Analyzes Kodaira-Iitaka dimension and multiplicity using intersection theory.
problem Understanding Kodaira-Iitaka dimension and multiplicity in analytic terms.
method Expresses dimensions and multiplicity in terms of intersection theory of plurisubharmonic envelopes.
result Introduces non-pluripolar numerical Kodaira-Iitaka dimension and shows it dominates the classical dimension.
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.
Study long-term flow on singular varieties, generalizing previous work.
problem Behavior of Kähler-Ricci flow on mildly singular varieties.
method Normalized Kähler-Ricci flow on mildly singular varieties of positive Kodaira dimension.
result Generalize results of Song and Tian on smooth minimal models.
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.
Analyzes complex structure deformations using cohomology contraction methods.
problem Deforming complex structures and identifying obstructions.
method Refined power series method for (p,q)-forms and complex structures, using Frölicher spectral sequence. result All obstruction classes lie in the kernel of contraction maps under natural vanishing conditions.
The paper explores deformations of compact holomorphic Poisson submanifolds.
problem Deformations of compact holomorphic Poisson submanifolds.
method Study of deformations through Kodaira's series and algebraic methods.
result Identification of first-order deformations and obstructions.
Decomposes financial networks to reveal cause-effect hierarchies during crises.
problem Complex financial networks are hard to interpret due to Granger causality.
method Helmholtz-Hodge-Kodaira decomposition to separate networks into rotational and gradient components.
result Precious metals and pharmaceutical products are identified as causal drivers during crises.
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.
Extends Kodaira Spencer to higher-dimensional Calabi-Yau manifolds.
problem No specific problem stated; extension of functional.
method Extension to Calabi-Yau manifolds of arbitrary dimension.
result Extension of the Kodaira Spencer functional.
Survey on Kodaira dimension in 2D and 3D topology.
problem Various notions of Kodaira dimension in low dimensional topology.
method Survey and review of existing literature.
result Progress in understanding Kodaira dimension in low dimensions.
Study on Kodaira dimension of almost Kähler manifolds and their curvature.
problem Understanding Kodaira dimension in almost Kähler manifolds.
method Explicit computation and analysis of curvature of the canonical connection.
result Ricci curvature vanishes for members of the family of almost Kähler manifolds.
The paper develops a deformation theory for Dolbeault cohomology classes.
problem Understanding the variations of Dolbeault cohomology classes.
method Established a deformation theory using the power series method and proved the extension equation.
result Proved the existence and unobstructedness of deformations under certain conditions.
Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.
problem Characterizing Vaisman solvmanifolds and their properties.
method Analyzing fundamental groups and quotient structures.
result Every Vaisman solvmanifold is a finite quotient of a Kodaira-Thurston manifold.
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 -∞.
The abstract presents power series proofs for local stabilities of Kähler and balanced structures.
problem Local stabilities of Kähler and balanced structures on complex manifolds.
method Power series method applied to a natural map of complex differential forms.
result New local stability theorems for balanced structures and p-Kähler structures.
Study on Kodaira fibrations with nontrivial cohomology, proving properties of their structure.
problem Characterizing Kodaira fibrations with specific cohomology properties.
method Analyzing invariant rational cohomology and properties of holomorphic sections.
result Kodaira fibrations with invariant cohomology admit specific coverings and monodromies.
Automorphisms of Kodaira surfaces are shown to be affine transformations.
problem Characterizing automorphisms of Kodaira surfaces.
method Analyzing lifts to the universal cover and conditions on affine transformations.
result Precise description of Kodaira surfaces' automorphism groups.
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.
In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.
Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.
problem Investigate non-abelian finite quotients of surface braid groups and double Kodaira fibrations with small signature.
method Introduced diagonal double Kodaira structures to study finite quotients of pure braid groups and constructed double Kodaira fibrations.
result Proved that if a finite group admits a diagonal double Kodaira structure, then its order is at least 32, with equality if and only if the group is extra-special.
We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of Kähler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of ho…
Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.
problem Characterizing and preserving Vaisman metrics on Kodaira-Thurston surface.
method Characterization of T2-invariant Vaisman metrics, analysis of pluriclosed flow behavior. result Pluriclosed flow preserves Vaisman condition on Kodaira-Thurston surface, including non-constant scalar curvature.
We bound the index of a subgroup in iterated Kodaira fibrations.
problem Bounding the index of a subgroup in iterated Kodaira fibrations.
method Passing to a finite index subgroup of π_1(X) to achieve the desired structure.
result We provide a bound on the index of such a group.
New findings on Frobenius structures on Kodaira manifolds.
problem Understanding Frobenius structures on Kodaira manifolds.
method Extended deformation theory and Frobenius structures.
result Frobenius structure on Kodaira manifolds is trivial on degree-2 component.
Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.
problem Extending complex geometry results to Lie algebroids.
method Using local coordinate calculations to generalize Kahler identities.
result Kernel of Lie algebroid Laplace operator vanishes for sufficiently large p+q.
We define the Kodaira dimension for 3-dimensional manifolds through Thurston's eight geometries, along with a classification in terms of this Kodaira dimension. We show this is compatible with other existing Kodaira dimensions and the partial order defined by non-zero degree maps. For higher dimensions, we explore th…
The paper generalizes a key theorem in complex geometry.
problem Complex geometry theorems and their generalizations.
method Analytical methods
result Generalization of the Kodaira embedding theorem
New groups without finite classifying spaces found in Kodaira fibrations.
problem Finding groups without finite classifying spaces.
method Constructing Kähler groups as fundamental groups of Kodaira fibrations.
result Found Kähler groups that are not commensurable to surface groups and do not have finite classifying spaces.
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.
The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.
problem Generalizing Kodaira vanishing theorems for non-abelian settings.
method Non-abelian Hodge theory and Mixed Twistor D-modules.
result Generalized Kodaira vanishing theorems for various settings.
Method solves ∂ˉ-harmonic forms on Kodaira-Thurston manifold.
problem Finding ∂ˉ-harmonic forms on Kodaira-Thurston manifold. method Weil-Brezin transform, linear ODE systems, fundamental problem solving.
result Dimension of almost complex ∂ˉ-Hodge numbers can be arbitrarily large. We define analogue of theta-functions on the Kodaira--Thurston manifold which is a compact 4-dimensional symplectic manifold and use them to construct canonical symplectic embedding of the Kodaira--Thurston manifold into the complex projective space (analogue of the Lefshetz theorem).
We provide infinitely many examples of pairs of diffeomorphic, non simply connected K\" ahler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of non deformation equivalent, diffeomorphic K\" ahler threefolds.
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.
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.
Abstract: Almost Kähler Hodge numbers vary with metric choices.
problem Variation of almost Kähler Hodge numbers with metrics.
method Analysis of almost complex Hodge numbers under different almost Kähler metrics.
result The almost Kähler Hodge number h0,1 varies with metric choices. The paper explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.
problem Investigating Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds.
method Analyzing compact almost Hermitian manifolds in the Gray-Hervella class and Hermitian manifolds with nonnegative scalar curvature.
result For compact almost Hermitian manifolds with nonnegative scalar curvature, the Kodaira dimension is either -∞ or 0, with specific conditions.
The fundamental group of a Kodaira fibration is constrained by specific conditions.
problem Understanding the fundamental group of Kodaira fibrations and its constraints.
method Analyzing the extension of surface groups and conditions for the fundamental group of a Kodaira fibration.
result The fundamental group of a Kodaira fibration is constrained by the relative irregularity, providing examples of symplectic 4-manifolds without Kähler structures.
Study of section conjecture analogues over complex numbers and Kodaira fibrations.
problem Investigate Grothendieck's section conjecture over complex numbers and Kodaira fibrations.
method Topological and Hodge theoretic analogues of the section conjecture over complex numbers, studied in the context of Kodaira fibrations and families of Jacobians.
result Both topological and Hodge-theoretic analogues of the injectivity part of the section conjecture hold for families of curves, but the topological analogue of the surjectivity part does not hold in general.
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.
Compact manifolds with positive scalar curvature have negative Kodaira dimension.
problem Understanding the properties of compact manifolds with positive scalar curvature.
method Analyzing the canonical bundle and complex structures on compact Riemannian manifolds.
result Compact manifolds with positive scalar curvature have negative Kodaira dimension.
Yamabe invariants of certain non-Kähler surfaces are zero.
problem Determining the sign of Yamabe invariants for non-Kähler surfaces.
method Analyzing Inoue surfaces and Kodaira surfaces, their blowups, and applying Seiberg-Witten theory.
result Yamabe invariants of Inoue surfaces and their blowups are all zero.
Study groups of order 64 and non-homeomorphic double Kodaira fibrations with same invariants.
problem Investigate finite quotients of braid groups and their quotients.
method Use algebraic and geometric methods to classify groups and construct fibrations.
result Prove that for groups of order 64, if not 32, then order is at least 64, and classify cases where equality holds.
Holomorphic tensors on Vaisman manifolds are invariant under the Lee field.
problem Characterizing holomorphic tensors on Vaisman manifolds.
method Using the parallelism of the Lee form and properties of the Lee field.
result The Kodaira dimension of Vaisman manifolds is invariant under certain quotients.
Paper introduces a new type of canonical metric for varieties with intermediate Kodaira dimension.
problem Finding canonical metrics for varieties with intermediate Kodaira dimension.
method Introducing a new notion of canonical metric and proving conditions for existence of relative Kähler-Einstein metric.
result Established conditions for the existence of relative Kähler-Einstein metric.
In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
New examples of double Kodaira fibrations found using surface braid groups.
problem Finding new examples of double Kodaira fibrations.
method Using finite Galois covers of a product of curves and explicit group epimorphisms from surface braid groups to finite Heisenberg groups.
result Every curve of genus b is the base of a double Kodaira fibration; the number of non-isomorphic surfaces is at least $oldsymbolω(b+1)$.