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.
The Kähler-Ricci flow on certain manifolds collapses to a canonical metric.
problem Understanding the behavior of Kähler-Ricci flow on compact manifolds.
method Asymptotic expansion of evolving metrics and analysis of the Iitaka fibration.
result The flow collapses to a canonical metric on the base of the Iitaka fibration.
The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.
problem Estimating higher-order curvature along Kähler-Ricci flows on compact Kähler manifolds.
method Proving uniform bounds for Ricci curvature and scalar curvature in various orders and norms.
result A geometric obstruction causes a specific third-order derivative of Ricci curvature to blow up at rate et/2. 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.
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 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.
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 -∞.
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.
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.
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…
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.
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 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.
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.
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.
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.
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.
This is a survey on the various notions of Kodaira dimension in low dimensional topology. The focus is on progress after the 2006 survey [78].
We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so…
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.
Study of twisted Kähler-Einstein metrics on Calabi-Yau spaces with singularities.
problem Understanding the collapsed Gromov-Hausdorff limits of Calabi-Yau spaces.
method Analyzing the geometry of twisted Kähler-Einstein metrics on holomorphic fiber spaces.
result Proving the existence of conical-type singularities in the base of fiber spaces.
The paper studies Kähler-Einstein metrics on fiber spaces with positive Kodaira dimension.
problem Understanding Kähler-Einstein metrics on fiber spaces with positive Kodaira dimension.
method Analyzes the properties of singular Kähler-Einstein metrics and their curvature.
result The fiberwise singular Kähler-Einstein metric induces a semipositively curved metric on the relative canonical bundle.
Symplectic 4-manifolds with Kodaira dimension zero can be viewed as symplectic Calabi-Yau surfaces. We are able to completely determine their Betti numbers by proving two general results on quaternionic vector bundles.
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.
Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particul…
Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension
problem Convergence of scalar curvature in Kähler-Ricci flow
method Uniform μ-entropy or uniform Sobolev inequality result Scalar curvature converges to negative Kodaira dimension
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.
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. Study on Kähler-Ricci flow's infinite-time singularities.
problem Understanding singularities in Kähler-Ricci flow.
method Relates flow's singularity type to fibration's indexes.
result Observation of singularity type's relation to fibration indexes.
The paper explores Newton-Cartan structures with torsion on Kodaira moduli spaces.
problem Understanding Newton-Cartan spacetimes and their deformations.
method Construction of connections and frames on Kodaira moduli spaces, generalizing canonical connections to include torsion.
result Novel twistor theories of Newton-Cartan spacetimes in three and five dimensions, including torsion.
The existence of Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the Kähler-Ri…
We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Koda…
We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension −∞. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although th…
Studying the behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampère equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized Kähler-Ricci flow on mildly singula…
We prove that if a closed oriented 4-manifold X fibers over a 2- or 3-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4-manifold X which is not a tower of torus bundles and fibering over a 2- or 3-dimensional manifold does not admit a torsion sy…
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.
This is a survey on symplectic birational geometry. In arbitrary dimension, this subject is centered around the notion of uniruledness. In low dimensions, we will also discuss Kodaira dimension and minimality.
The paper studies complex Finsler metrics and curvature inequalities.
problem Analyzing holomorphic sectional curvature in complex Finsler manifolds.
method Developed an inequality relating holomorphic sectional curvature and proved Schwarz Lemma.
result Complex Finsler manifolds with semi-positive but not identically zero holomorphic sectional curvature have negative Kodaira dimension under certain conditions.
New invariant distinguishes real algebraic surfaces.
problem Distinguishing real algebraic surfaces up to birational diffeomorphism.
method Introducing real (logarithmic)-Kodaira dimension.
result Constructs infinite families of non-birationally diffeomorphic surfaces.
Researchers solve Calabi-Yau equation on Kodaira-Thurston manifold.
problem Solving the Calabi-Yau equation on a specific manifold.
method Using an invariant almost-Kaehler structure and an ansatz to reduce the problem to a Monge-Ampère equation.
result The problem is reduced to a Monge-Ampère equation under certain conditions.
Study new invariant to measure flat directions on complex manifolds.
problem Investigate numerical rank invariant on projective Kähler manifolds with semi-negative holomorphic sectional curvature.
method Introduce and analyze a new differential geometric numerical rank invariant.
result Bounding the new invariant by nef dimension and numerical Kodaira dimension.
The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integ…
The paper studies Ricci curvature on Kähler-Ricci flow.
problem Analyzing Ricci curvature on Kähler-Ricci flow.
method Examining n-dimensional compact Kähler manifolds with semi-ample canonical line bundles under Kähler Ricci Flow.
result Ricci curvature converges to negative of generalized Kähler Einstein metric ωB locally away from singular set. Study on compact toric locally conformally Kähler manifolds, finding specific properties.
problem Characterizing properties of compact toric locally conformally Kähler manifolds.
method Analyzing Kodaira dimension, using specific examples and mappings.
result Kodaira dimension is -∞ for underlying complex manifolds, and specific properties for surfaces and Vaisman manifolds.