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Asymptotic invariants of linear systems

Final Activity Report Summary - ASYMPLIN (Asymptotic invariants of linear systems)

The project concerns certain central areas of higher-dimensional geometry, which is a part of algebraic geometry with close ties to fields outside mathematics, e.g. theoretic physics (string theory), control theory and cryptography. We were primarily interested in some of the fundamental objects of geometry: the behaviour of linear systems (interesting loci and functions on our spaces), their various invariants (cohomology, base loci among others), and the singular (i.e. non-smooth) points of the underlying spaces (varieties).

Such questions lie at the very heart of today's view of our physical world; according to wide-spread conjectures among physicists, the small-scale structure of our universe is given in terms of a certain 'Calabi-Yau' variety, a (real) six-dimensional space coming from higher-dimensional complex geometry. Thus being able to do geometry on these and similar geometric objects is a necessary prerequisite for understanding the fundamental structure of the world around us.

To put it more precisely, our main research objectives were (i) to give a characterisation of positive line bundles in terms of their asymptotic behaviour, (ii) to prove that stable base loci of line bundles grow at rational values, and (iii) to determine if among the singularities occurring on varieties of given dimension there is a 'least singular' (but not smooth) type.

All three of these goals were known to be difficult and ambitious, with the last one being a long-standing open problem and as such promising to be exceptionally demanding.
The first goal was achieved in its entirety working together with Tommaso de Fernex (University of Utah), and Robert Lazarsfeld (University of Michigan). We used a combination of sophisticated linear algebra and multiplier ideal methods. The resulting paper has already appeared in the prestigious mathematical journal Mathematische Annalen.

The probably most important piece of work done during this one year was our investigation of the interior structure of the cone of effective divisors on a projective variety with Endre Szabo. In order to find out about the growth of the stable base locus of a divisor while one wanders about in the Neron-Severi space, we have found interesting substructures in the effective cone associated to a projective variety. This discovery promises to have many significant consequences. Among others it has opened new vistas regarding the third goal of the project concerning log canonical thresholds of singularities.