A variational approach to complex Monge-Ampère equations
Berman, Robert J.1 ; Boucksom, Sébastien2 ; Guedj, Vincent3 ; Zeriahi, Ahmed4
1 Chalmers Techniska Högskola, Chalmers University of Technology and University of Gothenburg Göteborg Sweden
2 Institut de Mathématiques, CNRS-Université Pierre et Marie Curie 75252, Paris Cedex France
3 I.M.T., Université Paul Sabatier and Institut Universitaire de France 31062, Toulouse Cedex 09 France
4 I.M.T., Université Paul Sabatier 31062, Toulouse Cedex 09 France
Publications Mathématiques de l'IHÉS, Tome 117 (2013), pp. 179-245.

We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.

DOI : 10.1007/s10240-012-0046-6
Berman, Robert J. 1 ; Boucksom, Sébastien 2 ; Guedj, Vincent 3 ; Zeriahi, Ahmed 4

1 Chalmers Techniska Högskola, Chalmers University of Technology and University of Gothenburg Göteborg Sweden
2 Institut de Mathématiques, CNRS-Université Pierre et Marie Curie 75252, Paris Cedex France
3 I.M.T., Université Paul Sabatier and Institut Universitaire de France 31062, Toulouse Cedex 09 France
4 I.M.T., Université Paul Sabatier 31062, Toulouse Cedex 09 France 
@article{PMIHES_2013__117__179_0,
     author = {Berman, Robert J. and Boucksom, S\'ebastien and Guedj, Vincent and Zeriahi, Ahmed},
     title = {A variational approach to complex {Monge-Amp\`ere} equations},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {179--245},
     publisher = {Springer-Verlag},
     volume = {117},
     year = {2013},
     doi = {10.1007/s10240-012-0046-6},
     zbl = {1277.32049},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-012-0046-6/}
}
TY  - JOUR
AU  - Berman, Robert J.
AU  - Boucksom, Sébastien
AU  - Guedj, Vincent
AU  - Zeriahi, Ahmed
TI  - A variational approach to complex Monge-Ampère equations
JO  - Publications Mathématiques de l'IHÉS
PY  - 2013
SP  - 179
EP  - 245
VL  - 117
PB  - Springer-Verlag
UR  - http://archive.numdam.org/articles/10.1007/s10240-012-0046-6/
DO  - 10.1007/s10240-012-0046-6
LA  - en
ID  - PMIHES_2013__117__179_0
ER  - 
%0 Journal Article
%A Berman, Robert J.
%A Boucksom, Sébastien
%A Guedj, Vincent
%A Zeriahi, Ahmed
%T A variational approach to complex Monge-Ampère equations
%J Publications Mathématiques de l'IHÉS
%D 2013
%P 179-245
%V 117
%I Springer-Verlag
%U http://archive.numdam.org/articles/10.1007/s10240-012-0046-6/
%R 10.1007/s10240-012-0046-6
%G en
%F PMIHES_2013__117__179_0
Berman, Robert J.; Boucksom, Sébastien; Guedj, Vincent; Zeriahi, Ahmed. A variational approach to complex Monge-Ampère equations. Publications Mathématiques de l'IHÉS, Tome 117 (2013), pp. 179-245. doi : 10.1007/s10240-012-0046-6. http://archive.numdam.org/articles/10.1007/s10240-012-0046-6/

[Ale38] Aleksandrov, A. D. On the theory of mixed volumes of convex bodies III: Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sb., Volume 3 (1938), pp. 27-44 (Russian). [English translation available in Selected Works Part I: Selected Scientific Papers, Gordon and Breach]

[AT84] Alexander, H. J.; Taylor, B. A. Comparison of two capacities in C n , Math. Z., Volume 186 (1984), pp. 407-417 | DOI | MR | Zbl

[Aub84] Aubin, T. Réduction du cas positif de l’équation de Monge-Ampère sur les variétés kählériennes compactes à la démonstration d’une inégalité, J. Funct. Anal., Volume 57 (1984), pp. 143-153 | DOI | MR | Zbl

[BM87] Bando, S.; Mabuchi, T. Uniqueness of Einstein Kähler metrics modulo connected group actions, Algebraic Geometry (Adv. Stud. Pure Math., 10), Kinokuniya, Tokyo (1987), pp. 11-40 | Zbl

[BT82] Bedford, E.; Taylor, B. A. A new capacity for plurisubharmonic functions, Acta Math., Volume 149 (1982), pp. 1-40 | DOI | MR | Zbl

[BT87] Bedford, E.; Taylor, B. A. Fine topology, Šilov boundary, and (dd c ) n , J. Funct. Anal., Volume 72 (1987), pp. 225-251 | DOI | MR | Zbl

[BGZ09] Benelkourchi, S.; Guedj, V.; Zeriahi, A. Plurisubharmonic functions with weak singularities, Complex Analysis and Digital Geometry (Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist, 86), Uppsala Universitet, Uppsala (2009), pp. 57-74 | Zbl

[Berm09] Berman, R. Bergman kernels and equilibrium measures for line bundles over projective manifolds, Am. J. Math., Volume 131 (2009), pp. 1485-1524 | DOI | Zbl

[BB10] Berman, R.; Boucksom, S. Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math., Volume 181 (2010), pp. 337-394 | DOI | MR | Zbl

[BD12] Berman, R.; Demailly, J.-P. Regularity of plurisubharmonic upper envelopes in big cohomology classes, Perspectives in Analysis, Geometry, and Topology (Progr. Math., 296), Birkhäuser/Springer, New York (2012), pp. 39-66 | DOI | Zbl

[BBEGZ11] R. Berman, S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, Kähler-Ricci flow and Ricci iteration on log-Fano varieties, preprint (2011), . | arXiv

[Bern09a] Berndtsson, B. Curvature of vector bundles associated to holomorphic fibrations, Ann. Math., Volume 169 (2009), pp. 531-560 | DOI | MR | Zbl

[Bern09b] Berndtsson, B. Positivity of direct image bundles and convexity on the space of Kähler metrics, J. Differ. Geom., Volume 81 (2009), pp. 457-482 | MR | Zbl

[BCHM10] Birkar, C.; Cascini, P.; Hacon, C.; McKernan, J. Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010), pp. 405-468 | DOI | MR | Zbl

[Bło09] Błocki, Z. On geodesics in the space of Kähler metrics, Advances in Geometric Analysis (Advanced Lectures in Mathematics, 21), International Press, Somerville (2012), pp. 3-20 Proceedings of the Conference in Geometry dedicated to Shing-Tung Yau (Warsaw, April 2009) | Zbl

[BK07] Błocki, Z.; Kołodziej, S. On regularization of plurisubharmonic functions on manifolds, Proc. Am. Math. Soc., Volume 135 (2007), pp. 2089-2093 | DOI | Zbl

[Bou90] Bouche, T. Convergence de la métrique de Fubini-Study d’un fibré linéaire positif, Ann. Inst. Fourier, Volume 40 (1990), pp. 117-130 | DOI | Numdam | MR | Zbl

[Bou04] Boucksom, S. Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Super., Volume 37 (2004), pp. 45-76 | Numdam | MR | Zbl

[BEGZ10] Boucksom, S.; Eyssidieux, P.; Guedj, V.; Zeriahi, A. Monge-Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010), pp. 199-262 | DOI | MR | Zbl

[Cat99] Catlin, D. The Bergman kernel and a theorem of Tian, Analysis and Geometry in Several Complex Variables (Trends Math.), Birkhäuser, Boston (1999), pp. 1-23 | DOI | Zbl

[Ceg98] Cegrell, U. Pluricomplex energy, Acta Math., Volume 180 (1998), pp. 187-217 | DOI | MR | Zbl

[Che00] Chen, X. X. The space of Kähler metrics, J. Differ. Geom., Volume 56 (2000), pp. 189-234 | Zbl

[CGZ08] Coman, D.; Guedj, V.; Zeriahi, A. Domains of definition of Monge-Ampère operators on compact Kähler manifolds, Math. Z., Volume 259 (2008), pp. 393-418 | DOI | MR | Zbl

[Dem92] Demailly, J. P. Regularization of closed positive currents and intersection theory, J. Algebr. Geom., Volume 1 (1992), pp. 361-409 | MR | Zbl

[Dem91] J. P. Demailly, Potential theory in several complex variables, survey available at http://www-fourier.ujf-grenoble.fr/~demailly/books.html.

[Din09] Dinew, S. Uniqueness and stability in ${\mathcal{E}}(X,\omega)$ , J. Funct. Anal., Volume 256 (2009), pp. 2113-2122 | DOI | MR | Zbl

[Ding88] Ding, W.-Y. Remarks on the existence problem of positive Kähler-Einstein metrics, Math. Ann., Volume 282 (1988), pp. 463-471 | DOI | MR | Zbl

[Don01] Donaldson, S. K. Scalar curvature and projective embeddings I, J. Differ. Geom., Volume 59 (2001), pp. 479-522 | MR | Zbl

[Don05a] Donaldson, S. K. Scalar curvature and projective embeddings II, Q. J. Math., Volume 56 (2005), pp. 345-356 | DOI | MR | Zbl

[Don09] Donaldson, S. K. Some numerical results in complex differential geometry, Pure Appl. Math. Q., Volume 5 (2009), pp. 571-618 (Special Issue: In honor of Friedrich Hirzebruch. Part 1) | MR | Zbl

[Don99] Donaldson, S. et al. Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar (AMS Translations Series 2, 196), AMS, Providence (1999), pp. 13-33 | Zbl

[EGZ09] Eyssidieux, P.; Guedj, V.; Zeriahi, A. Singular Kähler-Einstein metrics, J. Am. Math. Soc., Volume 22 (2009), pp. 607-639 | DOI | MR | Zbl

[GZ05] Guedj, V.; Zeriahi, A. Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal., Volume 15 (2005), pp. 607-639 | DOI | MR | Zbl

[GZ07] Guedj, V.; Zeriahi, A. The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal., Volume 250 (2007), pp. 442-482 | DOI | MR | Zbl

[Kel09] Keller, J. Ricci iterations on Kähler classes, J. Inst. Math. Jussieu, Volume 8 (2009), pp. 743-768 | DOI | MR | Zbl

[Koł98] Kołodziej, S. The complex Monge-Ampère equation, Acta Math., Volume 180 (1998), pp. 69-117 | DOI | MR | Zbl

[LV11] L. Lempert and L. Vivas, Geodesics in the space of Kähler metrics, preprint (2011), . | arXiv | Zbl

[LT83] Levenberg, N.; Taylor, B. A. Comparison of capacities in C n , Complex Analysis (Lecture Notes in Math., 1094), Springer, Berlin (1984), pp. 162-172 | Zbl

[Mab86] Mabuchi, T. K-energy maps integrating Futaki invariants, Tohoku Math. J., Volume 38 (1986), pp. 575-593 | DOI | MR | Zbl

[Mab87] Mabuchi, T. Some symplectic geometry on compact Kähler manifolds, Osaka J. Math., Volume 24 (1987), pp. 227-252 | MR | Zbl

[Nak04] Nakayama, N. Zariski Decompositions and Abundance, MSJ Memoirs, 14, Mathematical Society of Japan, Tokyo, 2004 (xiv+277 pp) | MR | Zbl

[PSSW08] Phong, D. H.; Song, J.; Sturm, J.; Weinkove, B. The Moser-Trudinger inequality on Kähler-Einstein manifolds, Am. J. Math., Volume 130 (2008), pp. 1067-1085 | DOI | MR | Zbl

[Rai69] Rainwater, R. J. A note on the preceding paper, Duke Math. J., Volume 36 (1969), pp. 799-800 | DOI | MR | Zbl

[ST] Saff, E. B.; Totik, V. Logarithmic Potentials with Exterior Fields, Springer, Berlin, 1997 (with an appendix by T. Bloom) | MR

[Sem92] Semmes, S. Complex Monge-Ampère and symplectic manifolds, Am. J. Math., Volume 114 (1992), pp. 495-550 | DOI | MR | Zbl

[Sic81] Siciak, J. Extremal plurisubharmonic functions in C n , Ann. Pol. Math., Volume 39 (1981), pp. 175-211 | MR | Zbl

[Siu08] Siu, Y. T. Finite generation of canonical ring by analytic method, Sci. China Ser. A, Volume 51 (2008), pp. 481-502 | DOI | MR | Zbl

[Sko72] Skoda, H. Sous-ensembles analytiques d’ordre fini ou infini dans C n , Bull. Soc. Math. Fr., Volume 100 (1972), pp. 353-408 | Numdam | MR | Zbl

[SoTi08] Song, J.; Tian, G. Canonical measures and Kähler-Ricci flow, J. Am. Math. Soc., Volume 25 (2012), pp. 303-353 | DOI | MR | Zbl

[SoTi09] J. Song and G. Tian, The Kähler-Ricci flow through singularities, preprint (2009), . | arXiv | MR

[SzTo11] Székelyhidi, G.; Tosatti, V. Regularity of weak solutions of a complex Monge-Ampère equation, Anal. PDE, Volume 4 (2011), pp. 369-378 | DOI | MR | Zbl

[Tia90] Tian, G. On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom., Volume 32 (1990), pp. 99-130 | MR | Zbl

[Tia97] Tian, G. Kähler-Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997), pp. 239-265 | DOI | MR | Zbl

[Tian] Tian, G. Canonical Metrics in Kähler Geometry, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2000 | DOI | MR | Zbl

[Tsu10] Tsuji, H. Dynamical construction of Kähler-Einstein metrics, Nagoya Math. J., Volume 199 (2010), pp. 107-122 | MR | Zbl

[Wan05] Wang, X. Canonical metrics on stable vector bundles, Commun. Anal. Geom., Volume 13 (2005), pp. 253-285 | MR | Zbl

[Yau78] Yau, S. T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Commun. Pure Appl. Math., Volume 31 (1978), pp. 339-411 | DOI | MR | Zbl

[Zel98] Zelditch, S. Szegö kernels and a theorem of Tian, Int. Math. Res. Not., Volume 6 (1998), pp. 317-331 | DOI | MR | Zbl

[Zer01] Zeriahi, A. Volume and capacity of sublevel sets of a Lelong class of psh functions, Indiana Univ. Math. J., Volume 50 (2001), pp. 671-703 | DOI | MR | Zbl

Cité par Sources :