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 -∞.
Study Kodaira dimensions on almost complex manifolds, proving integrability and structural descriptions.
problem Understanding Kodaira dimensions on almost complex manifolds.
method Using pseudoholomorphic pluricanonical maps, defining new dimensions, and applying probabilistic combinatorics.
result Almost complex structures with top Kodaira dimension are integrable, and for compact 4-manifolds, they have elliptic fibration structures.
This is the first of a series of papers, in which we study the plurigenera, the Kodaira dimension and more generally the Iitaka dimension on compact almost complex manifolds. Based on the Hodge theory on almost complex manifolds, we introduce the plurigenera, Kodaira dimension and Iitaka dimension on compact almost com…
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.
We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.
Classified spaces in low dimensions.
problem Irreducible homogeneous almost Hermite-Lorentz spaces in low dimensions.
method Classification through complex dimension 3.
result Classification of spaces in low dimensions.
Study shows Roller compactification's median graph has limited asymptotic dimension.
problem Understanding the asymptotic dimension of Roller compactifications.
method Proved using finite dimensional CAT(0) cube complexes and Borel median graph.
result Borel asymptotic dimension is bounded by the complex's dimension.
The paper explores Kodaira dimension on almost complex manifolds.
problem Understanding Kodaira dimension on non-integrable almost complex manifolds.
method Generalization of Kodaira dimension to almost complex manifolds and study of its behavior under deformations.
result Kodaira dimension is invariant under holomorphic deformations for smooth projective manifolds but not for non-projective manifolds.
Study on Kodaira dimension of specific solvmanifolds without complex structures.
problem Analyzing Kodaira dimension for almost complex 4D solvmanifolds without integrable structures.
method Classification of solvmanifolds and computation of Kodaira dimension for specific structures.
result Showed that Kodaira dimension is not a deformation invariant for some solvmanifolds.
The paper examines Kodaira dimensions of specific complex 4-manifolds with torsion first Chern class.
problem Investigating the Kodaira dimension of almost complex 4-manifolds with torsion first Chern class.
method Developed theory of pseudoholomorphic structures on vector bundles, computed tangent spaces of infinitesimal deformations, and proved unobstructedness theorems.
result Proved that Kodaira dimension can only be 0 or -∞ for tamed almost complex structures.
New research determines the optimal sample complexity for multiclass and list learning.
problem Determining the optimal sample complexity for multiclass classification.
method Algebraic characterization of multiclass hypothesis classes in terms of their DS dimension.
result Proves a longstanding conjecture and determines the optimal dependence of sample complexity on DS dimension.
Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The spectral dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the s…
The paper constructs a complex for the Dirac operator in 4 dimensions.
problem Constructing a complex for the Dirac operator in 4 dimensions.
method Using the Penrose transform, the paper constructs a relative BGG complex and its direct image.
result An explicit construction of a complex starting with the Dirac operator in any number of variables.
New findings on embedding simplicial complexes, showing instability under joins.
problem Conditions for embedding simplicial complexes into double dimension.
method Study of van Kampen obstructions and Smith classes.
result Smith index is not stable under joins, leading to new embeddability results.
Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimensi…
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
Study on metric bubbles in complex dimensions 1 and 2.
problem Understanding degenerations of Kähler-Einstein metrics.
method Investigation of metric bubble trees for non-collapsing cases.
result Description of a conjectural higher-dimensional picture.
The study classifies complex parallelisable nilmanifolds with unobstructed deformations.
problem Characterizing complex parallelisable nilmanifolds with unobstructed deformations.
method Analyzing Lie algebras associated with nilmanifolds and their verbal ideals.
result There are finitely many complex homotopy types of unobstructed complex parallelisable nilmanifolds up to dimension 19, and infinitely many in dimension 20.
New contractible complex shows virtual cohomological dimension of RAAGs.
problem Determining virtual cohomological dimension of RAAGs.
method Equivariant deformation retraction of spine to contractible cube complex.
result Dimension of new complex realises virtual cohomological dimension of RAAGs.
Improved uniform convergence bound with fat-shattering dimension reduces sample complexity gap.
problem Gap between upper and lower bounds on sample complexity for fat-shattering dimension.
method Provided an improved uniform convergence bound.
result Closed the gap between existing upper and lower bounds on sample complexity.
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.
Paper extends Kodaira dimension's role in Yamabe invariant for most complex surfaces.
problem Determining the sign of Yamabe invariant for compact complex surfaces.
method Analyzing Kodaira dimension and using simplified proof techniques.
result Pattern of Yamabe invariant sign depends on Kodaira dimension for most surfaces.
Study on the Kodaira dimension of real parallelizable manifolds with almost complex structures.
problem Understanding the Kodaira dimension of real parallelizable manifolds with specific almost complex structures.
method Conditions and examples provided for calculating the Kodaira dimension of manifolds.
result Conditions under which the Kodaira dimension of a real parallelizable manifold is zero.
We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension 4n+4 from those of dimension 4n. We specialize this construction to the nilpotent case and apply complex symplec…
We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension n=3, whose group of holomorphic automorphisms has dimension n2+1 and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel s…
Study on Hodge theory for almost complex manifolds.
problem Determining Hodge numbers for almost complex manifolds.
method Review and analysis of recent developments in Hodge theory for almost complex manifolds.
result Hodge numbers are almost complex, almost Kähler, or birational invariants in dimension four.
The study classifies stable submanifolds in product spaces of projective spaces.
problem Classifying stable submanifolds in product spaces of projective spaces.
method Provided a classification theorem for compact stable minimal immersions in product spaces of projective spaces.
result Characterized complex minimal immersions in the product of two complex projective spaces.
We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.
A Hermitian metric on a complex manifold of complex dimension n is called {\em astheno-Kähler} if its fundamental 2-form F satisfies the condition ∂∂Fn−2=0. If n=3, then the metric is {\em strong KT}, i.e. F is ∂∂-closed. By using blow-ups and the …
Four-dimensional Einstein Dehn filling is impossible.
problem Complex-hyperbolic Einstein Dehn filling in four dimensions.
method Proof of impossibility.
result Complex-hyperbolic Einstein Dehn filling cannot be performed in dimension four.
Division algorithm for surface group rings yields standard complexes and cohomological dimensions.
problem Understanding cohomological dimensions of surface group actions.
method Division algorithm for group rings of surface groups.
result Some 2-complexes with surface fundamental groups are standard.
We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension n=2, we prove an optimal dimension reduction theorem for complete translating Kähler-Ricci solitons with nonnegative bisectional curvature. We also prove a general dimension reducti…
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.
On real hypersurfaces in complex space forms many results are proven. In this paper we generalize some results concerning extrinsic geometry of real hypersurfaces, to CR submanifolds of maximal CR dimension in complex space forms.
The action dimension of a discrete group G is the minimum dimension of contractible manifold that admits a proper G-action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including: 1) Artin groups, 2) graph products of groups, and 3) fundamental gr…
Determines higher smooth surgery structure sets of complex projective spaces.
problem Understanding the higher smooth surgery structure sets of complex projective spaces.
method Analyzes the free subgroup and torsion in low dimensions.
result Obtains information in all dimensions for the free subgroup.
A complex orthogonal (geometric) structure on a complex manifold is a geometric structure locally modelled on a non-degenerate quadric. One of the first examples of such a structure on a compact manifold of dimension three was constructed by Guillot. In this paper, we show that the same manifold carries a family of uni…
It has been proved that there are no real hypersurfaces satisfying RA = 0 in non-flat complex space forms. In this paper we prove that the same is true in the case of CR submanifolds of maximal CR dimension, that is there are no CR submanifolds of maximal CR dimension satisfying RA = 0 in non-flat complex space forms.
Characterizes the sample complexity of list regression tasks.
problem Understanding the sample complexity of list learning tasks in regression.
method Introducing two combinatorial dimensions: k-OIG dimension and k-fat-shattering dimension.
result These dimensions characterize realizable and agnostic k-list regression.
Training neural networks is hard in fixed dimensions.
problem Training two-layer neural networks is computationally hard in fixed dimensions.
method Parameterized complexity analysis considering dimension and number of neurons.
result Training two-layer neural networks is NP-hard for two dimensions.
The paper extends complex structure existence to manifolds of dimension 8.
problem Existence of complex structures on open manifolds of various dimensions.
method Construction of Γ_n^C structures on CP^n and application of obstruction theory.
result The homology of BΓ_n^C is derived, leading to a theorem about complex structures.
We prove that if G=G1×⋯×Gn acts essentially, properly and cocompactly on a CAT(0) cube complex X, then the cube complex splits as a product. We use this theorem to give various examples of groups for which the minimal dimension of a cube complex the group acts on is strictly larger than that of the…
Research confirms a conjecture about complex manifolds with total Betti number three.
problem Understanding the minimal total Betti number of closed almost complex manifolds.
method Analyzing properties of almost complex manifolds and using topological results.
result The only simply connected closed complex manifold with total Betti number three is the complex projective plane.
Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperkähler manifolds over a compact simply connected base has essential dimension not greater than 1. A similar result about families of complex tori is also…
For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is non-trivi…
Any Kaehler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dime…
Estimates intrinsic dimension of data for GANs.
problem Estimating intrinsic dimension of high-dimensional data.
method Uses Wasserstein distances for estimation.
result Provides sample complexity bounds for GANs.