The paper explores curvature positivity on Kähler and quasi-Kähler flag manifolds.
problem Analyzing curvature positivity on specific geometric structures.
method Investigation of Griffiths and dual-Nakano positivity for curvature of Chern connections on Kähler and quasi-Kähler flag manifolds.
result Classification of Kähler flag manifolds with Griffiths semi-positive curvature and restrictions for quasi-Kähler flag manifolds.
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.
Paper proves direct image sheaf positivity for certain Kähler fibrations.
problem Proving positivity of direct image sheaves for Kähler fibrations.
method Uses singular Hermitian line bundles and Narasimhan-Simha metric.
result Direct image sheaf is singular Nakano semi-positive.
The paper proves extension theorems for complex manifolds with Levi q-concave domains.
problem Holomorphic extension theorems for complex manifolds with Levi q-concave domains. method The proof relies on holomorphic Morse inequalities, the Kohn-Rossi extension theorem, and a general Nakano-Griffiths inequality.
result Holomorphic extension theorems for (0,ℓ)-forms on Levi q-concave domains. Introduce generalized Ueda obstruction classes for line bundles and apply them to non-semi-positivity.
problem Non-semi-positivity of line bundles
method Introduce generalized Ueda obstruction classes and use Dolbeault resolution
result Recover classical examples and provide new examples of nef but not semi-positive line bundles
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.
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…
Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.
problem Investigate positivity of Bismut curvature in non-Kähler manifolds.
method Analyze Bismut-Griffiths-positivity under Hermitian curvature flows.
result Identify HCFs that do not preserve Bismut-Griffiths-positivity.
Unique solution found for Demailly's equation on stable bundles.
problem Existence of a Griffiths positively curved metric on Hartshorne ample vector bundles.
method Proved an essentially unique solution to a Hermitian-Einstein-type equation for stable bundles.
result The proposed approach by Demailly must be modified to tackle the conjecture.
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…
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. The paper studies Kähler manifolds with partially semi-positive curvature and rational connectedness.
problem Analyzing compact Kähler manifolds with partially semi-positive curvature and rational connectedness.
method Proving rational connectedness for manifolds with BC-p positive tangent bundles, and applying these results to curvature conditions. result Confirming a conjecture and generalizing results on rational connectedness and curvature conditions.
Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge--Ampère type, whose correspondi…
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.
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.
Metrics are semipositively curved if they meet a specific asymptotic condition.
problem Characterizing semipositively curved metrics in Hermitian geometry.
method Proving metrics are semipositively curved if and only if they satisfy an asymptotic extension property.
result Proves a specific condition for Griffiths semipositively curved metrics.
The Griffiths conjecture asserts that every ample vector bundle E over a compact complex manifold S admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on OP(E∗)(1) to induce a Griffiths …
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.
In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…
In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the L2-norm of the scalar curvature along the Kähler-Ricci flow.
Study on algebraic curves' invariants and vanishing criteria.
problem Vanishing criteria for Griffiths infinitesimal invariants of algebraic curves.
method Analysis of moduli space of smooth genus 4 curves, study of normal functions.
result Vanishing criteria for the Griffiths infinitesimal invariants of Ceresa normal function.
Study of weakly Kähler hyperbolic manifolds, proving Lang and Green-Griffiths conjectures.
problem Verifying Lang and Green-Griffiths conjectures for a new class of manifolds.
method Introducing weakly Kähler hyperbolic manifolds and investigating their spectral properties.
result Proving weakly Kähler hyperbolic manifolds are of general type and verifying various aspects of Lang and Green-Griffiths conjectures.
The paper examines positivity properties of singular Hermitian metrics.
problem Investigating positivity for singular Hermitian metrics.
method Exploring Griffiths, ω-trace, and RC positivity.
result These positivity notions imply cohomology vanishing and rational connectedness.
In this paper we study a particular version of the Hermitian curvature flow (HCF) over a compact complex Hermitian manifold (M,g,J). We prove that if the initial metric has Griffiths positive (non-negative) Chern curvature Ω, then this property is preserved along the flow. On a manifold with Griffiths non-negative …
We exhibit examples of projective varieties with degenerate Gauss mappings and determine numerical invariants of such varieties. Our examples provide counter-examples to an asserted structure theorem of Griffiths and Harris (Ann. Sci. ENS 1979).
Proves a conjecture about Riemann surfaces using PDEs.
problem Griffiths' conjecture on holomorphic vector bundles on compact Riemann surfaces.
method Combines techniques from Uhlenbeck-Yau and Pingali's reduction to prove a system of PDEs.
result Analytic proof of Griffiths' conjecture on compact Riemann surfaces.
The paper proves a structure theorem for compact Kähler manifolds with semi-positive holomorphic sectional curvature.
problem Understanding the fundamental groups of compact Kähler manifolds with specific curvature properties.
method Analyzing foliations and topological properties to prove a locally trivial fibration structure.
result Compact Kähler manifolds with semi-positive holomorphic sectional curvature admit a locally trivial fibration structure.
We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the re…
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.
Among other results, a compact almost Kähler manifold is proved to be Kähler if the Ricci tensor is semi-negative and its length coincides with that of the star Ricci tensor or if the Ricci tensor is semi-positive and its first order covariant derivatives are Hermitian. Moreover, it is shown that there are no compact a…
Griffiths' first obstruction formula for vector bundles is derived.
problem Extending holomorphic vector bundles from submanifolds.
method Explicit formula using Atiyah class.
result Formula for the first obstruction.
In this paper, we establish a structure theorem for a smooth projective variety X with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural generalization of the structure theorem proved by Howard-Smyth-Wu and Mok for holom…
In this paper, with the aim of establishing a structure theorem for a compact Kähler manifold X with semi-positive holomorphic sectional curvature, we study a morphism φ:X→Y to a compact Kähler manifold Y with pseudo-effective canonical bundle. We prove that the morphism φ is always smooth (that is, a subm…
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.
Two remarks on curvature properties of Kähler manifolds.
problem Curvature properties of Kähler manifolds.
method Analyzing semi-positive holomorphic sectional curvature and quasi-negative k-Ricci curvature. result For semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration equals the number of non-truly-flat directions. For quasi-negative k-Ricci curvature, the canonical bundle is ample. The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements abou…
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.
Given a holomorphic family f:X→S of compact complex manifolds and a relative ample line bundle L→X, the higher direct images Rn−pf∗ΩX/Sp(L) carry a natural hermitian metric. Using the explicit formula for the curvature tensor of these direct images, we prove that the d…
Geometric quantization results for Riemann surfaces with semi-positive line bundles.
problem Analyzing geometric quantization for Riemann surfaces with semi-positive line bundles.
method Exploring the Bergman kernel expansion and related results for induced Fubini-Study metrics, Toeplitz operators, and holomorphic torsion.
result Asymptotic results for holomorphic torsion and random sections.
Solved Demailly systems for Vortex bundles on manifolds.
problem Equivalence of Hartshorne ampleness and Griffiths positivity for vector bundles.
method Applied the continuity method to prove smooth solutions for the Vortex bundle.
result Smooth solutions exist for the proposed Demailly systems.
The paper studies m-positive currents and line bundles on complex manifolds.
problem Understanding m-positive currents and their properties on complex manifolds. method Introducing m-plurisubharmonic functions, proving vanishing theorems, and regularisation theorems using viscosity solutions. result Global and local regularisation theorems for m-semi-positive currents. Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.
problem Analyzing the Hermitian Calabi functional on complexified orbits of symplectic manifolds.
method Explicit formula for Hessian of Hermitian Calabi functional, semi-positive definiteness proof, and weak parabolicity of Hermitian Calabi flow.
result Hessian of Hermitian Calabi functional is semi-positive definite on complexified orbits.
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.
In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degener…
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.
Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.
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.
In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern-Ricci two-form generate a holomorphic, integrable distribution. This dis…