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.
Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
New geometric structure on surfaces generalizing complex structures.
problem Defining and analyzing new geometric structures on surfaces.
method Using the punctual Hilbert scheme of the plane to define higher complex structures.
result Moduli space of higher complex structures is a generalization of Teichmüller space and conjecturally isomorphic to Hitchin's component.
This note constructs complex structures on specific isoparametric hypersurfaces.
problem Building complex structures on isoparametric hypersurfaces.
method Constructing almost or complex structures on isoparametric hypersurfaces in unit spheres.
result Complex structures on S1imesS7imesS6 and S1imesS3imesS2 are built. Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.
problem Understanding the Kodaira dimension of almost complex manifolds with SU(m)-structures.
method Introduced almost complex structure of splitting type and associated SU(m)-structure. Provided constructions for non-invariant almost complex structures with specific Kodaira dimensions.
result Found non-invariant almost complex structures with Kodaira dimensions 0 and -∞.
New complex orthogonal structures found on a 3D manifold.
problem Finding new geometric structures on a 3D manifold.
method Constructing uniformizable complex orthogonal and projective structures.
result A manifold has multiple uniformizable complex orthogonal structures.
Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
The paper studies lifts of complex structures on a manifold.
problem Understanding higher-order lifts of extended almost complex structures.
method Proved theorems on Nijenhuis tensor and introduced a new tensor field.
result Basic results on almost analytic complex vectors are investigated.
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.
The study classifies complex symplectic structures on 4D Lie algebras and constructs hypersymplectic structures.
problem Classifying and constructing complex symplectic structures on 4D Lie algebras.
method Interpreting complex symplectic and pseudo-Kähler structures, developing a method for constructing hypersymplectic structures.
result Obtained an example of a hypersymplectic structure on a 4-step nilmanifold.
Study complex Engel structures on complex surfaces, classifying homogeneous examples.
problem Classify homogeneous complex Engel structures on complex surfaces.
method Solve equivalence problems, use structure equations, classify examples.
result Determine all compact, homogeneous complex Engel structures.
Study shows almost complex structures with certain tensor properties are prevalent.
problem Characterizing almost complex structures with specific tensor properties.
method Analyzes the space of almost complex structures on compact manifolds.
result The space of almost complex structures with rank at least k Nijenhuis tensor is either empty or dense in each component.
The paper explores complex Poisson structures on smooth functions in complex manifolds.
problem Exploring complex Poisson structures on smooth functions in complex manifolds.
method Considering structures of complex Poisson brackets generated by a (1,1)-form. result Examples of complex Poisson structures are provided in $\C^\ast$.
Almost complex structures found on many homotopy complex projective spaces.
problem Finding almost complex structures on homotopy complex projective spaces.
method New proof using Chern classes and homotopy properties.
result Classification of almost complex structures on homotopy CPn for 3≤n≤6. Odd m-fold connected sums of complex projective spaces admit almost complex structures.
problem Existence of almost complex structures on connected sums of complex projective spaces.
method Analyzing the m-fold connected sum m#CP2n for m odd or even. result Only odd m-fold connected sums of complex projective spaces admit almost complex structures.
The paper examines complex and para-complex structures on pseudo-Riemannian spheres.
problem Exploring non-integrable Cayley structures on pseudo-Riemannian spheres.
method Study of Cayley structures on six-dimensional pseudo-Riemannian spheres.
result Existence of complex and para-complex structures on pseudospheres.
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.
The paper explores p-Kähler structures on complex manifolds and their properties.
problem Addressing conjectures on specific types of complex manifolds and their structures.
method Analyzing (n−2)-Kähler nilmanifolds and holomorphically parallelizable nilmanifolds, deriving necessary conditions for smooth curves of p-Kähler structures, studying cohomology classes, and providing examples. result Derives necessary conditions for the existence of smooth curves of p-Kähler structures and provides examples of compact complex manifolds with p-Kähler structures. New proof shows 4-manifolds can't support complex structures.
problem Proving 4-manifolds can't support complex structures.
method New proof of complex structure non-existence for real-hyperbolic 4-manifolds.
result Extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure.
The paper describes complex structures on Oeljeklaus-Toma manifolds.
problem Investigating different complex structures on Oeljeklaus-Toma manifolds.
method Algebraic description of biholomorphic manifolds obtained by conjugating complex embeddings.
result Compact non-Kähler manifolds with multiple rigid complex structures.
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. Two exceptional flag manifolds' complex structures are studied, proving rigidity for one.
problem Proving rigidity of complex structures on two exceptional flag manifolds.
method Homogeneous Kähler manifolds under G2 group, relating to sphere complex structures. result Rigidity of complex structure proved for one manifold.
The paper explores linear generalised complex structures over vector bundles.
problem Understanding holomorphic vector bundles in a generalized geometry context.
method Adapted linear splitting and equivalence to C-multiplication and C-Lie algebroid structure. result Generalised complex Lie algebroids are expressed as complex conjugated Lie bialgebroids.
This note is concerned in so called harmonic complex structures introduced by the author previously. I will recall some previous results and emphasize the motivation: Provide an attempt to a fundamental problem in geometry--determining the complex structures on an almost complex manifold. I also discuss the almost-Herm…
The paper constructs complex structures on specific manifolds using isoparametric theory.
problem Constructing explicit complex structures on closed manifolds.
method Isoparametric theory applied to specific manifolds.
result The construction of complex structures on isoparametric hypersurfaces and focal submanifolds.
Study on compact spaces of complex structures orthogonal to a Riemannian metric.
problem Compactness of spaces of k-Gauduchon orthogonal complex structures. method Focus on torsion, different from Salamon's curvature approach.
result Spaces of k-Gauduchon orthogonal complex structures are pre-compact. The paper constructs Levi flat structures using structure sheaves and differential complexes.
problem Global solvability and regularity of Levi flat structures.
method Employing formal integrability and differential complexes, the paper constructs a resolution for the structure sheaf.
result Global exactness and Sobolev regularity of the differential complex for Levi flat structures.
The paper studies almost complex structures on ACH Einstein manifolds and finds deformation results.
problem The study of canonical almost complex structures on ACH Einstein manifolds.
method Variational problem with the Dolbeault Laplacian acting on (0,1)-forms. result Deformation results of Einstein ACH metrics associated with critical almost complex structures.
Classifies complex structures on 6D Lie algebras as products.
problem Identifying all complex structures on 6D Lie algebras.
method Examined 6D Lie algebras of the form gimesg for integrable complex structures. result Listed all integrable complex structures on these Lie algebras.
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. 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 complex structures on geodesic and harmonic map spaces.
problem Existence and uniqueness of compatible complex structures.
method Definition and study of adapted complex and involutive structures.
result Existence and uniqueness of such structures are established.
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 on biharmonic almost complex structures on compact manifolds.
problem Existence and regularity of biharmonic almost complex structures.
method Analyzes biharmonic almost complex structures on compact almost Hermitian manifolds, focusing on dimension four.
result Existence of energy-minimizing biharmonic almost complex structures for various topologies and homotopy classes.
We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to double covering, as G/L where G is a simply connected solvable Lie group and L is a …
A complex structure on a subset of S^6 cannot be extended to a global integrable structure.
problem Non-integrability of complex structures on subsets of S^6.
method Proving the non-integrability of extensions of a specific complex structure on a subset of S^6.
result It is impossible to deform a non-integrable structure to an integrable one on S^6 while fixing it on a subset.
New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.
problem Stable generalized complex structures in higher dimensions with self-crossing singularities.
method Extending stable generalized complex structures to include anticanonical sections with normal self-crossings.
result Construction of large families of stable generalized complex manifolds in four dimensions.
New complex structures found on tangent bundles of Lie groups.
problem Finding integrable complex structures on tangent bundles of Lie groups.
method Inspired by Samelson's construction, a left-invariant integrable almost complex structure is defined on the tangent bundle of any compact Lie group.
result Tangent bundles of compact Lie groups admit left-invariant integrable almost complex structures.
Study on energy-minimizing structures in complex geometry.
problem Existence and regularity of harmonic almost complex structures.
method Inspired by harmonic map theory, proving results similar to Schoen-Uhlenbeck and Cheeger-Naber.
result Proved existence and regularity similar to harmonic map theory.
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 structures on Courant algebroids, linking to Poisson structures.
problem Characterizing generalised complex structures on transitive Courant algebroids.
method Analyzing components and integrability equations in a specific splitting.
result Generalised complex structures imply non-degenerate Poisson structures on the base manifold.
Stable generalized complex structures on certain surfaces are constant.
problem Existence of stable generalized complex structures on ruled surfaces.
method Analysis of sphere bundles over surfaces of genus ≥2.
result Stable generalized complex structures on these surfaces are of constant type.
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.
New findings on complex structures in nilpotent Lie algebras and pseudo-Kähler geometry.
problem Classifying complex structures on nilpotent Lie algebras.
method Analysis of Lie algebras in dimensions eight and higher, identifying structures with pseudo-Kähler metrics.
result Identification of pseudo-Kähler nilmanifolds with invariant complex structures, providing counterexamples and topological restrictions.
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…
In this paper, we introduce a new concept so called harmonic complex structure by using harmonic theory for vector bundle-valued differential forms. It is a new structure intermediates between complex structure and Kähler structure. From differential geometric viewpoint, it is a natural generalization of Kähler structu…
No direct generalized complex structure can be induced from S6's nearly Kähler structure.
problem Existence of generalized complex structures on S6. method Defined integrability in terms of the Dorfman bracket and studied S6's nearly Kähler structure. result No generalized complex structure can be induced from S6's nearly Kähler structure. We introduce integrable complex structures on twistor spaces fibered over complex manifolds. We then show, in particular, that the twistor spaces associated with generalized Kahler, SKT and strong HKT manifolds all naturally admit complex structures. Moreover, in the strong HKT case we construct a metric and three comp…