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Complex geometry and Bergman kernel asymptotics for line bundles

Final Activity Report Summary - CPL GEOM BERGMAN-KER (Complex geometry and Bergman kernel asymptotics for line bundles)

At the heart of mathematics and its applications to other sciences lies the possibility of solving algebraic equations. In high school, one is taught to deal with the simplest such equations: those of degree 1 and 2. However, for higher degrees the situation is far more complicated - as the degree increases the number of solutions increases too - and there are no practical ways of explicitly solving the equations 'by hand'. One well-known way to confront this problem is to adapt a statistical/probabilistic point of view; one tries to find the most probable distribution of the solutions of the equation. Quite remarkable it has been shown that the solutions tend to distribute themselves as if they were a gas of interacting particles; the most probable distribution of solutions corresponds to the distribution of the gas in its equilibrium state. This analogy has been very well-studied in mathematics and physics in the one-dimensional case, i.e. for algebraic equations in one variable. In this project, tools have been developed and many results obtained concerning the case of higher dimensions, i.e. for equations in several variables, where there were very few previous results. As is well-known one has to look not only for ordinary real solutions; but also for complex solutions, consisting of so called complex numbers.

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.