Final Activity Report Summary - CPL GEOM BERGMAN-KER (Complex geometry and Bergman kernel asymptotics for line bundles)
One of the main points of the project has been to put the previous problems into a geometric framework called 'complex geometry' which deals with geometric objects such as 'line bundles over complex manifolds' (these are used to represent the graphs of the polynomials describing the equations). This geometric frame work is also used in physics to describe quantum electrons in strong magnetic fields. One of the most important outcomes of this project is that we have shown that seemingly very different mathematical and physical objects tend to distribute themselves on an 'equilibrium measure' living on a complex manifold (in a certain precise sense).
This goes for example for the algebraic solutions in several complex variables referred to above, as well as for algebraic integers of small height (studied by number theorists) and electrons subject to a strong magnetic field. To this end, we have used a powerful tool called the Bergman kernel, which is an efficient way to encode all the information of the algebraic solutions into one single object. Other important results of this project concern a geometric frame work where one tries to go beyond the 'line bundles' and graphs referred to above to deal with more transcendental objects called 'currents'. In this direction, we are now close to obtaining so called 'transcendental holomorphic Morse inequalities'. These are crucial in order to control the behaviour of the currents.