Holomorphic Morse Inequalities on Manifolds with Boundary
Annales de l'Institut Fourier, Volume 55 (2005) no. 4, p. 1055-1103
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.
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.
DOI : https://doi.org/10.5802/aif.2121
Classification:  32A25,  32L10,  32L20
Keywords: Line bundles, cohomology, harmonic forms, holomorphic sections, Bergman kernel
@article{AIF_2005__55_4_1055_0,
     author = {Berman, Robert},
     title = {Holomorphic Morse Inequalities on Manifolds with Boundary},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {4},
     year = {2005},
     pages = {1055-1103},
     doi = {10.5802/aif.2121},
     mrnumber = {2157164},
     zbl = {1082.32001},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2005__55_4_1055_0}
}
Berman, Robert. Holomorphic Morse Inequalities on Manifolds with Boundary. Annales de l'Institut Fourier, Volume 55 (2005) no. 4, pp. 1055-1103. doi : 10.5802/aif.2121. https://aif.centre-mersenne.org/item/AIF_2005__55_4_1055_0/

[1] A. Andreotti Théorèmes de dépendance algébrique sur les espaces complexes pseudoconcaves, Bull. Soc. Math. France, Tome 91 (1963), pp. 1-38 | Numdam | MR 152674 | Zbl 0113.06403

[2] A. Andreotti; H. Grauert Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, Tome 90 (1962), pp. 193-259 | Numdam | MR 150342 | Zbl 0106.05501

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

[4] R. Berman Bergman kernels and local holomorphic Morse inequalities, Math Z., Tome 248 (2004) no. 2, pp. 325-344 | MR 2088931 | Zbl 1066.32002

[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, Amer. Math. Soc., Providence, RI, Contemp. Math., Tome 332 (2003) | MR 2016088 | Zbl 1038.32003

[7] T. Bouche Inégalité de Morse pour la d '' -cohomologie sur une variété non-compacte, Ann. Sci. École Norm. Sup, Tome 22 (1989), pp. 501-513 | Numdam | MR 1026747 | Zbl 0693.32016

[8] S.S. Chern; J.K. Moser Real hypersurfaces in complex manifolds, Acta Math., Tome 133 (1974), pp. 219-271 | Article | MR 425155 | Zbl 0302.32015

[9] J.-P. Demailly Champs magnétiques et inégalité de Morse pour la d '' -cohomologie, Ann. Inst. Fourier, Tome 355 (1985), pp. 185-229 | Numdam | MR 799607 | Zbl 0565.58017

[10] J.-P. Demailly Holomorphic Morse inequalities, Santa Cruz, CA, Tome 2 (1989), pp. 93-114 | Zbl 0755.32008

[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, Springer-Verlag (Lecture Notes in Mathematics) Tome 1646 (1994), pp. 4-12

[12] Y. Eliashberg A few remarks about symplectic filling, Geometry and topology, Tome 8 (2004) no. 6, pp. 277-293 | MR 2023279 | Zbl 1067.53070

[13] C. Epstein Geometric bounds on the relative index, J. Inst. Math. Jussieu, Tome 1 (2002) no. 3, pp. 441–465 | MR 1956056 | Zbl 1038.32030

[14] G.B. Folland J.J. Kohn The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Annals of Math. Studies, Tome 75 (1972) | MR 461588 | Zbl 0247.35093

[15] E. Getzler An analogue of Demailly's inequality for strictly pseudoconvex CR manifolds, J. Differential Geom., Tome 29 (1989) no. 2, pp. 231-244 | MR 982172 | Zbl 0714.58053

[16] P. Griffiths; J. Harris Principles of algebraic geometry, John Wiley & Sons, Inc., New York, Wiley Classics Library (1994) | MR 1288523 | Zbl 0836.14001

[17] M. Gromov Kähler hyperbolicity and L 2 -Hodge theory, J. Differential Geom., Tome 33 (1991) no. 1, pp. 263-292 | MR 1085144 | Zbl 0719.53042

[18] G. Henkin; C. Epstein Stability of embeddings for prseudoconcave surfaces and their boundaries, Acta Math., Tome 185 (2000) no. 2, pp. 161-237 | Article | MR 1819994 | Zbl 0983.32035

[19] L. Hörmander L 2 estimates and existence theorems for the ¯-operator, Acta Math., Tome 113 (1965), pp. 89-152 | Article | MR 179443 | Zbl 0158.11002

[20] R. Lazarsfeld Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Springer-Verlag, Berlin, A series of modern surveys in Mathematics, Tome 48 (2004) | MR 2095471 | Zbl 02134816

[21] G. Marinescu Asymptotic Morse inequalities for Pseudoconcave manifolds, Ann. Scuola. Norm. Sup. Pisa CL Sci., Tome 23-1 (1996) no. 4, pp. 27-55 | Numdam | MR 1401416 | Zbl 0867.32004

[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), Springer-Verlag, New York (1965), pp. 242-256 | Zbl 0143.30301

[24] W. Rudin Real and complex analysis, McGraw-Hill Book Company, international edition (1987) | MR 924157 | Zbl 0925.00005

[25] Y.T. Siu Some recent results in complex manifold theory related to vanishing theorems for the semipositive case, Workshop Bonn 1984 (Bonn, 1984), Springer, Berlin (Lecture Notes in Math.) Tome 1111 (1985), pp. 169-192 | Zbl 0577.32032

[26] Y.T. Siu A vanishing theorem for semipositive line bundles over non-Kähler manifolds, J. Differential Geom., Tome 19 (1984) no. 2, pp. 431-452 | MR 755233 | Zbl 0577.32031

[27] R.O. Wells Jr. Differential analysis on complex manifolds, Springer-Verlag, New York-Berlin, Graduate Texts in Mathematics, Tome 65 (1980) | MR 608414 | Zbl 0435.32004

[28] E. Witten Supersymmetry and Morse theory, J. Differential Geom., Tome 17 (1982) no. 4, pp. 661-692 | MR 683171 | Zbl 0499.53056