Date of Award
5-11-2025
Date Published
June 2025
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
Advisor(s)
Dan Coman
Keywords
Analysis;Complex Analysis;Complex Geometry;Differential Geometry;Pluripotential Theory;Several Complex Variables
Subject Categories
Mathematics | Physical Sciences and Mathematics
Abstract
Let $(X,\omega)$ be a Kähler manifold, and $L_p$ be a sequence of line bundles on $X$ equipped with continuous hermitian metrics $h_p$. Let $\gamma_p$ be the Fubini-Study current and $h_p^{eq}$ the equilibrium metric corresponding to the metric $h_p$. Our goal is to generalize a theorem of Tian in Complex Geometry to a setting using arbitrary line bundles equipped with metrics with non-positive curvature. This amounts to finding sufficient conditions for $\gamma_p-c_1(L_p,h_p^{eq})$, when appropriately scaled, to converge to $0$ in the sense of currents. To achieve this goal, we imply methods of Demailly and the Oshawa-Takegoshi Extension Theorem to bound a scaled potential for $\gamma_p-c_1(L_p,h_p^{eq})$, and show that it converges in $L^1$. After we establish sufficient conditions for the aforementioned convergence, we ask the question of when do $\gamma_p$ and $c_1(L_p,h_p^{eq})$ converge. To simplify our investigation, we restrict ourselves to the case where $c_1(L_p,h_p)$ converges. We establish sufficient conditions for the convergences of $\gamma_p$ using methods similar to our previous results. In this case, a construction of the limit is given, which is defined as a particular upper quasi-plurisubharmonic envelope. Lastly, we establish application of our results to special cases. In particular, we discuss the cases where $(L_p,h_p)$ is a tensor product of powers of line bundles, that is $(L_p,h_p)$ is of the form $(F^{\otimes m_{1,p}}_1\otimes \cdots \otimes F_k^{\otimes m_{k,p}},h_1^{\otimes m_{1,p}}\otimes \cdots \otimes h_k^{\otimes m_{k,p}})$ for some metrics $(F_j,h_j)$ and $m_{j,p}\in \mathbb{N}$. This particular case is handled using the fact that the weights of $h_1^{\otimes m_{1,p}}\otimes \cdots \otimes h_k^{\otimes m_{k,p}}$ are a sum of the weights of the metrics $h_1,\ldots, h_k$.
Access
Open Access
Recommended Citation
Wolff, Melody Ann, "Asymptotics of Fubini-Study Currents for Sequences of Line Bundles" (2025). Dissertations - ALL. 2116.
https://surface.syr.edu/etd/2116
