Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
Study on invariant almost complex structures on real flag manifolds.
problem Existence of invariant almost complex structures on real flag manifolds.
method Analysis of real flag manifolds associated to split real forms of complex simple Lie algebras.
result Some real flag manifolds do not admit invariant almost complex structures.
The paper explores invariant vs non-invariant complex structures on Lie groups.
problem Understanding complex structures on Lie groups and their properties.
method Analysis of invariant and non-invariant almost complex structures on compact quotients of Lie groups.
result New computations of Kodaira dimension for invariant and non-invariant structures.
The study characterizes real flag manifolds with invariant generalized almost complex structures.
problem Characterizing real flag manifolds with invariant generalized almost complex structures.
method Characterization through invariant B-transformations and classification of structures. result No GM2-maximal real flag manifolds admit integrable invariant generalized almost complex structures. Classifies complex structures on SL(2,C), finding new non-regular ones.
problem Classifying complex structures on SL(2,C) up to automorphisms.
method Classification via Lie group automorphisms and topological analysis.
result Found one new, non-regular complex structure.
New subalgebra concept helps classify invariant complex structures.
problem Classifying invariant generalized complex structures on Lie groups.
method Introducing subregular subalgebras and using them to construct invariant structures.
result Computed all invariant generalized complex structures for real forms of G2.
The paper classifies invariant generalized complex structures on specific flag manifolds.
problem Classifying invariant generalized complex structures on partial flag manifolds.
method Proved that invariant generalized almost complex structures are constant in each component of the isotropy representation.
result All invariant generalized complex structures on partial flag manifolds with at most four isotropy summands are classified.
This research classifies invariant complex structures and Kähler metrics on principal bundles.
problem Classifying invariant complex structures and Kähler metrics on principal bundles.
method Using Wang's theory of invariant connections and the Levi-Civita connection, the study provides direct geometric proofs and extends the classification from Hermitian to general symmetric spaces.
result The invariant integrable complex structures are unique in the reduced frame bundles of the upper half-plane and complex projective spaces.
Classifies and computes cohomologies of complex structures on Lie groups.
problem Classifying and computing cohomologies of complex structures on Lie groups.
method Complete classification and computation of invariant cohomologies for left invariant structures.
result Computed invariant cohomologies for various generalized complex and Kähler structures.
This paper connects complex hyperkähler structures to Donaldson-Thomas invariants.
problem Describing geometric structures on spaces of stability conditions.
method Introducing Joyce structures and showing their relation to complex hyperkähler structures.
result A Joyce structure on a complex manifold defines a complex hyperkähler structure on its tangent bundle.
Study invariant structures on flag manifolds using transformations and pure spinors.
problem Understanding invariant generalized complex and Kähler structures on flag manifolds.
method Description of moduli spaces using invariant structures, Weyl group action, and pure spinors.
result Alternative description and cell decomposition of moduli spaces.
Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
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.
problem Calculating dimensions of complex structures on manifolds.
method Developed computational techniques to calculate Kodaira dimension and Dolbeault harmonic forms.
result Computed dimensions of left-invariant almost complex structures.
Counterexample found for Stein property of certain solvable Lie groups.
problem Stein property of simply connected unimodular solvable Lie groups with left-invariant complex structures.
method Constructing a solvable Lie group with specific properties.
result A simply connected solvable Lie group with a left-invariant complex structure whose universal cover is not Stein.
The paper studies invariant generalized complex structures on flag manifolds.
problem Classifying invariant generalized complex structures on flag manifolds.
method Analyzing invariant 4-dimensional generalized almost complex structures restricted to each root space, and studying the Nijenhuis operator for a triple of roots. result Classification of integrable and Ω-integrable generalized complex structures. 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.
problem Understanding the space of Lie algebras with complex structures and their transformations.
method Identifying invariants that remain consistent under degenerations and applying to four-dimensional case.
result Found invariants that help in understanding the behavior of Lie algebras under complex structures.
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
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…
The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.
problem Characterizing and understanding nilpotent complex structures on stratified Lie algebras.
method Introduced a new descending series pj to prove a new characterization of nilpotent complex structures and examined whether these structures preserve the strata. result Found that there exists a J-invariant stratification on a step 2 nilpotent Lie algebra with a complex structure. We prove that any quasitoric manifold M2n admits a Tn-invariant almost complex structure if and only if M admits a positive omniorientation. In particular, we show that all obstructions to existence of Tn-invariant almost complex structure on M2n arise from cohomology of underlying polytope - and henc…
Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.
problem Study of Arnold-type invariants of immersed curves and surfaces.
method Framework on dual complexes, locally normalized maps, finite-difference structures, and Shumakovitch-type identities.
result Unified evaluation of Arnold-type invariants St(1) and St(2) on dual skeleta. Study anti-invariant cohomology on almost complex manifolds, showing infinite and finite dimensions.
problem Understanding the cohomology of anti-invariant forms on almost complex manifolds.
method Construction of specific almost complex structures and analysis of cohomology groups.
result Found manifolds with anti-invariant cohomology of infinite, 0-1, and 2 dimensions.
Study complex structure deformations on Lie algebras and Dolbeault cohomology.
problem Deformations of complex structures on Lie algebras and their associated Dolbeault cohomology.
method Construct a complete deformation of complex structures similar to the Kuranishi family, showing extension isomorphism validity.
result Analytic open subset of deformations where Dolbeault cohomology can be computed by left invariant tensor fields.
Geometric compactification for complex structures on Lie groups.
problem Compactifying moduli stack of complex structures on Lie groups.
method Describes a geometric compactification using CR structures transverse to a real foliation.
result Extra points represent CR structures transverse to a real foliation.
The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.
problem Classifying six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.
method Complete classification of six-dimensional solvable strongly unimodular Lie algebras admitting complex structures, identifying those with non-invariant holomorphic sections of their canonical bundle.
result Construction of a new six-dimensional solvmanifold with non-invariant holomorphic sections of its canonical bundle.
The study of invariant SKT structures on nilmanifolds, focusing on 2-step cases.
problem Existence of invariant SKT structures on complex nilmanifolds.
method Construction of examples and negative answer to the existence of invariant SKT structures on higher-step nilmanifolds.
result Negative result on the existence of invariant SKT structures on k-step (k>2) complex nilmanifolds. Proves a complex structure conjecture for a specific type of Lie groups.
problem Proving a conjecture about left-invariant complex structures on nilpotent Lie groups.
method Analyzes simply connected, nilpotent Lie groups of dimension 2n.
result Proves biholomorphism to C^n for the specified Lie groups.
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 …
Study semi-Kähler structures on specific Lie groups without symplectic structures.
problem Exploring structures on Lie groups without symplectic structures.
method Defined semi-Kähler and almost para-semi-Kähler structures on specific Lie groups.
result Geometric properties of these structures are studied.
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.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.
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.
Characterizes complex structures on hermitian symmetric spaces.
problem Understanding invariant complex structures on principal bundles.
method Using Jordan algebraic approach for curvature computations.
result Complete characterization of integrable complex structures.
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.
problem Computing invariants on six-dimensional solvmanifolds.
method Computed almost-complex and almost-Hermitian invariants on families of solvmanifolds.
result Provides obstructions to symplectic structures on compact almost-complex manifolds.
Let M=Γ\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.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
The paper generalizes the number of complex structures on metric Lie algebras.
problem How many orthogonal bi-invariant complex structures exist on metric Lie algebras?
method Developed a unique orthogonal decomposition into irreducible factors for metric Lie algebras.
result There are either 0 or 2^k such complex structures, with k the number of irreducible factors.
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 C(g) of invariant complex structures on M, the Dolbeault cohomology of M is isomorphic to the one of the differential bigraded algebra ass…
New algebra structure for curvature measures in complex space forms.
problem Understanding curvature measures in complex space forms.
method Explicitly describing the algebra structure of dual unitarily invariant curvature measures.
result Characterization of invariant valuations on complex space forms.
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.
problem Understanding cohomotopy classes for families of 4-manifolds with complex spin structures.
method Using Bauer--Furuta invariants in parametrised stable homotopy theory.
result Definition of characteristic cohomotopy classes on Thom spectra.
Defines invariants for reflection groups and connects them to Frobenius structures.
problem Understanding invariants for reflection groups and their relation to Frobenius structures.
method Defines good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants for reflection groups lead to Frobenius structure constants.
Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form ∂∂ˉ-closed, i.e. to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed …
The harmonic volume is a complex analytic invariant that captures detailed complex structure information.
problem Capturing detailed complex structure information of compact Riemann surfaces.
method Defined using Chen's iterated integrals, the harmonic volume extends the period.
result Enabled a quantitative study of the local structure of the moduli space.