×

zbMATH — the first resource for mathematics

Hodge theory for twisted differentials. (English) Zbl 1320.32027
Summary: We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class \(\mathcal C\) of Fujiki. We give a Hodge-theoretical proof of the characterization of solvmanifolds in class \(\mathcal C\) of Fujiki, first stated by D. Arapura.

MSC:
32M10 Homogeneous complex manifolds
53C30 Differential geometry of homogeneous manifolds
58A14 Hodge theory in global analysis
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] A. Aeppli, On the cohomology structure of Stein manifolds, Proc. Conf. Complex Analysis (Minneapolis, Minn., 1964), Springer, Berlin, 1965, pp. 58-70.
[2] D. Angella, The cohomologies of the Iwasawa manifold and of its small deformations, J. Geom. Anal. 23 (2013), no. 3, 1355- 1378. · Zbl 1278.32013
[3] D. Angella, Cohomologies of certain orbifolds, J. Geom. Phys. 171 (2013), 117-126. · Zbl 1281.55008
[4] D. Angella, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arXiv:1212.5708v3 · Zbl 1384.22004
[5] .
[6] D. Angella, A. Tomassini, On the @@-Lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. 1, 71-81. · Zbl 1271.32011
[7] D. Angella, A. Tomassini, Inequalities à la Frölicher and cohomological decompositions, to appear in J. Noncommut. Geom.. · Zbl 1325.32018
[8] D. Arapura, Kähler solvmanifolds, Int. Math. Res. Not. 2004 (2004), no. 3, 131-137.
[9] W. L. Baily, The decomposition theorem for V-manifolds, Amer. J. Math. 78 (1956), no. 4, 862-888. · Zbl 0173.22705
[10] W. L. Baily, On the quotient of an analytic manifold by a group of analytic homeomorphisms, Proc. Nat. Acad. Sci. U. S. A. 40 (1954), no. 9, 804-808. · Zbl 0056.16901
[11] G. Bharali, I. Biswas, M. Mj, The Fujiki class and positive degree maps, arXiv:1312.5655v1 · Zbl 1320.32023
[12] .
[13] F. A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), no. 1, 1-40. · Zbl 0988.32017
[14] Ch. Benson, C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513-518. · Zbl 0672.53036
[15] R. Bott, S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965), no. 1, 71-112. · Zbl 0148.31906
[16] S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), no. 2, 111-124. · Zbl 1028.58024
[17] P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yaumanifolds, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, 1281-1296. · Zbl 1127.53065
[18] P. Deligne, Ph. Griffiths, J. Morgan, D. P. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245-274. · Zbl 0312.55011
[19] A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641-644.
[20] A. Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225-258. · Zbl 0367.32004
[21] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65-71. · Zbl 0691.53040
[22] K. Hasegawa, A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), no. 1, 131-135. · Zbl 1105.32017
[23] H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z. 273 (2013), no. 1-2, 437-447. · Zbl 1261.22009
[24] H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, J. Geom. Phys. 76 (2014), 61-65. · Zbl 1279.22010
[25] H. Kasuya, Flat bundles and Hyper-Hodge decomposition on solvmanifolds, arXiv:1309.4264v1 · Zbl 1327.53065
[26] , To appear in Int. Math. Res. Not. IMRN.
[27] K. Kodaira, Complex manifolds and deformation of complex structures, Translated from the 1981 Japanese original by Kazuo Akao, Reprint of the 1986 English edition, Classics in Mathematics, Springer-Verlag, Berlin, 2005. · Zbl 1058.32007
[28] K. Kodaira, D. C. Spencer, On deformations of complex analytic structures. III. Stability theorems for complex structures, Annals of Math. (2) 71 (1960), no. 1, 43-76. · Zbl 0128.16902
[29] B. G. Moˇıšezon, On n-dimensional compact complexmanifolds having n algebraically independent meromorphic functions. I, II, III, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), no. 1-2-3, 133-174, 345-386, 621-656. Translation in Am. Math. Soc., Transl., II. Ser. 63 (1967), 51-177.
[30] I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Differ. Geom. 10 (1975), no. 1, 85-112. · Zbl 0297.32019
[31] L. Ornea, M. Verbitsky, Morse-Novikov cohomology of locally conformally Kählermanifolds, J. Geom. Phys. 59 (2009), no. 3, 295-305. · Zbl 1161.57015
[32] I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), no. 6, 359-363. · Zbl 0074.18103
[33] M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528v1
[34] .
[35] C. T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. No. 75 (1992), 5-95. · Zbl 0814.32003
[36] C. Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002.
[37] R. O. Wells, Jr., Comparison of de Rham and Dolbeault cohomology for proper surjectivemappings, Pacific J.Math. 53 (1974), no. 1, 281-300. · Zbl 0261.32005
[38] R O., Wells, Jr., Differential analysis on complex manifolds, Third edition,With a new appendix by Oscar Garcia-Prada, Graduate Texts in Mathematics, 65, Springer, New York, 2008.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.