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QGM seminar by Martin de Borbon (Aarhus University)

Title: Blow-up analysis for Kahler-Einstein metrics with cone singularities along a divisor in two complex dimensions

2017.09.08 | Jane Jamshidi

Date Wed 13 Sep
Time 16:30 17:30
Location 1531-215 (Aud-D3)


Around  the '90 the  celebrated orbifold compactness theorem for non-collapsed sequences of Einstein metrics on four-manifolds was established (through the work Anderson, Tian, Bando-Kasue-Nakajima, ..): in a non-collapsed regime a sequence of smooth solutions to the Einstein equations can degenerate only by developing isolated quotient singularities and the blow-up limits of the sequence at these points are the so called ALE gravitational instantons (classified by Kronheimer in the simply connected Kahler case).



In the talk we outline (at some conjectural level) a parallel theory for the case of Kahler-Einstein (KE) metrics with conical singularities along a smooth complex curve. A main new feature arises  when the conical set develops an isolated plane complex curve singularity and in particular we show existence of models for blow-up limits at an ordinary d-tuple point.  Our speculations are highly inspired by recent developments in Kahler geometry relating metric degenerations of KE metrics with singularities of algebraic varieties.