We study plurisubharmonic functions and their applications to K\"ahler geometry. We begin by studying regularity of envelopes of plurisubharmonic functions, particularly when the reference form is degenerate. This is then applied to show regularity of geodesic of K\"ahler metrics on singular varieties, as well as regularity of certain geodesic rays. In particular, we are able to answer a question of Ross--Witt-Nystr\"om concerning regularity of the Hele-Shaw flow on a compact Riemann surface. We then study Seshadri constants, generalizing an inequality of McKinnon-Roth to transcendental K\"ahler classes on compact K\"ahler manifolds. After that, we use the polar transform of Lehmann-Xiao to look at the Seshadri constant $S_x$ of curve classes. This necessitates that we study the Nakayama constant as well. We conclude by characterizing the vanishing locus of $S_x$, in a manner analogous to the regular Seshdari constant. Finally, we consider Fano manifolds that do not admit a K\"ahler-Einstein metric. In this case, Tian and Yau expressed the expectation that the solutions to the Aubin-Yau continuity path would limit to a pluricomplex Green's-like function on $X$ with weakly analytic singularities along a subvariety $V$. We confirm their expectation, by showing that any limiting function will have vanishing Monge-Amp\`{e}re mass on $X\setminus V$.

Creator

• McCleerey, Nicholas James

DOI

• https://doi.org/10.21985/n2-sw3z-bw49

Subject

• Mathematics

Language

• en

Alternate Identifier

• etdadmin_upload_763826
• http://dissertations.umi.com/northwestern:15252

Keyword

• Kahler Geometry
• Plurisubharmonic Functions

Date created

• 2020-01-01

Resource type

• Dissertation

Rights statement

• In Copyright