arXiv research

Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.

The paper explores invariant vs non-invariant complex structures on Lie groups.

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

Classifies complex structures on SL(2,C), finding new non-regular ones.

In this work we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures.…

This research classifies invariant complex structures and Kähler metrics on principal bundles.

Classifies and computes cohomologies of complex structures on Lie groups.

This paper connects complex hyperkähler structures to Donaldson-Thomas invariants.

Classifies complex Dirac structures with invariants and local structure.

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homo…

Computational techniques calculate dimensions of complex structures.

Counterexample found for Stein property of certain solvable Lie groups.

We show that all 6-dimensional nilmanifolds admit generalized complex structures. This includes the five classes of nilmanifold which admit no known complex or symplectic structure. Furthermore, we classify all 6-dimensional nilmanifolds according to which of the four types of left-invariant generalized complex structu…

Study Lie algebras with complex structures, focusing on degenerations and deformations.

The paper classifies invariant structures on complex almost Abelian groups.

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…

We prove that any quasitoric manifold $M^{2n}$ admits a $T^n$-invariant almost complex structure if and only if $M$ admits a positive omniorientation. In particular, we show that all obstructions to existence of $T^n$-invariant almost complex structure on $M^{2n}$ arise from cohomology of underlying polytope - and henc…

The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar…

Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.

Geometric compactification for complex structures on Lie groups.

Study complex structure deformations on Lie algebras and Dolbeault cohomology.

Proves a complex structure conjecture for a specific type of Lie groups.

We introduce the notion of a subregular subalgebra, which we believe is useful for classification of subalgebras of Lie algebras. We use it to construct a non-regular invariant generalized complex structure on a Lie group. As an illustration of the study of invariant generalized complex structures, we compute them all …

The study of invariant SKT structures on nilmanifolds, focusing on 2-step cases.

The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.

We describe moduli spaces of invariant generalized complex structures and moduli spaces of invariant generalized Kähler structures on maximal flag manifolds under $B$-transformations. We give an alternative description of the moduli space of generalized complex structures using pure spinors, and describe a cell decompo…

We study the space of closed anti-invariant forms on an almost complex manifold, possibly non compact. We construct families of (non integrable) almost complex structures on $\R^4$, such that the space of closed $J$-anti-invariant forms is infinite dimensional, and also $0$- or $1$-dimensional. In the compact case, we …

It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description …

Let $G$ be a complex semi-simple Lie group and form its maximal flag manifold $\mathbb{F}=G/P=U/T$ where $P$ is a minimal parabolic subgroup, $U$ a compact real form and $T=U\cap P$ a maximal torus of $U$. The aim of this paper is to study invariant generalized complex structures on $\mathbb{F}$. We describe the invari…

Study semi-Kähler structures on specific Lie groups without symplectic structures.

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of …

The paper explores geometric structures on Hom-Lie groups and algebras.

In this paper we describe the algebra of differential invariants for GL(n,C)-structures. This leads to classification of almost complex structures of general positions. The invariants are applied to the existence problem of higher-dimensional pseudoholomorphic submanifolds.

The study of quasi-Kähler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kähler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structur…

Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.

Let $M = Γ\backslash G$ be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result of C. Maclaughlin, H. Pedersen, Y.S. Poon and S. Salamon for 2-step nilmanifolds. We characterize small def…

Study of complex and Hermitian structures on specific Lie groups.

The paper generalizes the number of complex structures on metric Lie algebras.

Let $M= G/Γ$ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space ${\cal C} ({\frak g})$ of invariant complex structures on $M$, the Dolbeault cohomology of $M$ is isomorphic to the one of the differential bigraded algebra ass…

We discuss the known evidence for the conjecture that the Dolbeault cohomology of nilmanifolds with left-invariant complex structure can be computed as Lie-algebra cohomology and also mention some applications.

We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an anal…

Study cohomotopy classes for 4-manifolds using complex spin structures.

Defines invariants for reflection groups and connects them to Frobenius structures.

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial \bar \partial$-closed, i.e. to strong Kähler with torsion (${\rm SKT}$) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed …

The natural bundle $π:E\to M$ of almost-complex structures is considered. The action of the pseudogroup of all diffeomorphisms of $M$ on the total space $E$ is investigated. A nontrivial 1-st order differential invariant of this action is constructed. It is proved that the Nijenhuise tensor of an almost-complex structu…

We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a r…

The paper develops algorithms and topological invariants for distinguishing dynamic systems.