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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...

Improvement of Grauert-Riemenschneider's theorem for a normal surface

Jean Giraud (1982)

Annales de l'institut Fourier

Let X ˜ be a desingularization of a normal surface X . The group Pic ( X ˜ ) is provided with an order relation L _ 0 , defined by L . V 0 for any effective exceptional divisor V . Comparing to the usual order relation we define the ceiling of L which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which is not a Cartier...

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