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Corrigendum to: Holomorphic Morse inequalities on manifolds with boundary

Robert Berman — 2008

Annales de l’institut Fourier

A statement in the paper “Holomorphic Morse inequalities on manifolds with boundary” saying that the holomorphic Morse inequalities for an hermitian line bundle L over X are sharp as long as L extends as semi-positive bundle over a Stein-filling is corrected, by adding certain assumptions. A more general situation is also treated.

Holomorphic Morse Inequalities on Manifolds with Boundary

Robert Berman — 2005

Annales de l’institut Fourier

Let X be a compact complex manifold with boundary and let L k be a high power of a hermitian holomorphic line bundle over X . When X has no boundary, Demailly’s holomorphic Morse inequalities give asymptotic bounds on the dimensions of the Dolbeault cohomology groups with values in L k , in terms of the curvature of L . We extend Demailly’s inequalities to the case when X has a boundary by adding a boundary term expressed as a certain average of the curvature of the line bundle and the Levi curvature of the...

Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties

Robert J. BermanBo Berndtsson — 2013

Annales de la faculté des sciences de Toulouse Mathématiques

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P . Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties ( X , Δ ) saying that ( X , Δ ) admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and...

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