arXiv research

Study finds numerical moduli in special 2-flags of length 5.

Study invariant structures on flag manifolds using transformations and pure spinors.

Study Legendrian surfaces using N-graphs and flag moduli.

Study geometrically characterizes piecewise circular curves with decreasing curvature.

We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds $W^6$, $W^{12}$ and $W^{24}$, which are respectively the manifolds of complete flags in $\mathbb C^3$, $\mathbb H^3$ and $\mathbb{Ca}^3$. Together with our earlier work, this concl…

Study of holomorphic triples on surfaces leads to Vafa-Witten invariants.

The paper connects Nahm's equations to rational maps between projective spaces.

In the recent years, a number of issues concerning distributions generating 1- flags (called also Goursat flags) has been analyzed. Presently similar questions are discussed as regards distributions generating multi-flags. (In fact, only so-called special multi-flags, to avoid functional moduli.) In particular and fore…

Groups $Π_k(X;σ)$ of "flagged homotopies" are introduced of which the usual (abelian for $k>1$) homotopy groups $π_k(X;p)$ is the limit case for flags $σ$ contracted to a point $p$. Calculus of exterior forms with values in algebra $A$ is developped of which the limit cases are differential forms calculus (for $A=\bb R…

We define convex projective structures on 2D surfaces with holes and investigate their moduli space. We prove that this moduli space is canonically identified with the higher Teichmuller space for the group PSL_3 defined in our paper math/0311149. We define the quantum version of the moduli space of convex projective s…

The paper maps two types of hyperkähler manifolds and identifies their symplectic structures.

Sequences of Hitchin representations on surfaces are studied to describe their limits on trees.

The moduli space of solutions to Nahm's equations of rank (k,k+j) on the circle, and hence, of SU(2) calorons of charge (k,j), is shown to be equivalent to the moduli of holomorphic rank 2 bundles on P^1xP^1 trivialized at infinity with c_2=k and equipped with a flag of degree j along P^1x{0}. An explicit matrix descri…

In the paper we discuss certain classes of vector distributions in the tangent bundles to manifolds, obtained by series of applications of the so-called generalized Cartan prolongations (gCp). The classical Cartan prolongations deal with rank-2 distributions and are responsible for the appearance of the Goursat distrib…

Study of generalized double Bruhat cells and their integrations.

The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6…

We consider some classical fibre bundles furnished with almost complex structures of twistor type, deduce their integrability in some cases and study \textit{self-holomorphic} sections of a \textit{symplectic} twistor space. With these we define a moduli space of $ω$-compatible complex structures. We recall the theory …

Study of parabolic Higgs bundles on curves with special fixed points.

Paper generalizes Higgs bundle limits to parabolic setting.

Instanton bundles on $\mathbb{P}^3$ have been at the core of the research in Algebraic Geometry during the last thirty years. Motivated by the recent extension of their definition to other Fano threefolds of Picard number one, we develop the theory of instanton bundles on the complete flag variety $F:=F(0,1,2)$ of poin…

Introduces generalized hyperpolygons and their geometric and algebraic properties.

The book develops a new bordism-theoretic approach to understanding orientations of moduli spaces.

A Hermitian metric $ω$ on a complex manifold is called SKT or pluriclosed if $dd^cω=0$. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is Kähler, hence isomorphic to $\C P^3$ or a flag space. This result is obtained from r…

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

This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $¶^1$, and the generalized quiver representations) were included. The main goal is the construction of gauge th…

Cominuscule subvarieties found in flag varieties.

Investigates secant dimensions and identifiability in flag varieties.

This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannia…

Study of weighted nonlinear flags in symplectic geometry.

New algorithm computes flag mean and median on flag manifolds.

The paper studies Finsler manifolds with a new curvature concept.

Study nonlinear flags as coadjoint orbits of Hamiltonian diffeomorphisms.

Suppose $(X, g)$ is a compact, spin Riemannian 7-manifold, with Dirac operator $D$. Let $G$ be SU$(m)$ or U$(m)$, and $E\to X$ be a rank $m$ complex bundle with $G$-structure. Write ${\mathcal B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There is a natural principal ${\mathbb Z}_…

Minimal dimensions found for flag manifolds embeddings.

The paper introduces geometric surgeries for flag structures and provides examples of uniformizable and non-uniformizable types.

Researchers prove formulas for flag area measures, extending previous work.

Extends residue theory to flags of holomorphic distributions.

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

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

The paper calculates expected distances on partially oriented flag manifolds.

Research characterizes intersection cohomology groups of gauge theories and cotangent bundles.

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

The paper classifies complex Dirac structures on flag manifolds.

The aim of this paper is to show that any stable complete Riemannian flag on a compact and connected manifold is conjugated to a flag of homogenus foliations (see Definitions). Also, we give a characterization of Riemannian flags that homogenus. This result is a step toward the classification of Riemannian flags.

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.