Weak transcendental holomorphic Morse inequalities on compact Kähler manifolds
Annales de l'Institut Fourier, Volume 65 (2015) no. 3, p. 1367-1379

Transcendental holomorphic Morse inequalities aim at characterizing the positivity of transcendental cohomology classes of type (1,1). In this paper, we prove a weak version of Demailly’s conjecture on transcendental Morse inequalities on compact Kähler manifolds. And as a consequence, we partially improve a result of Boucksom-Demailly-Paun-Peternell.

Les inégalités de Morse holomorphes transcendantes caractérisent la positivité des classes cohomologiques transcendantes de type (1,1). Dans ce papier, nous démontrons une version faible d’une conjecture de Demailly sur les inégalités de Morse holomorphes transcendantes sur les variétés kähleriennes. En conséquence, nous améliorons partiellement un résultat de Boucksom-Demailly-Paun-Peternell.

DOI : https://doi.org/10.5802/aif.2959
Classification:  32C30,  32Q15
Keywords: Transcendental holomorphic Morse inequalities, positivity of cohomology classes, Kähler manifolds
@article{AIF_2015__65_3_1367_0,
     author = {Xiao, Jian},
     title = {Weak transcendental holomorphic Morse inequalities on compact K\"ahler manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {3},
     year = {2015},
     pages = {1367-1379},
     doi = {10.5802/aif.2959},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_3_1367_0}
}
Xiao, Jian. Weak transcendental holomorphic Morse inequalities on compact Kähler manifolds. Annales de l'Institut Fourier, Volume 65 (2015) no. 3, pp. 1367-1379. doi : 10.5802/aif.2959. http://www.numdam.org/item/AIF_2015__65_3_1367_0/

[1] Boucksom, Sébastien; Demailly, Jean-Pierre; Păun, Mihai; Peternell, Thomas The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., Tome 22 (2013) no. 2, pp. 201-248 | Article | MR 3019449 | Zbl 1267.32017

[2] Buchdahl, Nicholas On compact Kähler surfaces, Ann. Inst. Fourier (Grenoble), Tome 49 (1999) no. 1, pp. vii, xi, 287-302 | Article | Numdam | MR 1688136 | Zbl 0926.32025

[3] Buchdahl, Nicholas A Nakai-Moishezon criterion for non-Kähler surfaces, Ann. Inst. Fourier (Grenoble), Tome 50 (2000) no. 5, pp. 1533-1538 | Article | Numdam | MR 1800126 | Zbl 0964.32014

[4] Chiose, I. The Kähler rank of compact complex manifolds (http://arxiv.org/abs/1308.2043)

[5] Demailly, Jean-Pierre Champs magnétiques et inégalités de Morse pour la d '' -cohomologie, Ann. Inst. Fourier (Grenoble), Tome 35 (1985) no. 4, pp. 189-229 | Article | Numdam | MR 812325 | Zbl 0565.58017

[6] Demailly, Jean-Pierre Holomorphic Morse inequalities, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 52 (1991), pp. 93-114 | MR 1128538 | Zbl 0755.32008

[7] Demailly, Jean-Pierre A numerical criterion for very ample line bundles, J. Differential Geom., Tome 37 (1993) no. 2, pp. 323-374 http://projecteuclid.org/euclid.jdg/1214453680 | MR 1205448 | Zbl 0783.32013

[8] Demailly, Jean-Pierre Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q., Tome 7 (2011) no. 4, Special Issue: In memory of Eckart Viehweg, pp. 1165-1207 | Article | MR 2918158

[9] Demailly, Jean-Pierre; Paun, Mihai Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2), Tome 159 (2004) no. 3, pp. 1247-1274 | Article | MR 2113021 | Zbl 1064.32019

[10] Dinew, Sławomir; Kołodziej, Sławomir Pluripotential estimates on compact Hermitian manifolds, Advances in geometric analysis, Int. Press, Somerville, MA (Adv. Lect. Math. (ALM)) Tome 21 (2012), pp. 69-86 | MR 3077248

[11] Gauduchon, Paul Le théorème de l’excentricité nulle, C. R. Acad. Sci. Paris Sér. A-B, Tome 285 (1977) no. 5, p. A387-A390 | MR 470920 | Zbl 0362.53024

[12] Lamari, A. Le cône kählérien d’une surface, J. Math. Pures Appl. (9), Tome 78 (1999) no. 3, pp. 249-263 | Article | MR 1687094 | Zbl 0941.32007

[13] Lamari, Ahcène Courants kählériens et surfaces compactes, Ann. Inst. Fourier (Grenoble), Tome 49 (1999) no. 1, pp. vii, x, 263-285 | Article | Numdam | MR 1688140 | Zbl 0926.32026

[14] Popovici, D. An Observation Relative to a Paper by J. Xiao (http://arxiv.org/abs/1405.2518)

[15] Siu, Yum Tong 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 | Article | MR 797421 | Zbl 0577.32032

[16] Sullivan, Dennis Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., Tome 36 (1976), pp. 225-255 | Article | MR 433464 | Zbl 0335.57015

[17] Tosatti, Valentino; Weinkove, Ben The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc., Tome 23 (2010) no. 4, pp. 1187-1195 | Article | MR 2669712 | Zbl 1208.53075