arXiv research

Study on $p$-Kähler structures on fibrations and Lie groups.

Deformation retracts Baumslag-Solitar representations onto a simpler subgroup.

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …

Monodromy map from differential systems to character variety is generically immersive for complex $G$-representations.

Develops quantum character theory for complex reductive groups.

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…

Let $E_G$ be a stable principal $G$--bundle over a compact connected Kaehler manifold, where $G$ is a connected reductive linear algebraic group defined over the complex numbers. Let $H\subset G$ be a complex reductive subgroup which is not necessarily connected, and let $E_H\subset E_G$ be a holomorphic reduction of s…

Algebraic treatment of connection reduction over a special disc.

Let $G$ be a complex reductive connected algebraic group equipped with the Sklyanin bracket. A classification of Poisson homogeneous $G$-spaces with connected isotropy subgroups is given. This result is based on Drinfeld's correspondence between Poisson homogeneous $G$-spaces and Lagrangian subalgebras in the double $D…

Study Lie algebroid connections on principal bundles over complex projective varieties.

The paper computes characteristic classes for Lie group representations.

Introduces new algebraic structures for relational groupoids and proves a reduction theorem.

This document contains tables with the classification of prehomogeneous modules for reductive algebraic groups with up to two simple factors due to Sato, Kimura and many others, as well as corresponding tables of the étale modules appearing in this list, determined by the author. It is intended as a convenient referenc…

The study finds invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.

Computes the component group of real reductive groups.

Let $G$ be a connected reductive complex affine algebraic group and $K\subset G$ a maximal compact subgroup. Let $M$ be a compact complex torus equipped with a flat Kähler structure and $(E_G ,θ)$ a polystable Higgs $G$-bundle on $M$. Take any $C^\infty$ reduction of structure group $E_K \subset E_G$ to the subgroup $K…

We prove a version of the affine Kempf-Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkahler quotients of T*G, where G is a complex reductive group.

A complex vector space $V$ is a prehomogeneous $G$-module if $G$ acts rationally on $V$ with a Zariski-open orbit. The module is called etale if $\dim V=\dim G$. We study etale modules for reductive algebraic groups $G$ with one-dimensional center. For such $G$, even though every etale module is a regular prehomogeneou…

Jacobi operators on certain naturally reductive spaces have constant coefficient ODEs.

We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.

For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…

Characterizes complex structures on specific Lie groups.

Geometrically revisits and models homogeneous spaces of compact Lie group $G_2$.

New method for constructing real algebraic surfaces from complex tropical hypersurfaces.

Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the GIT quotient space Hom(A,G)//G. As a corollary, we determine necessa…

The paper investigates gradings of complex simple Lie algebras, focusing on $|3|$-gradings and their algebraic structures.

Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.

Motivated by Demailly's strategy towards the Kobayashi hyperbolicity conjecture, we study the action on the k-jets of germs of holomorphic discs in a complex manifold X of the reparametrization group of k-jets of germs of biholomorphisms of the source. This reparametrization group is a subgroup of the general linear gr…

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…

The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.

Let G be a complex reductive linear algebraic group and let K be a maximal compact subgroup of G. Given a nilpotent group Γgenerated by r elements, we consider the representation spaces Hom(Γ,G) and Hom(Γ,K) with the natural topology induced from an embedding into G^r and K^r respectively. The goal of this paper is to …

This paper constructs and proves the uniqueness of pluriharmonic maps to Euclidean buildings.

Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle …

Introduces a new method for symplectic reduction along submanifolds.

Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.

A connected Fano complex-contact manifold is isomorphic to the kaehlerian C-space of Boothby type with a natural complex-contact structure corresponding to a non-abelian simple complex Lie algebra if the contact line bundle is very ample. A. Beauville relaxed the provision to two assumptions that the contact line bundl…

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary fiel…

Study on special Lie groups with Lorentzian metrics.

We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be viewed as the real points of a complex reductive group $G^\mathbb C$ which acts on $V…

The paper classifies test configurations and derives a criterion for uniform K-stability of certain algebraic varieties.

Study of wild mapping class groups and their cabled braids.

Generalizes reductive homogeneous spaces to arbitrary Lie groups using gauge theory.

Reductive quotients preserve klt singularities in algebraic geometry.

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 …

Study on semistable points and convexity of gradient maps for group actions.

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topolo…

Investigates properties of moment maps and stratifications on Lie groups.