Proves CM line bundle is ample on moduli space of KSBA stable varieties.
problem Ensuring the CM line bundle is ample on moduli space.
method Proved using KSBA stability.
result CM line bundle is ample on moduli space.
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48 results for “Ample bundle”
New system solves curvature for ample vector bundles, proving Griffiths conjecture.
problem Proving Griffiths conjecture on vector bundle positivity.
method Proposes Hermitian-Yang-Mills elliptic system for curvature.
result Solutions provide metrics with positive curvature in Griffiths sense.
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.
Solves Demailly's system for direct sums of ample line bundles on Riemann surfaces.
problem Proving the existence of smooth solutions for Demailly's system.
method Used Demailly's system and Leray-Schauder degree theory to reduce the problem.
result Proved existence of smooth solutions for direct sums of ample line bundles.
Explains differences and similarities of strictly nef and ample vector bundles.
problem Characterizing geometry of projective manifolds with strictly nef bundles.
method Brief exposition on strictly nef and ample vector bundles.
result Differences and similarities between strictly nef and ample vector bundles.
The paper studies volumes of direct images for high tensor powers of ample bundles.
problem Understanding asymptotics of Monge-Ampère volumes for high tensor powers of ample line bundles.
method Analyzes the leading term of asymptotics and classifies bundles saturating a topological bound.
result Provides a characterization of bundles admitting projectively flat Hermitian structures in the case of high symmetric powers of ample vector bundles.
Proves polynomial injectivity of Fubini-Study map for ample line bundles.
problem Injectivity of Fubini-Study map for ample line bundles.
method Polynomial injectivity proof with polynomial dependence on ample line bundle exponent.
result Quantitative version of injectivity proved, polynomial in ample line bundle exponent.
Shows CM line bundles are ample on K-stable varieties.
problem Ensuring CM line bundles are ample on K-stable varieties.
method Analyzes CM line bundles on K-stable varieties and their families.
result CM line bundles are ample on K-stable varieties with maximal variation.
Study on curvature of direct image bundle for higher powers of relatively ample line bundles.
problem Analyzing the curvature of direct image bundles associated with higher powers of relatively ample line bundles.
method Obtained asymptotic of curvature for -metric and Qullien metric on direct image bundles up to lower order terms.
result The analytic torsion satisfies .
In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if is an ample vector bundle over a compact Kähler manifold , $S^kE\…
Proves complex Finsler bundles with positive curvature are ample and biholomorphic to projective space.
problem Solving a 1975 problem posed by S. Kobayashi about complex Finsler vector bundles with positive Kobayashi curvature.
method Analyzes properties of complex Finsler vector bundles with positive Kobayashi curvature.
result Complex Finsler vector bundles with positive Kobayashi curvature are ample and biholomorphic to projective space.
Continuous metrics on ample bundles lie in infinite-dimensional cones.
problem Understanding the structure of positive metrics on ample line bundles.
method Analyzing bounded graded filtrations and embedding into Mabuchi-flat cones.
result Continuous metrics embed isometrically into the space of positive metrics.
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.
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.
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.
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.
The paper proves conditions for vector bundles to be Kobayashi and Griffiths positive.
problem Conditions for vector bundles to be Kobayashi and Griffiths positive.
method Comparing the curvature of and for large and using duality of convex Finsler metrics.
result Conditions for vector bundles to be Kobayashi and Griffiths positive.
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 paper extends positivity results from vector bundles to Kobayashi positive ones.
problem Extending positivity results from vector bundles to Kobayashi positive ones.
method Using convexity of Kobayashi positive Finsler metrics and duality for convex Finsler metrics.
result The quotient and tensor product of Kobayashi positive vector bundles are also Kobayashi positive.
It is conjectured that the moduli b-divisor of the Kawamata-Kodaira canonical bundle formula associated to a klt-trivial fibration 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…
The paper studies geodesic-Einstein metrics on line bundles over fibration.
problem Characterizing geodesic-Einstein metrics on line bundles over holomorphic fibrations.
method Introduced geodesic-Einstein flow and defined S-classes and C-classes to study the properties of geodesic-Einstein metrics.
result Proved that is nonlinear semistable under certain conditions.
We prove that the metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle ten…
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 study of Chern numbers on vector bundles uses combinatorial methods to establish bounds and ordering.
problem Establishing bounds and ordering for Chern numbers on vector bundles.
method Combinatorial ideas to study Chern numbers on ample and numerically effective vector bundles.
result An effective lower bound for Chern numbers of ample vector bundles and reverse dominance ordering for nef vector bundles.
Characterizes projective submanifolds in high dimensions.
problem Characterizing projective submanifolds of specific codimensions.
method Uses Chern classes and ample line bundles.
result Generalizes earlier findings for hypersurfaces.
Uniformizes singular varieties in higher dimensions.
problem Problems in uniformizing higher-dimensional minimal varieties.
method Introduces technical concepts and sketches proofs.
result Recent uniformization theorems for singular varieties.
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.
The unit ball is characterized by a Kähler-Einstein potential.
problem Characterizing the unit ball in complex geometry.
method Using a global potential function of the Kähler-Einstein metric.
result A compact Kähler manifold with an ample canonical bundle is the unit ball if it has a specific potential function.
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.
Proves projectivity and ampleness of a Kähler manifold using complex Monge-Ampère equation.
problem Projectivity and ampleness of a compact Kähler manifold with negative Ricci curvature.
method An alternate proof using a -estimate for a complex Monge-Ampère equation.
result Proves projectivity and ampleness of the Kähler manifold.
The paper proves a CM line bundle is ample on K-moduli spaces.
problem Proving positivity of the CM line bundle on K-moduli spaces.
method Developed a new invariant for filtrations to test K-stability.
result Proves the CM line bundle is ample on reduced uniformly K-stable Fano varieties.
Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfacesmath.AG
Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.
problem Characterizing curves that can be vanishing cycles in degenerations of linear systems.
method Computing mapping class group-valued monodromy and identifying it with r-spin mapping class groups.
result Identifies simple closed curves as vanishing cycles and provides characterizations of discriminants and Lefschetz fibrations.
Let X be a smooth complex projective variety of dimension d. It is classical that ample line bundles on X satisfy many beautiful geometric, cohomological, and numerical properties that render their behavior particularly tractable. By contrast, examples due to Cutkosky and others have led to the common impression that t…
The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.
problem Understanding the ampleness of canonical line bundles for Kähler manifolds with specific curvature properties.
method Introducing a new notion of almost quasi-negative holomorphic sectional curvature and extending the theorem to this setting.
result The theorem is extended to compact Kähler manifolds with almost quasi-negative holomorphic sectional curvature, and a gap-type theorem is derived.
Formula found for minimum Ricci curvature in Fano bundles.
problem Finding the minimum Ricci curvature in Fano homogeneous toric bundles.
method Derived a formula for the greatest lower bound on Ricci curvature.
result Formula for the minimum Ricci curvature in Fano bundles.
The paper examines conditions for representing Chern-Weil forms in two scenarios.
problem When can a Chern-Weil form be represented by a given form with a suitable metric?
method Investigates semi-stable Hartshorne-ample vector bundles and a specific rank-2 bundle over a product of a Riemann surface and the sphere.
result Evidence provided for a conjecture of Griffiths by producing metrics whose Chern forms are positive.
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.
Study of monodromy and vanishing cycles for complete intersection curves.
problem Computing topological monodromy of complete intersection curves.
method Innovative tools for studying monodromy of tensor products of very ample line bundles, induction on multi-degree.
result Answer given by the r-spin mapping class group associated to the maximal root of the adjoint line bundle.
The log term of Szego Kernelmath.DG
In this paper, we study the relations between the log term of the Szegö kernel of the unit circle bundle of the dual line bundle of an ample line bundle over a compact Kählermanifold. We proved a local rigidity theorem. The result is related to the classical Ramadanov Conjecture.
The paper establishes conditions for optimal sampling configurations on complex manifolds.
problem Finding optimal sampling configurations on complex manifolds.
method Analyzes point configurations on compact complex manifolds using tensor powers of Hermitian ample line bundles.
result Necessary and sufficient conditions for the existence of asymptotically Fekete sequences.
Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavio…
Let be a smooth projective complex variety with an ample line bundle , and let be a simple normal crossing divisor. We establish the Kobayashi-Hitchin correspondence between tame harmonic bundles on and -stable parabolic -flat bundles with trivial characteristic numbers on . Especially, …
Uniform scalar curvature bound for conical Kähler-Ricci flow solutions.
problem Bounding scalar curvature for conical Kähler-Ricci flow solutions.
method Uniform scalar curvature bound proof with semi-ampleness condition and uniform estimates for potentials and their derivatives.
result Uniform scalar curvature bound established for conical Kähler-Ricci flow solutions.
On characteristic forms of positive vector bundles, mixed discriminants and pushforward identitiesmath.AG
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
The spectrum of the Laplace-Dolbeault operator for any line bundle with parallel curvature on a flat complex torus is computed. The Ray-Singer analytic torsion is then deduced, generalizing thus Bost's result for ample line bundles and Ray-Singer's ones for flat bundles, of which we a geometric interpretation is given.
We introduce several families of filtrations on the space of vector bundles over a smooth projective variety. These filtrations are defined using the large k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted by the kth power of an ample line bundle. The filtrations measure the failure of the…
The paper establishes inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.
problem Establishing inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.
method Applying effective very ampleness of adjoint bundles, log-concavity, and Khovanskii-Teissier inequalities.
result For any projective manifold X and ample line bundle L, there exists a universal bivariate polynomial Q_λ(x, y) with deg Q ≤ d, such that the inequality holds.