CIMPA

Introductory course 1: Introduction to Complex Analysis in Several Variables

by Judith Brinkschulte

The basic notions in several complex variables will be given a detailed presentation : holomorphic
functions of several variables, Cauchy formula, the Analytic Continuation Theorem and Montel’s
Theorem.
The Hartog’s phenomena will be explained and the notion of domains of holomorphy will be given
as well as their characterization in terms of holomorphic convexity.
Then the notion of plurisubharmonic function and pseudconvexity will be introduced and the Levi
problem stated. The Levi problem will be solved using Hörmander’s L^2 methods for solving the
Cauchy-Rieman equations with L^2 estimates will be presented.
Then the Oshawa-Takegoshi extentension theorem will be proved and some of its applications will
be sketched . Demailly’s approximation theorem for plurisubharmonic functions, Siu’s analyticity
theorem, and the solution by Q. Guan and X. Zhou to the openess conjecture of Demailly and
Kollar.
This course is intended to serve as a basis for virtually all the other courses, both introductory and
advanced, that will form part of this school. For this reason, we do not pair it with any particular
advanced course.

 

 

Introductory course 2: Introduction to Pluripotential Theory

by Azimbay Sadullaev

The notion of subharmonic function will be presented first, starting from the construction of the
Green kernel, the Green-Riesz representation formula and the Poisson-Jensen formula relating the
average value of a function on a sphere to the average on the ball of the Laplacian of that function.

After the basic properties of subharmonic functions have been thoroughly studied, including the
maximum principle, the stability under non-increasing passages to the limit and the positivity in the
sense of measures of their Laplacians.
Then the study will be refined by considering plurisubharmonic (psh) functions and the positivity
property of their complex hessian. Basic facts about postive currents will be presented and many
examples given, namely the current of integration on (singular) analytic sets that provides an
important link with complex geometry.
The complex Monge-Ampère operator will be introduced starting from the original definition of
Bedford-Taylor in the case of locally bounded plurisubharmonic functions.
A general maximum principle as well as various comparison principles for the complex Monge-
Ampère operator will be proved and the Dirichlet problem for the complex Monge-Ampère
operator on strongly pseudoconvex domains will be solved using the Perron method.
If time permits, some facts on pluripotential Theory on comapct Kähler manifolds will be given as
well as some applications to Kähler Geometry.

 

 

Introductory course 3: Introduction to Hodge Theory

by Dan Popovici

 

This course will present the basic notions in the Hodge Theory of compact Hermitian and
Kähler manifolds. It will start by introducing the notion of elliptic differential operators and by
proving Garding’s Inequality and its main geometric consequence on compact manifolds for these
operators. The usual Laplace-type operators will then be introduced on both real and complex
manifolds, they will be proved to be elliptic and the Hodge Isomorphism Theorem will then be
deduced. No Kaehlerianity asumption will be needed so far.
The notion of Kähler metric on a complex manifold will then be introduced and examples of both
Kähler (including projective) and non-Kähler manifolds will be provided. The Kähler commutation
relations will then be proved and the resulting ddbar-Lemma, Hodge Decomposition and Hodge
Symmetry on compact Kähler manifolds will be deduced. If time permits, some basic notions in the
Theory of Deformations of Complex Structures will also be presented along with the Kodaira-
Spencer Semicontinuity Theorem in holomorphic families of compact complex manifolds.

 

Introductory course 4: Introduction to Complex Dynamics

by Jasmin Raissy

The first part of this course will deal with iterations of rational maps of projective space
(indeterminacy set, degrees of iterates, algebraically stable maps), as well as with Fatou sets and
Julia sets. The notion of Green current associated with a meromorphic map will also be studied.
The second part of the course will deal with polynomial automorphisms of \C^k and the notions of
periodic points, entropy and basin of attraction.
The third part of the course will deal with holomorphic endomorphisms of projective space :
exterior powers of certain currents, mixing and Lyapunov exponents.

 

 

Advanced course 1: Complex Monge-Ampère Flows and Singular Kähler-Einstein Metrics

by Vincent Guedj

The aim of this mini-course is to explain recent developments in
the theory of degenerate complex Monge-Ampère equations, with applications
towards the construction of canonical metrics and the study of the
Kaehler-Ricci flow on mildly singular projective algebraic varieties.
Complex Monge-Ampère equations have been one of the most powerful tools in
Kaehler geometry since Aubin and Yau’s classical works, culminating in Yau’s
solution to the Calabi conjecture. A notable application is the construction
of Kaehler-Einstein metrics on compact Kaehler manifolds.
Whereas their existence on manifolds with trivial or ample canonical class
was settled as a corollary of the Calabi conjecture, determining necessary
and sufficient conditions on a Fano manifold to carry a Kaehler-Einstein
metric has only been solved recently by Chen-Donaldson-Sun.

Following Tsuji’s pioneering work, degenerate complex Monge-Ampère equations
have been intensively studied by many authors in the last decade. In
relation to the Minimal Model Program, they led to the construction of
singular Kaehler-Einstein metrics with zero or negative Ricci curvature or,
more generally, of canonical volume forms on compact Kaehler manifolds with
nonnegative Kodaira dimension. Making sense of and constructing singular
Kaehler-Einstein metrics on singular Fano varieties requires more advanced
tools in the study of degenerate complex Monge-Ampère equations.
We shall survey the theory of Kaehler-Einstein metrics/currents, explain
their equivalent formulation in terms of degenerate complex Monge-Ampère
equations, and present the variational approach developed by
Berman-Boucksom-Guedj-Zeriahi, as well as its application to the study of
the normalized Kaehler-Ricci flow. This allows in patricular to generalize
deep results of Perelman-Tian-Zhu. If time permits, we plan to discuss the
most recent pluripotential parabolic approach of Guedj-Lu-Zeriahi.

 

Advanced course 2: Real and Complex Brunn-Minkowski Theory

by Bo Berndtsson

The course will start with a recapitulation of the classical Brunn-Minkowski theorem and its
functional version, Prékopa’s theorem. Proofs will be based on the approach by Brascamp and Lieb,
using a real variable version of Hormander’s L^2-estimates.
Then the Prékopa-Leindler theorem (a partly non-convex version of Prékopa’s theorem) will be
discussed. The proof of this also has a complex counterpart, as it uses a real variable analog of Yau’s
solution to the Calabi conjecture.
After that, we review the Bergman kernel, and its interpretation as a metric on a line bundle, and
give the first version of a complex Brunn-Minkowski theorem. Then we discuss direct images
bundles, and different notions of positivity of their curvature. Finally, some applications to Kähler
geometry will be sketched.

 

Advanced course 3: Dynamics in several complex variables

by Charles Favre

The idea is to survey some developments in the study of the dynamics of holomorphic self-maps of
the complex projective space. We will explain how tools from analysis in several complex variables
can be used to construct invariant measures whose support is the Julia set, that is the locus where
the dynamics is the most unstable.
We shall also examine the Fatou set over which the dynamics is stable, and discuss some of its
geometric properties.

 

Training Session 1: Applications of Pluripotential Theory to Geometry

by Eleonora Di Nezza

A detailed proof of Demailly’s regularisation-of-currents theorem will be presented as an
application of the introductory courses 1 (L2 methods, especially Hörmander’s L2 estimates, Skoda’s
L2 Division Theorem and the Ohsawa-Takegoshi L2 Extension Theorem) and 2 (Monge-Ampère
currents and masses, weak convergence of currents, plurisubharmonic functions). Many of the
intermediate results needed will be presented in the form of exercises that have been split up into
successive stages and provided with detailed hints that will guide the students through the proof
while enabling them to discover by themselves quite a number of basic techniques.

 

Training Session 2: Applications of Hodge Theory to Deformations of Complex Structures

by Simona Myslivets

This session is intended as a complement to the classical Kodaira-Spencer results on deformations
of complex structures. The Bott-Chern and Aeppli cohomologies, together with the corresponding
4-th order Laplace-type operators that provide a Hodge theory thereof, will be presented.
Schweizer’s Aeppli Laplacian and simplification of the Kodaira-Spencer proof of the deformation
openness of the Kähler property of compact complex manifolds, will be presented in the form of
exercises with detailed hints guiding the students through the proof. A similar approach will be
adopted with respect to the new Hodge theory, using recently introduced pseudo-differential
Laplace-type operators, for all the pages of the Frölicher spectral sequence of a compact complex
manifolds and its applications to deformation theory.