The paper develops residue currents for cohesive modules and proves a generalized Poincaré-Lelong formula.
problem Analyzing coherent sheaves on complex manifolds using global analytic methods.
method Developing residue currents for cohesive modules and proving their properties.
result Proves a generalized Poincaré-Lelong formula for cohesive modules.
Generalizes double transgression formulas on complex manifolds.
problem Currential double transgression formulas on complex manifolds.
method General framework using Bott-Chern Duality.
result Complements existing transgression formulas and provides new applications.
This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps α:E→F which, for smooth connections on E and F, establishes formulas of the type $$ φ\ = \ \text{\rm Res}_φΣ…
In this paper, we develop a method of solving the Poincaré-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on (1,1)-forms. The method is effective in proving an optimal result when M has nonnegative bisectional curvature. It also provides …
In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR (2n+1)-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this sol…
Extends Lelong number theory to positive plurisubharmonic currents.
problem Lack of in-depth exploration of Lelong number theory for positive plurisubharmonic currents.
method Introduces generalized Lelong numbers and studies their properties using Lelong-Jensen formulas for the normal bundle.
result Shows the top degree Lelong number of a positive plurisubharmonic current is totally intrinsic.
The paper introduces generalized Lelong numbers for currents and their applications in intersection theory.
problem Defining and studying generalized Lelong numbers for currents in intersection theory.
method Formulating generalized Lelong numbers for closed smooth (j,j)-forms, defining horizontal dimension, and establishing properties and formulas.
result Effective sufficient conditions for defining and continuity of intersections of positive closed currents.
Kähler-Ricci flow smooths out positive closed currents with divisorial singularities
problem Geometric regularization of positive closed currents
method Kähler-Ricci flow
result Gradual replacement of divisorial singularities by Poincaré type ones
We show that the solution constructed in an earlier work of Y-G. Shi and the authors can be used to obtain sharp gradient estimates for the Kaehler-Ricci flow which achieves equality on a steady soliton. The estimate can be applied to obtain a long time existence of the Kaehler-Ricci flow. In the second part of the pap…
Study shows Lelong numbers vanish for certain currents in weakly hyperbolic foliations.
problem Analyzing Lelong numbers for currents in weakly hyperbolic foliations.
method Local and global analysis of directed positive harmonic currents and currents directed by foliations.
result Lelong numbers of currents at the singularity vanish.
Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.
problem Analyzing the residual Monge-Ampère mass of symmetric plurisubharmonic functions.
method Proved zero mass for functions with zero Lelong number at origin and S1-invariance. result Zero mass conjecture answered for symmetric functions.
The paper studies Fubini-Study metrics and zero distributions on CR manifolds.
problem Understanding Fubini-Study metrics on CR manifolds.
method Asymptotic analysis of Toeplitz operators and pull-back metrics.
result Established the distribution of zero divisors of random CR functions.
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.
The aim of this paper is to study the Lelong number, the integrability index and the Monge-Ampère mass at the origin of an S1-invariant plurisubharmonic function on a balanced domain in Cn under the Schwarz symmetrization. We prove that n times the integrability index is exactly the Lelong number of th…
Our main results are: (1) The complex a Lagrangian points of a non-complex Lagrangian 2n-dimensional submanifold $F:M\ra N$, immersed with parallel mean curvature and with equal Kaehler angles into a Kaehler-Einstein manifold (N,J,g) of complex dimension 2n, are zeros of finite order of sin2θ and cos2θ re…
For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincaré-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincaré-Hopf index formula to the case of complex vector fi…
Paper derives a formula for renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
problem Calculating the renormalized area of minimal hypersurfaces in 5D Poincaré-Einstein spaces.
method Derives a Gauss-Bonnet formula for renormalized area in terms of scalar invariants and characterizes minimal hypersurfaces in terms of conformal geometry.
result Derives a formula expressing renormalized area in terms of integrals of scalar Riemannian invariants.
Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.
problem Estimating the residual Monge-Ampère mass of symmetric plurisubharmonic functions with isolated singularities.
method Utilized Sasakian geometry to derive estimates on the residual mass in relation to Lelong numbers.
result Partially resolved the zero mass conjecture by Guedj and Rashkovskii.
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.
We compute renormalized curvature integrals on Poincaré-Einstein manifolds.
problem Computing renormalized curvature integrals on Poincaré-Einstein manifolds.
method General procedure that connects Gauss-Bonnet-type formulas and identifies scalar conformal invariants.
result Explicit conformally invariant Gauss--Bonnet-type formulas for compact Einstein manifolds.
For Poincare series of binary polyhedral groups and Coxeter polynomials there are obtained statements close to the Euclid algorithm and orthogonal polynomials theory: generalized Ebeling formula, decompositions into ramified continued fractions, Christoffel-Darboux identity, combinatorial formula. Known results about t…
For two complex vector bundles admitting a homomorphism between them, a Poincaré-Hopf formula for the difference of the Chern character numbers of these two vector bundles with isolated singularities is established by Huitao Feng, Weiping Li and Weiping Zhang. This article extend their reslut about Poincaré-Hopf type f…
The paper derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
Derives formulas for differential forms on weighted manifolds.
problem Developing formulas for differential forms on weighted manifolds.
method Derives a Reilly formula and explores its applications.
result Proves a Poincaré-type inequality and obtains new eigenvalue estimates.
Let X be a compact Kähler manifold. We prove that the Kähler-Ricci flow starting from arbitrary closed positive (1,1)-currents is smooth outside some analytic subset. This regularity result is optimal meaning that the flow has positive Lelong numbers for short time if the initial current does. We also prove that th…
Paper shows regularizing flow for conical Kähler-Ricci equations.
problem Regularizing property of conical Kähler-Ricci flow.
method Regularizing property of the twisted conical Kähler-Ricci flow from a positive closed current with zero Lelong number.
result Extends regularizing property to conical singularity case.
The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain Ω must necessarily be asymptotically totally geodesic. A…
Let M be a compact, holomorphically symplectic Kahler manifold, and η a (1,1)-current which is nef (a limit of Kahler forms). Assume that the cohomology class of η is parabolic, that is, its top power vanishes. We prove that all Lelong sets of η are coisotropic. When M is generic, this is used to show that all Le…
Formula derived for renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
problem Calculating the renormalized area of minimal submanifolds in Poincaré-Einstein manifolds.
method Decomposition of extrinsic Q-curvature and application to renormalized area. result Renormalized area formula expressed as a linear combination of Euler characteristic and scalar conformal submanifold invariant.
Study eight categorifications of colored Jones polynomial, verifying physics conjectures.
problem Categorification of colored Jones polynomial and its applications.
method Comparison of eight finite-dimensional categorifications and verification of conjectures.
result Isomorphic results over a field of characteristic zero and closed formula for Poincaré series.
The Kähler-Ricci flow smooths positive currents on Kähler manifolds.
problem Geometric regularization of positive closed currents on Kähler manifolds.
method Kähler-Ricci flow on compact Kähler manifolds.
result Local Arnold multiplicities linearly decrease to zero under the flow.
In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and ind…
Abstract: Determinants and formulas for operators on various spaces.
problem Determinants and formulas for operators on different algebras and spaces.
method Use of Poincaré type determinants, invariant operators, and full matrix-symbols.
result Explicit formulas for determinants of elliptic operators and periodic pseudo-differential operators.
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.
problem Proving inequalities for convex hypersurfaces.
method Introducing a flat logarithmic centro-affine geometry and using Bochner formulas.
result Established new Poincaré and Brunn-Minkowski inequalities.
The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …
The study establishes inequalities on path space for sub-Riemannian manifolds.
problem Understanding functional inequalities on path space for sub-Riemannian manifolds.
method Derivative and integration by parts formulae on path space with respect to a natural gradient operator, showing bounds of horizontal Ricci curvature.
result Established functional inequalities on path space analogous to Riemannian geometry.
We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the expectation of all the Lipschitz--Killing curvatures of these sets. Whereas our motivation is primarily probabilistic, with statis…
The paper studies metrics on vector bundles with singularities and their associated forms.
problem Analyzing singular Hermitian metrics on vector bundles and their associated forms.
method Defines and analyzes the Segre and Chern forms of singular metrics, proving properties of their Lelong numbers.
result Lelong numbers of the associated forms are integers if singularities are integral.
New Poincaré inequality for differential forms on manifolds.
problem Developing inequalities for differential forms on manifolds.
method Proving a new Poincaré-type inequality and deriving new inequalities involving mean and scalar curvatures.
result Characterized the limiting case of a new inequality involving mean and scalar curvatures of the boundary.
The paper explores the topology of polygonal meshes and their properties.
problem Understanding the topological properties of polygonal meshes.
method Overview of topological concepts, definitions of intrinsic and extrinsic topology, proofs of Euler and Euler-Poincaré formulas, and discussion on cutting meshes.
result Detailed understanding and definitions of polygonal mesh topology, including intrinsic and extrinsic properties.
We calculate a wall crossing formula for 4-dimensional Poincare-Einstein metrics, through a wall made of orbifold Poincare-Einstein metrics with A1 singularities. This is based on a formalism which enables to deal with higher order terms of the Einstein equation in this setting. Some other consequences are deduced.
Study on renormalized volume of minimal submanifolds in Poincare-Einstein manifolds.
problem Analyzing minimal submanifolds in Poincare-Einstein manifolds.
method Deriving formulae for first and second variations of renormalized volume, proving asymptotic descriptions, and deriving inner-product relationships.
result Existence of asymptotic description and L2-inner-product relationship for specific cases. Holomorphic families yield metrics with explicit curvature formulas.
problem Positivity of direct images with Poincaré type singularities.
method Analyzes holomorphic families and line bundles with Poincaré type singular metrics.
result Explicit formula for curvature of direct image metrics.
The paper studies Gauss sums and their applications in algebra and topology.
problem Computing signatures of bilinear forms and Poincaré spaces.
method Investigates properties of Gauss sums and applies them to algebraic and topological problems.
result Reproves and generalizes results on signatures mod 8 of bilinear and Poincaré spaces.
We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functi…
Study vector fields with complex singularities, proving bounds and formulas.
problem Understanding the Milnor number of vector fields with specific singularities.
method Global and local formulas expressing Milnor/Poincare-Hopf contributions, sharp lower bounds under perturbations.
result Sharp lower bounds for Milnor number contributions under holomorphic perturbations.