On cohomogeneity one Hermitian non-Kähler metrics. (English) Zbl 1519.53059
In differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. The geometry of Hermitian manifolds has had an important role in several classical problems of geometric measure theory and complex geometry. Several authors studied the geometry of Hermitian manifolds, e.g., [A. Gray, Tohoku Math. J., II. Ser. 28, 601–612 (1976; Zbl 0351.53040); Michigan Math. J. 12, 273–287 (1965; Zbl 0132.16702); Illinois J. Math. 10, 353–366 (1966; Zbl 0183.50803); C. J. Yu, Sci. China, Math. 60, No. 2, 285–300 (2017; Zbl 1367.53027); S. S. Chern, Ann of Math 47, 85–121 (1946; Zbl 0060.41416); C. Koca and M. Lejmi, Kodai Math. J. 43, No. 3, 409–430 (2020; Zbl 1454.53059); F. Podesta, Transform. Groups 23, No. 4, 1129–1147 (2018; Zbl 1405.32020); B. Yang and F. Zheng, Commun. Anal. Geom. 26, No. 5, 1195–1222 (2018; Zbl 1408.53031)].
The principal objective in this paper is to investigate the curvature and the properties of Hermitian non-Kähler manifolds with large isometry groups. The authors prove four results given in Theorems A, B, C and D investigating the existence of special non-Kähler Hermitian metrics, and prove that all the Bérard-Bergery standard cohomogeneity-one complex manifolds admit complete, non-Kähler metrics of cohomogeneity one with constant Chern-scalar curvature.
The principal objective in this paper is to investigate the curvature and the properties of Hermitian non-Kähler manifolds with large isometry groups. The authors prove four results given in Theorems A, B, C and D investigating the existence of special non-Kähler Hermitian metrics, and prove that all the Bérard-Bergery standard cohomogeneity-one complex manifolds admit complete, non-Kähler metrics of cohomogeneity one with constant Chern-scalar curvature.
Reviewer: Lakehal Belarbi (Mostaganem)
MSC:
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 53C30 | Differential geometry of homogeneous manifolds |
| 32Q55 | Topological aspects of complex manifolds |
Citations:
Zbl 0351.53040; Zbl 0132.16702; Zbl 0183.50803; Zbl 1367.53027; Zbl 0060.41416; Zbl 1454.53059; Zbl 1405.32020; Zbl 1408.53031References:
| [1] | Alekseevsky, D. V. and Perelomov, A. M.. Invariant Kaehler-Einstein metrics on compact homogeneous spaces. Funct. Anal. Appl. 20 (1986), 171-182. · Zbl 0641.53050 |
| [2] | Alekseevsky, D. V. and Podestà, F.. Homogeneous almost-Kähler manifolds and the Chern-Einstein equation. Math. Z. 296 (2020), 831-846. · Zbl 1446.53037 |
| [3] | Alekseevsky, D. V. and Spiro, A.. Invariant CR structures on compact homogeneous manifolds. Hokkaido Math. J. 32 (2003), 209-276. · Zbl 1053.32021 |
| [4] | Alekseevsky, D. V. and Zuddas, F.. Cohomogeneity one Kähler and Kähler-Einstein manifolds with one singular orbit I. Ann. Global Anal. Geom. 52 (2017), 99-128. · Zbl 1428.53081 |
| [5] | Alekseevsky, D. V. and Zuddas, F.. Cohomogeneity one Kähler and Kähler-Einstein manifolds with one singular orbit II. Ann. Global Anal. Geom. 57 (2020), 153-174. · Zbl 1433.53097 |
| [6] | Angella, D., Calamai, S. and Spotti, C.. On the Chern-Yamabe problem. Math. Res. Lett. 24 (2017), 645-677. · Zbl 1388.53022 |
| [7] | Angella, D., Calamai, S. and Spotti, C.. Remarks on Chern-Einstein Hermitian metrics. Math. Z. 295 (2020), 1707-1722. · Zbl 1446.53014 |
| [8] | Aubin, Th.. Équations du type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. (2)102 (1978), 63-95. · Zbl 0374.53022 |
| [9] | Bérard-Bergery, L.. Sur de nouvelles variétès riemanniennes d’Einstein. Institut Élie Cartan6 (1982), 1-60. Inst. Élie Cartan, 6, Univ. Nancy, Nancy, 1982. |
| [10] | Besse, A. L., Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 10 (Springer-Verlag, Berlin, 1987). · Zbl 0613.53001 |
| [11] | Boothby, W. M.. Hermitian manifolds with zero curvature. Michigan Math. J. 5 (1958), 229-233. · Zbl 0084.37501 |
| [12] | Bordermann, M., Forger, M. and Römer, H.. Homogeneous Kähler Manifolds: paving the way towards new supersymmetric Sigma Models. Commun. Math. Phys. 102 (1986), 605-647. · Zbl 0585.53018 |
| [13] | Borel, A. and Hirzebruch, F.. Characteristic classes and homogeneous spaces. I. Am. J. Math. 80 (1958), 458-538. |
| [14] | Böhm, C.. Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. Invent. Math. 134 (1998), 145-176. · Zbl 0965.53033 |
| [15] | Böhm, C.. Non-compact cohomogeneity one Einstein manifolds. Bull. Soc. Math. France127 (1999), 135-177. · Zbl 0935.53021 |
| [16] | Bredon, G. E., Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46 (Academic Press, New York-London, 1972). · Zbl 0246.57017 |
| [17] | Dancer, A. and Wang, M. Y.. Kähler Einstein metrics of cohomogeneity one and bundle construction for Einstein Hermitian metrics. Math. Ann. 312 (1998), 503-526. · Zbl 0965.53034 |
| [18] | Derdziński, A. and Maschler, G.. Local classification of conformally-Einstein Kähler metrics in higher dimensions. Proc. London Math. Soc. 87 (2003), 779-819. · Zbl 1049.53017 |
| [19] | Dragomir, S. and Ornea, L., Locally conformal Kähler geometry, Progress in Mathematics, Vol. 155 (Birkhäuser Boston, Inc., Boston, MA, 1998). · Zbl 0887.53001 |
| [20] | Eschenburg, J.-H. and Wang, Mck. Y.. The initial value problem for cohomogeneity one Einstein metrics. J. Geom. Anal. 10 (2000), 109-137. · Zbl 0992.53033 |
| [21] | Gauduchon, P.. Le théorème de l’excentricité nulle. C. R. Acad. Sci. Paris Sér. A-B285 (1977), A387-A390. · Zbl 0362.53024 |
| [22] | Gauduchon, P.. La topologie d’une surface hermitienne d’Einstein. C. R. Acad. Sci. Paris Sér. A-B290 (1980), A509-A512. · Zbl 0435.53036 |
| [23] | Gauduchon, P.. \(La 1\)-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267 (1984), 495-518. · Zbl 0523.53059 |
| [24] | Gauduchon, P. and Ivanov, S.. Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension \(4\). Math. Z. 226 (1997), 317-326. · Zbl 1006.53061 |
| [25] | Gauduchon, P., Moroianu, A. and Ornea, L.. Compact homogeneous lcK manifolds are Vaisman. Math. Ann. 361 (2015), 1043-1048. · Zbl 1319.53081 |
| [26] | Gray, A.. Curvature identities for Hermitian and almost Hermitian manifolds. Tohoku Math. J. (2)28 (1976), 601-612. · Zbl 0351.53040 |
| [27] | Grove, K. and Ziller, W.. Cohomogeneity one manifolds with positive Ricci curvature. Invent. Math. 149 (2002), 619-646. · Zbl 1038.53034 |
| [28] | Guan, D. and Chen, X.X.. Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one, Loo-Keng Hua: a great mathematician of the twentieth century. Asian J. Math. 4 (2000), 817-829. · Zbl 1003.32003 |
| [29] | Hasegawa, K. and Kamishima, Y.. Compact homogeneous locally conformally Kähler manifolds. Osaka J. Math. 53 (2016), 683-703. · Zbl 1361.32027 |
| [30] | Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Corrected reprint of the 1978 original. Graduate Studies in Mathematics, Vol. 34 (American Mathematical Society, Providence, RI, 2001). · Zbl 0993.53002 |
| [31] | Huckleberry, A. and Snow, D.. Almost-homogeneous Kahler manifolds with hypersurface orbits. Osaka J. Math. 19 (1982), 763-786. · Zbl 0507.32023 |
| [32] | Kobayashi, S. and Nomizu, K., Foundations of differential geometry. Vol. II, Reprint of the 1969 original, Wiley Classics Library, A Wiley-Interscience Publication (John Wiley & Sons, Inc., New York, 1996). |
| [33] | Koca, C. and Lejmi, M.. Hermitian metrics of constant Chern scalar curvature on ruled surfaces. Kodai Math. J. 43 (2020), 409-430. · Zbl 1454.53059 |
| [34] | Koiso, N. and Sakane, Y., Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds, Curvature and topology of Riemannian manifolds (Katata, 1985), 165-179, Lecture Notes in Math., Vol. 1201 (Springer, Berlin, 1986). · Zbl 0591.53056 |
| [35] | Koiso, N. and Sakane, Y.. Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds. II. Osaka J. Math. 25 (1988), 933-959. · Zbl 0704.53052 |
| [36] | Lebrun, C., Einstein metrics on complex surfaces, Geometry and physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math., Vol. 184 (Dekker, New York, 1997), 167-176. · Zbl 0876.53024 |
| [37] | Liu, K.-F. and Yang, X.-K.. Geometry of Hermitian manifolds. Int. J. Math. 23 (2012), 1250055. · Zbl 1248.53055 |
| [38] | Madani, F., Moroianu, A. and Pilca, M.. Conformally related Kähler metrics and the holonomy of lcK manifolds. J. Eur. Math. Soc. (JEMS)22 (2020), 119-149. · Zbl 1475.53028 |
| [39] | Malgrange, B.. Sur les points singuliers des équations différentielles. Enseign. Math. (2)20 (1974), 147-176. · Zbl 0299.34011 |
| [40] | Michelsohn, M. L.. On the existence of special metrics in complex geometry. Acta Math. 149 (1982), 261-295. · Zbl 0531.53053 |
| [41] | Mostert, P. S.. On a compact Lie group acting on a manifold. Ann. Math. (2)65 (1957), 447-455. Errata, ‘On a compact Lie group acting on a manifold’, Ann. Math. 66 (1957), 589. · Zbl 0080.16702 |
| [42] | Nishiyama, M.. Classification of invariant complex structures on irreducible compact simply connected coset spaces. Osaka J. Math. 21 (1984), 39-58. · Zbl 0553.32020 |
| [43] | O’Neill, B.. The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459-469. · Zbl 0145.18602 |
| [44] | Ornea, L. and Verbitsky, M.. An immersion theorem for Vaisman manifolds. Math. Ann. 332 (2005), 121-143. · Zbl 1082.53074 |
| [45] | Page, D. N.. A compact rotating gravitational instanton. Phys. Lett. B79 (1978), 235-238. |
| [46] | Page, D. N. and Pope, C. N.. Inhomogeneous Einstein metrics on complex line bundles. Classical Q. Gravity4 (1987), 213-225. · Zbl 0613.53020 |
| [47] | Podestá, F.. Homogeneous Hermitian manifolds and special metrics. Transform. Groups23 (2018), 1129-1147. · Zbl 1405.32020 |
| [48] | Podestà, F. and Spiro, A.. Kähler manifolds with large isometry group. Osaka J. Math. 36 (1999), 805-833. · Zbl 0981.53066 |
| [49] | Podestà, F. and Spiro, A.. New examples of almost homogeneous Kähler-Einstein manifolds. Indiana Univ. Math. J. 52 (2003), 1027-1074. · Zbl 1080.53037 |
| [50] | Sakane, Y.. Examples of compact Kähler-Einstein manifolds with positive Ricci curvature. Osaka Math. J. 23 (1986), 585-616. · Zbl 0636.53068 |
| [51] | Streets, J. and Tian, G.. Hermitian curvature flow. J. Eur. Math. Soc. (JEMS)13 (2011), 601-634. · Zbl 1214.53055 |
| [52] | Tosatti, V., Non-Kähler Calabi-Yau manifolds, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, 261-277, Contemp. Math., Vol. 644 (Am. Math. Soc., Providence, RI, 2015). · Zbl 1341.53108 |
| [53] | Tosatti, V. and Weinkove, B.. The complex Monge-Ampère equation on compact Hermitian manifolds. J. Am. Math. Soc. 23 (2010), 1187-1195. · Zbl 1208.53075 |
| [54] | Verdiani, L. and Ziller, W., Smoothness Conditions in Cohomogeneity manifolds, arXiv:1804.04680, to appear in Transf. Groups. |
| [55] | Wang, H. C.. Closed manifolds with homogeneous complex structures. Am. J. Math. 76 (1954), 1-32. · Zbl 0055.16603 |
| [56] | Wang, Mck. Y.-K., Einstein metrics from symmetry and bundle constructions, Surveys in differential geometry: essays on Einstein manifolds, 287-325, Surv. Differ. Geom., Vol. 6 (Int. Press, Boston, MA, 1999). · Zbl 1003.53037 |
| [57] | Wang, Mck. Y.-K., Einstein metrics from symmetry and bundle constructions: a sequel, Differential geometry, 253-309, Adv. Lect. Math. (ALM), Vol. 22 (Int. Press, Somerville, MA, 2012). · Zbl 1262.53044 |
| [58] | Yang, B. and Zheng, F.. On curvature tensors of Hermitian manifolds. Comm. Anal. Geom. 26 (2018), 1195-1222. · Zbl 1408.53031 |
| [59] | Yau, S.-T.. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31 (1978), 339-411. · Zbl 0369.53059 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
