### Polyhedral Kahler cone metrics on **C**^{n}

Geometry and Topology Seminar

6th December 2022, 2:00 pm – 3:00 pm

Fry Building, 2.04

The talk is based on joint work with Dmitri Panov. We study flat torsion free meromorphic connections on **C**^{n} with simple poles at hyperplane arrangements which are invariant by scalar multiplication. Our main result is that, if the holonomy is unitary, then the metric completion (of the flat Kahler metric on the arrangement complement) is a polyhedral cone with vertex at 0. Taking the quotient by scalar multiplication leads to new interesting "polyhedral Fubini-Study" metrics on projective space. In the particular case of the braid arrangement, our result extends to higher dimensions the well-known existence criterion for spherical metrics on the Riemann sphere with three (non-integer) cone points.

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