Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler. (English) Zbl 1484.53084
Let \(M\) be a closed almost Kähler 4-manifold. The authors study the conditions such that \(M\) is Kähler. It is shown that among these conditions the assumptions for holomorphic sectional curvature play an important role. Assuming that \(M\) is of pointwise constant Hermitian holomorphic sectional curvature \(k\geq 0\), the authors prove that \(M\) is Kähler-Einstein, holomorphically isometric to \(\mathbb{C}P\) with the Fubini-Study metric (for \(k>0\)) and a complex torus or a hyperelliptic curve with the Ricci-flat Kähler metric (for \(k=0\)). Furthermore, assuming that \(k<0\) and the Ricci tensor is \(J\)-invariant they prove that \(M\) is Kähler-Einstein, holomorphically isometric to a compact quotient of the complex hyperbolic ball \(\mathbb{B}^4\) with the Bergman metric.
To prove previously mentioned results the authors use algebraic methods at the beginning, upon that they exploit the index theorems for the signature and the Euler characteristic using Chern-Weil theory, as well as some results from Seiber-Witten theory, and formulas for the Bach tensor.
At the end some open problems concerning non-compact manifolds, Lie groups, counterexamples to Schur’s theorem, the twistor space approach are formulated.
To prove previously mentioned results the authors use algebraic methods at the beginning, upon that they exploit the index theorems for the signature and the Euler characteristic using Chern-Weil theory, as well as some results from Seiber-Witten theory, and formulas for the Bach tensor.
At the end some open problems concerning non-compact manifolds, Lie groups, counterexamples to Schur’s theorem, the twistor space approach are formulated.
Reviewer: Neda Bokan (Beograd)
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
Keywords:
canonical Hermitian connection; holomorphic sectional curvature; Chern-Weil theory; Seiberg-Witten theory; Bach tensorReferences:
| [1] | V. Apostolov, J. Armstrong and T. Drăghici, Local models and integrability of certain almost Kähler 4-manifolds, Math. Ann. 323 (2002), no. 4, 633-666. · Zbl 1015.53011 |
| [2] | V. Apostolov, D. M. J. Calderbank and P. Gauduchon, The geometry of weakly self-dual Kähler surfaces, Compositio Math. 135 (2003), no. 3, 279-322. · Zbl 1031.53045 · doi:10.1023/A:1022251819334 |
| [3] | V. Apostolov, J. Davidov and O. Muškarov, Compact self-dual Hermitian surfaces, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3051-3063. · Zbl 0880.53053 |
| [4] | V. Apostolov and T. Drăghici, Almost Kähler 4-manifolds with \(J\)-invariant Ricci tensor and special Weyl tensor, Q. J. Math. 51 (2000), no. 3, 275-294. · Zbl 0983.53046 |
| [5] | J. Armstrong, On four-dimensional almost Kähler manifolds, Q. J. Math. 48 (1997), no. 192, 405-415. · Zbl 0899.53038 |
| [6] | A. Balas, Compact Hermitian manifolds of constant holomorphic sectional curvature, Math. Z. 189 (1985), no. 2, 193-210. · Zbl 0575.53044 · doi:10.1007/BF01175044 |
| [7] | A. Balas and P. Gauduchon, Any Hermitian metric of constant nonpositive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z. 190 (1985), no. 1, 39-43. · Zbl 0549.53063 · doi:10.1007/BF01159161 |
| [8] | A. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 10, Springer-Verlag, Berlin, 1987. · Zbl 0613.53001 |
| [9] | D. E. Blair, Nonexistence of 4-dimensional almost Kaehler manifolds of constant curvature, Proc. Amer. Math. Soc. 110 (1990), no. 4, 1033-1039. · Zbl 0718.53028 |
| [10] | S. G. Chiossi and P.-A. Nagy, Systems of symplectic forms on four-manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. 12 (2013), no. 3, 717-734. · Zbl 1295.53083 |
| [11] | T. Drăghici, On some 4-dimensional almost Kähler manifolds, Kodai Math. J. 18 (1995), no. 1, 156-168. · Zbl 0836.53028 |
| [12] | T. Drăghici, Almost Kähler 4-manifolds with \(J\)-invariant Ricci tensor, Houston J. Math. 25 (1999), no. 1, 133-145. · Zbl 0983.53045 |
| [13] | M. Falcitelli, A. Farinola and O. T. Kassabov, Almost Kähler manifolds whose antiholomorphic sectional curvature is pointwise constant, Rend. Mat. Appl. 18 (1998), no. 1, 151-166. · Zbl 0917.53009 |
| [14] | P. Gauduchon, Calabi’s extremal Kähler metrics: An elementary introduction, Book project in preparation, 2010. |
| [15] | S. I. Goldberg, Integrability of almost Kaehler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96-100. · Zbl 0174.25002 · doi:10.1090/S0002-9939-1969-0238238-1 |
| [16] | A. Gray, Classification des variétés approximativement kählériennes de courbure sectionnelle holomorphe constante, C. R. Acad. Sci. Paris Sér. A 279 (1974), 797-800. · Zbl 0301.53016 |
| [17] | A. Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tôhoku Math. J. 28 (1976), no. 4, 601-612. · Zbl 0351.53040 |
| [18] | A. Gray and L. Vanhecke, Almost Hermitian manifolds with constant holomorphic sectional curvature, Časopis Pěst. Mat. 104 (1979), no. 2, 170-179. · Zbl 0413.53011 · doi:10.21136/CPM.1979.118013 |
| [19] | N. S. Hawley, Constant holomorphic curvature, Canadian J. Math. 5 (1953), no. 5, 53-56. · Zbl 0050.16203 · doi:10.4153/CJM-1953-007-1 |
| [20] | J. Igusa, On the structure of a certain class of Kaehler varieties, Amer. J. Math. 76 (1954), 669-678. · Zbl 0058.37901 · doi:10.2307/2372709 |
| [21] | O. T. Kassabov, Almost Kähler manifolds of constant antiholomorphic sectional curvature, Serdica 9 (1984), no. 4, 372-376. · Zbl 0537.53058 |
| [22] | S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II, John Wiley & Sons, Inc., New York, 1996. · Zbl 0119.37502 |
| [23] | F. Lalonde and D. McDuff, \(J\)-curves and the classification of rational and ruled symplectic 4-manifolds, in Contact and symplectic geometry, vol. 8, pages 3-42, Cambridge Univ. Press, 1996. · Zbl 0867.53028 |
| [24] | C. LeBrun, On the topology of self-dual 4-manifolds, Proc. Amer. Math. Soc. 98 (1986), no. 4, 637-640. · Zbl 0606.53029 |
| [25] | C. LeBrun, Weyl curvature, Del Pezzo surfaces, and almost-Kähler geometry, J. Geom. Anal. 25 (2015), no. 3, 1744-1772. · Zbl 1325.53060 · doi:10.1007/s12220-014-9492-3 |
| [26] | M. Lejmi and L. Vezzoni, 4-dimensional almost-Kähler Lie algebras of constant Hermitian holomorphic sectional curvature are Kähler, J. Lie Theory 29 (2019), no. 1, 181-190. · Zbl 1412.53103 |
| [27] | A. Liu, Some new applications of general wall crossing formula, Gompf’s conjecture and its applications, Math. Res. Lett. 3 (1996), no. 5, 569-585. · Zbl 0872.57025 · doi:10.4310/MRL.1996.v3.n5.a1 |
| [28] | J. W. Milnor and J. D. Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press, Princeton, N. J., 1974. · Zbl 0298.57008 |
| [29] | H. Ohta and K. Ono, Notes on symplectic 4-manifolds with \(b^+_2=1\). II, Internat. J. Math. 7 (1996), no. 6, 755-770. · Zbl 0873.53019 |
| [30] | T. Sato, An example of an almost Kähler manifold with pointwise constant holomorphic sectional curvature, Tokyo J. Math. 23 (2000), no. 2, 387-401. · Zbl 0981.53026 |
| [31] | K. Sekigawa, On some 4-dimensional compact Einstein almost Kähler manifolds, Math. Ann. 271 (1985), no. 3, 333-337. · Zbl 0562.53032 |
| [32] | C. H. Taubes, The Seiberg-Witten and Gromov invariants, Math. Res. Lett. 2 (1995), no. 2, 221-238. · Zbl 0854.57020 · doi:10.4310/MRL.1995.v2.n2.a10 |
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