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.
The paper defines new types of positivity and proves properties of Schur forms for vector bundles.
problem Defining and characterizing new types of positivity for vector bundles.
method Introducing and characterizing two types of strongly decomposable positivity, proving properties of Schur forms.
result Schur forms of strongly decomposable positive vector bundles are positive or weakly positive, answering a question of Griffiths.
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.
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 E is an ample vector bundle over a compact Kähler manifold X, $S^kE\…
Study on positivity properties of vector bundle Monge-Ampère equation.
problem Analyzing positivity in vector bundle Monge-Ampère equation.
method Investigates MA-positivity and MA-semi-positive solutions for different ranks of holomorphic bundles over complex surfaces and manifolds.
result Positivity preservation in rank-two holomorphic bundles but not in higher ranks.
Survey on positivity in vector bundles, inspired by Brunn-Minkowski theorems.
problem Positivity of vector bundles in complex analysis, Kähler geometry, and algebraic geometry.
method Inspiration from Brunn-Minkowski theorems.
result Survey results on positivity in vector bundles.
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.
Solves a Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
problem Solving Monge-Ampère type equations for Nakano positive curvature tensors of holomorphic vector bundles.
method Solves the Monge-Ampère type equation in the conformal class of a Nakano positive Hermitian metric.
result Solves the Monge-Ampère type equation for Nakano positive curvature tensors of holomorphic vector bundles.
The paper proves positivity of third Chern form for certain vector bundles.
problem Proving positivity of third Chern form for Griffiths positive vector bundles.
method Analyzing mixed discriminants and Schur forms.
result Positivity of third Chern form for Griffiths positive vector bundles.
Characterizes curvature positivity for Riemannian metrics on flat vector bundles.
problem Characterizing Nakano positivity of Riemannian flat vector bundles.
method Using solvability of the d equation with specific L2 estimates and inspired by recent works on Hermitian holomorphic vector bundles. result Alternative proof of matrix-valued Prekopa's theorem.
Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.
problem Positivity of characteristic forms in vector bundles.
method Explicit computation of Chern curvature and use of flag bundles.
result Positivity of polynomials in Chern forms for Griffiths semipositive bundles.
The paper defines positivity for singular metrics on vector bundles and proves related theorems.
problem Positivity of singular Hermitian metrics for holomorphic vector bundles.
method The method of Berndtsson and Lempert, along with a Berndtsson-type positivity theorem for holomorphic vector bundles.
result Sharp L2 extension theorem for holomorphic vector bundles. 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 (detE∗)k and SkE for large k and using duality of convex Finsler metrics. result Conditions for vector bundles to be Kobayashi and Griffiths positive.
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.
We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M with positive holomorphic sectional curvature and any holomorphic vector bundle E over M, the projectivized vector bundle P(E) admits a Kähler metric with positive holomorphic sectional curvature.
Alternative metric defined on vector bundles, proving vanishing theorem.
problem Defining singular Hermitian metrics on vector bundles.
method Alternative definition of singular Hermitian metric, discussing Griffiths and Nakano positivities.
result Generalised Griffiths' vanishing theorem proved.
Constructs a convex Finsler metric on vector bundles under specific conditions.
problem Creating a convex Finsler metric on vector bundles with positive curvature.
method Uses the negativity of direct image bundles and Minkowski inequality for norms.
result Shows how to upgrade a Kobayashi positive Finsler metric to a convex one.
The paper proves a condition for a metric on a vector bundle to be Griffiths positive.
problem Proving that a hermitian metric on a vector bundle is Griffiths positive.
method Defined a flow on the projective bundle and argued for its convergence.
result The relative Kähler-Ricci flow on the projective bundle converges.
The paper proves a pointwise Gysin formula for vector bundles and applies it to show positivity of polynomials.
problem Positivity of polynomials in Chern forms of Griffiths semi-positive vector bundles.
method Develops a pointwise Gysin formula for hermitian vector bundles and applies it to show positivity.
result Positivity of several polynomials in Chern forms of Griffiths semi-positive vector bundles.
Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.
problem Establishing conditions for optimal L2 extension in complex geometry.
method Analyzing singular Nakano positivity and applying L2 extension theorem.
result Necessary condition for equality in optimal L2 extension theorem.
Proves generic surjectivity of vector bundles via degeneration.
problem Understanding generic surjectivity of vector bundles.
method Uses degeneration argument, Berndtsson's theorem, and Lempert's proof.
result Generalizes previous work on L2 division theorem. Proves curvature positivity of invariant direct images in complex geometry.
problem Curvature positivity of invariant direct images in complex geometry.
method Compact group action and Hörmander's L2 theory of ∂ˉ. result Direct image of Nakano positive vector bundle is Nakano positive.
We prove that the L2 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…
The paper studies curvature properties of vector bundles and their applications to quasi-Fuchsian space.
problem Curvature positivity of Griffiths negative vector bundles and its implications.
method Analyzes Griffiths and Nakano positivity, calculates curvature, and estimates curvature operators.
result Constructs a Kähler metric on quasi-Fuchsian space extending the Weil-Petersson metric.
The paper introduces RC-positivity for vector bundles and proves manifold properties.
problem Understanding rational connectedness and positivity in complex manifolds.
method Introducing RC-positivity and proving properties of vector bundles and manifolds.
result Compact Kähler manifolds with positive holomorphic sectional curvature are projective and rationally connected.
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.
The paper characterizes positivity of holomorphic vector bundles via Lp-estimates and extensions.
problem Characterizing positivity of holomorphic vector bundles using Lp-estimates and extensions. method Introducing four conditions for Hermitian (or Finsler) vector bundles and characterizing Nakano and Griffiths positivity.
result Characterization of Nakano and Griffiths positivity via specific Lp-conditions. The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.
problem Analyzing vector bundles with singular Hermitian metrics and positivity.
method Develops harmonic theory and extends results from complex geometry.
result Extends Nakano's vanishing theorem to vector bundles with singular metrics.
New characterizations of curvature operators for specific forms via L2-estimates.
problem Characterizing semi-positive and semi-negative curvature operators for (n,q) and (p,n)-forms. method Using L2-estimates to characterize curvature operators for (n,q) and (p,n)-forms. result New characterizations of Nakano semi-positivity and semi-negativity.
The study explores the implications of curvature positivity in vector bundles and Hodge theory.
problem Understanding the existence of sections in semi-positive but not strictly positive vector bundles and the nature of Hodge metrics.
method Analyzes various positivity measures and their implications in algebraic geometry and Hodge theory.
result Provides insights into the existence of sections in semi-positive vector bundles and the nature of Hodge metrics.
New stability criteria for vector bundles linked to Hermite-Einstein geometry.
problem Stability of higher-rank vector bundles and their moduli spaces.
method Introducing m-positivity and a smooth function for coherent subbundles, linking to Hermite-Einstein geometry. result Hermite-Einstein bundles are uniformly semi-stable, and new stability conditions are established.
Constructs metrics for surfaces to show uniform positive scalar curvature.
problem Proving positive scalar curvature on surfaces and their bundles.
method Constructs complete Riemannian metrics on total spaces of vector bundles.
result Shows total space of tangent bundles on non-torus surfaces admit uniform positive scalar curvature.
Paper proves positivity of Chern-Weil forms for certain vector bundles.
problem Proving positivity of Chern-Weil forms for Griffiths semipositive vector bundles.
method Analyzing characteristic differential forms and Schur polynomials.
result Positivity of c1(E,h)∧c2(E,h)−c3(E,h) established. We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of nonnegative sectional curvature.
This paper addresses Cheeger and Gromoll's question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle which admits a metric of nonnegative curvature must admit a c…
We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then the product of E …
This paper is a sequel to \cite{Berndtsson}. In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kähler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Naka…
The paper connects curvature positivity to rational connectedness in complex geometry.
problem Establishing a geometric criterion for rational connectedness.
method Uhlenbeck-Yau's continuity method applied to mean curvature positivity.
result Holomorphic tangent bundle mean curvature positivity is equivalent to rational connectedness of compact Kähler manifolds.
We prove the logarithmic divergence of equivariant analytic torsion for one-parameter degenerations of projective algebraic manifolds, when the coefficient vector bundle is given by a Nakano semi-positive vector bundle twisted by the relative canonical bundle.
Introduces a new equation for complex surfaces, proving stability and inequalities.
problem Stability conditions involving higher Chern forms on complex surfaces.
method Vector bundle version of the complex Monge-Ampere equation, positivity condition (MA positivity), stability and inequalities.
result Proves stability and a Kobayashi-Lubke-Bogomolov-Miyaoka-Yau type inequality for positively curved solutions.
Uniform RC-positivity results for direct image bundles.
problem Understanding the relation between rational connectedness and RC-positivity.
method Analyzing vector bundles and their direct images, using weak RC-positivity as a starting point.
result Uniform RC-positivity of direct image bundles under weak RC-positivity conditions.
We classify compact Kähler manifolds with semi-positive holomorphic bisectional and big tangent bundles. We also classify compact complex surfaces with semi-positive tangent bundles and compact complex 3-folds of the form P(T∗X) whose tangent bundles are nef. Moreover, we show that if X is a Fano manifold such t…
The paper studies curvature properties of direct image bundles.
problem Investigating curvature properties of direct image bundles.
method Using subharmonic metrics and mean curvature analysis.
result Direct image bundles carry metrics with positive mean curvature.
New characterization of Riemannian metric positivity and L2 estimates for d operator.
problem Characterize positivity of Riemannian metrics and L2 estimates for d operator. method Apply L2 technique developed by Deng-Ning-Wang-Zhou, new characterizations given. result Prove new results parallel to Liu-Yang-Zhou's answer to Lempert's question.
Proves stability of certain vector bundles on Kähler surfaces.
problem Stability of rank 2 holomorphic vector bundles on Kähler surfaces.
method Proves existence of Z-positive and Z-critical metrics leading to bundle stability. result Proves stability results for deformed Hermitian Yang-Mills and almost Hermite-Einstein equations for rank 2 bundles.
The Euler class vanishes under certain curvature conditions.
problem Conditions for the vanishing of the Euler class in positive scalar curvature manifolds.
method Generalization of Lichnerowicz vanishing theorem for flat vector bundles.
result The Euler class vanishes for specific flat vector bundles over manifolds with positive scalar curvature.
Solves Lempert's question on Nakano semi-positivity preservation.
problem Preserving Nakano semi-positivity under limits of metrics.
method Establishes L2 extension theorem and uses multiplier submodule sheaves. result Affirmative solution to Lempert's question on Nakano semi-positivity.
Extends classical stability results to new geometric settings.
problem Stability of holomorphic vector bundles on complex manifolds.
method Introduces (ω,Ω)-Hermite-Einstein and (ω,Ω)-stable conditions. result Generalised Hermite-Einstein condition implies (ω,Ω)-semi-stability.