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.

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...

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 ${\mathbb{R}}^{n}$ 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 $(X,\Delta )$ saying that $(X,\Delta )$ 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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