Deformation of Sasakian metrics

Nozawa, Hiraku

doi:10.1090/S0002-9947-2013-06020-5

Deformation of Sasakian metrics
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by Hiraku Nozawa PDF

Trans. Amer. Math. Soc. 366 (2014), 2737-2771 Request permission

Abstract:

Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics. We show that the triviality of the $(0,2)$-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.

References

Yasuo Akizuki and Shigeo Nakano, Note on Kodaira-Spencer’s proof of Lefschetz theorems, Proc. Japan Acad. 30 (1954), 266–272. MR 66694
Jesús A. Alvarez López, On Riemannian foliations with minimal leaves, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 1, 163–176 (English, with French summary). MR 1056780
Jesús A. Alvarez López, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom. 10 (1992), no. 2, 179–194. MR 1175918, DOI 10.1007/BF00130919
W. L. Baily, On the imbedding of $V$-manifolds in projective space, Amer. J. Math. 79 (1957), 403–430. MR 100104, DOI 10.2307/2372689
Florin Alexandru Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), no. 1, 1–40. MR 1760667, DOI 10.1007/s002080050357
David E. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, vol. 203, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1874240, DOI 10.1007/978-1-4757-3604-5
Charles P. Boyer and Krzysztof Galicki, Sasakian geometry, hypersurface singularities, and Einstein metrics, Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 57–87. MR 2152356
Charles P. Boyer and Krzysztof Galicki, Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008. MR 2382957
Charles P. Boyer, Krzysztof Galicki, and Michael Nakamaye, On the geometry of Sasakian-Einstein 5-manifolds, Math. Ann. 325 (2003), no. 3, 485–524. MR 1968604, DOI 10.1007/s00208-002-0388-3
Charles P. Boyer, Krzysztof Galicki, and Michael Nakamaye, On positive Sasakian geometry, Geom. Dedicata 101 (2003), 93–102. MR 2017897, DOI 10.1023/A:1026363529906
Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
Yves Carrière, Flots riemanniens, Astérisque 116 (1984), 31–52 (French). Transversal structure of foliations (Toulouse, 1982). MR 755161
C. van Coevering, Stability of Sasaki-extremal metrics under complex deformations, to appear in Internat. Math. Res. Notices, DOI 10.1093/imrn/rns210
J.-P. Demailly, Transcendental proof of a generalized Kawamata-Viehweg vanishing theorem, Geometrical and algebraical aspects in several complex variables (Cetraro, 1989) Sem. Conf., vol. 8, EditEl, Rende, 1991, pp. 81–94. MR 1222208
Jean-Pierre Demailly, Multiplier ideal sheaves and analytic methods in algebraic geometry, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 1–148. MR 1919457
J.-P. Demailly, Complex analytic and differential geometry, Preprint (2009) available at http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
T. Duchamp and M. Kalka, Holomorphic foliations and deformations of the Hopf foliation, Pacific J. Math. 112 (1984), no. 1, 69–81. MR 739141
Aziz El Kacimi-Alaoui, Sur la cohomologie feuilletée, Compositio Math. 49 (1983), no. 2, 195–215 (French). MR 704391
Aziz El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990), no. 1, 57–106 (French, with English summary). MR 1042454
A. El Kacimi-Alaoui and G. Hector, Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 207–227 (French, with English summary). MR 865667
Joan Girbau, Sur le théorème de Le Potier d’annulation de la cohomologie, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 6, Aiv, A355–A358. MR 430333
J. Girbau, A. Haefliger, and D. Sundararaman, On deformations of transversely holomorphic foliations, J. Reine Angew. Math. 345 (1983), 122–147. MR 717890, DOI 10.1515/crll.1983.345.122
Xavier Gómez-Mont, Transversal holomorphic structures, J. Differential Geometry 15 (1980), no. 2, 161–185 (1981). MR 614365
R. Goto, Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities, J. Math. Soc. Japan 64, no. 3 (2012), 1005–1052.
Hans Grauert and Oswald Riemenschneider, Kählersche Mannigfaltigkeiten mit hyper-$q$-konvexem Rand, Problems in analysis (Lectures Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 61–79 (German). MR 0355107
André Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geometry 15 (1980), no. 2, 269–284 (1981). MR 614370
A. Haefliger, Deformations of transversely holomorphic flows on spheres and deformations of Hopf manifolds, Compositio Math. 55 (1985), no. 2, 241–251. MR 795716
Yoji Hatakeyama, Some notes on differentiable manifolds with almost contact structures, Tôhoku Math. J. (2) 15 (1963), 176–181. MR 0150718, DOI 10.2748/tmj/1178243844
G. Hector, E. Macías, and M. Saralegi, Lemme de Moser feuilleté et classification des variétés de Poisson régulières, Publ. Mat. 33 (1989), no. 3, 423–430 (French, with English summary). MR 1038481, DOI 10.5565/PUBLMAT_{3}3389_{0}4
Franz W. Kamber and Philippe Tondeur, Duality for Riemannian foliations, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 609–618. MR 713097
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2) 67 (1958), 328–466. MR 112154, DOI 10.2307/1970009
K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures, Ann. of Math. (2) 71 (1960), 43–76. MR 115189, DOI 10.2307/1969879
Xosé Masa, Duality and minimality in Riemannian foliations, Comment. Math. Helv. 67 (1992), no. 1, 17–27. MR 1144611, DOI 10.1007/BF02566486
Maung Min-Oo, Ernst A. Ruh, and Philippe Tondeur, Vanishing theorems for the basic cohomology of Riemannian foliations, J. Reine Angew. Math. 415 (1991), 167–174. MR 1096904
Pierre Molino, Riemannian foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston, Inc., Boston, MA, 1988. Translated from the French by Grant Cairns; With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu. MR 932463, DOI 10.1007/978-1-4684-8670-4
Pierre Molino and Vlad Sergiescu, Deux remarques sur les flots riemanniens, Manuscripta Math. 51 (1985), no. 1-3, 145–161 (French, with English summary). MR 788676, DOI 10.1007/BF01168350
H. Nozawa, Continuity of the Álvarez class under deformations, J. Reine Angew. Math. 673 (2012), 125–159.
Liviu Ornea and Misha Verbitsky, Structure theorem for compact Vaisman manifolds, Math. Res. Lett. 10 (2003), no. 5-6, 799–805. MR 2024735, DOI 10.4310/MRL.2003.v10.n6.a7
Liviu Ornea and Misha Verbitsky, Locally conformal Kähler manifolds with potential, Math. Ann. 348 (2010), no. 1, 25–33. MR 2657432, DOI 10.1007/s00208-009-0463-0
José Ignacio Royo Prieto, The Euler class for Riemannian flows, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 1, 45–50 (English, with English and French summaries). MR 1805626, DOI 10.1016/S0764-4442(00)01770-5
Hansklaus Rummler, Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts, Comment. Math. Helv. 54 (1979), no. 2, 224–239 (French). MR 535057, DOI 10.1007/BF02566270
M. Saralegui, The Euler class for flows of isometries, Differential geometry (Santiago de Compostela, 1984) Res. Notes in Math., vol. 131, Pitman, Boston, MA, 1985, pp. 220–227. MR 864872
Dennis Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv. 54 (1979), no. 2, 218–223. MR 535056, DOI 10.1007/BF02566269
Tsutomu Yamazaki, A construction of $K$-contact manifolds by a fiber join, Tohoku Math. J. (2) 51 (1999), no. 4, 433–446. MR 1725620, DOI 10.2748/tmj/1178224713