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.

[1] A. Andreotti Théorèmes de dépendance algébrique sur les espaces complexes pseudoconcaves, Bull. Soc. Math. France, Volume 91 (1963), pp. 1-38 | Numdam | MR | Zbl

[2] A. Andreotti; H. Grauert Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, Volume 90 (1962), pp. 193-259 | Numdam | MR | Zbl

[3] V.I. Arnold Symplectic Geometry, Dynamical systems IV (Encyclopaedia Math. Sci.), Volume 4 (2001), pp. 1-138

[4] R. Berman Bergman kernels and local holomorphic Morse inequalities, Math Z., Volume 248 (2004) no. 2, pp. 325-344 | MR | Zbl

[5] R. Berman Super Toeplitz operators on holomorphic line bundles (arXiv.org/ abs/math.CV/0406032, http://arxiv.org/abs/math.CV/0406032)

[6] B. Berndtsson Bergman kernels related to Hermitian line bundles over compact comlex manifolds, Contemp. Math., 332, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[7] T. Bouche Inégalité de Morse pour la $d^{\text{'}\text{'}}$-cohomologie sur une variété non-compacte, Ann. Sci. École Norm. Sup, Volume 22 (1989), pp. 501-513 | Numdam | MR | Zbl

[8] S.S. Chern; J.K. Moser Real hypersurfaces in complex manifolds, Acta Math., Volume 133 (1974), pp. 219-271 | DOI | MR | Zbl

[9] J.-P. Demailly Champs magnétiques et inégalité de Morse pour la $d^{\text{'}\text{'}}$-cohomologie, Ann. Inst. Fourier, Volume 355 (1985), pp. 185-229 | Numdam | MR | Zbl

[10] J.-P. Demailly Holomorphic Morse inequalities, Volume 2 (1989), pp. 93-114 | Zbl

[11] J.-P. Demailly Introduction à la théorie de Hodge, Transcendental methods in algebraic geometry. Lectures given at the 3rd C.I.M.E. Session held in Cetraro, July 4-12, 1994 (Lecture Notes in Mathematics), Volume 1646 (1994), pp. 4-12

[12] Y. Eliashberg A few remarks about symplectic filling, Geometry and topology, Volume 8 (2004) no. 6, pp. 277-293 | MR | Zbl

[13] C. Epstein Geometric bounds on the relative index, J. Inst. Math. Jussieu, Volume 1 (2002) no. 3, pp. 441–465 | MR | Zbl

[14] G.B. Folland J.J. Kohn The Neumann problem for the Cauchy-Riemann complex, Annals of Math. Studies, 75, Princeton University Press, 1972 | MR | Zbl

[15] E. Getzler An analogue of Demailly's inequality for strictly pseudoconvex CR manifolds, J. Differential Geom., Volume 29 (1989) no. 2, pp. 231-244 | MR | Zbl

[16] P. Griffiths; J. Harris Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994 | MR | Zbl

[17] M. Gromov Kähler hyperbolicity and $L^2$-Hodge theory, J. Differential Geom., Volume 33 (1991) no. 1, pp. 263-292 | MR | Zbl

[18] G. Henkin; C. Epstein Stability of embeddings for prseudoconcave surfaces and their boundaries, Acta Math., Volume 185 (2000) no. 2, pp. 161-237 | DOI | MR | Zbl

[19] L. Hörmander $L^2$ estimates and existence theorems for the $\overline{\partial }$-operator, Acta Math., Volume 113 (1965), pp. 89-152 | DOI | MR | Zbl

[20] R. Lazarsfeld Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, A series of modern surveys in Mathematics, 48, Springer-Verlag, Berlin, 2004 | MR | Zbl

[21] G. Marinescu Asymptotic Morse inequalities for Pseudoconcave manifolds, Ann. Scuola. Norm. Sup. Pisa CL Sci., Volume 23-1 (1996) no. 4, pp. 27-55 | Numdam | MR | Zbl

[22] G. Marinescu Existence of holomorphic sections and perturbation of positive line bundles over $q$-concave manifolds (arXiv.org/abs/math.CV/0402041, http://arxiv.org/abs/math.CV/0402041)

[23] H. Rossi Attaching analytic spaces to an analytic space along a pseudoconcave boundary, Proc. Conf. Complex Manifolds (Minneapolis) (1965), pp. 242-256 | Zbl

[24] W. Rudin Real and complex analysis, McGraw-Hill Book Company, international edition, 1987 | MR | Zbl

[25] Y.T. Siu Some recent results in complex manifold theory related to vanishing theorems for the semipositive case, Workshop Bonn 1984 (Bonn, 1984) (Lecture Notes in Math.), Volume 1111 (1985), pp. 169-192 | Zbl

[26] Y.T. Siu A vanishing theorem for semipositive line bundles over non-Kähler manifolds, J. Differential Geom., Volume 19 (1984) no. 2, pp. 431-452 | MR | Zbl

[27] R.O. Wells Jr. Differential analysis on complex manifolds, Graduate Texts in Mathematics, 65, Springer-Verlag, New York-Berlin, 1980 | MR | Zbl

[28] E. Witten Supersymmetry and Morse theory, J. Differential Geom., Volume 17 (1982) no. 4, pp. 661-692 | MR | Zbl