Seminar Algebra and Geometry: Mirko Mauri (Imperial College London)

The essential skeletons of pairs and the geometric P=W conjecture

The geometric P=W conjecture is a conjectural description of the asymptotic behavior of a celebrated correspondence in non-abelian Hodge theory. In particular, it is expected that the dual boundary complex of the compactification of character varieties is a sphere. In a joint work with Enrica Mazzon and Matthew Stevenson, we manage to compute the first non-trivial examples of dual complexes in the compact case. This requires to develop a new theory of essential skeletons over a trivially-valued field. As a byproduct, inspired by these constructions, we show that certain character varieties appear in degenerations of compact hyper-Kähler manifolds.  In this talk we will explain how these new non-archimedean techniques can shed new light into classical algebraic geometry problems. 

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