## Dissertations - ALL

May 2018

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Dan Coman

John Laiho

#### Keywords

Complex Analysis, Currents, Lelong Numbers, Plurisubharmonic Functions

#### Subject Categories

Physical Sciences and Mathematics

#### Abstract

Let $T$ be a positive closed bidegree $(p,p)$ current in $\mathbb{P}^n$. In this thesis, our goal is to understand more about the geometric properties of the sets of highly singular points of the current $T$. Lelong numbers will be the main tool used for determining how singular a point of a current is. For the first main result of this thesis, we let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on the complex projective space $\mathbb P^2$. For $\alpha > 2/5$ and $\beta = (2-2\alpha)/3$ we show that if $T$ has four points with Lelong number at least $\alpha$, the upper level set $E_{\beta}^+ (T)$ of points of $T$ with Lelong number strictly larger than $\beta$ is contained within a conic with the exception of at most one point.

Afterwards, we will let $T$ be a positive closed current of bidimension $(p,p)$ with unit mass on the complex projective space $\mathbb P^n$. Our aim here is to generalize some results of D. Coman as well as look at the result in the previous paragraph in a more generalized setting. For certain values of $\alpha$ and $\beta = \beta(p, \alpha)$ we show that if $T$ has enough points where the Lelong number is at least $\alpha$, then the upper level set $E_{\beta}^+ (T)$ has certain geometric properties, in particular it will be contained in either a complex line $L$ except for exactly $p$ points of the upper level set that are not contained on the line, or the upper level set will be contained in a $p$-dimensional linear subspace.

Open Access

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