Holomorphic Morse Inequalities on Manifolds with Boundary

Berman, Robert

Annales de l'Institut Fourier, Tome 55 (2005) no. 4, pp. 1055-1103.

Soit $X$ une variété complexe compacte à bord et soit $L^k$ une grande puissance d’un fibré en droites hermitien holomorphe sur $X$. Quand $X$ n’a pas de bord, les inégalités de Morse holomorphes de Demailly donnent des estimations asymptotiques des dimensions des groupes de cohomologie de Dolbeault à valeurs dans $L^k$, en termes de la courbure de $X$. On étend les inégalités de Demailly au cas où $X$ a un bord, en ajoutant un terme au bord exprimé comme une certaine moyenne de la courbure du fibré et de la courbure de Levi du bord. Nous donnons des exemples qui montrent que les inégalités sont optimales.

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.

Classification : 32A25, 32L10, 32L20
Keywords: Line bundles, cohomology, harmonic forms, holomorphic sections, Bergman kernel
Mot clés : fibrés en droites, cohomologie, formes harmoniques, sections holomorphes, noyaux de Bergman

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