100, rue des maths 38610 Gières / GPS : 45.193055, 5.772076 / Directeur : Louis Funar

Mattew Stover

Orbifold uniformization of varieties
Mercredi, 5 Juin, 2024 - 16:45
Résumé : 

Classifying smooth projective curves by their genus goes back to the
19th century, and by the uniformization theorem (one of the most
fundamental theorems in mathematics) they all admit a Riemannian metric
of constant curvature. A variant I quite like is that they all admit an
orbifold metric of constant curvature -1, and hence are dominated by a
hyperbolic surface. Amazingly the classification of smooth projective
surfaces remains open: the analogues of "genus 0" and "genus 1" are
enumerated, but the "higher genus" box may is a¯\_(ツ)_/¯ with tons
of examples. A question due independently to Farb and Gromov could
actually make the "higher genus" box a primitive recursive enumeration
problem using orbifolds of holomorphic sectional curvature -1,
succinctly generalizing orbifold uniformization of curves. My talk will
be dedicated to setting the stage for their question and using simple
topological arguments (and a giant uniformization hammer) to give some
evidence for it, spinning examples due to Deligne-Mostow and Hirzebruch
(and perhaps my own, if I have time) in this light.

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