relation between the $\bar\partial_M$ cohomology of a minimal orbit $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a flag manifold $G/Q$ and the Dolbeault cohomology of the Matsuki dual open orbit $X$ of the complexification $K$ of a maximal compact subgroup $K_0$ of $G_0$, under the assum…

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost …

In this work we prove a Baum-Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum-Bott type rationality conjecture in this …

Random matrix ensembles yield uniform distributions on manifolds.

Condition for intersection of real flag manifolds in complex flag manifold.

Let $\mathbb{F}_{Θ}=G/P_{Θ}$ be a generalized flag manifold, where $G$ is a real noncompact semi-simple Lie group and $P_{Θ}$ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow $\mathbb{F}_Θ$ with a cellular CW structure. In this paper we exhibit explicit …

We show that the notion of $3$-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct poly…

Geometric structures on 5-manifolds from surface group representations of G2'.

Abstract: Study cohomology of flag bundles over compact Hermitian locally symmetric spaces.

New algorithm computes flag mean and median on flag manifolds.

Minimal dimensions found for flag manifolds embeddings.

Study equigeodesics on $G_2$-type flag manifolds, splitting tangent spaces.

Study of weighted nonlinear flags in symplectic geometry.

The paper studies Finsler manifolds with a new curvature concept.

We compute the Hochschild-Kostant-Rosenberg decomposition of the Hochschild cohomology of generalised Grassmannians, i.e. partial flag varieties associated to maximal parabolic subgroups in a simple algebraic group. We explain how the decomposition is concentrated in global sections for so-called (co)minuscule and (co)…

The paper examines the geometry of specific submanifolds in flag manifolds.

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivarian…

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Ri…

We show that the totally nonnegative part of a partial flag variety $G/P$ (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.

The paper classifies complex Dirac structures on flag manifolds.

The paper studies totally nonnegative parts of flag varieties and their topologies.

Study nonlinear flags as coadjoint orbits of Hamiltonian diffeomorphisms.

Classifies minimal immersions from $S^2$ into specific flag manifolds.

Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.

The paper explores curvature positivity on Kähler and quasi-Kähler flag manifolds.

In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.

This research connects quantum spectra of flag bundles to prime factorization of integers.

Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.

Holomorphic structures on quantum flag manifolds uniquely defined.

Study and classify totally geodesic submanifolds in nearly Kaehler flag manifold.

Embeds flag manifolds into classical ones, proving rigidity in Kähler geometry.

Geodesic orbit metrics on real flag manifolds identified.

Characterizes invariant spinors on flag manifolds.

The study characterizes real flag manifolds with invariant generalized almost complex structures.

Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.

In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi ansatz technique. We use this approach to provide several e…

Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.