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From complex to non-archimedean geometry

Final Report Summary - NONARCOMP (From complex to non-archimedean geometry)

The main aim of the project was to bring together two seemingly different geometries, namely the classical study of complex analytic varieties and
its non-Archimedean analog, that is the study of analytic varieties defined over fields endowed with a norm satisfying the strong triangle
inequality | a + b| <= max { |a|, |b| }. The interest in developing such non-Archimedean geometries grew up from two main sources of applications, firstly to degeneration problems in complex geometry in which case one argues over the field of Laurent series C((t)); and secondly to problems in arithmetic geometry which are treated using adelic methods
by considering all completions of the field of rational numbers, i.e. R and Q_p for all prime integers p.

The project was focused on two aspects of the interactions between Archimedean and non-Archimedean geometry, more specifically on pluripotential analysis
which lies at the heart of modern complex geometry;
and on algebraic dynamics which deals with problems of algebraic nature concerning dynamical systems defined by systems of polynomials. Several important progress have been made on these two aspects by the members of the team.

A major achievement was the extension to a non-Archimedean context by the P.I. S. Boucksom and M. Jonsson of a celebrated theorem of Yau on the existence of special metrics
having a prescribed curvature volume form. The general strategy was to follow the variational approach originally developed in the complex setting by Berman, Boucksom, Guedj and Zeriahi. To be able to do so several key properties for the space of all semi-positive metrics were crucial to uncover such as its compactness and the differentiability of a suitable projection operator defined in terms of envelopes. The proofs of these properties turned out to quite deep and technical and relied in the end on new estimates for intersection numbers of b-divisors on algebraic varieties.

Several key results have also been obtained in algebraic dynamics. This line of research is very recent and has grown very rapidly during the lifetime of the project. Several instances of famous conjectures in the field have been tackled by the members of the project, and these results have received a great recognition from the community.
J. Xie has a given a proof of the dynamical Mordell-Lang conjecture for all polynomial maps of the affine plane. His delicate proof ranges over 100 pages and
relies on an interesting new technic to build auxiliary polynomials. C. Favre and T. Gauthier have proved the dynamical André-Oort conjecture for cubic polynomials,
pushing to its limits the original arguments of Baker and DeMarco, thereby obtaining a very detailed description of the distribution of PCF polynomials in the moduli space.

It is also important to note that several results obtained during the project have opened new unexpected perspectives. Beside the ones already mentioned above, R. Rodriguez Vazquez has studied compactness properties in spaces of non-Archimedean analytic maps which hints at a new notion of hyperbolicity in non-Archimedean geometry; and L. Fantini has introduced an analytic structure on non-Archimedean links which make them amenable to new technics of investigations.