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.
","abstract_html":"<p>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 P2. For α > 2/5 and β = (2-2α)/3 we show that if $T$ has four points with Lelong number at least α, the upper level set Eβ+ (T) of points of $T$ with Lelong number strictly larger than β is contained within a conic with the exception of at most one point.</p> <p>Afterwards, we will let $T$ be a positive closed current of bidimension $(p,p)$ with unit mass on the complex projective space \\mathbb Pn. 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 α and β = β(p, α) we show that if $T$ has enough points where the Lelong number is at least α, then the upper level set Eβ+ (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.</p>","abstract_has_math":true,"creators":["Heffers, James"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dan Coman","John Laiho"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-05-13T07:00:00Z","date_published":"2018-05-13T07:00:00Z","updated_at":"2026-07-24T04:55:27Z","subjects":["Complex Analysis","Currents","Lelong Numbers","Plurisubharmonic Functions","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/852","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dan Coman","John Laiho"]},{"key":"dc:creator","label":"Author","values":["Heffers, James"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Complex Analysis","Currents","Lelong Numbers","Plurisubharmonic Functions","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://surface.syr.edu/etd/852"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Lelong Numbers and Geometric Properties of Upper Level Sets of Currents on Projective Space"]}]}],"canonical_facts":{"dc:contributor":["Dan Coman","John Laiho"],"dc:creator":["Heffers, James"],"dc:description.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.
"],"dc:identifier":["https://surface.syr.edu/etd/852"],"dc:subject":["Complex Analysis","Currents","Lelong Numbers","Plurisubharmonic Functions","Physical Sciences and Mathematics"],"dc:title":["Lelong Numbers and Geometric Properties of Upper Level Sets of Currents on Projective Space"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:55:27Z"}