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\,,…
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.
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 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.
Compact Hermitian manifolds with quasi-negative curvature have ample canonical line bundles.
problem Determining conditions for ample canonical line bundles in Hermitian manifolds.
method Hermitian curvature flow with specific curvature conditions.
result Canonical line bundle is ample under given curvature conditions.
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. The paper connects bundle curvature to random zero currents.
problem Understanding the relationship between bundle curvature and random zero currents.
method Heat flow on Hermitian line bundles over Riemannian manifolds.
result Random zero currents connect bundle curvature to ground state zero current.
We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.
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 observe that the line bundle associated to the tame symbol of two invertible holomorphic functions also carries a fairly canonical hermitian metric, hence it represents a class in a Hermitian holomorphic Deligne cohomology group. We put forward an alternative definition of hermitian holomorphic structure on a gerbe …
The paper establishes conditions for optimal sampling configurations on complex manifolds.
problem Finding optimal sampling configurations on complex manifolds.
method Analyzes point configurations on compact complex manifolds using tensor powers of Hermitian ample line bundles.
result Necessary and sufficient conditions for the existence of asymptotically Fekete sequences.
In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao an…
Study of moduli spaces for special harmonic maps from projective line to quadrics.
problem Characterizing moduli spaces of Einstein-Hermitian harmonic maps.
method Using vector bundles and representation theory, the authors describe the moduli spaces of these maps.
result The dimension of the moduli spaces is independent of the Einstein-Hermitian constant.
Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.
problem Defining and analyzing volume functionals for Hermitian connections on manifolds.
method Introduces the line bundle mean curvature flow and relates it to deformed connections and special submanifolds.
result Proves the mirror equality for mSpin(7)-dDT connections and deduces their properties. 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 heat kernel asymptotics for high power line bundles on complex manifolds.
problem Proving heat kernel asymptotics for Kodaira Laplacians of high power line bundles.
method Scaling technique applied to both compact and non-compact manifolds.
result Direct proof of holomorphic Morse inequalities and generalization to vector bundles.
We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.
The paper studies geodesic-Einstein metrics on line bundles over fibration.
problem Characterizing geodesic-Einstein metrics on line bundles over holomorphic fibrations.
method Introduced geodesic-Einstein flow and defined S-classes and C-classes to study the properties of geodesic-Einstein metrics.
result Proved that (X,L) is nonlinear semistable under certain conditions. 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.
Study on line bundle flow on Kähler surfaces converging to a singular solution.
problem Analyzing the mean curvature flow on Kähler surfaces.
method Investigates the flow under hypercritical phase and semipositivity conditions.
result The flow converges to a singular solution away from curves of negative self-intersection.
Study of Fubini-Study forms on surfaces with punctures.
problem Analyzing Fubini-Study forms on surfaces with punctures.
method Using Hermitian metrics, holomorphic line bundles, and Kodaira maps.
result Fubini-Study forms grow polynomially near punctures.
We find the entropy's infinite-size behavior in complex manifold sections.
problem Determining entropy behavior in complex manifold sections.
method Analyzing entanglement entropy in tensor powers of hermitian line bundles.
result Asymptotic formula for expected entanglement entropy.
Let X→S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over C. Let L→X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing $\la…
Study gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.
problem Understanding gaps and clusters in eigenvalues of magnetic Laplacian on manifolds.
method Analyzes high tensor powers of Hermitian line bundles with non degenerate curvature, proving Riemann-Roch numbers for eigenvalue clusters and describing spectral projectors.
result Clusters and gaps in eigenvalues are described by Riemann-Roch numbers and have pointwise kernel descriptions.
Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
problem Finding conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.
method Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields.
result Provides necessary and sufficient conditions for the functional to attain a minimum.
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…
Defines geodesic-Einstein metrics and their relation to nonlinear stabilities.
problem Nonlinear stabilities of line bundles over holomorphic fibrations.
method Introduces geodesic-Einstein metrics and a Donaldson type functional.
result Geodesic-Einstein metrics minimize the Donaldson type functional.
Study shows how random sections distribute on Kähler manifolds.
problem Distribution of random sections on Kähler manifolds.
method Normalized currents of integration and Hölder singular metrics.
result Currents distribute asymptotically to curvature currents.
Study shows zeros of random sections are uniformly distributed.
problem Distribution of zeros in random holomorphic sections.
method Equidistribution and moment assumptions for singular Hermitian line bundles.
result Asymptotic distribution of zeros is independent of probability measure.
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.
It is shown that the determinant line bundle associated to a family of Dirac operators over a closed partitioned manifold has a canonical Hermitian metric with compatible connection whose curvature satisfies an additivity formula with contributions from the families of Dirac operators over the two halves. This curvatur…
Let X be a compact Kähler manifold, E→X a Hermitian vector bundle and L→X an ample line bundle. We construct a non-linear heat flow corresponding to the almost Hermitian-Einstein equation introduced by N.C. Leung, and prove that the solution exists for a short time. We also construct a potential function $D…
The paper proves properties of line bundles on certain spaces.
problem Investigating properties of line bundles on specific spaces.
method Analyzing curvature and injectivity radius conditions to prove properties.
result Proves properties of line bundles L on a tower of spaces. 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.
Let X be a non-compact geometrically finite hyperbolic 3-manifold without cusps of rank 1. The deformation space $\mc{H}$ of X can be identified with the Teichmüller space $\mc{T}$ of the conformal boundary of X as the graph of a section in $T^*\mc{T}$. We construct a Hermitian holomorphic line bundle $\mc{L}$ on…
Diagonal metrics solve Hermitian-Einstein equations for decomposed Higgs bundles.
problem Existence of diagonal pluriharmonic metrics in G-Higgs bundles. method Analyzes Higgs bundles over compact Kähler manifolds, decomposes vector bundles, and uses torus action to relate stability and conditions.
result Necessary and sufficient conditions for the existence of diagonal metrics solving Hermitian-Einstein equations.
The mirror of a projective toric manifold XΣ is given by a Landau-Ginzburg model (Y,W). We introduce a class of Lagrangian submanifolds in (Y,W) and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over XΣ. Through this ge…
We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curva…
We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using …
The abstract discusses embedding theorems for pseudo-Kähler manifolds.
problem Embedding theorems for pseudo-Kähler manifolds.
method Using quantizable pseudo-Kähler manifolds and Hermitian line bundles, the asymptotic expansion of Bergman kernels is analyzed.
result The asymptotic expansion of Bergman kernels implies analogues of Kodaira embedding theorem and Tian's almost-isometry theorem.
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. 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.
The paper studies singularities in a complex flow related to mean curvature.
problem Investigating singularities in a complex flow related to mean curvature.
method Constructing two distinct examples of singularities using the line bundle mean curvature flow.
result Found a finite time singularity, ruling out long time existence of the flow.
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
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…
Analyzes Quillen norm on determinant line bundle for curves with cusps.
problem Analyzing Quillen norm on determinant line bundle for curves with cusps.
method Studies Quillen norm on determinant line bundle for complex curves with cusps, using analytic torsion.
result Derives explicit formula for curvature current, refining Riemann-Roch-Grothendieck theorem.
Study shows Bergman kernel quotient approaches one for punctured surfaces.
problem Analyzing Bergman kernels on punctured Riemann surfaces.
method Examined a punctured Riemann surface with a specific metric and line bundle, calculating quotient of Bergman kernels.
result The quotient of Bergman kernels tends to one as tensor power increases.
Study the Bochner-Schrödinger operator's trace in semiclassical limit.
problem Trace formula for Bochner-Schrödinger operator on tensor powers of line and vector bundles.
method Semiclassical analysis of the Bochner-Schrödinger operator Hp on tensor powers of a Hermitian line bundle and vector bundle. result Complete asymptotic expansion of the trace of φ(Hp) in the semiclassical limit po∞.