A six-dimensional hyperkähler manifold has at most 23 second Betti number.
problem Bounding the second Betti number of hyperkähler manifolds.
method Assumption on Looijenga-Lunts-Verbitsky decomposition of cohomology.
result The second Betti number is at most 23.
Survey on hypothetical complex structure on 6-sphere.
problem Understanding the algebraic dimension and biholomorphisms of a hypothetical complex 6-sphere.
method Discussion of existing results and examples.
result Overview of Peternell--Campana--Demailly's result on algebraic dimension and Huckleberry--Kebekus--Peternell's on biholomorphisms.
We generalise the notion of contact manifold by allowing the contact distribution to have codimension two. There are special features in dimension six. In particular, we show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution.
Study of complex structures on specific solvmanifolds, proving existence and non-existence results.
problem Classification and properties of complex structures on six-dimensional solvmanifolds.
method Classification of Lie algebras, analysis of complex structures, study of Hermitian metrics.
result Determination of new balanced solvmanifolds and confirmation of conjectures.
The study shows conditions for almost complex structures on rational homology spheres and limits on their Betti numbers.
problem Conditions for almost complex structures on rational homology spheres.
method Analyzes rational homology spheres and provides examples of manifolds with almost complex structures.
result Closed almost complex manifolds with sum of Betti numbers three have dimensions that are powers of two.
In this note, the geography problem in dimension four is reviewed and then its extension to dimension six for the symplectic case is explained. Finally some examples in dimension six are provided.
Characterizes algebraic squares of irreducible complex spinors in various dimensions.
problem Understanding the relationship between spinors and exterior forms in different dimensions.
method Formalism using geometric product and algebraic relations.
result General correspondence between irreducible complex spinors and algebraically constrained exterior forms.
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.
Local supertwistors help study 6D conformal supergravity.
problem Understanding conformal supergravity in 6D.
method Local supertwistor formalism with a superconformal connection.
result Derived geometry of (1,0) and (2,0) supergravity multiplets.
Classifies rationally elliptic manifolds and proves geometric formality properties.
problem Classifying and understanding geometric formality in rationally elliptic manifolds.
method Cohomology structure analysis and geometric formality criteria.
result Geometrically formal six-dimensional manifolds with specific Betti numbers are identified.
Investigates hypothetical complex structures on S6 and their Hodge numbers.
problem Determining the existence of a complex structure on S6. method Analyzes Hodge numbers and spectral sequences for S6 with hypothetical complex structures. result Calculates Hodge numbers and dimensions of spectral sequence pages.
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…
Lower bounds on cone density for nontrivial complements in low dimensions.
problem Finding density limits for minimal cones with nontrivial complements.
method Proving lower bounds on cone density for cones of dimensions less than seven with nontrivial complements.
result Established lower bounds on cone density for minimal cones with nontrivial complements in dimensions less than seven.
Proves conjecture about compatible SKT and balanced metrics on compact solvmanifolds.
problem Compact complex manifolds with both SKT and balanced metrics.
method Shear construction and classification of two-step solvable Lie algebras.
result Proves conjecture for compact two-step solvmanifolds with invariant complex structures.
Topological field theories of 2- and 3-forms in 6D, showing no propagating degrees of freedom.
problem Exploring field theories of 2- and 3-forms in six dimensions.
method Analyzing field theories with kinetic term BdC and varying potential terms. result The theories remain topological even with added potential terms, and their reductions yield 3D gravity.
Researchers found solutions to minimal surface equations in 6D.
problem Equations of minimal surface type in six dimensions.
method Constructed nonlinear entire solutions using parametric elliptic functionals.
result Found solutions to minimal surface equations in 6D.
Study Kodaira dimensions on compact almost complex manifolds.
problem Understanding invariants on almost complex manifolds.
method Introduce plurigenera, Kodaira dimension, and Iitaka dimension based on Hodge theory. Prove Hartogs extension theorem using foliation-by-disks technique.
result Show that plurigenera and Kodaira dimension are birational invariants in almost complex category, especially in dimension 4.
Proof of existence of a complex structure on the six-sphere, followed by an explicit computation of its underlying integrable almost complex tensor by the aid of inner automorphisms of the octonions, is exhibited. Both are elementary and self-contained however the size and complexity of the emerging almost complex tens…
Develops complex spinorial forms for all dimensions and signatures, proving Brinkmann waves in supergravity.
problem Study of constrained parallelicity conditions for irreducible complex spinors.
method Differential theory of complex spinorial forms, reformulating conditions as equivalent differential systems.
result Every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to a family of Brinkmann waves.
Proves Dolbeault cohomology conjecture for complex nilmanifolds foliated by tori.
problem Computing Dolbeault cohomology of complex nilmanifolds.
method Generalizes previous methods by assuming foliation in toroidal groups.
result Conjecture holds in real dimension up to six.
Study homology manifolds using spectral sheaves and spectral six functor formalism.
problem Characterize and understand homology manifolds through spectral sheaves.
method Adapt six functor formalism to spectral sheaves on locally compact Hausdorff spaces.
result Prove that compact ANR homology manifolds are Poincaré duality complexes.
No almost complex structures on the six-sphere satisfy certain commutation conditions.
problem Existence of almost complex structures on the six-sphere with specific properties.
method Refined differential inequalities and almost-complex analogue of the Gauss map.
result No integrable almost complex structures on the six-sphere satisfy the given conditions.
This paper classifies Hermitian structures on 6-dimensional nilmanifolds M=G/L for which the fundamental 2-form is d d-bar closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obta…
Study almost complex structures on six-manifolds using twistor spaces.
problem Understanding the space of almost complex structures on six-dimensional manifolds.
method Using twistor spaces and rational homotopy theory, compute the space of almost complex structures and their homological properties.
result Computed the rational homotopy theoretic minimal model of components of almost complex structures satisfying a Chern number condition.
New geometric structures defined on six-dimensional nilpotent Lie groups.
problem Absence of symplectic and complex structures on six-dimensional nilpotent Lie groups.
method Definition of new left-invariant geometric structures.
result Obtained examples of multiparametric families of metrics and almost para-complex pseudo-Riemannian half-flat structures.
Study shows existence of Strominger system solutions is not stable under complex structure deformations.
problem Stability of Strominger system solutions under complex structure deformations.
method Analyzes stability of solutions to the Strominger system in dimensions six, considering both positive and negative slope parameters.
result Existence of solutions to the Strominger system is neither open nor closed under holomorphic deformations of the complex structure.
Study para-complex structures on specific Lie groups, finding explicit forms and properties.
problem Characterizing para-Kähler structures on six-dimensional nilpotent Lie groups.
method Examined left-invariant para-complex structures on six-dimensional nilpotent Lie groups, obtained explicit expressions and investigated curvature properties.
result Para-complex structures are nilpotent and para-Kähler metrics are Ricci-flat.
We study the special algebraic properties of alternating 3-forms in 6 and 7 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed manifold M in these dimensions. Restricting the functional to closed forms in a fixed cohomology class, we find that a critical poin…
We show that the heterotic supersymmetry (Killing spinor equations) and the anomaly cancellation imply the heterotic equations of motion in dimensions five, six, seven, eight if and only if the connection on the tangent bundle is an instanton. For heterotic compactifications in dimension six this reduces the choice of …
Researchers calculate superalgebras for six-dimensional Lorentzian manifolds.
problem Understanding rigidly supersymmetric geometries in six dimensions.
method Calculating Spencer cohomology of (1,0) Poincaré superalgebras. result Maximally supersymmetric backgrounds identified, including Lorentzian Lie groups.
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.
Global obstructions found for conformally Einstein metrics in 6D.
problem Obstructing the existence of conformally Einstein metrics in six dimensions.
method Presentation of global conformal invariants.
result Nontrivial global conformal invariant obstructing conformally Einstein metrics.
Criterion found for blowing down in 6D symplectic geometry.
problem Blowing down criterion in 6D symplectic geometry.
method Criterion for blowing down in 6D symplectic geometry.
result Criterion established for blowing down in 6D symplectic geometry.
This paper deals with complex structures on Lie algebras $\ct_π \hh=\hh \ltimes_π V$, where π is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\ct_π \hh$ for $\hh$ a three dimensional real Lie algebra. First it was proposed the study of complex …
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…
The paper explores SKT, balanced, and generalized Kähler structures on specific Lie groups.
problem Investigating invariant SKT, balanced, and generalized Kähler structures on compact quotients of almost nilpotent Lie groups.
method Characterization and classification of Hermitian almost nilpotent Lie algebras, study of structures under flows, and non-existence results.
result Construction of new compact SKT manifolds and examples of non-split generalized Kähler structures.
We prove the existence of invariant almost complex structure on any positively omnioriented quasitoric orbifold. We construct blowdowns. We define Chen-Ruan cohomology ring for any omnioriented quasitoric orbifold. We prove that the Euler characteristic of this cohomology is preserved by a crepant blowdown. We prove th…
Paper proves existence of compatible Lefschetz fibrations on 6-ball and Stein domains.
problem Existence of compatible Lefschetz fibrations on Stein domains.
method Topological proof for 6-ball fibrations, construction of relative Stein pairs.
result Existence of compatible Lefschetz fibrations on Stein domains of dimension six.
Results of R. Stanley and M. Masuda completely characterize the h-vectors of simplicial posets whose order complexes are spheres. In this paper we examine the corresponding question in the case where the order complex is a ball. Using the face rings of these posets, we develop a series of new conditions on their h-vect…
The study explores geometric formality in specific dimensions of rationally elliptic manifolds.
problem Investigating geometric formality for rationally elliptic manifolds of dimensions 6 and 7.
method Analyzing real cohomology and homotopy properties to determine geometric formality.
result Geometrically formal six-dimensional biquotients with b2=3 have symmetric space cohomology. Study shows a copy of 7D real projective space in the space of almost complex structures on 6-sphere.
problem Understanding the topology of almost complex structures on the 6-sphere.
method Analyzes the fundamental and rational homotopy groups, computes homotopy fiber and groups, and generalizes to 6-manifolds.
result Induces isomorphism on fundamental and rational homotopy groups of the 6-sphere.
We study the problem of extending a complex structure to a given Lie algebra g, which is firstly defined on an ideal h of g. We consider the next situations: h is either complex or it is totally real. The next question is to equip g with an additional structure, such as a (non)-definite metric or a symplectic structure…
Analyze boundary terms in conformal anomaly calculations.
problem Structure of boundary terms in conformal anomaly calculations.
method Analyzing anomalies of type B in six dimensions and deriving boundary terms.
result Boundary terms are necessary for a well-defined variational procedure.
Reviews six finance topics, including 'radical complexity'.
problem None explicitly stated, focuses on research directions.
method Informal review and discussion of open questions.
result No specific key result mentioned, focuses on research directions.
Constructs six-dimensional braid group representations for knot detection.
problem Detecting braid vertibility of knots and links.
method Constructs six-dimensional block representations of B3 and uses them to detect braid vertibility. result Some representations can detect braid vertibility of known knots and links.
Researchers glue and deform Calabi-Yau 3-folds, proving local diffeomorphisms.
problem Deforming and gluing asymptotically cylindrical Calabi-Yau 3-folds.
method Developed new proofs for gluing and deformation, extending results to the asymptotically cylindrical case.
result Proved the gluing map is a local diffeomorphism, connecting moduli spaces of Calabi-Yau structures.
Proves properties of CMC hypersurfaces in specific spaces.
problem Properties of CMC hypersurfaces in Riemannian products.
method Analyzes complete finite index immersed hypersurfaces in Riemannian products.
result Proves compactness or minimality of CMC hypersurfaces.
In this paper we give a full diffeomorphism characterization of compact simply connected cohomogeneity one manifolds in dimension six.