Abstract:
本文研究了五阶图与圈图的联图交叉数.利用假设法和比较法等方法,得到了W4 ∨ Cn的交叉数为Z(5,n)+n+？(n)/(2)」+4,并推广了联图交叉数的结果与方法. In the paper, we study the crossing numbers of the join graph of order flve with cycle. Based on the hypothesis and comparison, we get the results of the crossing number of W4 ∨ Cn is Z(5,n) + n + ？(n)/(2)」 + 4, which extends the results and methods of the crossing numbers of the join graph

Abstract:
Given a positive closed (1,1)-current $T$ defined on the regular locus of a projective variety $X$ with bounded mass near the singular part of $X$ and $Y$ an irreducible algebraic subset of $X$, we present uniform estimates for the locus inside $Y$ where the Lelong numbers of $T$ are larger than the generic Lelong number of $T$ along $Y$.

Abstract:
We establish plurisubharmonicity of the envelope of Lelong functional on almost complex manifolds of real dimension four, thereby we generalize the corresponding result for complex manifolds.

Abstract:
We look for pointwise bounds on a plurisubharmonic function near its singularity point, given the value of its generalized Lelong number with respect to a plurisubharmonic weight. To this end, an extremal problem is considered. In certain cases, the problem is solved explicitly.

Abstract:
We introduce a quantity which measures the singularity of a plurisubharmonic function f relative to another plurisubharmonic function g, at a point a. This quantity, which we denote by $\nu_{a,g}(f)$, can be seen as a generalization of the classical Lelong number, in a natural way. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form $\{z: \nu_{z,g}(f) \geq c > 0 \}$, are in fact analytic sets, under certain conditions on the weight g.

Abstract:
In this note we study the existence of the Lelong-Demailly number of a negative plurisubharmonic current with respect to a positive plurisubharmonic function on an open subset of $\C^n$. Then we establish some estimates of the Lelong-Demailly numbers of positive or negative plurisubharmonic currents.

Abstract:
Generalized Lelong numbers of plurisubharmonic functions with respect to plurisubharmonic weights (due to Demailly) are specified for weights with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the directional Lelong numbers of the functions and the Newton diagrams of the weights. An extension of Demailly's approximation theorem is proved as well.

Abstract:
One of Demailly's characterizations of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note sections of multiples of the line bundle are used to produce such metrics and then to deduce another formula for Seshadri constants. It is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata's conjecture.