Holomorphic Morse Inequalities on Manifolds with Boundary

Robert Berman[1]

  • [1] Chalmers University of Technology, Department of Mathematics, Eklandag. 86, 412 96 Göteborg (Suède)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 4, page 1055-1103
  • ISSN: 0373-0956

Abstract

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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 boundary. Examples are given that show that the inequalities are sharp.

How to cite

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Berman, Robert. "Holomorphic Morse Inequalities on Manifolds with Boundary." Annales de l’institut Fourier 55.4 (2005): 1055-1103. <http://eudml.org/doc/116214>.

@article{Berman2005,
abstract = {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 boundary. Examples are given that show that the inequalities are sharp.},
affiliation = {Chalmers University of Technology, Department of Mathematics, Eklandag. 86, 412 96 Göteborg (Suède)},
author = {Berman, Robert},
journal = {Annales de l’institut Fourier},
keywords = {Line bundles; cohomology; harmonic forms; holomorphic sections; Bergman kernel; line bundles},
language = {eng},
number = {4},
pages = {1055-1103},
publisher = {Association des Annales de l'Institut Fourier},
title = {Holomorphic Morse Inequalities on Manifolds with Boundary},
url = {http://eudml.org/doc/116214},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Berman, Robert
TI - Holomorphic Morse Inequalities on Manifolds with Boundary
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 4
SP - 1055
EP - 1103
AB - 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 boundary. Examples are given that show that the inequalities are sharp.
LA - eng
KW - Line bundles; cohomology; harmonic forms; holomorphic sections; Bergman kernel; line bundles
UR - http://eudml.org/doc/116214
ER -

References

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