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\centerline{\bf J.P. Demailly's address to the G\"ottingen Academy of Sciences}
\centerline{\bf on the occasion of the Dannie Heineman prize attribution}
\centerline{\bf November 15, 1991}
\vfill
\centerline{\bf Analytic techniques of Complex Geometry}
\vskip20pt
First I would like to express my deep gratitude to the Academy for the
attribution of the Dannie Heineman prize rewarding my mathematical 
work.  This is certainly the greatest honour I have ever had in my
mathematical career. In this address to the Academy, I will try to outline
the historical development of ideas which gradually led to the modern
concepts of analytic and algebraic geometry, in particular those
connected with my personal work.

Of course, the prehistory of algebraic geometry traces back to the work of
the ancient Greeks, through the study of conic curves by Apollonius and the
solution of geometric questions such as duplication of cubes and division of 
angles in equal parts. However, analytic geometry could not be developed
on its own stand before the invention of cartesian coordinates by Fermat and
Descartes in the first part of the 17th century.  In the previous
century, imaginary numbers had been introduced by Tartaglia and Cardano
as a way to circumvent the nonexistence of real solutions to algebraic
equations of degree $2$ or more.  These numbers remained mythical
quantities for more than two hundred years.  The clear picture everyone
has in mind today that imaginary numbers are in one-to-one
correspondence with points of a plane was surprisingly discovered only
at the end of the 18th century by Gauss, Argand and Wessel. Roughly at
the same period took place an important development of projective geometry
which culminated with the work of Poncelet. Originally, projective geometry
was designed to describe properties of figures depending only on incidence
properties of their constituents, namely points, lines, curves, etc. For that
purpose, the introduction of points at infinity appeared as a very efficient
way of avoiding exceptional cases in the general statements of intersection
theory, for example in Bezout's theorem that two plane algebraic curves of
degrees $p$ and $q$ meet in $p\times q$ points exactly. Since conic curves
are of degree~$2$, Bezout's theorem tells us for instance that two conic
curves should always intersect in four points.
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\parshape=1 2cm 9cm{\noindent
The figure shows that the intersection of a circle with an ellipse
always has four points, provided points are counted with multiplicities
and imaginary points are taken into account (they are hidden from our eyes
in some third or fourth dimension). In the case of two circles, the
imaginary points of intersection are located at infinity in the directions
corresponding to slopes $\pm\sqrt{-1}$}
\vskip15pt
So, in some sense, it became more or less clear to geometers around 1850
that a nice theory could only be made on what we now call in modern
language ``compact complex manifolds'', that is, generalized surfaces of
arbitrary dimension, on which coordinates are allowed to take imaginary
values, including points at infinity.

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\parshape=1 2cm 9cm{\noindent
A manifold is a composite object obtained by attaching open
coordinate patches together.  The manifold is said to be compact if it
has no boundary points at finite or infinite distance, as a sphere or a
torus for instance, in contrast with a disk or a ball which do have
boundary points.}
\vskip15pt
As a culmination of Euler's achievements, analytic function theory
was developed during the 19th century by several prominent analysts
among which Legendre, Abel, Cauchy, Gauss, Jacobi.  Finally Riemann
realized the very rich interplay between analytic function theory
and geometry.  Especially, he was the first mathematician to have a
clear picture of the general concept of manifolds of arbitrary
dimension, although the correct definition could only be introduced
much later by Hermann Weyl.  Riemann was also the first to make use of
the basic properties of harmonic functions in a geometric context.  
It was not before 1900 that Hilbert could finally establish on solid
grounds Riemann's intuitive application of the Dirichlet principle.
As the audience has probably noticed, a large part of the mathematicians
involved in these crucial achievements lived or worked in G\"ottingen.

Another strong impetus in the interconnection between geometry and 
analysis was inspired by Physics. This impetus came in particular through 
Maxwell's theory of electromagnetism and the introduction of the concept
of space-time continuum by Einstein at the turn of the 20th century, 
following previous works by Lorentz and Minkowski. According to Einstein's
general relativity theory, the gravitational force can be described
in terms of the curvature of the space-time continuum via the so
called Einstein equations
$$
R^i_{i\,\alpha\beta}=\lambda g_{\alpha\beta},
$$
where $g$ denotes the Minkowski metric and $R$ its curvature tensor. 
The other forces can be also described in terms of vector or scalar
fields satisfying suitable partial diffe\-rential equations. In quantum 
field theory, the behaviour of any particle is characterized by its 
wave function, which is obtained by computing solutions of the Schr\"odinger 
equation. From a geometric point of view, a magnetic field can be also seen
as the \hbox{curvature} of a geometric object called a vector bundle, representing
all possible variations of the state of the particle in function of
its position $p$ in the space-time continuum.

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\bigskip

In quantum field theory, there is a corresponding energy
functional for the particle submitted to the electromagnetic field.
In 1985, I developed a new theory which is now rewarded by the Dannie
Heineman prize, according to which it is possible to obtain
asymptotic estimates of the energy eigenstates of the particle when
the electromagnetic field becomes very large. All this has striking
consequences when the universe is taken to be a compact
complex manifold, that is, includes imaginary points and points at infinity.
In fact complex analytic functions on a manifold are
solutions of the Cauchy-Riemann $\overline\partial$ operator
and the obstructions to the solvability of the corresponding equation
are computed by what mathematicians call analytic cohomology groups.
These groups were introduced around 1950 and studied by many prominent
mathematicians, among whom I should mention Kodaira, Cartan, Serre,
Grothendieck, Prof. Friedrich Hirzebruch from the Max Planck Institut
in Bonn and Prof. Hans Grauert from G\"ottingen; some other important
contributions to analytic geometry were also made by Prof. C.L.~Siegel
in G\"ottingen. In the 1985 work already mentioned,  I obtained a
rather general formula relating the curvature tensor to the analytic
cohomology groups, that is, to the ground state energy levels of the
particle. 

More recently, during a visit to Bayreuth university in 1989, 
I realized that a modified version of Einstein's gravitational equations
could be used to solve a long standing problem of algebraic geometry
concerning $3$ or higher dimensional varieties of general type. The
idea is to use Einstein's equations for a large pointwise mass
distribution: physicists would probably imagine a black hole in that
context (in their terminology, a black hole is a big star collapsed
onto itself). Similar ideas have led physicists to introduce new models
of our universe called Calabi-Yau manifolds, which are supposed to
achieve the long-awaited unified theory of forces. Shing-Tung Yau is a
famous Chinese-American mathematician who received the Fields medal in
mathematics in 1978 for his solution of Calabi's conjectures on
Einstein equations.

I hope that these rough indications will contribute to give an idea of
the very rich interplay existing today between analysis, geometry and
physical theories. I would like to conclude by saying that the role of 
past and present mathematicians working in G\"ottingen has been extremely
influential in all these domains.
\vskip30pt
{\noindent\hangindent4cm\hangafter-1
{\sl Jean-Pierre Demailly

}}

{\noindent\hangindent4cm\hangafter-4{\sl
Universit\'e de Grenoble I,\hfill\break
Institut Fourier, BP 74,\hfill\break
URA 188 du C.N.R.S.\hfill\break
F-38402 Saint-Martin d'H\`eres

}}
\end 
