The paper characterizes projective spaces and quadrics using strictly nef bundles.
problem Characterizing projective spaces and quadrics using geometric properties of bundles.
method Analyzing strictly nef bundles on smooth projective varieties and curves.
result Strictly nef bundles on smooth projective varieties and curves have specific geometric properties.
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 smooth projective varieties with strictly nef tangent bundles and their properties.
problem Understanding smooth projective varieties with strictly nef tangent bundles.
method Analyzing the tangent bundle properties and using rational connectedness.
result Smooth projective varieties with strictly nef tangent bundles are rationally connected.
Projective manifolds with specific bundles are isomorphic to simpler spaces.
problem Characterizing projective manifolds with tangent bundles containing strictly nef subsheaves.
method Analyzing the structure of the tangent bundle and using properties of strictly nef subsheaves.
result Projective manifolds with the described bundles are isomorphic to projective bundles over hyperbolic manifolds or projective spaces.
The paper proves structures for complex projective varieties with certain tangent bundle properties.
problem Characterizing complex projective varieties based on properties of their tangent bundles.
method Analyzing the positivity of exterior powers of tangent bundles and using étale covers.
result Complex projective varieties with nef exterior powers of tangent bundles are Fano fiber spaces over Abelian varieties.
The paper studies algebraic fibre spaces with specific properties and proves key results about their structure.
problem Analyzing properties of algebraic fibre spaces with strictly nef relative anti-log canonical divisors.
method Using projective klt pairs and fibration techniques, the paper proves locally constant fibration properties and rational connectedness.
result The fibration is locally constant with rationally connected fibers, and the base is a canonically polarized hyperbolic projective manifold.
Study positive characteristic Fano 4-folds with nef tangent bundles.
problem Positive characteristic version of the Campana-Peternell conjecture for Fano 4-folds.
method Analyzes Fano 4-folds with nef tangent bundles in positive characteristic.
result Affirmative answer for Fano 4-folds with Picard number > 1 and nef tangent bundle.
Homogeneous Fano manifolds with specific bundle properties proven.
problem Characterizing Fano manifolds with nef tangent bundles.
method Proving homogeneity for specific Fano manifolds.
result Any Fano manifold of coindex three with nef tangent bundle is homogeneous.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…
Fundamental groups of certain Kähler orbifolds have polynomial growth.
problem Understanding the fundamental groups of specific types of orbifolds.
method Analyzing the orbifold fundamental group with respect to the nef anticanonical bundle.
result The orbifold fundamental group has polynomial growth.
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.
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.
Study the Albanese map for Kähler manifolds with nef anticanonical bundle.
problem Characterize the structure of the Albanese map for Kähler manifolds with nef anticanonical bundle.
method Analyze two cases: general fiber is Calabi-Yau or projective space. Provide proofs for both cases.
result For the case where the general fiber is Calabi-Yau, the manifold itself must be Calabi-Yau.
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.
The paper establishes inequalities for Chern classes and numbers on polarized manifolds and nef vector bundles.
problem Chern class and number inequalities on polarized manifolds and nef vector bundles.
method Sharp inequalities derived from polarized pairs and nef vector bundles.
result Bounding Chern numbers of nef vector bundles and classifying compact Kähler 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 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.
Study compact Kähler manifolds with pseudo-effective tangent bundles.
problem Characterize compact Kähler manifolds with strongly pseudo-effective tangent bundles.
method New proofs and characterizations of properties of vector bundles.
result Only projective spaces have big tangent bundles.
The paper studies foliations on smooth projective varieties and their properties.
problem Characterizing and understanding foliations on smooth projective varieties.
method Develops a structure theorem for smooth projective varieties with almost nef regular foliations, using a smooth morphism and MRC fibration.
result An almost nef regular foliation on a smooth projective variety can be decomposed into a numerically flat regular foliation and a smooth morphism.
Study of tangent bundle positivity on complex projective varieties.
problem Positivity of the second exterior power of tangent bundles on smooth complex projective varieties.
method Analyzes properties of tangent bundles and uses algebraic geometry techniques.
result Proves that up to a finite cover, the Albanese map is a locally trivial fibration with nef fibers.
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.
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.
Alternative proof of semipositivity and nefness for K-semistable log-Fano pairs.
problem Semipositivity and nefness of Chow-Mumford line bundle for K-semistable log-Fano pairs.
method Alternative proof using families of K-semistable log-Fano pairs.
result Proof of semipositivity and nefness for K-semistable log-Fano pairs.
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.
No non-constant holomorphic maps between certain complex manifolds with specific properties.
problem Existence of non-constant holomorphic maps between complex manifolds.
method Analyzing properties of tangent and cotangent bundles, pseudo-effectiveness, and nefness.
result Holomorphic maps between certain complex manifolds are constant.
Researchers prove existence of cscK metrics on smooth minimal models.
problem Existence of constant scalar curvature Kähler metrics on compact Kähler manifolds.
method Direct proof showing existence on smooth minimal models and blowups.
result Compact Kähler manifolds with nef canonical bundle always admit cscK metrics.
The paper studies metrics on complex hyperbolic manifolds with cusps and their properties.
problem Analyzing metrics on complex hyperbolic manifolds with cusps and their properties.
method Using singular metrics and curvature conditions to study the bigness and nefness of tangent and cotangent bundles.
result Effective ramification orders for covers to have big cotangent bundles and high enough ramification makes the tangent bundle nef.
Two types of nonvanishing results are presented for compact Kähler varieties.
problem Deriving geometric consequences from numerical information in Kähler geometry.
method Analyzing non-uniruled varieties and hyperkähler manifolds to establish nonvanishing results for adjoint and nef bundles.
result Strong abundance-type results are obtained in dimension 4.
Study on a weaker curvature condition for Kähler manifolds.
problem Understanding Kähler manifolds with nonpositive k-Ricci curvature. method Introducing almost nonpositive k-Ricci curvature and analyzing the twisted Kähler-Ricci flow. result Compact Kähler manifolds with almost nonpositive k-Ricci curvature have nef canonical line bundles. Miyaoka-Yau inequality proven for smooth minimal models.
problem Proving the Miyaoka-Yau inequality for smooth minimal models.
method Existence of cscK metrics in a neighborhood of the canonical class.
result Miyaoka-Yau inequality holds for compact Kähler manifolds with nef canonical bundle.
The study introduces a new concept of almost nonpositivity for holomorphic sectional curvature and proves its implications on nefness and Miyaoka-Yau inequalities.
problem Understanding the behavior of holomorphic sectional curvature on compact Kähler manifolds.
method Introducing a new notion of almost nonpositivity for holomorphic sectional curvature and proving its implications.
result Compact Kähler manifolds with almost nonpositive holomorphic sectional curvature have nef canonical line bundles, contain no rational curves, and satisfy Miyaoka-Yau type inequalities.
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
Estimates Lelong numbers of Monge-Ampère products for Kähler manifolds.
problem Estimating Lelong numbers of Monge-Ampère products on Kähler manifolds.
method Analyzes generalized Monge-Ampère products and applies estimates to Chern and Segre currents of pseudoeffective vector bundles.
result Generalizes a recent result about pseudoeffective vector bundles and their nefness.
Paper proves Hermitian-Yang-Mills metrics for stable Kähler bundles.
problem Existence of Hermitian-Yang-Mills metrics for Kähler vector bundles.
method Analyzes slope polystability and uses Kähler currents with singularities.
result Validates existence of Hermitian-Yang-Mills metrics for stable bundles and nef-big currents.
The study of projective varieties with nef anticanonical divisors and log terminal singularities.
problem Understanding the structure and properties of projective varieties with specific divisor conditions.
method Analyzing the Albanese map and MRC fibration for klt projective varieties, showing locally constant fibrations and product decompositions.
result Generalization of results for smooth projective varieties to the klt case, including decomposition into rationally connected and projective varieties with trivial canonical divisor.
Compact surfaces' quotient schemes lack semipositive Ricci curvature.
problem Geometric properties of quotient schemes on compact surfaces.
method Analyzing the anticanonical line bundle of quotient schemes.
result The anticanonical line bundle is not nef, indicating lack of semipositive Ricci curvature.
We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the pro…
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, …
Let X be a smooth variety and let L be an ample line bundle on X. If π1alg(X) is large, we show that the Seshadri constant ε(p∗L) can be made arbitrarily large by passing to a finite étale cover p:X′→X. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an …
The paper establishes lower bounds on Yang-Mills functionals for fibrations.
problem Analyzing the stability and nefness of direct image sheaves in fibrations.
method Generalizing mean curvature and Harder-Narasimhan filtrations to arbitrary polarized fibrations.
result Optimal lower bounds on fibered Yang-Mills functionals in terms of direct image sheaves.
In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, w…
The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.
problem Understanding the structure of projective klt varieties with nef tangent sheaves.
method Developing theory of positivity of coherent sheaves and proving structure theorem.
result Projective klt varieties with specific tangent sheaf properties admit rationally connected fibrations onto abelian varieties.
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…
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.
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.
We generalize the maximal time existence of Kähler-Ricci flow in Tian-Zhang and Song-Tian to conical case. Furthermore, if the twisted canonical bundle KM+(1−β)[D] is big or big and nef, we can expect more on the limit behaviors of such conical Kähler-Ricci flow. Moreover, the results still hold for simple normal …
Study on volumes and scalar curvature in complex geometry, proving new bounds and conditions.
problem Understanding volumes and scalar curvature in complex geometry.
method Asymptotic behavior of Bergman kernel, Kähler-Ricci flow, singular Kähler-Einstein metrics, and gluing techniques.
result New bounds and conditions for volumes and scalar curvature in complex manifolds.
Solves Teissier problem for nef classes on compact Kähler manifolds.
problem Characterizing equality in Khovanskii-Teissier inequality for nef classes.
method Constructing examples and extending previous results.
result Optimal results for nef classes and some non-nef classes.