数学科学研究所
Insitute of Mathematical Science

Mini Course: Global pluripotential theory on compact Kähler manifolds

Seminar| Institute of Mathematical Sciences

TimeJuly 8th, 10th, 12th, 13rd,  2023 , 15:00-16:30
Location:IMS, RS408

Speaker: Mingchen Xia, Institut de Mathématiques de Jussieu-Paris Rive Gauche



Organizer: Long Li

AbstractPluripotential theory is the study of plurisubharmonic functions on domains and complex manifolds. Traditionally, one begins with the local setting of domains and studies plurisubharmonic functions and their Monge—Ampère operators. In this course, we will be interested in the global setting, namely studying (quasi-)subharmonic functions on compact Kähler manifolds. In this setting, unlike on domains, the geometry of the manifold plays an essential role. We will explore the interplay between (quasi-)subharmonic functions and the algebraic/differential geometry of the manifold.

The course consists of four lectures. The first two lectures are general introduction to the pluripotential theory on compact Kähler manifolds. In the last two lectures, I will talk about more specific topics: the theory of I-good singularities and their applications developed by Tamás Darvas and myself in the last few years.


Schedule

July 8th, 15:00-16:30

Lecture 1: Bedford—Taylor theory

In this lecture, I will give a general introduction to the  plurisubharmonic functions. We will define the Monge—Ampère operator of bounded plurisubharmonic functions following Bedford—Taylor.


References: 

Bedford—Taylor, Fine topology, Silov boundary, and (ddc)n

Guedj—Zeriahi, Degenerate complex Monge-Ampère equations


July 10th, 15:00-16:30

Lecture 2: Non-pluripolar products


In this lecture, I will introduce the most useful extension of Bedford—Taylor theory to unbounded plurisubharmonic functions, the non-pluripolar product. I will explain the basic properties of the non-pluripolar product developed in the last decade, including the monotonicity theorem, the comparison principle, the semicontinuity theorem etc.


Refereces: 

Guedj—Zeriahi, Degenerate complex Monge-Ampère equations

Boucksom—Eyssidieux—Guedj—Zeriahi, Monge-Ampère equations in big cohomology classes

Darvas—Di Nezza—Lu, Relative pluripotential theory on compact Kähler manifolds 


July 12th, 15:00-16:30

Lecture 3: Singularities of plurisubharmonic functions


In this lecture, I will explain how to study the singularities of plurisubharmonic functions. I will introduce the d_S pseudometric on the space of plurisubharmonic functions describing how close two singularities are. Then I will proceed to define the notion of I-good singularities and prove a singular transcendental Morse inequality. I will also discuss the theory of b-divisors.


References:

Darvas—Di Nezza—Lu, The metric geometry of singularity types

Darvas—Xia, The closures of test configurations and algebraic singularity types

Darvas—Xia, The volume of pseudoeffective line bundles and partial equilibrium

Xia, Restrictions of currents and transcendental partial Okounkov bodies (unpublished, available upon request)


July 13rd, 15:00-16:30

Lecture 4: Applications of I-good singularities


In this lecture, I will discuss the applications of the tools developed in the previous lectures, especially the I-good singularities. I will first introduce the Ross—Witt Nyström correspondence and the Berman—Boucksom—Jonsson embedding. We will see how I-good singularities are related to the non-Archimedean pluripotential theory developed by Boucksom—Jonsson. Secondly, I will introduce the partial Okounkov bodies associated with a line bundle endowed with a singular metric.


References:

Ross—Witt Nyström, Analytic test configurations and geodesic rays

Darvas—Xia, The closures of test configurations and algebraic singularity types

Berman—Boucksom—Jonsson, A variational approach to the Yau—Tian—Donaldson conjecture

Darvas—Xia—Zhang, A transcendental approach to non-Archimedean metrics of pseudoeffective classes

Xia, Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics

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