Geometric pluripotential theory on Kähler manifolds

Abstract

Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Ampère equations. The purpose of this survey is to describe these developments from basic principles. Smith, V. Tosatti and the anonymous referees for their suggested corrections, careful remarks, and precisions. Also, I thank the students of MATH868D at the University of Maryland for their intriguing questions and relentless interest throughout the Fall of 2016. Chapter 3 is partly based on work done as a graduate student at Purdue University, and I am indebted to L. Lempert for encouragement and guidance. Chapter 4 surveys to some extent joint work with Y. Rubinstein, and I am grateful for his mentorship over the years.

Quantization in Geometric Pluripotential Theory

“…For a general Young weight χ ∈ W + p , let χ k be a sequence of smooth strictly convex weights that converge to χ uniformly on compact intervals. Such sequence can always be found by [28,Proposition 1.7]. By the above we have that…”

“…Here we recall some of the main points on the L p Finsler geometry of the space of Kähler potentials. For a detailed exposition, we refer to [28,Chapter 3], as well as the original articles [26,27]. As follows from the definition, the space of Kähler potentials H ω is a convex open subset of C ∞ (X), hence one can think of it as a trivial Fréchet manifold.…”

“…The fact that χ is smooth plays an essential role here. As a courtesy to the reader, we provide the argument that is essentially identical to the proof of [28,Proposition 3.7].…”

“…, e tλn ), t ∈ [0, 1]. Next we provide the finite-dimensional analog of a result of [27], that in turn builds on calculations of [20] in case of the L 2 geometry (see also [28,Proposition 3.11]): Proposition 3.3. Let χ be a smooth strictly convex weight.…”

“…Clearly the standard L p weight is an element of W + p . In the context of Kähler geometry these weights were introduced by Guedj-Zeriahi [40], and to learn more about their use we refer to [28,Chapter 1]. Theorem 3.5.…”