Study on zero sets of sections of pseudo-effective line bundles on Kähler manifolds.
problem Distribution of common zero sets of sections of pseudo-effective line bundles.
method Analyzing the wedge product of curvature currents and approximating them by analytic cycles.
result Sufficient conditions for approximating wedge products of curvature currents by analytic cycles.
Generalizes Kollár's results to pseudo-effective line bundles.
problem Injectivity and vanishing theorems for pseudo-effective line bundles.
method Analytic approach using L2-harmonic forms and multiplier ideal sheaves. result Establishes a generalization of Kollár's injectivity theorem for adjoint bundles.
The paper proves new injectivity theorems for complex geometry.
problem Injectivity of higher direct image sheaves under Kähler morphisms.
method Transcendental methods, focusing on Kähler morphisms and singular hermitian metrics.
result Established new injectivity theorems for canonical bundles and their twists.
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 classifies surfaces with pseudo-effective tangent bundles.
problem Understanding projective manifolds with pseudo-effective tangent bundles.
method Developed singular hermitian metrics on vector bundles and applied to projective manifolds.
result Projective manifolds with pseudo-effective tangent bundles admit a smooth fibration to a flat projective manifold.
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.
Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.
problem Compact Kähler manifolds with pseudo-effective tangent bundles.
method Smooth or locally constant rationally connected fibration onto a quotient of a compact complex torus.
result Compact Kähler manifolds with pseudo-effective tangent bundles admit a fibration structure.
The study constructs universal invariants for non-Archimedean metrics on projective varieties.
problem Understanding the singularity of non-Archimedean metrics on projective varieties.
method Constructing partial Okounkov bodies and Duistermaat--Heckman measures for non-Archimedean metrics.
result Generalization of Duistermaat--Heckman measures to finite energy metrics on Berkovich analytifications.
The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the h…
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.
The paper shows how certain complex projective varieties can be broken down into simpler types.
problem Understanding the structure of complex projective varieties with pseudo-effective tangent sheaves.
method Developed a theory of pseudo-effective sheaves and applied the minimal model program.
result Projective klt varieties with pseudo-effective tangent sheaves can be decomposed into Fano varieties and Q-abelian varieties.
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 paper proves injectivity and vanishing theorems on compact Kahler manifolds.
problem Injectivity and vanishing theorems on compact Kahler manifolds.
method Hodge theory, Bochner-Kodaira-Nakano identity, analytic method, transcendental method, Demailly-Peternell-Schneider equisingular approximation theorem, Hormander L2 estimates.
result The main injectivity theorem implies several Nadel type vanishing theorems.
Compact manifolds with positive scalar curvature have negative Kodaira dimension.
problem Understanding the properties of compact manifolds with positive scalar curvature.
method Analyzing the canonical bundle and complex structures on compact Riemannian manifolds.
result Compact manifolds with positive scalar curvature have negative Kodaira dimension.
The paper proves a logarithmic partial derivative lemma and applies it to several geometric problems.
problem Proving a logarithmic partial derivative lemma for compact Kähler manifolds.
method Developed a new ∂∂ˉ-type lemma for logarithmic differential forms. result Confirmed a conjecture by X. Wan and derived several geometric applications.
Study on m-positivity in Kähler manifolds with new Monge-Ampère-type equation.
problem Exploring m-positivity in Kähler manifolds and its geometric applications. method Generalizing pseudo-effective and big Bott-Chern cohomology classes, proposing a new Monge-Ampère-type equation.
result Proof of a form of uniqueness for solutions of the Monge-Ampère-type equation.
Let φ:Cn→X a holomorphic map to an n-dimensional connected compact complex manifold X. We establish links between the positivity properties of the canonical bundle of X and the rate of growth of φ which extend results of Kodaira and Kobayashi-Ochiai. For example: if the average degree of φ on balls…
Study shows non-polyhedral structure in moduli spaces for n≥8.
problem Identifying non-polyhedral structure in moduli spaces of pointed stable curves.
method Constructing an extremal non-polyhedral ray via maps on meromorphic strata of differentials.
result Moduli spaces are not Mori Dream Spaces for n≥8.
The paper proves conditions for positive scalar curvature on complex manifolds.
problem Conditions for the existence of metrics with positive scalar curvature on compact complex manifolds.
method Proof based on properties of the canonical bundle and recent solutions to related conjectures.
result Conditions for the existence of metrics with positive scalar curvature on compact complex manifolds.
We describe the positive cone and the pseudo-effective cone of a non-Kählerian surface. We use these results for two types of applications: - Describe the set σ(X) of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-Kähhlerian surface, and decide whether the assignment $…
We explain the bundle structures of the {\it Determinant line bundle} and the {\it Quillen determinant line bundle} considered on the connected component of the space of Fredholm operators including the identity operator in an intrinsic way. Then we show that these two are isomorphic and that they are non-trivial line …
Logarithmic Picard algebroids solve meromorphic line bundle prequantization.
problem Classifying meromorphic line bundles on smooth projective varieties.
method Introducing logarithmic Picard algebroids and using their cohomology.
result Logarithmic Picard algebroids classify meromorphic line bundles.
Classifies discrete vector bundles over simplicial complexes, generalizing Weil's theorem.
problem Classification of discrete vector bundles over simplicial complexes.
method Discrete Differential Geometry approach, including classification theorem and curvature association.
result Discrete hermitian line bundles with curvature have a unique piecewise-smooth counterpart.
Estimates cohomology dimensions for Nakano q-semipositive line bundles.
problem Estimating cohomology dimensions for Nakano q-semipositive line bundles.
method Asymptotic estimates for high tensor powers of semipositive line bundles over q-convex manifolds and various complex manifolds.
result Optimal order estimates for cohomology dimensions.
Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.
problem Understanding projective flat holomorphic vector bundles over Riemann surfaces.
method Assigning Wronskian line bundles to vector bundles and interpreting Abel's identity.
result Abel's identity is the first Chern class of the Wronskian line bundle.
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 proves Hodge symmetry on Oeljeklaus-Toma manifolds with line bundles.
problem Hodge symmetry on complex manifolds with line bundles.
method Analyzes Dolbeault cohomology of Oeljeklaus-Toma manifolds with holomorphic line bundles.
result Proves Hodge symmetry and vanishing/non-vanishing of Dolbeault cohomology.
Geometric quantization extended to big line bundles.
problem Quantization of line bundles with large curvature.
method Proving asymptotic isometry and submultiplicative norms equivalence, showing Mabuchi geodesic rays.
result Bounded submultiplicative filtrations on big line bundles lead to Mabuchi geodesic rays.
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.
Computes the decomposition of rank-three bundles over the projective line with three marked points.
problem Decomposing rank-three bundles over the projective line with three marked points.
method Using the monodromy derivative to compute the roots of the bundles.
result Computes the exact decomposition of rank-three bundles for m=3. The paper estimates the dimension of cohomology for semipositive line bundles on various manifolds.
problem Estimating the dimension of cohomology for semipositive line bundles over different types of manifolds.
method Asymptotic estimates for harmonic (0,q)-forms with high tensor powers of semipositive line bundles. result Asymptotic estimates for the dimension of cohomology of semipositive line bundles on various manifolds.
Study differential invariants for line bundle operators.
problem Conditions for equivalence of differential operators in line bundles.
method Use differential invariants and groups of automorphisms.
result Find conditions for equivalence of differential operators.
Let M be an irreducible smooth complex projective variety equipped with an action of a compact Lie group G, and let (L,h) be a G-equivariant holomorphic Hermitian line bundle on M. Given a compact connected Riemann surface X, we construct a G-equivariant holomorphic Hermitian line bundle $(L\,,…
Paper proves ε-regularity for line bundle mean curvature flow.
problem Proving regularity for a specific type of geometric flow.
method Develops a scale-invariant monotone quantity and defines self-shrinkers.
result Establishes ε-regularity theorem for line bundle mean curvature flow.
Proves Chern-Weil theory for line bundles with a specific metric.
problem Extending a result on Jacobi forms to line bundles on families of curves.
method Uses the notion of a b-divisor.
result Generalizes a result on line bundles of Jacobi forms to a broader class of line bundles.
Study stability thresholds of big line bundles, proving bounds and generalizing results.
problem Stability thresholds of big line bundles and their asymptotic behavior.
method Explicit bounds on error terms, using quasi-monomial valuations to compute stability thresholds.
result Proves Jin--Rubinstein--Tian's questions affirmatively.
Quantization of fermions yields determinant line bundle.
problem Quantizing fermions to derive mathematical structures.
method Batalin-Vilkovisky quantization of massless free fermions.
result Determinant line bundle derived from fermion quantization.
In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line b…
The paper calculates the asymptotic of holomorphic analytic torsion forms for line bundles.
problem Calculating the asymptotic of holomorphic analytic torsion forms for line bundles.
method Using asymptotic formula and Toeplitz operators, the paper generalizes the results for direct images of line bundles.
result Generalized asymptotic formula for holomorphic torsion forms of line bundles and direct images.
Formula compares metrics on branched coverings of line bundles.
problem Comparing metrics on branched coverings of line bundles.
method Formula to compare Quillen metrics on branched coverings.
result Formula for comparing Quillen metrics on branched coverings.
Estimates Bergman kernel for positive line bundles.
problem Estimating the Bergman kernel for positive line bundles.
method Explicit estimate using positive line bundles.
result Explicit lower bound of the Bergman kernel.
The paper examines the stability of a specific flow on complex manifolds.
problem Stability of line bundle mean curvature flow on complex manifolds.
method Analyzes the convergence of the line bundle mean curvature flow to a deformed Hermitian-Yang-Mills metric.
result The flow converges exponentially to the deformed Hermitian-Yang-Mills metric in the C∞ sense. We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possi…
Existence and uniqueness of solitons on complex line bundles.
problem Existence and uniqueness of gradient steady Ricci solitons on complex line bundles.
method One-parameter family of smooth complete U(1)-invariant gradient steady Ricci solitons. result Existence and uniqueness of solitons on the total space of any complex line bundle over a Fano Kähler-Einstein base.
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.
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.
Oeljeklaus-Toma manifolds are rigid and line bundles are flat.
problem Rigidity and flatness of Oeljeklaus-Toma manifolds.
method Proving rigidity and flatness of line bundles.
result Oeljeklaus-Toma manifolds of simple type are rigid.
Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.
problem Estimating ∂-operators for flat line bundles. method Uniform L2-estimates for ∂-operators on Kähler manifolds. result Recovers Ueda's lemma for compact Kähler manifolds and generalizes to Ricci-flat manifolds.