Pluripotential theory on compact Hermitian manifolds

Sławomir Dinew

doi:10.5802/afst.1488

Sławomir Dinew ¹

¹ Department of Mathematics and Computer Science, Jagiellonian University, 30-409 Kraków, ul. Lojasiewicza 6, Poland

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 1, pp. 91-139.

Abstract
Abstract

In this survey article we collect the basic results in pluripotential theory in the setting of compact Hermitian manifolds. In particular we discuss in detail the corresponding capacity theory, various comparison principles, and the solution of the Hermitian counterpart of the Calabi-Yau equation.

Dans cet article nous collectons des résultats fondamentaux de la théorie du potentiel sur des variétés hermitiennes compactes. En particulier, nous discutons en détail la théorie de la capacité, plusieurs principes de comparaison, et la résolution de l’équation de Calabi-Yau sur les variétés hermitiennes compactes.

Received: 2014-12-02
Accepted: 2015-06-15
Published online: 2016-02-29

MR Zbl

DOI: 10.5802/afst.1488

Author's affiliations:

Sławomir Dinew ¹

¹ Department of Mathematics and Computer Science, Jagiellonian University, 30-409 Kraków, ul. Lojasiewicza 6, Poland

@article{AFST_2016_6_25_1_91_0,
     author = {S{\l}awomir Dinew},
     title = {Pluripotential theory on compact {Hermitian} manifolds},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {91--139},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {1},
     year = {2016},
     doi = {10.5802/afst.1488},
     zbl = {1342.32022},
     mrnumber = {3485292},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1488/}
}

TY  - JOUR
AU  - Sławomir Dinew
TI  - Pluripotential theory on compact Hermitian manifolds
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2016
SP  - 91
EP  - 139
VL  - 25
IS  - 1
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1488/
DO  - 10.5802/afst.1488
LA  - en
ID  - AFST_2016_6_25_1_91_0
ER  -

%0 Journal Article
%A Sławomir Dinew
%T Pluripotential theory on compact Hermitian manifolds
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2016
%P 91-139
%V 25
%N 1
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1488/
%R 10.5802/afst.1488
%G en
%F AFST_2016_6_25_1_91_0

Sławomir Dinew. Pluripotential theory on compact Hermitian manifolds. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 1, pp. 91-139. doi : 10.5802/afst.1488. https://afst.centre-mersenne.org/articles/10.5802/afst.1488/

References
Cited by

[1] Alessandrini (L.) and Bassanelli (G.).— Modifications of compact balanced manifolds, C. R. Acad. Sci. Paris 320, p. 1517-1522 (1995). | Zbl

[2] Bedford (E.) and Taylor (B. A.).— The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37, no. 1, p. 1-44 (1976). | DOI | Zbl

[3] Bedford (E.) and Taylor (B. A.).— A new capacity for plurisubharmonic functions, Acta Math. 149, no. 1-2,p. 1-40 (1982). | DOI | MR | Zbl

[4] Blanchard.— Sur les vatiétés analytiques complexes, Anal. Sci. Ecole Norm. Sup. 73, p. 157-202 (1956). | DOI | Zbl

[5] Blocki (Z.).— On the uniform estimate in the Calabi-Yau theorem, II. Sci. China Math. 54, no. 7, p. 1375-1377 (2011). | DOI | MR | Zbl

[6] Boucksom (S.), Demailly (J.P.), Paun (M.) and Peternell (T.).— The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Alg. Geom. 22, p. 201-248 (2013). | DOI | Zbl

[7] Cao (H. D.).— Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81, no. 2, p. 359-372 (1985). | DOI | Zbl

[8] Cascini (P.).— Rational curves on complex manifolds. Milan J. Math. 81, no. 2, p. 291-315 (2013). | DOI | MR

[9] Cherrier (P.).— Équations de Monge-Ampère sur les variétés hermitiennes compactes. (French) [Monge-Ampére equations on compact Hermitian manifolds] Bull. Sci. Math. (2) 111, no. 4, p. 343-385 (1987). | Zbl

[10] Chern (S.S.), Levine (H.I.) and Nirenberg (L.).— Intrinsic norms on a complex manifold (1969). Global Analysis (Papers in Honor of K. Kodaira) p. 119-139 Univ. Tokyo Press, Tokyo | DOI

[11] Demailly (J.P.).— Champs magnetiques et inegalities de Morse pour la $\bar{\partial}$ -cohomologie, Ann. Inst. Fourier 35, p. 189-229 (1985). | DOI | Zbl

[12] Demailly (J.P.).— A numerical criterion for very ample line bundles. J. Differential Geom. 37, no. 2, p. 323-374 (1993). | DOI | MR | Zbl

[13] Demailly (J.P.).— Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure and Appl. Math. Quarterly 7, p. 1165-1208 (2011). | DOI | MR | Zbl

[14] Demailly (J.P.).— Complex Analytic and Differential geometry, self published e-book.

[15] Delanoë (P.).— Équations du type de Monge-Ampère sur les variétés riemanniennes compactes. II. (French) [Monge-Ampère equations on compact Riemannian manifolds. II] J. Funct. Anal. 41, no. 3, p. 341-353 (1981). | DOI | Zbl

[16] Dinew (S.) and Kolodziej (S.).— Pluripotential estimates on compact Hermitian Manifolds, Advances in geometric analysis, 69-86, Adv. Lect. Math. (21) (2012). | DOI | MR | Zbl

[17] Dloussky (G.), Oeljeklaus (K.) and Toma (M.).— Class $V I I_{0}$ surfaces with $b_{2}$ curves, Tohoku Math. J. (2) 55 p. 283-309 (2003). | DOI

[18] Eyssidieux (P.), Guedj (V.) and Zeriahi (A.).— Singular Kähler-Einstein metrics. J. Amer. Math. Soc. 22, p. 607-639 (2009). | DOI

[19] Fernandez (M.), Ivanov (S.), Ugarte (L.) and Villacampa (R.).— Non Kaehler heterotic string compactifications with non-zero fluxes and constant dilation, Comm. Math. Phys. 288, p. 677-697 (2009). | DOI | MR | Zbl

[20] Fino (A.), Parton (M.) and Salamon (S.).— Families of strong KT structures in six dimensions. Comment. Math. Helv. 79, p. 317-340 (2004). | DOI | MR

[21] Fu (Y.), Li (J.) and Yau (S. T.).— Balanced metrics on non-Kähler Claabi-Yau threefolds. J. Diff. Geom. 90, p. 81-130 (2012). | DOI

[22] Fino (A.), Tomassini (G.).— On Astheno-Kähler metrics, J. Lond. Math. Soc. 83, no. 2, p. 290-308 (2011). | DOI | Zbl

[23] Gauduchon (P.).— Le théorème de l’excentricitè nulle. (French) C. R. Acad. Sci. Paris 285, p. 387-390 (1977).

[24] Gill (M.).— Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. Comm. Anal. Geom. 19, no. 2, p. 277-303 (2011). | DOI | Zbl

[25] Guan (B.) and Li (Q.).— Complex Monge-Ampère equations and totally real submanifolds. Adv. Math. 225, p. 1185-1223 (2010). | DOI | Zbl

[26] Guedj (V.) and Zeriahi (A.).— Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15, no. 4, p. 607-639 (2005). | DOI | Zbl

[28] Guedj (V.) and Zeriahi (A.).— The weighted Monge-Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250, no. 2, p. 442-482 (2007). | DOI | Zbl

[29] Hopf (H.).— Zur Topologie der komplexen Mannigfaltigkeiten, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, Interscience Publishers, Inc., New York, p. 167-185 (1848).

[30] Hörmander (L.).— Notions of convexity. Progress in Mathematics, 127. Birkhäuser Boston, Inc., Boston, MA, viii+414 pp. (1994). | Zbl

[31] Inoue (M.).— On surfaces of Class $V I I_{0}$ . Invent. Math. 24 (1974), p. 269-310. | DOI | Zbl

[32] Jost (J.) and Yau (S. T.).— A non linear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geoemtry. Acta Math. 170, p. 221-254 (1993). | DOI | MR | Zbl

[33] Kato (M.).— Compact complex manifolds containing “global” spherical shells. I, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Tokyo.— Kinokuniya Book Store, p. 45-84.

[34] Kodaira (K.).— On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 p. 751-798 (1964). | DOI | MR | Zbl

[35] Kodaira (K.).— On the structure of compact complex analytic surfaces. II, Amer. J. Math. 88 p. 682-721 (1966). | DOI | MR | Zbl

[36] Kodaira (K.).— On the structure of compact complex analytic surfaces. III, Amer. J. Math. 90 p. 55-83 (1968). | DOI | MR

[37] Kodaira (K.).— On the structure of complex analytic surfaces. IV, Amer. J. of Math. 90 p. 1048-1066 (1968). | DOI | MR | Zbl

[38] Kodaira (K.) and . Spencer (D.C.).— On deformations of complex analytic structures III. Stability theorems for complex structures. Ann. of Math. (2) 71 p. 43-76 (1960). | DOI | MR

[39] Kolodziej (S.).— The complex Monge-Ampère equation. Acta Math. 180, no. 1, p. 69-117 (1998). | DOI | Zbl

[40] Kolodziej (S.).— The Monge-Ampère equation on compact Kähler manifolds. Indiana Univ. Math. J. 52, no.3, p. 667-686 (2003). | DOI | Zbl

[41] Nguyen (N. C.) and Kolodziej (S.).— Weak solutions to the complex Monge-Ampère equation on compact Hermitian manifolds, Contemp. Math. 644, p. 141-158 (2015). | DOI | Zbl

[42] Kolodziej (S.), Tian (G.).— A uniform $L^{\infty}$ estimate for complex Monge-Ampère equations. Math. Ann. 342, no. 4, p. 773-787 (2008). | DOI | Zbl

[43] Lübke (M.), Teleman (A.).— The Kobayashi-Hitchin correspondence, World Scientific publishing Co. (1995). | DOI | Zbl

[44] Mitsuo (K.).— Astheno-Kähler structures on Calabi-Eckman manifolds. Colloq. Math. 155, p. 33-39 (2009). | DOI

[45] Mori (S.).— Projective manifolds with ample tangent bundles. Ann. of Math. (2) 110, no. 3, p. 593-606 (1979). | DOI | MR | Zbl

[46] Nakamura (I.).— On surfaces of class $V I I_{0}$ with curves, Invent. Math. 78 (3), p. 393-443 (1984). | DOI | Zbl

[47] Popovici (D.).— Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics. Invent. Math. 194, no. 3, p. 515-534 (2013). | DOI | MR | Zbl

[48] Popovici (D.).— An Observation Relative to a Paper by J. Xiao, preprint arXiv:1405.2518, to appear in Math. Ann. as Sufficient bigness criterion for differences of two nef classes. | DOI | MR

[49] Popovici (D.).— Deformation openness and closedness of various classes of compact complex manifolds; examples, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13, no. 2, p. 255-305 (2014). | Zbl

[50] Siu (Y. T.).— Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds, preprint arXiv:0902.2809. | Zbl

[51] Song (J.) and Tian (G.).— The Kähler-Ricci flow through singularities, preprint arXiv:0909.4898. | DOI

[52] Streets (J.) and Tian (G.).— Hermitian curvature flow. J. Eur. Math. Soc. (JEMS) 13, no. 3, p. 601-634 (2011). | DOI | Zbl

[53] Streets (J.) and Tian (G.).— Regularity results for pluriclosed flow. Geom. Topol. 17, no. 4, p. 2389-2429 (2013). | DOI | MR | Zbl

[54] Teleman (A.).— Instantons and curves on class $V I I$ surfaces. Ann. of Math. (2) 172, no. 3, p. 1749-1804 (2010). | DOI | MR | Zbl

[55] Tosatti (V.) and Weinkove (B.).— Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds. Asian J. Math. 14, no. 1, p. 19-40 (2010). | DOI | Zbl

[56] Tosatti (V.) and Weinkove (B.).— The complex Monge-Ampère equation on compact Hermitian manifolds. J. Amer. Math. Soc. 23, no. 4, p. 1187-1195 (2010). | DOI | Zbl

[57] Tosatti (V.) and Weinkove (B.).— Plurisubharmonic functions and nef classes on complex manifolds. Proc. Amer. Math. Soc. 140, no. 11, p. 4003-4010 (2012). | DOI | MR | Zbl

[58] Tosatti (V.) and Weinkove (B.).— The Chern-Ricci flow on complex surfaces. Compos. Math. 149, no. 12, p. 2101-2138 (2013). | DOI | MR | Zbl

[59] Tosatti (V.), Y. WANG, Weinkove (B.) and Yang (X.).— $𝒞^{2, α}$ estimates for nonlinear elliptic equations in complex and almost complex geometry, Calc. Var. PDE 54, p. 431-453 (2015). | DOI | MR

[60] Tian (G.) and Zhang (Z.).— On the Kähler-Ricci flow on projective manifolds of general type. Chinese Ann. Math. Ser. B 27, no. 2, p. 179-192 (2006). | DOI | Zbl

[61] Xiao (J.).— Weak trancendental holomorphic Morse inequalities on compact Kähler manifolds, preprint arXiv1308.2878, Ann. Inst. Fourier 65, p. 1367-1379 (2015). | DOI | Zbl

[62] Yau (S.T.).— On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31, no. 3, p. 339-411 (1978). | DOI | Zbl

Cited by Sources: