Hodge theory for twisted differentials

Daniele Angella; Hisashi Kasuya

Complex Manifolds (2014)

  • Volume: 1, Issue: 1, page 64-85, electronic only
  • ISSN: 2300-7443

Abstract

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We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class C of Fujiki. We give a Hodgetheoretical proof of the characterization of solvmanifolds in class C of Fujiki, first stated by D. Arapura.

How to cite

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Daniele Angella, and Hisashi Kasuya. "Hodge theory for twisted differentials." Complex Manifolds 1.1 (2014): 64-85, electronic only. <http://eudml.org/doc/276960>.

@article{DanieleAngella2014,
abstract = {We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class C of Fujiki. We give a Hodgetheoretical proof of the characterization of solvmanifolds in class C of Fujiki, first stated by D. Arapura.},
author = {Daniele Angella, Hisashi Kasuya},
journal = {Complex Manifolds},
keywords = {twisted differential; local system; Dolbeault cohomology; Bott-Chern cohomology; Hodge decomposition; solvmanifolds; class C of Fujiki; complex manifolds with twisted differentials; Hodge theory; manifolds of class of Fujiki},
language = {eng},
number = {1},
pages = {64-85, electronic only},
title = {Hodge theory for twisted differentials},
url = {http://eudml.org/doc/276960},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Daniele Angella
AU - Hisashi Kasuya
TI - Hodge theory for twisted differentials
JO - Complex Manifolds
PY - 2014
VL - 1
IS - 1
SP - 64
EP - 85, electronic only
AB - We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class C of Fujiki. We give a Hodgetheoretical proof of the characterization of solvmanifolds in class C of Fujiki, first stated by D. Arapura.
LA - eng
KW - twisted differential; local system; Dolbeault cohomology; Bott-Chern cohomology; Hodge decomposition; solvmanifolds; class C of Fujiki; complex manifolds with twisted differentials; Hodge theory; manifolds of class of Fujiki
UR - http://eudml.org/doc/276960
ER -

References

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  1. [1] A. Aeppli, On the cohomology structure of Stein manifolds, Proc. Conf. Complex Analysis (Minneapolis, Minn., 1964), Springer, Berlin, 1965, pp. 58–70. 
  2. [2] D. Angella, The cohomologies of the Iwasawa manifold and of its small deformations, J. Geom. Anal. 23 (2013), no. 3, 1355– 1378. [WoS][Crossref] Zbl1278.32013
  3. [3] D. Angella, Cohomologies of certain orbifolds, J. Geom. Phys. 171 (2013), 117–126. Zbl1281.55008
  4. [4] D. Angella, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arXiv:1212.5708v3 
  5. [math.DG]. 
  6. [5] D. Angella, A. Tomassini, On the @@-Lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. 1, 71–81. [WoS] Zbl1271.32011
  7. [6] D. Angella, A. Tomassini, Inequalities à la Frölicher and cohomological decompositions, to appear in J. Noncommut. Geom.. Zbl1325.32018
  8. [7] D. Arapura, Kähler solvmanifolds, Int. Math. Res. Not. 2004 (2004), no. 3, 131–137. 
  9. [8] W. L. Baily, The decomposition theorem for V-manifolds, Amer. J. Math. 78 (1956), no. 4, 862–888. [Crossref] Zbl0173.22705
  10. [9] 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. [Crossref] Zbl0056.16901
  11. [10] G. Bharali, I. Biswas, M. Mj, The Fujiki class and positive degree maps, arXiv:1312.5655v1 Zbl1320.32023
  12. [math.GT]. 
  13. [11] F. A. Belgun, On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), no. 1, 1–40. Zbl0988.32017
  14. [12] Ch. Benson, C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518. [Crossref] Zbl0672.53036
  15. [13] 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. Zbl0148.31906
  16. [14] S. Console, A. Fino, Dolbeault cohomology of compact nilmanifolds, Transform. Groups 6 (2001), no. 2, 111–124. [Crossref] Zbl1028.58024
  17. [15] P. de Bartolomeis, A. Tomassini, On solvable generalized Calabi-Yaumanifolds, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, 1281–1296. Zbl1127.53065
  18. [16] P. Deligne, Ph. Griffiths, J. Morgan, D. P. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. [Crossref] Zbl0312.55011
  19. [17] A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641–644. [Crossref] Zbl0065.16502
  20. [18] A. Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225–258. [Crossref] Zbl0367.32004
  21. [19] K. Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71. Zbl0691.53040
  22. [20] K. Hasegawa, A note on compact solvmanifolds with Kähler structures, Osaka J. Math. 43 (2006), no. 1, 131–135. Zbl1105.32017
  23. [21] H. Kasuya, Techniques of computations of Dolbeault cohomology of solvmanifolds, Math. Z. 273 (2013), no. 1-2, 437–447. [WoS] Zbl1261.22009
  24. [22] H. Kasuya, Hodge symmetry and decomposition on non-Kähler solvmanifolds, J. Geom. Phys. 76 (2014), 61–65. Zbl1279.22010
  25. [23] H. Kasuya, Flat bundles and Hyper-Hodge decomposition on solvmanifolds, arXiv:1309.4264v1 Zbl1327.53065
  26. [math.DG], To appear in Int. Math. Res. Not. IMRN. 
  27. [24] 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. Zbl1058.32007
  28. [25] 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. Zbl0128.16902
  29. [26] 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. [27] I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Differ. Geom. 10 (1975), no. 1, 85–112. Zbl0297.32019
  31. [28] L. Ornea, M. Verbitsky, Morse-Novikov cohomology of locally conformally Kählermanifolds, J. Geom. Phys. 59 (2009), no. 3, 295–305. [Crossref] Zbl1161.57015
  32. [29] I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), no. 6, 359–363. [Crossref] Zbl0074.18103
  33. [30] M. Schweitzer, Autour de la cohomologie de Bott-Chern, arXiv:0709.3528v1 
  34. [math.AG]. 
  35. [31] C. T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. No. 75 (1992), 5–95. Zbl0814.32003
  36. [32] C. Voisin, Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002. 
  37. [33] R. O. Wells, Jr., Comparison of de Rham and Dolbeault cohomology for proper surjectivemappings, Pacific J.Math. 53 (1974), no. 1, 281–300. Zbl0261.32005
  38. [34] 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. 

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