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On the stability of locally conformal Kähler structures. (English) Zbl 1310.53062

If \(X\) a complex manifold and \(L\) a flat line bundle over \(X\), then we say that \((X,L)\) satisfies the \(\partial\bar{\partial}\)-lemma at degree \((p,q)\) if there is an \(L\)-valued form \(\gamma\) of type \((p-1,q-1)\) such that \(\partial_L\bar{\partial}_L\gamma=\partial_L\alpha\in\Gamma(X,L\otimes\Lambda^{p,q})\), for every \(\bar{\partial}_L\)-closed \(L\)-valued form \(\alpha\) of type \((p-1,q)\). A locally conformal Kähler (LCK) manifold is a complex Hermitian manifold, with a Hermitian form \(\omega\) satisfying \(d\omega=\theta\wedge\omega\), where \(\eta\) is a closed, non-exact 1-form, called the Lee form of the manifold.
In this paper, the author obtains the following criterion for the stability of LCK structures: “If \(L\) is a flat line bundle corresponding to an LCK structure \(\omega_0\) on \(X\) and \((X,L)\) satisfies the \(\partial\bar{\partial}\)-lemma at degree \((1,2)\), then the stability of LCK structures holds, that is, every small deformation \(X_t\) of \(X\) admits an LCK structure \(\omega_t\) where \(L\) is still the corresponding flat line bundle to the deformed structure \(\omega_t\)”. Moreover, the author shows that there are obstructions to the stability of LCK structures under deformations of flat line bundles on Inoue surfaces with \(b_2=0\) and disproves a conjecture of L. Ornea and M. Verbitsky [J. Geom. Phys. 59, No. 3, 295–305 (2009; Zbl 1161.57015)].

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
32G05 Deformations of complex structures

Citations:

Zbl 1161.57015
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References:

[1] F. A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann., 317 (2000), 1-40. · Zbl 0988.32017
[2] P. C. Boyer and K. Galicki, Sasakian Geometry, Oxford Math. Monogr., Oxford Science Publications, Oxford University Press, Oxford, 2008.
[3] M. Brunella, Locally conformally Kähler metrics on certain non-Kählerian surfaces, Math. Ann., 346 (2010), 629-639. · Zbl 1196.32015
[4] M. Brunella, Locally conformally Kähler metrics on Kato surfaces, Nagoya Math. J., 202 (2011), 77-81. · Zbl 1227.32026
[5] S. Dragomir and L. Ornea, Locally Conformal Kähler Geometry, Progr. Math., 155 , Birkhäuser Boston, Inc., Boston, MA, 1998.
[6] A. Fujiki and M. Pontecorvo, Anti-self-dual bihermitian structures on Inoue surfaces, J. Differential Geom., 85 (2010), 15-71. · Zbl 1206.53077
[7] R. Goto, Moduli spaces of topological calibrations, Calabi-Yau, hyper-Kähler, \(G_2\) and Spin\((7)\) structures, Internat. J. Math., 15 (2004), 211-257. · Zbl 1046.58002
[8] R. Goto, On deformations of generalized Calabi-Yau, hyperKähler, \(G_2\) and Spin\((7)\) structures.. · Zbl 0527.13012
[9] R. Goto, Deformations of generalized complex and generalized Kähler structures, J. Differential Geom., 84 (2010), 525-560. · Zbl 1201.53085
[10] R. Goto, Poisson structures and generalized Kähler submanifolds, J. Math. Soc. Japan, 61 (2009), 107-132. · Zbl 1160.53014
[11] R. Goto, Unobstructed K-deformations of generalized complex structures and bi-Hermitian structures, Adv. Math., 231 (2012), 1041-1067. · Zbl 1252.53095
[12] P. Gauduchon and L. Ornea, Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier (Grenoble), 48 (1998), 1107-1127. · Zbl 0917.53025
[13] M. Inoue, On surfaces of Class \({\mathrm VII}_{0}\), Invent. Math., 24 (1974), 269-310. · Zbl 0283.32019
[14] M. Ise, On the geometry of Hopf manifolds, Osaka Math. J., 12 (1960), 387-402. · Zbl 0108.36001
[15] K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Grundlehren Math. Wiss., 283 , Springer-Verlag, New York, 1986. · Zbl 0581.32012
[16] K. Kodaira and D. C. Spencer, On deformations of complex, analytic structures. I, II, Ann. Math. (2), 67 (1958), 328-466. · Zbl 0128.16901
[17] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. III, stability theorems for complex structures, Ann. Math. (2), 71 (1960), 43-76. · Zbl 0128.16902
[18] M. de León, B. López, J. C. Marrero and E. Padrón, On the computation of the Lichnerowicz-Jacobi cohomology, J. Geom. Phys., 44 (2003), 507-522. · Zbl 1092.53060
[19] D. Mall, The cohomology of line bundles on Hopf manifolds, Osaka J. Math., 28 (1991), 999-1015. · Zbl 0759.32007
[20] L. Ornea and M. Verbitsky, Locally conformal Kähler manifolds with potential, Math. Ann., 348 (2010), 25-33. · Zbl 1213.53090
[21] L. Ornea and M. Verbitsky, Morse-Novikov cohomology of locally conformally Kähler manifolds, J. Geom. Phys., 59 (2009), 295-305. · Zbl 1161.57015
[22] F. Tricerri, Some examples of locally conformal Kähler manifolds, Rend. Sem. Mat. Univ. Politec. Torino, 40 (1982), 81-92. · Zbl 0511.53068
[23] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata, 13 (1982), 231-255. · Zbl 0506.53032
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