Weighted Bott-Chern and Dolbeault cohomology for LCK-manifolds with potential. (English) Zbl 1390.32019

Let \(M\) be a locally conformal Kähler (LCK) manifold with universal covering \(\widetilde{M}\), i.e., \(M\) is a complex manifold with a Kähler structure on \(\widetilde{M}\) and with a deck transform group acting on \(\widetilde{M}\) by holomorphic homothetic maps and which is hence naturally equipped with a Hermitian form \(\omega\) (called the Lee form) such that \(d\omega = \omega \wedge \theta\) for some closed 1-form \(\theta\) with values in a local system \(L\) (called conformal weight bundle). The main result, Theorem 3.2, establishes generic vanishing properties for the weighted Dolbeault cohomology and serves in particular as the basis for the proof of an analogue of the \(dd^c\)-lemma on Kähler manifolds, in the setting of \(L\)-valued (Morse-Novikov) cohomology: Let \((M,\omega,\theta)\) be a compact LCK manifold, and let \(L\) its weight bundle. Then there exists a countable discrete subset \(S\) of the complex plane \(\mathbb C\) such that, for each \(\alpha\in\mathbb C\setminus S\), if \(L_\alpha\) is the flat line bundle on \(M\) corresponding to \(\alpha\cdot \theta\), then \(H^q(M,\Omega_M^p \otimes L_\alpha) =0\) for all \(q\in\mathbb N\). As a consequence one can prove the following geometrically far reaching result (in particular a true \(dd^c\)-conjecture version in Vaisman manifolds with coefficients in a sufficiently large power of \(L\)): If \(M\) is an LCK manifold with proper potential, then \(H^{p,q}_{BC}(M,L\alpha)=0\) for all \(\alpha\in\mathbb C\setminus S\) (Bott-Chern cohomology).


32Q15 Kähler manifolds
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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[1] A. Banyaga, Examples of non \(d_ω\)-exact locally conformal symplectic forms, \doihref10.1007/s00022-006-1849-8J. Geom., 87 (2007), 1-13. · Zbl 1157.53040
[2] F. A. Belgun, On the metric structure of non-Kähler complex surfaces, \doihref10.1007/s002080050357Math. Ann., 317 (2000), 1-40.
[3] Ch. P. Boyer and K. Galicki, Sasakian Geometry, \doihref10.1093/acprof:oso/9780198564959.001.0001Oxford Math. Monographs, Oxford Univ. Press, 2008.
[4] S. Dragomir and L. Ornea, Locally conformal Kähler geometry, \doihref10.1007/978-1-4612-2026-8Progress in Math., 155, Birkhäuser, Boston, Basel, 1998.
[5] D. Eisenbud, Commutative algebra with a view towards algebraic geometry, \doihref10.1007/978-1-4612-5350-1GTM, 150, Springer, 1994.
[6] P. Gauduchon and L. Ornea, Locally conformally Kähler metrics on Hopf surfaces, \doihref10.5802/aif.1651Ann. Inst. Fourier, 48 (1998), 1107-1127. · Zbl 0917.53025
[7] R. Goto, On the stability of locally conformal Kähler structures, \doihref10.2969/jmsj/06641375J. Math. Soc. Japan, 66 (2014), 1375-1401. · Zbl 1310.53062
[8] C. Huneke, Lectures on Local Cohomology, \doihref10.1090/conm/436Appendix 1 by Amelia Taylor. Contemp. Math., 436, Interactions between homotopy theory and algebra, Amer. Math. Soc., Providence, RI, 2007, 51-99.
[9] M. Inoue, On surfaces of class \(VII_0\), \doihref10.1007/BF01425563Invent. Math., 24 (1974), 269-310.
[10] M. Ise, On the geometry of Hopf manifolds, Osaka J. Math., 12 (1960), 387-402. · Zbl 0108.36001
[11] M. de León, B. López, J. C. Marrero and E. Padrón, On the computation of the Lichnerowicz-Jacobi cohomology, \doihref10.1016/S0393-0440(02)00056-6J. Geom. Phys., 44 (2003), 507-522.
[12] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, \doihref10.4310/jdg/1214433987J. Diff. Geom., 12 (1977), 253-300.
[13] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), \doihref10.1007/BF01244301Invent. Math., 113 (1993), 41-55. · Zbl 0795.13004
[14] D. Mall, The cohomology of line bundles on Hopf manifolds, Osaka J. Math., 28 (1991), 999-1015. · Zbl 0759.32007
[15] D. V. Millionshchikov, Cohomology of solvmanifolds with local coefficients and problems in the Morse-Novikov theory, \doihref10.1070/RM2002v057n04ABEH000545Russian Math. Surveys, 57 (2002), 813-814. · Zbl 1051.57036
[16] S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory (Russian), Uspekhi Mat. Nauk, 37 (1982), 3-49.
[17] K. Oeljeklaus and M. Toma, Non-Kähler compact complex manifolds associated to number fields, \doihref10.5802/aif.2093Ann. Inst. Fourier, 55 (2005), 1291-1300. · Zbl 1071.32017
[18] L. Ornea and M. Verbitsky, Morse-Novikov cohomology of locally conformally Kähler manifolds, \doihref10.1016/j.geomphys.2008.11.003J. Geom. Phys., 59 (2009), 295-305. · Zbl 1161.57015
[19] L. Ornea and M. Verbitsky, Locally conformally Kähler manifolds with potential, \doihref10.1007/s00208-009-0463-0Math. Ann., 348 (2010), 25-33.
[20] L. Ornea and M. Verbitsky, Topology of Locally Conformally Kähler Manifolds with Potential, \doihref10.1093/imrn/rnp144Int. Math. Res. Notices, 4 (2010), 717-726. · Zbl 1188.53080
[21] L. Ornea and M. Verbitsky, Locally conformally Kähler manifolds admitting a holomorphic conformal flow, \doihref10.1007/s00209-012-1022-zMath. Z., 273 (2013), 605-611. · Zbl 1276.53077
[22] L. Ornea and M. Verbitsky, Locally conformally Kähler metrics obtained from pseudoconvex shells, \doihref10.1090/proc12770Proc. Amer. Math. Soc., 144 (2016), 325-335. · Zbl 1327.53098
[23] L. Ornea and M. Verbitsky, LCK rank of locally conformally Kähler manifolds with potential, \doihref10.1016/j.geomphys.2016.05.011J. Geom. Phys., 107 (2016), 92-98. · Zbl 1347.53058
[24] A. V. Pajitnov, Exactness of Novikov-type inequalities for the case \(π_1(M)= \mathbb{Z}^m\) and for Morse forms whose cohomology classes are in general position, Soviet Math. Dokl., 39 (1989), 528-532.
[25] A. Ranicki, Circle valued Morse theory and Novikov homology, Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), 539-569, ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002. · Zbl 1068.57031
[26] S. Rollenske, The Frölicher spectral sequence can be arbitrarily non-degenerate, \doihref10.1007/s00208-007-0206-zMath. Ann., 341 (2008), 623-628; erratum ibid. \doihref10.1007/s00208-013-0996-0 358 (2014), 1119-1123. · Zbl 1188.53083
[27] F. Tricerri, Some examples of locally conformal Kähler manifolds, Rend. Sem. Mat. Univ. Politec. Torino, 40 (1982), 81-92. · Zbl 0511.53068
[28] M. Verbitsky, Theorems on the vanishing of cohomology for locally conformally hyper-Kähler manifolds, Proc. Steklov Inst. Math., 246 (2004), 54-78.
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