Ricci flat Kähler metrics on rank two complex symmetric spaces
Journal de l’École polytechnique — Mathématiques, Volume 6 (2019), pp. 163-201.

We obtain Ricci flat Kähler metrics on complex symmetric spaces of rank two by using an explicit asymptotic model whose geometry at infinity is interpreted in the wonderful compactification of the symmetric space. We recover the metrics of Biquard-Gauduchon in the Hermitian case and obtain in addition several new metrics.

Nous obtenons des métriques kählériennes Ricci plates sur les espaces symétriques complexes de rang 2 à partir d’un modèle asymptotique explicite, dont la géométrie à l’infini s’interprète en termes de la compactification magnifique de l’espace symétrique. Dans le cas hermitien, on retrouve les métriques de Biquard-Gauduchon mais on produit aussi des métriques nouvelles.

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Accepted:
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DOI: 10.5802/jep.91
Classification: 53C25, 14J32, 14M27
Keywords: Calabi-Yau metric; asymptotically conical metric; complex symmetric space; wonderful compactification
Mot clés : Métrique de Calabi-Yau ; métrique asymptotiquement conique ; espace symétrique complexe ; compactification magnifique

Olivier Biquard 1; Thibaut Delcroix 2

1 Sorbonne Université and École Normale Supérieure, PSL University 45 rue d’Ulm, 75005 Paris, France
2 Université de Strasbourg, IRMA 7 rue René Descartes, 67000 Strasbourg, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Olivier Biquard; Thibaut Delcroix. Ricci flat Kähler metrics on rank two complex symmetric spaces. Journal de l’École polytechnique — Mathématiques, Volume 6 (2019), pp. 163-201. doi : 10.5802/jep.91. https://jep.centre-mersenne.org/articles/10.5802/jep.91/

[AL92] H. Azad & J.-J. Loeb - “Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces”, Indag. Math. (N.S.) 3 (1992) no. 4, p. 365-375 | DOI | Zbl

[BBE + 16] R. J. Berman, S. Boucksom, P. Eyssidieux, V. Guedj & A. Zeriahi - “Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties”, J. reine angew. Math. (2016), doi:10.1515/crelle-2016-0033 | Zbl

[BG96] O. Biquard & P. Gauduchon - “La métrique hyperkählérienne des orbites coadjointes de type symétrique d’un groupe de Lie complexe semi-simple”, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996) no. 12, p. 1259-1264 | Zbl

[CDR16] R. J. Conlon, A. Degeratu & F. Rochon - “Quasi-asymptotically conical Calabi-Yau manifolds” (2016), arXiv:1611.04410 | Zbl

[CH13] R. J. Conlon & H.-J. Hein - “Asymptotically conical Calabi-Yau manifolds. I”, Duke Math. J. 162 (2013) no. 15, p. 2855-2902 | DOI | MR | Zbl

[CH15] R. J. Conlon & H.-J. Hein - “Asymptotically conical Calabi-Yau metrics on quasi-projective varieties”, Geom. Funct. Anal. 25 (2015) no. 2, p. 517-552 | DOI | MR | Zbl

[DCP83] C. De Concini & C. Procesi - “Complete symmetric varieties”, in Invariant theory (Montecatini, 1982), Lect. Notes in Math., vol. 996, Springer, Berlin, 1983, p. 1-44 | DOI | MR | Zbl

[Del16] T. Delcroix - “K-Stability of Fano spherical varieties” (2016), arXiv:1608.01852, to appear in Ann. Sci. École Norm. Sup. (4)

[Del17a] T. Delcroix - “Kähler-Einstein metrics on group compactifications”, Geom. Funct. Anal. 27 (2017) no. 1, p. 78-129 | DOI | MR | Zbl

[Del17b] T. Delcroix - “Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional” (2017), arXiv:1712.00221

[Don08] S. K. Donaldson - “Kähler geometry on toric manifolds, and some other manifolds with large symmetry”, in Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, p. 29-75 | Zbl

[GH15] G. Gagliardi & J. Hofscheier - “Gorenstein spherical Fano varieties”, Geom. Dedicata 178 (2015), p. 111-133 | DOI | MR | Zbl

[Gho02] M. Ghomi - “The problem of optimal smoothing for convex functions”, Proc. Amer. Math. Soc. 130 (2002) no. 8, p. 2255-2259 | DOI | MR | Zbl

[Hei10] H.-J. Hein - On gravitational instantons, Ph. D. Thesis, Princeton University, 2010 | MR

[Hel78] S. Helgason - Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc., New York-London, 1978 | MR | Zbl

[Hum78] J. E. Humphreys - Introduction to Lie algebras and representation theory, Graduate Texts in Math., vol. 9, Springer-Verlag, New York-Berlin, 1978 | MR | Zbl

[Li17] Y. Li - “A new complete Calabi-Yau metric on n (2017), arXiv:1705.07026

[LM85] R. B. Lockhart & R. C. McOwen - “Elliptic differential operators on noncompact manifolds”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), p. 409-447 | MR | Zbl

[Ruz12] A. Ruzzi - “Fano symmetric varieties with low rank”, Publ. RIMS, Kyoto Univ. 48 (2012) no. 2, p. 235-278 | DOI | MR | Zbl

[Ste93] M. B. Stenzel - “Ricci-flat metrics on the complexification of a compact rank one symmetric space”, Manuscripta Math. 80 (1993) no. 2, p. 151-163 | DOI | MR | Zbl

[Sze17] G. Szekelyhidi - “Degenerations of n and Calabi-Yau metrics” (2017), arXiv:1706.00357

[TY90] G. Tian & S.-T. Yau - “Complete Kähler manifolds with zero Ricci curvature. I”, J. Amer. Math. Soc. 3 (1990) no. 3, p. 579-609 | Zbl

[TY91] G. Tian & S.-T. Yau - “Complete Kähler manifolds with zero Ricci curvature. II”, Invent. Math. 106 (1991) no. 1, p. 27-60 | DOI | Zbl

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