Ambitoric geometry II:  Extremal toric surfaces  and Einstein 4-orbifolds

Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul

doi:10.24033/asens.2266

Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds
[Géométrie ambitorique II : surfaces toriques complexes extrémales et orbifolds d'Einstein de dimension 4]

Apostolov, Vestislav ; Calderbank, David M. J. ; Gauduchon, Paul

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1075-1112.

Résumé
Abstract

Nous donnons une solution complète et explicite du problème d'existence de métriques kählériennes extrémales sur un orbifold torique $M$ de dimension réelle 4, dont le nombre de Betti $b_{2} (M)$ est égal à 2. Nous montrons plus précisément que $M$ admet de telles métriques si et seulement si son polytope de Delzant rationnel — qui est alors un quadrilatère étiqueté — est $K$ -polystable, suivant la théorie générale développée dans le cas torique par S. K. Donaldson, E. Legendre, G. Székelyhidi et al., et que ces métriques sont alors ambitoriques, donc complètement explicites d'après la classification figurant dans la première partie de ce travail. Notre approche donne de surcroît une façon effective de tester la stabilité des quadrilatères étiquetés. Parmi les métriques kählériennes construites dans cet article figurent celles dont le tenseur de Bach est nul, qui sont à la fois extrémales et conformément Einstein. Nous obtenons ainsi, en dimension 4, de nouveaux exemples explicites d'orbifolds d'Einstein compacts ou de variétés d'Einstein non-compactes, complètes et lisses.

We provide an explicit resolution of the existence problem for extremal Kähler metrics on toric 4-orbifolds $M$ with second Betti number $b_{2} (M) = 2$ . More precisely we show that $M$ admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Székelyhidi et al.). Furthermore, in this case, the extremal Kähler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kähler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals.

Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kähler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kähler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.

Publié le : 2015-09-24

MR Zbl

DOI : 10.24033/asens.2266

Classification : 53C55, 53C25.
Keywords: Extremal Kähler metrics, toric geometry, Einstein 4-orbifolds.
Mot clés : Métriques kählériennes extrémales, géométrie torique, orbifolds d'Einstein de dimension 4.

@article{ASENS_2015__48_5_1075_0,
     author = {Apostolov, Vestislav and Calderbank, David M. J. and Gauduchon, Paul},
     title = {Ambitoric geometry {II:}  {Extremal} toric surfaces  and {Einstein} 4-orbifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1075--1112},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {5},
     year = {2015},
     doi = {10.24033/asens.2266},
     mrnumber = {3429476},
     zbl = {1346.32007},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2266/}
}

TY  - JOUR
AU  - Apostolov, Vestislav
AU  - Calderbank, David M. J.
AU  - Gauduchon, Paul
TI  - Ambitoric geometry II:  Extremal toric surfaces  and Einstein 4-orbifolds
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2015
SP  - 1075
EP  - 1112
VL  - 48
IS  - 5
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2266/
DO  - 10.24033/asens.2266
LA  - en
ID  - ASENS_2015__48_5_1075_0
ER  -

%0 Journal Article
%A Apostolov, Vestislav
%A Calderbank, David M. J.
%A Gauduchon, Paul
%T Ambitoric geometry II:  Extremal toric surfaces  and Einstein 4-orbifolds
%J Annales scientifiques de l'École Normale Supérieure
%D 2015
%P 1075-1112
%V 48
%N 5
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2266/
%R 10.24033/asens.2266
%G en
%F ASENS_2015__48_5_1075_0

Apostolov, Vestislav; Calderbank, David M. J.; Gauduchon, Paul. Ambitoric geometry II:  Extremal toric surfaces  and Einstein 4-orbifolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1075-1112. doi : 10.24033/asens.2266. http://www.numdam.org/articles/10.24033/asens.2266/

Bibliographie
Cité par

Abreu, M. Kähler metrics on toric orbifolds, J. Differential Geom., Volume 58 (2001), pp. 151-187 http://projecteuclid.org/euclid.jdg/1090348285 (ISSN: 0022-040X) | MR | Zbl

Abreu, M. Kähler geometry of toric varieties and extremal metrics, Internat. J. Math., Volume 9 (1998), pp. 641-651 (ISSN: 0129-167X) | DOI | MR | Zbl

Apostolov, V.; Calderbank, D. M. J.; Gauduchon, P. Ambitoric geometry, I: Einstein metrics and extremal ambikähler structures (preprint arXiv:1302.6975, to appear in J. reine angew. Math., doi:10.1515/crelle-2014-0060 ) | MR

Apostolov, V.; Calderbank, D. M. J.; Gauduchon, P. The geometry of weakly self-dual Kähler surfaces, Compositio Math., Volume 135 (2003), pp. 279-322 (ISSN: 0010-437X) | DOI | MR | Zbl

Apostolov, V.; Calderbank, D. M. J.; Gauduchon, P. Hamiltonian 2-forms in Kähler geometry. I. General theory, J. Differential Geom., Volume 73 (2006), pp. 359-412 http://projecteuclid.org/euclid.jdg/1146169934 (ISSN: 0022-040X) | MR | Zbl

Apostolov, V.; Calderbank, D. M. J.; Gauduchon, P.; Tønnesen-Friedman, C. W. Hamiltonian 2-forms in Kähler geometry. II. Global classification, J. Differential Geom., Volume 68 (2004), pp. 277-345 http://projecteuclid.org/euclid.jdg/1115669513 (ISSN: 0022-040X) | MR | Zbl

Apostolov, V.; Calderbank, D. M. J.; Gauduchon, P.; Tønnesen-Friedman, C. W. Hamiltonian 2-forms in Kähler geometry. III. Extremal metrics and stability, Invent. Math., Volume 173 (2008), pp. 547-601 (ISSN: 0020-9910) | DOI | MR | Zbl

Anderson, M. T., Perspectives in Riemannian geometry (CRM Proc. Lecture Notes), Volume 40, Amer. Math. Soc., Providence, RI, 2006, pp. 1-26 | DOI | MR | Zbl

Aubin, T. Équations du type Monge-Ampère sur les variétés kählériennes compactes, C. R. Acad. Sci. Paris Sér. A-B, Volume 283 (1976), pp. 119-121 | MR | Zbl

Bérard-Bergery, L. Sur de nouvelles variétés riemanniennes d'Einstein, Inst. Élie Cartan, Volume 6, Univ. Nancy, Nancy (1982), pp. 1-60 | MR | Zbl

Boyer, C. P.; Galicki, K., Oxford Mathematical Monographs, Oxford Univ. Press, Oxford, 2008, 613 pages (ISBN: 978-0-19-856495-9) | MR | Zbl

Boyer, C. P.; Galicki, K.; Mann, B. M.; Rees, E. G. Compact 3-Sasakian 7-manifolds with arbitrary second Betti number, Invent. Math., Volume 131 (1998), pp. 321-344 (ISSN: 0020-9910) | DOI | MR | Zbl

Boyer, C. P.; Galicki, K.; Simanca, S. R. Canonical Sasakian metrics, Comm. Math. Phys., Volume 279 (2008), pp. 705-733 (ISSN: 0010-3616) | DOI | MR | Zbl

Bryant, R. L. Bochner-Kähler metrics, J. Amer. Math. Soc., Volume 14 (2001), pp. 623-715 (ISSN: 0894-0347) | DOI | MR | Zbl

Calabi, E., Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton, N. J., 1957, pp. 78-89 | DOI | MR | Zbl

Calabi, E., Seminar on Differential Geometry (Ann. of Math. Stud.), Volume 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290 | MR | Zbl

Calabi, E., Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 95-114 | DOI | MR | Zbl

Cvetič, M.; Lü, H.; Page, D. N.; Pope, C. N. New Einstein-Sasaki spaces in five and higher dimensions, Phys. Rev. Lett., Volume 95 (2005), 071101 pages (ISSN: 0031-9007) | DOI | MR

Chen, B.; Li, A.-M.; Sheng, L. Extremal metrics on toric surfaces (preprint arXiv:1008.2607 ) | MR

Chen, X.; LeBrun, C.; Weber, B. On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc., Volume 21 (2008), pp. 1137-1168 (ISSN: 0894-0347) | DOI | MR | Zbl

Chen, X.; Tian, G. Geometry of Kähler metrics and foliations by holomorphic discs, Publ. Math. IHÉS, Volume 107 (2008), pp. 1-107 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Delzant, T. Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France, Volume 116 (1988), pp. 315-339 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Derdziński, A. Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math., Volume 49 (1983), pp. 405-433 (ISSN: 0010-437X) | Numdam | MR | Zbl

Derdzinski, A.; Maschler, G. A moduli curve for compact conformally-Einstein Kähler manifolds, Compositio Math., Volume 141 (2005), pp. 1029-1080 (ISSN: 0010-437X) | DOI | MR | Zbl

Donaldson, S. K. Scalar curvature and stability of toric varieties, J. Differential Geom., Volume 62 (2002), pp. 289-349 http://projecteuclid.org/euclid.jdg/1090950195 (ISSN: 0022-040X) | MR | Zbl

Donaldson, S. K. Interior estimates for solutions of Abreu's equation, Collect. Math., Volume 56 (2005), pp. 103-142 (ISSN: 0010-0757) | MR | Zbl

Donaldson, S. K. Lower bounds on the Calabi functional, J. Differential Geom., Volume 70 (2005), pp. 453-472 http://projecteuclid.org/euclid.jdg/1143642909 (ISSN: 0022-040X) | MR | Zbl

Donaldson, S. K. Extremal metrics on toric surfaces: a continuity method, J. Differential Geom., Volume 79 (2008), pp. 389-432 http://projecteuclid.org/euclid.jdg/1213798183 (ISSN: 0022-040X) | MR | Zbl

Donaldson, S. K. Constant scalar curvature metrics on toric surfaces, Geom. Funct. Anal., Volume 19 (2009), pp. 83-136 (ISSN: 1016-443X) | DOI | MR | Zbl

Donaldson, S. K., Fields Medallists' lectures (World Sci. Ser. 20th Century Math.), Volume 5, World Sci. Publ., River Edge, NJ, 1997, pp. 384-403 | DOI | MR

Futaki, A.; Mabuchi, T. Bilinear forms and extremal Kähler vector fields associated with Kähler classes, Math. Ann., Volume 301 (1995), pp. 199-210 (ISSN: 0025-5831) | DOI | MR | Zbl

Fujiki, A. The moduli spaces and Kähler metrics of polarized algebraic varieties, Sūgaku, Volume 42 (1990), pp. 231-243 ; translation: Sugaku Expositions 5 (1992), 173–191 (ISSN: 0039-470X) | MR | Zbl

Gauntlett, J. P.; Martelli, D.; Sparks, J.; Waldram, D. A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys., Volume 8 (2004), pp. 987-1000 http://projecteuclid.org/euclid.atmp/1143663673 (ISSN: 1095-0761) | DOI | MR | Zbl

Guan, D. On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles, Math. Res. Lett., Volume 6 (1999), pp. 547-555 (ISSN: 1073-2780) | DOI | MR | Zbl

Guillemin, V. Kaehler structures on toric varieties, J. Differential Geom., Volume 40 (1994), pp. 285-309 http://projecteuclid.org/euclid.jdg/1214455538 (ISSN: 0022-040X) | MR | Zbl

Guillemin, V., Progress in Math., 122, Birkhäuser, 1994, 150 pages (ISBN: 0-8176-3770-2) | DOI | MR | Zbl

Hwang, A. D.; Singer, M. A. A momentum construction for circle-invariant Kähler metrics, Trans. Amer. Math. Soc., Volume 354 (2002), pp. 2285-2325 (ISSN: 0002-9947) | DOI | MR | Zbl

LeBrun, C. On Einstein, Hermitian 4-manifolds, J. Differential Geom., Volume 90 (2012), pp. 277-302 http://projecteuclid.org/euclid.jdg/1335230848 (ISSN: 0022-040X) | MR | Zbl

LeBrun, C. Einstein manifolds and extremal Kähler metrics, J. reine angew. Math., Volume 678 (2013), pp. 69-94 (ISSN: 0075-4102) | MR | Zbl

LeBrun, C., Geometry and physics (Aarhus, 1995) (Lecture Notes in Pure and Appl. Math.), Volume 184, Dekker, New York, 1997, pp. 167-176 | MR | Zbl

Legendre, E. Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics, Compositio Math., Volume 147 (2011), pp. 1613-1634 (ISSN: 0010-437X) | DOI | MR | Zbl

Legendre, E. Toric geometry of convex quadrilaterals, J. Symplectic Geom., Volume 9 (2011), pp. 343-385 http://projecteuclid.org/euclid.jsg/1310388900 (ISSN: 1527-5256) | DOI | MR | Zbl

Lejmi, M. Extremal almost-Kähler metrics, Internat. J. Math., Volume 21 (2010), pp. 1639-1662 (ISSN: 0129-167X) | DOI | MR | Zbl

Lerman, E. Contact toric manifolds, J. Symplectic Geom., Volume 1 (2003), pp. 785-828 http://projecteuclid.org/euclid.jsg/1092749569 (ISSN: 1527-5256) | DOI | MR | Zbl

Lerman, E.; Tolman, S. Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc., Volume 349 (1997), pp. 4201-4230 (ISSN: 0002-9947) | DOI | MR | Zbl

Mabuchi, T. A stronger concept of K-stability (preprint arXiv:0910.4617 )

Martelli, D.; Sparks, J. Toric Sasaki-Einstein metrics on $S^{2} \times S^{3}$ , Phys. Lett. B, Volume 621 (2005), pp. 208-212 (ISSN: 0370-2693) | DOI | MR | Zbl

Page, D. N. A compact rotating gravitational instanton, Phys. Lett. B, Volume 79 (1978), pp. 235-238 | DOI

Ross, J.; Thomas, R. Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom., Volume 88 (2011), pp. 109-159 http://projecteuclid.org/euclid.jdg/1317758871 (ISSN: 0022-040X) | MR | Zbl

Sparks, J., Surveys in differential geometry. Volume XVI. Geometry of special holonomy and related topics (Surv. Differ. Geom.), Volume 16, Int. Press, Somerville, MA, 2011, pp. 265-324 | DOI | MR | Zbl

Stoppa, J.; Székelyhidi, G. Relative K-stability of extremal metrics, J. Eur. Math. Soc. (JEMS), Volume 13 (2011), pp. 899-909 (ISSN: 1435-9855) | DOI | MR | Zbl

Stoppa, J. K-stability of constant scalar curvature Kähler manifolds, Adv. Math., Volume 221 (2009), pp. 1397-1408 (ISSN: 0001-8708) | DOI | MR | Zbl

Székelyhidi, G. Extremal metrics and $K$ -stability (2006) | MR | Zbl

Székelyhidi, G. Extremal metrics and $K$ -stability, Bull. Lond. Math. Soc., Volume 39 (2007), pp. 76-84 (ISSN: 0024-6093) | DOI | MR | Zbl

Thurston, W. The Geometry and Topology of Three-Manifolds (1980) (mimeographed notes, http://library.msri.org/books/gt3m/PDF/1.pdf )

Tian, G. Kähler-Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997), pp. 1-37 (ISSN: 0020-9910) | DOI | MR | Zbl

Tønnesen-Friedman, C. W. Extremal Kähler metrics on minimal ruled surfaces, J. reine angew. Math., Volume 502 (1998), pp. 175-197 (ISSN: 0075-4102) | DOI | MR | Zbl

Webster, S. M. On the pseudo-conformal geometry of a Kähler manifold, Math. Z., Volume 157 (1977), pp. 265-270 (ISSN: 0025-5874) | DOI | MR | Zbl

Yau, S. T. Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A., Volume 74 (1977), pp. 1798-1799 (ISSN: 0027-8424) | DOI | MR | Zbl

Yau, S. T., Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) (Proc. Sympos. Pure Math.), Volume 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1-28 | MR | Zbl

Zhou, B.; Zhu, X. $K$ -stability on toric manifolds, Proc. Amer. Math. Soc., Volume 136 (2008), pp. 3301-3307 (ISSN: 0002-9939) | DOI | MR | Zbl

Zhou, B.; Zhu, X. Relative $K$ -stability and modified $K$ -energy on toric manifolds, Adv. Math., Volume 219 (2008), pp. 1327-1362 (ISSN: 0001-8708) | DOI | MR | Zbl

Cité par Sources :