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Göteborgs universitets publikationer

A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry

Författare och institution:
Bo Berndtsson (Institutionen för matematiska vetenskaper, matematik, Chalmers/GU)
Publicerad i:
Inventiones Mathematicae, 200 ( 1 ) s. 149-200
Artikel, refereegranskad vetenskaplig
Sammanfattning (abstract):
For ϕ a metric on the anticanonical bundle, −KX , of a Fano manifold X we consider the volume of X ∫Xe−ϕ. In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on −KX . Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on X , even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando–Mabuchi uniqueness theorem for Kähler–Einstein metrics. A generalization of this theorem to ‘twisted’ Kähler–Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than −KX , and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kähler–Ricci solitons.
Ämne (baseras på Högskoleverkets indelning av forskningsämnen):
Chalmers fundament:
Grundläggande vetenskaper
Postens nummer:
Posten skapad:
2014-12-02 12:54
Posten ändrad:
2016-09-14 16:09

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