论文标题
高兹布鲁克表面上的lelong数字集
Upper level sets of Lelong numbers on Hirzebruch surfaces
论文作者
论文摘要
令$ \ mathbb f_a $表示内姆布鲁克表面，$ \ nircal {t} _ {α，α，α^{\ prime}}（\ Mathbb {f} _ {a} _ {a}） IS是$αf+α^{\ prime} h $其中$ f $和$ h $生成$ \ mathbb f_a $的Picard组。 $ e^+_β（t）$表示Lelong数字$ν（t，x）$的上层集合$ t \ in \ Mathcal {t} _ {α，α，α^{\ prime}}（\ m马理bb {f} _ {f} _ {a}）$。当$ a = 0 $，（$ \ mathbb f_a = \ mathbb p^1 \ times \ times \ mathbb p^1 $），对于任何当前的$ t \ in \ mathcal t _ {α，α'}（\ mathbb p^1 \ times \ times \ mathb p^1）$，我们显示$ e^{+} _ {（α+α'）/3}（t）$包含在总学位$ 2 $的曲线中，可能是$ 1 $点。对于任何当前的$ t \ in \ Mathcal t_ {α，α'}（\ Mathbb f_a）$，我们表明$ e^{+}_β（t）$都包含在bidegreey $（0,1）$的曲线中（a+1）α^{\ prime}）/（a+2）$。
Let $\mathbb F_a$ denote the Hirzebruch surfaces and $\mathcal{T}_{α,α^{\prime}}(\mathbb{F}_{a})$ denotes the set of positive, closed $(1,1)$-currents on $\mathbb{F}_{a}$ whose cohomology class is $αF+α^{\prime} H$ where $F$ and $H$ generates the Picard group of $\mathbb F_a$. $E^+_β(T)$ denotes the upper level sets of Lelong numbers $ν(T,x)$ of $T\in \mathcal{T}_{α,α^{\prime}}(\mathbb{F}_{a})$. When $a=0$, ($\mathbb F_a=\mathbb P^1\times \mathbb P^1$), for any current $T\in \mathcal T_{α,α'}(\mathbb P^1\times \mathbb P^1)$, we show that $E^{+}_{(α+α')/3}(T)$ is contained in a curve of total degree $2$, possibly except $1$ point. For any current $T\in \mathcal T_{α,α'}(\mathbb F_a)$, we show that $ E^{+}_β(T)$ is contained in either in a curve of bidegree $(0,1)$ or in $a+1$ curves of bidegree $(1,0)$ where $β\geq (α+ (a+1)α^{\prime})/(a+2)$.
