In this project our focus will be on the study of Differential and Algebraic Geometry from a
Complex Analytic perspective. Certain Riemannian and semi-Riemannian objects, as well as
various properties of them in a complex context will be analysed.
Specifically, this school aims at providing the interested graduate students, at both a pre-
doctoral and a doctoral level, from India and other Asian countries, with the basic notions and
some recent ideas and techniques at an advanced stage of current research in a wide
spectrum of themes from classical complex analysis , pluripotential theory, geometry of
compact complex manifolds and special Hermitian metrics.
The training offer is divided into 4 introductory courses, 3 advanced courses and 2 groups of
training sessions. The first introductory course is intended to teach some basic material in
complex analysis of several variables upon which all the other courses will build.
The three main themes of this school are interrelated and each of them is first treated in an
introductory course that provides the basic notions, ideas and techniques of the subject, and
then is given an in-depth illustration through recent results at the forefront of current research
in pluripotential theory, with geometric applications, in Kähler and non-Kähler complex
geometry and, respectively, in the theory of deformations of complex structures. The planned
training sessions are intended to present further examples and exercises illustrating the
themes covered in the courses.
Our ambition is to foster the growth of complex analysis and geometry in a part of the world
where there has long been a keen interest in these subjects as attested by a strong local
tradition. We hope to channel pre-doctoral students into these subjects and to reveal new
avenues of research to doctoral students from India and Asia at large. The background we
expect the students attending this CIMPA School to have includes:
-basic notions in complex analysis in one variable (holomorphic functions; the Cauchy-
Riemann equations; the Cauchy formula and its consequences; meromorphic functions) that
are taught in the basic undergraduate course in complex analysis that mathematics students
typically take. Building on these notions, our Introductory Course 1 will present their
analogues in several complex variables as well as some of the new phenomena peculiar to
several variables.
-basic notions in differential geometry (real curves and surfaces; real smooth manifolds and
vector bundles; the notion of Riemannian metric; connections, such as the Levi-Civita
connection, and curvature; definition, basic properties and examples of real Lie groups) that
are taught in a typical introductory course to differential geometry in most universities around
the world. Building on these notions, our Introductory Courses 2 and 3 will introduce the
students to some of the complex counterparts of these notions: complex manifolds and
holomorphic vector bundles (Course 2) and Kähler metrics (Course 3).
-basic notions in functional analysis (Fréchet, Banach and Hilbert spaces; weak topologies), a
few rudiments in distribution theory and possibly a smattering of subharmonic functions and
the standard Laplacian. These will be developed in the Introductory Course 4. We will operate
the selection of the students attending this CIMPA School by assessing their background and
the extent to which they satisfy the above requirements.