New construction provides dense set of slopes for surfaces of Albanese dimension one.
problem Finding slopes for surfaces of Albanese dimension one.
method Second construction of ramified double coverings of cyclic covers.
result Provides a dense set of slopes in [8,9] for surfaces of Albanese dimension one.
Study shows how to separate jets on complex varieties using coverings.
problem Separating jets on complex varieties of maximal Albanese dimension.
method Using finite abelian étale covers and moving Seshadri constants.
result Existence of abelian covers that separate jets.
Study the Albanese map for Kähler manifolds with nef anticanonical bundle.
problem Characterize the structure of the Albanese map for Kähler manifolds with nef anticanonical bundle.
method Analyze two cases: general fiber is Calabi-Yau or projective space. Provide proofs for both cases.
result For the case where the general fiber is Calabi-Yau, the manifold itself must be Calabi-Yau.
A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form ω, for which the bilinear form ω(I⋅,⋅) is positive definite. In this work we prove ddc-lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifol…
Study BNS invariants to prove algebraic fibrations of certain groups.
problem Proving algebraic fibrations of group extensions.
method Study of BNS invariants to analyze group extensions.
result Groups with specific BNS invariants algebraically fiber.
Counterexample disproves log canonical Beauville--Bogomolov decomposition.
problem Disproving the log canonical Beauville--Bogomolov decomposition.
method Constructing a specific log canonical, K-trivial variety with non-birational fibers.
result Provides a counterexample to the Beauville--Bogomolov decomposition in the log canonical setting.
Defines basic Albanese maps for foliated Riemannian manifolds.
problem No specific problem stated; focuses on new concept definition.
method Introduces basic Albanese map using basic 1-forms.
result Relates basic Albanese map to classical Albanese map.
The paper studies Kähler groups and their virtual algebraic fibrations, finding implications for their fundamental groups.
problem Understanding the structure of Kähler groups and their finite index subgroups.
method Analyzing the virtual algebraic fibrations and Albanese dimension of Kähler groups.
result Groups with virtual Albanese dimension ≤ 1 have strong relations with surface groups and coincide with Green--Lazarsfeld sets.
Proves Singer conjecture for specific geometric varieties.
problem Singer conjecture for certain geometric varieties.
method Semismall Albanese map and residually finite fundamental group.
result Proves Singer conjecture for specified varieties.
Compact complex manifolds with trivial canonical bundle and ∂∂ˉ-Lemma have smooth deformations and a surjective Albanese map.
problem Understanding the structure and deformations of compact complex manifolds with specific properties.
method Analyzing the Kuranishi space, showing smooth deformations, and studying the Albanese map.
result Compact complex manifolds with trivial canonical bundle and ∂∂ˉ-Lemma have smooth deformations and a surjective Albanese map. Compact Kähler spaces with zero first Chern class have special geometric properties.
problem Characterizing Kähler spaces with zero first Chern class.
method Analyzing holomorphic tensors, decomposing tangent sheaf, and using Bochner principle.
result Spaces with zero first Chern class split off a complex torus and have specific holonomy representations.
Study proves Kotschick's conjecture for certain compact Kähler manifolds.
problem Proving a conjecture about one-forms without zeros on compact Kähler manifolds.
method Using a conjecture about homologically trivial fibrations and properties of Albanese torus.
result Proves Kotschick's conjecture for specific compact Kähler manifolds.
Iwasawa manifold self-duality proven via Albanese map and Hodge theory.
problem Proving self-duality of Iwasawa manifold.
method Deriving Hodge theory for sGG manifolds, constructing Albanese map.
result Iwasawa manifold self-duality proven.
Holomorphic symplectic structure on Lagrangian moduli space.
problem Understanding the structure of Lagrangian submanifolds in hyperKähler manifolds.
method Proving the existence of a natural holomorphic symplectic structure on the relative Albanese over the moduli space.
result The relative Albanese over the moduli space of complex Lagrangian submanifolds has a natural holomorphic symplectic structure.
The study of projective varieties with nef anticanonical divisors and log terminal singularities.
problem Understanding the structure and properties of projective varieties with specific divisor conditions.
method Analyzing the Albanese map and MRC fibration for klt projective varieties, showing locally constant fibrations and product decompositions.
result Generalization of results for smooth projective varieties to the klt case, including decomposition into rationally connected and projective varieties with trivial canonical divisor.
The study bounds growth of Hodge numbers and computes L2-Betti numbers for irregular varieties.
problem Bounding growth of normalized Hodge numbers and computing L2-Betti numbers for irregular varieties. method Analysis of abelian covers, weak generic Nakano vanishing theorem, and convergence of plurigenera.
result Optimal bounds on the growth of normalized Hodge numbers and computation of L2-Betti numbers. Study of complex projective manifolds using arithmetic lattices.
problem Holomorphic convexity for toroidal compactifications of ball quotients.
method Show that Albanese mapping on an étale covering space generates jets on the interior.
result Shafarevich conjecture on holomorphic convexity satisfied in dimension 2 for arithmetic lattices.
The note proves positive currents induced by VKE with mixed singularities.
problem Variation of Kahler-Einstein metrics with mixed singularities.
method Fiberation between compact Kahler manifolds with generic smooth log canonical pairs.
result Current induced by VKE with mixed cone and Poincare singularities is positive.
We prove that for a compact Kähler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every Kähler threefold of Kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modifi…
Study of tangent bundle positivity on complex projective varieties.
problem Positivity of the second exterior power of tangent bundles on smooth complex projective varieties.
method Analyzes properties of tangent bundles and uses algebraic geometry techniques.
result Proves that up to a finite cover, the Albanese map is a locally trivial fibration with nef fibers.
Given a (meromorphic) fibration f:X→Y where X and Y are compact complex manifolds of dimensions n and m, we define Lf to be the invertible subsheaf of the sheaf of holomorphic m-forms of X given by the saturation of f∗KY, where KY is the canonical sheaf of Y. We define the Kodaira dimension…
We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical …
The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.
problem Understanding the fundamental groups of compact Kahler varieties with specific properties.
method Application of geometric analytic theory of Kahler spaces and study of Albanese maps.
result The fundamental group of compact Kahler varieties with nef anti-canonical bundle is almost Abelian.
We show a non-existence result for some class of equivariant maps between sphere bundles over tori. The notion of equivariant KO-degree is used in the proof. As an application to Seiberg-Witten theory, for a connected closed oriented spin 4-manifold with indefinite intersection form, we have a new bound of the second B…
In this paper, we give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain four dimensional fiber bundles by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau. As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree o…
Take a riemanniann nilmanifold, lift its metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the laplacian for the dirichlet problem on them. We describe the asymptotic behaviour of the sp…
The paper studies moduli spaces of non-smooth metric structures with non-negative Ricci curvature.
problem Understanding moduli spaces of non-smooth metric structures with non-negative Ricci curvature.
method Relating convergence of RCD(0,N)-structures to their lifts, constructing Albanese and soul maps, proving their continuity, and constructing examples.
result Construction of moduli spaces with non-trivial rational homotopy groups.
New classification for Vaisman manifolds with specific properties.
problem Classifying Vaisman manifolds with large first Betti number and vanishing first basic Chern class.
method Analyzing properties and using diffeomorphism and complex structure invariance.
result Every Vaisman manifold with large first Betti number and vanishing first basic Chern class is diffeomorphic to a Kodaira-Thurston manifold.
Study geometric properties of log Calabi-Yau manifolds, focusing on Fano manifolds with smooth or two proportional components.
problem Geometric properties of log Calabi-Yau manifolds in two specific cases.
method Analysis of various geometric properties, focusing on Bochner principle, local triviality, polystability, and compactifiability of universal cover.
result The universal cover of X∖D is a Calabi-Yau manifold of infinite topological type when D has two components. The paper constructs families of algebraic and hyperbolic manifolds sharing invariants.
problem Constructing families of manifolds with shared algebraic and analytic invariants.
method Integral refinement of Gassman-Sunada construction.
result Arbitrarily large collections of manifolds with shared invariants.
Expanding on subgroup dimension, this paper covers new dimensions and compactness.
problem Understanding subgroup dimensions and their properties.
method Generalization of dimension datum, study of disconnected subgroups, and isospectral sets.
result New insights into subgroup dimensions and compactness of isospectral sets.
Direct product cohomological dimension equals twice the group's dimension.
problem Cohomological dimension of direct product groups
method Proved geometrically finite groups cohomological dimension
result Direct product cohomological dimension equals 2 times the group's dimension
Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.
problem Classifying Thurston geometries and understanding their properties.
method Introduces an axiomatic definition for the Kodaira dimension and studies its compatibility with traditional notions.
result Establishes a connection between the simplicial volume and the holomorphic Kodaira dimension, showing implications for smooth Kähler 3-folds.
Paper relates asymptotic dimension to cofinal dimension using coarse proximities.
problem Relating asymptotic dimension to cofinal dimension in metric spaces.
method Introducing coarse proximities and inverse limit constructions.
result Asymptotic dimension is bounded by coarse cofinal dimension and cofinal dimension of Higson corona.
We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the …
Our goal in this paper is to develop an effective estimator of fractal dimension. We survey existing ideas in dimension estimation, with a focus on the currently popular method of Grassberger and Procaccia for the estimation of correlation dimension. There are two major difficulties in estimation based on this method. …
We introduce a new measure of dimension for binary datasets.
problem Defining the effective dimensionality of sparse binary datasets.
method Adapted fractal dimension concept for binary data and introduced normalized fractal dimension.
result Normalized fractal dimension measures the degree of dependency structure of binary datasets.
Study on CR structures in 7D, proving maximal symmetry dimension.
problem Proving symmetry dimension bound for CR structures in 7D.
method Investigating homogeneous models and proving uniqueness.
result 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in 7D.
We establish cohomological and extension dimension versions of the Hurewicz dimension-raising theorem
Generic homeos on complex manifolds have full metric mean dimension.
problem Understanding the metric mean dimension of generic homeomorphisms.
method Analyzing C0-generic homeomorphisms on compact manifolds. result The metric mean dimension equals the manifold's dimension.
Groups act on Euclidean buildings with a minimum dimension bound.
problem Understanding the minimum dimension a group can act on properly.
method Analyzing actions on Euclidean buildings and S-arithmetic groups.
result Action dimension is at least twice the building's dimension.
We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^…
In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo \cite{Buyalo1}) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: micr…
Thurston's spine dimension exceeds virtual cohomological dimension.
problem Understanding the dimension of Thurston's spine in moduli space.
method Analyzing closed hyperbolic surfaces and their systoles.
result The dimension of the set of surfaces is at least \(5-\varepsilon\) for large genus.
Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.
problem Understanding the dimensionality of harmonic measures for random walks.
method Analyzing finite range random walks on Fuchsian Schottky groups.
result Harmonic measures have dimension strictly less than the limit set's Hausdorff dimension.
Short note shows unbounded dimensions in Fano K-moduli spaces.
problem Unboundedness of Fano K-moduli space dimensions.
method Analyzes Fano varieties and their K-moduli spaces.
result Dimension of K-moduli space can be unbounded.
Investigates CR structures in 7D, showing 8 is max symmetry dimension.
problem CR structures in 7D with intransitive symmetry.
method Various methods to bound symmetry dimension, demonstrating existence of non-equivalent models.
result Existence of infinitely many non-equivalent submaximally symmetric models.
Study finds all hyperbolic manifolds with specific automorphism group dimensions.
problem Classifying homogeneous Kobayashi-hyperbolic manifolds based on automorphism group size.
method Analyzing holomorphic automorphism groups of dimension n2−3 for manifolds of dimension n≥2. result All connected homogeneous Kobayashi-hyperbolic manifolds of dimension n≥2 with automorphism group of dimension n2−3 are identified.