Existence proven for special metrics on certain complex manifolds.
problem Existence of constant scalar curvature Kähler metrics.
method Compact Kähler manifolds with semi-ample canonical bundles.
result Existence of constant scalar curvature Kähler metrics proven.
It is conjectured that the moduli b-divisor of the Kawamata-Kodaira canonical bundle formula associated to a klt-trivial fibration (X,B)→Z is semi-ample. In this paper, we show the semi-ampleness of an arbitrarily small perturbation of the moduli b-divisor by a fixed appropriate divisor which roughly speaking come…
Study shows Kähler-Ricci flow singularity type is consistent over time.
problem Understanding singularity types in Kähler-Ricci flow.
method Analyzes the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles.
result Singularity type at infinity is consistent regardless of initial metric.
Estimates curvature for long-time continuity method solutions.
problem Curvature estimates for long-time continuity method solutions.
method Adapting arguments from Kähler-Ricci flow to semi-ample canonical line bundles.
result Derives curvature bounds for product manifolds.
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.
Study local curvature of Kähler-Ricci flow on semi-ample manifolds.
problem Local curvature estimates of long-time solutions to Kähler-Ricci flow.
method Local curvature estimates using semi-ample canonical line bundles.
result Set of singular fibers where curvature blows up is independent of initial metric.
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…
The paper proves conditions for minimal compact Kähler manifolds with vanishing second Chern class.
problem Conditions for minimal compact Kähler manifolds with vanishing second Chern class.
method Study of the abundance conjecture and associated Iitaka fibrations.
result For a minimal compact Kähler manifold, the second Chern class vanishes if and only if the cotangent bundle is nef and the canonical bundle has numerical dimension 0 or 1.
Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler-Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.
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. It is proved by Kawamata that the canonical bundle of a projective manifold is semi-ample if it is big and nef. We give an analytic proof using the Ricci flow, degeneration of Riemannian manifolds and L2-theory. Combined with our earlier results, we construct unique singular Kahler-Einstein metrics with a global Rie…
The Kähler-Ricci flow yields bounded diameter and Ricci curvature for minimal models.
problem Estimating the diameter and Ricci curvature of long-time solutions of the Kähler-Ricci flow.
method Analyzing the semi-ample canonical line bundle and using Perelman's estimates.
result Uniform bounds on diameter and Ricci curvature for long-time solutions.
Flow proves Kähler-Einstein on minimal elliptic surfaces.
problem Proving Kähler-Einstein on minimal elliptic surfaces.
method Conical Kähler-Ricci flow with semi-ampleness assumption.
result Flow converges to Kähler-Einstein on canonical model.
Uniform diameter bound for Ricci-flat Kahler metrics on projective manifolds.
problem Bounding the diameter of Ricci-flat Kahler metrics on projective manifolds.
method Proving uniform diameter bounds for long time solutions of the normalized Kahler-Ricci flow under specific curvature conditions.
result The normalized Kahler-Ricci flow converges to the canonical model of the manifold.
Compact metrics found near Kähler manifold's canonical class.
problem Finding unique cscK metrics near Kähler manifold's canonical class.
method Proved existence and uniqueness of cscK metrics for Kähler classes near canonical class.
result Metric spaces are pre-compact in Gromov-Hausdorff sense.
We show that the scalar curvature is uniformly bounded for the normalized Kahler-Ricci flow on a Kahler manifold with semi-ample canonical bundle. In particular, the normalized Kahler-Ricci flow has long time existence if and only if the scalar curvature is uniformly bounded, for Kahler surfaces, projective manifolds o…
Study shows scalar curvature of a specific type of manifold converges to -m outside singular points.
problem Analyzing scalar curvature of Kahler-Ricci flow on manifolds with positive Kodaira dimension.
method Calabi-Yau fibration and normalized Kahler-Ricci flow approach.
result Scalar curvature converges to -m outside singular points.
Study on Kähler-Ricci flow's Hölder regularity on compact manifolds.
problem Hölder regularity of Kähler-Ricci flow on compact Kähler manifolds.
method Adapting Hein-Tosatti's method for collapsing Calabi-Yau metrics, uniform spatial Hölder estimate obtained for all time.
result Uniform spatial Hölder estimate of Kähler-Ricci flow for all time.
The paper classifies minimal projective varieties satisfying a specific equality.
problem Classifying minimal projective varieties with a specific equality.
method Established a structure theorem for minimal projective klt varieties satisfying Miyaoka's equality.
result Minimal projective klt varieties with Miyaoka's equality have semi-ample canonical divisors and specific Kodaira dimensions.
Researchers prove existence of cscK metrics on smooth minimal models.
problem Existence of constant scalar curvature Kähler metrics on compact Kähler manifolds.
method Direct proof showing existence on smooth minimal models and blowups.
result Compact Kähler manifolds with nef canonical bundle always admit cscK metrics.
Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a no…
Defines plurisubharmonic metrics on hybrid spaces and proves their canonical extensions.
problem Defining and analyzing plurisubharmonic metrics on hybrid spaces.
method Introduces a class of plurisubharmonic metrics on hybrid spaces and proves their canonical extensions.
result Canonical plurisubharmonic extensions of metrics on hybrid spaces are continuous and can be described in terms of canonical models.
Sufficient condition for log-continuity of complex Monge-Ampère solutions.
problem Ensuring log-continuity of solutions to complex Monge-Ampère equations.
method Analyzing compact Kähler manifolds and line bundles, providing sufficient conditions for log-continuity.
result Log-continuity of solutions to complex Monge-Ampère equations with Lp right-hand sides. The Kähler-Ricci flow converges to a negative Kähler-Einstein metric under certain conditions.
problem Regularity of long-time solutions to the Kähler-Ricci flow on compact manifolds.
method Parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau metrics.
result The Ricci curvature is uniformly bounded on compact subsets away from singular fibers when generic fibers are biholomorphic.
The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extensi…
Study on algebraic fiber spaces and their anti-canonical divisors.
problem Understanding positivity conditions and base loci of algebraic fiber spaces.
method Algebraic and analytic methods for positivity of direct image sheaves.
result Algebraic fiber spaces with semi-ample relative anti-canonical divisor have a product structure.
Study orders of canonical bundles over graph configuration spaces.
problem Determining bundle orders for planar and nonplanar graphs.
method Analyzing configuration spaces of graphs to find bundle orders.
result Bundle orders are 2 for planar and 4 for nonplanar graphs.
Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.
problem Characterize and produce examples of complex solvmanifolds with trivial canonical bundle.
method Characterize invariant trivializing sections using Koszul 1-form, provide algebraic obstructions, and exhibit specific examples.
result New examples of complex solvmanifolds with trivial canonical bundle and algebraic obstructions for triviality.
The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…
Analyzes canonical bundle formula in algebraic geometry.
problem Analyzes the canonical bundle formula in algebraic geometry.
method Uses L2 metrics and valuative equivalence of plurisubharmonic singularities. result Identifies the singularity of the Ohsawa measure and gives a partial answer to a semipositivity question.
The paper studies positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.
problem Positivity properties of cotangent bundles in complex hyperbolic manifolds with cusps.
method Analyzes intrinsic positivity properties of cotangent bundles using toroidal compactifications and ample line bundles.
result The cotangent bundle is ample modulo the boundary divisor for sufficiently small rational numbers.
Formula found for Calabi-Yau metrics on flag variety bundles.
problem Finding unique asymptotically conical Calabi-Yau metrics on flag variety canonical bundles.
method Generalizing the Calabi Ansatz to Kähler classes, deriving a simple formula.
result Explicit formula for metrics on canonical bundles of flag varieties.
Paper proves structure of compact Kähler 3-folds with specific bundles.
problem Characterizing compact Kähler 3-folds with nef anti-canonical bundles.
method Minimal Model Program, positivity of direct image sheaves, Q-conic bundles, orbifold vector bundles.
result Compact Kähler 3-folds with nef anti-canonical bundles are essentially one of three types.
We describe an explicit open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of a compact surface.
Stein and Weinstein structures are described for disk cotangent bundles of surfaces.
problem Characterizing Stein and Weinstein structures on disk cotangent bundles.
method Using Legendrian handlebody diagrams and symplectic/contact mappings.
result The canonical contact structure on the unit cotangent bundle of S is obtained via surgery.
Study Miyaoka-Yau inequality for certain projective manifolds.
problem Proving Miyaoka-Yau inequality for specific types of manifolds.
method Using recent work by K.~Zhang and delta-invariant introduced by Fujita and Odaka.
result Established Miyaoka-Yau type inequality for projective manifolds with nef anti-canonical line bundle.
Uniformizes compact Sasakian manifolds into circle bundles.
problem Deforming compact Sasakian manifolds to locally isomorphic forms.
method Criterion based on circle bundles of anti-canonical bundles over Hermitian symmetric spaces.
result Sasakian manifolds can be deformed to locally isomorphic forms.
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.
Study locates divisors in Hodge bundle with specific properties.
problem Locating effective divisors in the projectivized Hodge bundle.
method Computing the class of closures of loci of canonical divisors with specific conditions.
result Strata of canonical and bicanonical divisors with double zeros span extremal rays of pseudoeffective cones.
The paper describes Calabi-Yau metrics on complex flag manifolds using Lie theory.
problem Finding complete Calabi-Yau metrics on canonical bundles of complex flag manifolds.
method Using Lie theory and the Calabi ansatz technique to provide explicit examples of noncompact complete Calabi-Yau manifolds.
result Explicit examples of noncompact complete Calabi-Yau manifolds, including canonical bundles of non-toric flag manifolds.
The article describes canonical metrics on holomorphic fibre bundles.
problem Existence of canonical metrics on isotrivial Kähler fibrations.
method Induced from Hermite--Einstein connections on holomorphic principal bundles.
result Existence of optimal symplectic connections when principal bundles are polystable.
Paper proves Miyaoka-Yau inequality for certain Kähler manifolds.
problem Proving Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle.
method Estimate of the L2-norm of the scalar curvature along the Kähler-Ricci flow. result Proves Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle.
We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …
The ∂ˉJ operator over an almost complex manifold induces canonical connections of type (0,1) over the bundles of (p,0)-forms. If the almost complex structure is integrable then the previous connections induce the canonical holomorphic structures of the bundles of (p,0)-forms. For p=1 we can …
We consider the general Kähler-Ricci flows which exist for all time. The zeroth order control on the flow metric potential for various infinite time singularities is the focus. The possible semi-amplness for numerically effective classes serves as the main motivation.
Gradient and eigenvalue estimates for Kähler manifolds' canonical bundle.
problem Estimating Hodge Laplacian on (m,0) forms for Kähler manifolds. method New Bochner type formula involving Ricci curvature and scalar curvature gradient.
result Gradient and eigenvalue estimates depend only on Ricci curvature bound.
Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…
Flow analysis leads to metric completion in Kähler geometry.
problem Analyzing Kähler-Ricci flows on compact manifolds.
method Normalized Kähler-Ricci flow convergence to Gromov-Hausdorff limits.
result Metric completion of twisted Kähler-Einstein metric.