Negative curvature implies ample canonical bundle on manifolds.
problem Conditions for ample canonical bundles on complex manifolds.
method Use of negative holomorphic sectional curvature to prove ample canonical bundle.
result Projective manifolds with negative holomorphic sectional curvature have ample canonical bundles.
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.
Compact Kahler manifolds with nonpositive holomorphic sectional curvature have nef canonical bundle.
problem Characterizing compact Kahler manifolds based on their curvature properties.
method Utilizing the recent solution of Wu-Yau's conjecture in the projective case.
result Compact Kahler manifolds with negative holomorphic sectional curvature have ample canonical bundle.
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.
Compact complex manifolds with specific curvature properties have positive canonical bundles.
problem Positivity of canonical bundles in complex geometry.
method Analysis of Kähler metrics with quasi-negative holomorphic sectional curvature.
result Compact complex manifolds with non-positive and strictly negative holomorphic sectional curvature have positive canonical bundles.
Describes a contact structure on a surface's cotangent bundle.
problem Contact structures on cotangent bundles of surfaces.
method Explicit open book decomposition.
result Explicit decomposition for canonical contact structure.
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 canonical 1-forms on Lie-Rinehart structures of Weil bundles.
problem Properties of differential operators on Weil bundles.
method Construction of canonical 1-forms for Lie-Rinehart structures on Weil bundles.
result Construction of canonical 1-forms associated with Lie-Rinehart structures on Weil bundles.
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.
Characterizes trivial canonical bundle on complex supermanifolds.
problem Triviality of canonical bundle on complex supermanifolds.
method Algebraic characterisation using Batalin-Vilkovisky superalgebra and complex integral forms.
result Explicit formula for Calabi-Yau case in terms of Levi-Civita connection.
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 …
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…
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.
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.
We give a differential geometric description of the Cartan (or tractor) bundle and its canonical connection in CR geometry, thus offering a direct, alternative, definition to the usual abstract approach.
Study on Kähler manifolds with non-positive mixed curvature and its implications.
problem Characterizing properties of Kähler manifolds with non-positive mixed curvature.
method Analyzing the canonical line bundle properties based on curvature types.
result Canonical line bundle is nef, big, and ample under certain curvature conditions.
We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…
Study compact Kähler manifolds with nonpositive holomorphic sectional curvature and their canonical bundles.
problem Characterizing compact Kähler manifolds with nonpositive holomorphic sectional curvature and properties of their canonical bundles.
method Analyzing properties of Hermitian metrics and canonical bundles on compact Kähler manifolds.
result Proves nefness of canonical bundle and ampleness in complex dimension two for negative holomorphic sectional curvature.
Geometric equation defines canonical metrics on vector bundle families.
problem Finding canonical metrics on families of holomorphic vector bundles.
method Introducing a geometric partial differential equation for families of holomorphic vector bundles.
result Construction of Hermite--Einstein metrics in adiabatic classes on product manifolds and proof of the existence of a unique solution for the Dirichlet problem.
Compact Hermitian manifolds with quasi-negative curvature have ample canonical line bundles.
problem Determining conditions for ample canonical line bundles in Hermitian manifolds.
method Hermitian curvature flow with specific curvature conditions.
result Canonical line bundle is ample under given curvature conditions.
We give a representation of canonical vector bundles over Grassmannian manifolds as non-compact affine symmetric spaces as well as their Cartan model in the group of the Euclidean motions.
Study on Kähler-Ricci flow on manifolds with big canonical bundle.
problem Analyzing convergence of Kähler-Ricci flow on manifolds of general type.
method Normalized Kähler-Ricci flow, viscosity theory for degenerate complex Monge-Ampère flows.
result Convergence to singular Kähler-Einstein metric in canonical class.
The paper explores geometric structures on Weil bundles and their canonical lifts.
problem Transfer of geometric structures from a manifold to its Weil bundle.
method Utilizes differential geometric properties and Weil projection to lift structures.
result Demonstrates canonical lifts of various geometric structures to Weil bundles.
Constructs a geometric representation for holomorphic vector bundles.
problem Representing the second Beilinson-Chern class of holomorphic vector bundles.
method Constructs holomorphic bundle 2-gerbes to represent the second Beilinson-Chern class.
result Establishes precise relationship between holomorphic and smooth gerbes.
Study log canonical thresholds on curved metrics on group compactifications.
problem Computing log canonical thresholds for curved metrics on group compactifications.
method Associated convex functions to metrics, using asymptotic behavior to determine thresholds.
result Formula for alpha invariant in terms of polytope associated to group compactification.
Classifies fat bundles over homogeneous spaces with conditions.
problem Classifying fat bundles over compact homogeneous spaces.
method Generalizing Berard-Bergery's classification for arbitrary G-structures.
result Necessary conditions for the existence of fat bundles.
Canonical maps connect complex structures to Hitchin components.
problem Connecting higher complex structures to Hitchin components.
method Equivariant vector bundle construction and canonical isomorphisms.
result Canonical diffeomorphisms between spaces of higher complex structures and Hitchin components.
In this article we construct a canonical Kähler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical Kähler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical Kähler-Einstein current the c…
We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.
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.
The paper defines and analyzes n-fold vector bundles and their decompositions.
problem Understanding and decomposing n-fold vector bundles. method Introducing n-fold vector bundles as functors and studying their cores and decompositions. result Any n-fold vector bundle admits a non-canonical isomorphism to a decomposed n-fold vector bundle. Constructs Ricci-flat metrics on vector bundles over flag manifolds.
problem Finding explicit complete Ricci-flat metrics on vector bundles over flag manifolds.
method Explicit construction of metrics on vector bundles over flag manifolds of SU(n).
result Natural generalizations of known metrics on specific bundles.
New equivalences found between graded supermanifolds and vector bundles.
problem Understanding equivalences between graded supermanifolds and vector bundles.
method Explicit geometric constructions using supergeometry tools.
result Desuperization equivalence functor as a composition of canonical equivalences.
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.
The paper shows how compact Kähler manifolds with a special bundle can be broken down into simpler parts.
problem Understanding compact Kähler manifolds with a specific type of anti-canonical bundle.
method Introduced a new approach to extend a strategy from smooth projective varieties to singular Kähler spaces, using a flatness criterion for pseudo-effective sheaves.
result Compact Kähler manifolds with a nef anti-canonical bundle admit a locally trivial fibration with rational connected fibers and a Calabi-Yau base.
A new proof connects curvature negativity to bundle positivity using Kähler-Ricci flow.
problem Negativity of holomorphic sectional curvature and positivity of canonical bundles for Kähler manifolds.
method Using the Kähler-Ricci flow approach.
result Established the connection between curvature negativity and bundle positivity.
Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.
problem Defining and rigorously studying higher differential geometric objects like stringor bundles.
method Developed a framework of 2-Hilbert bundles, including an associated bundle construction.
result Proves the Stolz-Teichner stringor bundle is canonically isomorphic to the 2-Hilbert bundle.