arXiv research

The paper introduces generalized Lelong numbers for currents and their applications in intersection theory.

Extends Lelong number theory to positive plurisubharmonic currents.

Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.

Study shows Lelong numbers vanish for certain currents in weakly hyperbolic foliations.

Estimates Lelong numbers of Monge-Ampère products for Kähler manifolds.

The aim of this paper is to study the Lelong number, the integrability index and the Monge-Ampère mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $\mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of th…

Study on residual Monge-Ampère mass for symmetric plurisubharmonic functions.

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…

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…

Paper shows regularizing flow for conical Kähler-Ricci equations.

The paper develops residue currents for cohesive modules and proves a generalized Poincaré-Lelong formula.

The paper studies metrics on vector bundles with singularities and their associated forms.

The Kähler-Ricci flow smooths positive currents on Kähler manifolds.

Solves a complex Monge-Ampère equation on compact Hermitian manifolds.

We study the regularizing properties of complex Monge-Ampère flows on a Kähler manifold $(X,ω)$ when the initial data are $ω$-psh functions with zero Lelong number at all points. We prove that the general Monge-Ampère flow has a solution which is immediately smooth. We also prove the uniqueness and stability of solutio…

Let $X$ be a compact Kähler manifold. We show that the Kähler-Ricci flow (as well as its twisted versions) can be run from an arbitrary positive closed current with zero Lelong numbers and immediately smoothes it.

We prove that a general complex Monge-Ampère flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generaliza…

Introduces trace operator for quasi-plurisubharmonic functions on Kähler manifolds.

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…

Generalizes double transgression formulas on complex manifolds.

Given a Kähler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular Kähler-Einstein metric induces a semipositively curved metric on the relative canonical bundle $K_{X/Y}$ of $p$. We also propose a conjectural generalization of this result for relative twisted Kähler-Eins…

Study singularities of $m$-subharmonic functions along submanifolds.

We establish plurisubharmonicity of the envelope of Poisson and Lelong functionals on almost complex manifolds. That is, we generalize the corresponding results for complex manifolds and almost complex manifolds of complex dimension two. We also provide some applications to the regularization of J-plurisubharmonic func…

Kähler-Ricci flow smooths out positive closed currents with divisorial singularities

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 $α: E \rightarrow F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $$ φ\ = \ \text{\rm Res}_φΣ…

Sharp inequalities for weighted log canonical thresholds derived.

Let $X$ be a compact Kähler manifold and $\{θ\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $θ$-plurisubharmonic functions with full mass a…

Study finite-energy metrics over complex manifold degenerations.

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…

We study conformal Fefferman-Lorentz manifolds introduced by Fefferman. To do so, we introduce Fefferman-Lorentz structure on (2n+2)-dimensional manifolds. By using causal conformal vector fields preserving that structure, we shall establish two theorems on compact Fefferman-Lorentz manifolds: One is the coincidence of…

Study on algebraic fiber spaces and their anti-canonical divisors.

Two types of nonvanishing results are presented for compact Kähler varieties.

Let $\overline{M}$ be a compact complex manifold with smooth Kähler metric $η$, and let $D$ be a smooth divisor on $\overline{M}$. Let $M=\overline{M}\setminus D$ and let $\hatω$ be a Carlson-Griffiths type metric on $M$. We study complete solutions to Kähler-Ricci flow on $M$ which are comparable to $\hatω$, starting …

The paper studies Fubini-Study metrics and zero distributions on CR manifolds.

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 $\sin^2θ$ and $\cos^2θ$ re…

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…

Paper characterizes relation numbers for parabolic two-generator groups.

Paper refines generating function for 2-bridge knot groups.

Jablan and Radović originally defined two invariants called the Meander number and OGC number of knots for certain classes of knots. We generalize these definitions to all knots and name the straight number and contained straight number of a knot, respectively, and prove they are well defined. We answer two questions a…

Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the sequence of torsion numbers satisfies a linear homogeneous recurrence relation …

Ascending numbers are determined for 64 knots with at most n=10 crossings. After proving the theorem about the signature of alternating knot families, we distinguished all families of knots obtained from generating alternating knots with at most 10 crossings, for which the unknotting number can be confirmed by using th…

We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numb…

Odd crossing numbers and even rotation numbers for cycles in plane immersions.

Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.

Study of generalized knots and links, proving inequality involving crossing number and braid index.

We give a tropical interpretation of Hurwitz numbers extending the one discovered in \cite{CJM}. In addition we treat a generalization of Hurwitz numbers for surfaces with boundary which we call open Hurwitz numbers.

Study of Fishburn numbers and their congruences for torus knots.