Upper level sets of Lelong numbers on Hirzebruch surfaces

Abstract

Let ${\mathbb{F}}_a$ denote the Hirzebruch surfaces and ${\mathcal{T}}_{\alpha ,{\alpha }^{\prime }}({\mathbb{F}}_a)$ denotes the set of positive, closed $(1,1)$-currents on ${\mathbb{F}}_a$ whose cohomology class is $\alpha F+{\alpha }^{\prime }H$ where $F$ and $H$ generates the Picard group of ${\mathbb{F}}_a$. $E_{\beta }^{+}(T)$ denotes the upper level sets of Lelong numbers $\nu (T,x)$ of $T\in {\mathcal{T}}_{\alpha ,{\alpha }^{\prime }}({\mathbb{F}}_a)$. When $a=0$, (${\mathbb{F}}_a={\mathbb{P}}^1\times {\mathbb{P}}^1$), for any current $T\in {\mathcal{T}}_{\alpha ,{\alpha }^{\prime }}({\mathbb{P}}^1\times {\mathbb{P}}^1)$, we show that $E_{(\alpha +{\alpha }^{\prime })/3}^{+}(T)$ is contained in a curve of total degree $2$, possibly except $1$ point. For any current $T\in {\mathcal{T}}_{\alpha ,{\alpha }^{\prime }}({\mathbb{F}}_a)$, we show that $E_{\beta }^{+}(T)$ is contained in either in a curve of bidegree $(0,1)$ or in $a+1$ curves of bidegree $(1,0)$ where $\beta \ge (\alpha +(a+1){\alpha }^{\prime })/(a+2)$.

Accepted to The Journal of Geometric Analysis