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…
The paper proves properties of image structures of fibrations with semi-positive curvature.
problem Structures and images of maximal rationally connected fibrations of projective manifolds with semi-positive holomorphic sectional curvature.
method Generalizes Yang's solution using RC positivity for Yau's conjecture.
result The canonical bundle of images of such fibrations is not big.
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.
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 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.
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 studies morphisms of compact Kähler manifolds with semi-positive holomorphic sectional curvature.
problem Establishing structure theorems for compact Kähler manifolds with semi-positive holomorphic sectional curvature.
method Analyzing morphisms to compact Kähler manifolds with pseudo-effective canonical bundles, proving smoothness and isomorphism of fibers, and applying maximal rationally connected fibrations.
result Compact Kähler manifolds with semi-positive holomorphic sectional curvature are rationally connected and have uniformization properties.
The paper solves Yau's conjecture and generalizes structure theorems for projective manifolds with semi-positive holomorphic sectional curvature.
problem Projective manifolds with semi-positive holomorphic sectional curvature.
method Combining previous work with the theory of holomorphic foliations, the authors prove that the universal cover of such manifolds is biholomorphic and isometric to a product of a flat metric and a Kähler metric.
result The structure theorem for projective manifolds with semi-positive holomorphic sectional curvature, including the solution for Yau's conjecture.
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.
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.
Proves CM line bundle positivity for K-stable klt Fano varieties.
problem Proving ampleness of the CM line bundle for K-stable klt Fano varieties.
method Algebraic approach, including probability theory for limit computations.
result Proves semi-positivity and positivity statements for K-semi-stable and uniform K-stable cases.
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…
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 generalizes Bochner Laplacian results to Riemann surfaces.
problem Analyzing curvature vanishing line bundles on Riemann surfaces.
method Exploiting the relation of Bochner Laplacian on tensor powers with sR Laplacian.
result Bergman kernel expansion for semi-positive line bundles.
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. 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.
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.
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.
We use Liu-Tian's virtual moduli cycle methods to construct detailedly the explicit isomorphism between Floer homology and quantum homology for any closed symplectic manifold that was first outlined by Piunikhin, Salamon and Schwarz for the case of the semi-positive symplectic manifolds.
For one-parameter degenerations of compact Kähler manifolds, we determine the asymptotic behavior of the first Chern form of the direct image of a Nakano semi-positive vector bundle twisted by the relative canonical bundle, when the direct image is equipped with the L2-metric.
Solves open problems on curved projective varieties.
problem Structure theorems for curved projective varieties.
method Supplements and proposes open problems.
result Provides new insights into structure of curved projective varieties.
We define higher genus Gromov-Witten invariants and establish a mathematical theory of sigma model coupled with gravity over any semi-positive symplectic manifolds. As applications, we verify the stablizing conjecture of symplectic 4-manifolds for simply connected elliptic surfaces and construct smooth 6-manifolds admi…
Generalized Tanaka prolongation ensures convergence of formal embeddings of complex manifolds.
problem Ensuring convergence of formal embeddings of complex manifolds under weaker conditions.
method Formulated and proved generalized Tanaka prolongation for geometric structures.
result Convergence of formal embeddings holds under weaker semi-positive normal bundle conditions.
For proper surjective holomorphic maps from K"ahler manifolds to analytic spaces, we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by Nakano semi-positive vector bundles by means of the higher direct image sheaves, by using the theory of harmonic integrals developed by Takegoshi…
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.
The paper studies complex Finsler metrics and curvature inequalities.
problem Analyzing holomorphic sectional curvature in complex Finsler manifolds.
method Developed an inequality relating holomorphic sectional curvature and proved Schwarz Lemma.
result Complex Finsler manifolds with semi-positive but not identically zero holomorphic sectional curvature have negative Kodaira dimension under certain conditions.
In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, …
Estimates for scalar curvature equations on Kähler manifolds with singularities.
problem Developing estimates for scalar curvature equations with singular metrics.
method Estimates and Laplacian estimates for scalar curvature equations of degenerate Kähler metrics.
result Derivation of estimates for singular constant scalar curvature Kähler metrics and singular Kähler-Einstein metrics.
Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are non-fillable. Using contact surgery on his examples we create on every sphere S^{2n-1}, n>1, an exotic contact structure ξ_- that also contains a plastikstufe. …
We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…
We establish a product formula for Gromov-Witten invariants for closed, connected, relatively semi-positive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps with marked poin…
Study of J-flow on Kähler manifolds confirms energy properness.
problem Properness of Mabuchi K-energy on Kähler manifolds.
method Degenerate twisted J-flow on compact Kähler manifolds.
result Flow converges to weak solution of degenerate twisted J-equation.
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.
Establishes metrics with positive curvature on projective line bundles.
problem Existence of complete Kähler metrics with semi-positive holomorphic sectional curvature.
method Calabi's Ansatz and product approach.
result Existence of complete Kähler metrics with many zeroes.
Master thesis proves Bergman kernel asymptotics for positive line bundles.
problem Proving asymptotic expansion of Bergman kernel for positive line bundles.
method Introduced a semi-classical symbol space and symbolic calculus.
result Established pointwise asymptotic expansion on positive parts of certain semi-positive line bundles.
Debt swaps improve financial networks by optimizing clearing payments and stability.
problem Improving financial network stability and efficiency through debt swaps.
method Analyzing computational complexity of debt swaps, focusing on semi-positive swaps and v-improving swaps.
result Polynomial length of sequences of semi-positive v-improving swaps for ranking-based clearing, but NP-hard for arbitrary v-improving swaps.
We propose a new point of view on quantum cohomology, strongly motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is the D-module which "quantizes" a commutative algebra associated to the (uncompactified) space of rational curves. A stand…
Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin-Toeplitz quantization of the open set of M where the curvature on L is non-degenerate. The quantum spaces are the spectral spaces corresponding to [0,k−N] (N>1 fixed), of the Kodaira…
In this paper we consider Riemannian manifolds (Mn,g) of dimension n≥5, with semi-positive Q-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Gree…
Defines complex structure for families of Hilbert spaces with reasonable curvature.
problem Curvature of families of Hilbert spaces not forming a holomorphic bundle.
method Defines a new complex analytic structure and curvature for families of Hilbert spaces.
result New proof of Berndtsson's theorem on curvature of direct images of semi-positively twisted relative canonical bundles.
Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.
problem Degeneration of asymptotically conical Ricci-flat Kähler metrics.
method Analysis of Kähler class degeneration and convergence of metrics.
result Construction of singular Calabi-Yau metrics and their metric geometry.
Compact metrics found for Riemannian manifolds with controlled curvature.
problem Finding compact metrics with controlled curvature.
method Analyzing metrics conformal to a standard sphere with bounded \(Q\)-curvature and eigenvalue.
result Precompactness result in \(C^α\)-Hölder topology.
The paper explores mixed curvature for Hermitian manifolds and its implications.
problem Investigating the properties of mixed curvature for Hermitian manifolds.
method Analyzing convex combinations of first Chern Ricci curvature and holomorphic sectional curvature.
result Compact Hermitian surfaces with constant mixed curvature are Kähler unless specific conditions are met.