A statement in the paper “Holomorphic Morse inequalities on manifolds with boundary” saying that the holomorphic Morse inequalities for an hermitian line bundle over are sharp as long as extends as semi-positive bundle over a Stein-filling is corrected, by adding certain assumptions. A more general situation is also treated.
Let be a compact complex manifold with boundary and let be a high power of a hermitian holomorphic line bundle over When has no boundary, Demailly’s holomorphic Morse inequalities give asymptotic bounds on the dimensions of the Dolbeault cohomology groups with values in in terms of the curvature of We extend Demailly’s inequalities to the case when 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...
In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle with non-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds.
We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in with exponential non-linearity and target a convex body is solvable iff is the barycenter of Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties saying that 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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