Quantitative and qualitative cohomological properties for non-Kähler manifolds

Angella, Daniele; Tardini, Nicoletta

doi:10.1090/proc/13209

Quantitative and qualitative cohomological properties for non-Kähler manifolds
HTML articles powered by AMS MathViewer

by Daniele Angella and Nicoletta Tardini PDF

Proc. Amer. Math. Soc. 145 (2017), 273-285 Request permission

Abstract:

We introduce a “qualitative property” for Bott-Chern cohomology of complex non-Kähler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the $\partial \overline \partial$-Lemma. This follows from a quantitative study of Bott-Chern cohomology. In this context, we also prove a new bound on the dimension of the Bott-Chern cohomology in terms of the Hodge numbers. We also give a generalization of this upper bound, with applications to symplectic cohomologies.

References

A. Aeppli, On the cohomology structure of Stein manifolds, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 58–70. MR 0221536
Daniele Angella, The cohomologies of the Iwasawa manifold and of its small deformations, J. Geom. Anal. 23 (2013), no. 3, 1355–1378. MR 3078358, DOI 10.1007/s12220-011-9291-z
D. Angella, On the Bott-Chern and Aeppli cohomology, in Bielefeld Geometry & Topology Days, https://www.math.uni-bielefeld.de/sfb701/suppls/ssfb15001.pdf, 2015, arXiv:1507.07112.
Daniele Angella, Georges Dloussky, and Adriano Tomassini, On Bott-Chern cohomology of compact complex surfaces, Ann. Mat. Pura Appl. (4) 195 (2016), no. 1, 199–217. MR 3453598, DOI 10.1007/s10231-014-0458-7
D. Angella, H. Kasuya, Bott-Chern cohomology of solvmanifolds, arXiv:1212.5708.
Daniele Angella and Adriano Tomassini, On the $\partial \overline {\partial }$-lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. 1, 71–81. MR 3032326, DOI 10.1007/s00222-012-0406-3
Daniele Angella and Adriano Tomassini, Inequalities à la Frölicher and cohomological decompositions, J. Noncommut. Geom. 9 (2015), no. 2, 505–542. MR 3359019, DOI 10.4171/JNCG/199
Daniele Angella and Adriano Tomassini, On Bott-Chern cohomology and formality, J. Geom. Phys. 93 (2015), 52–61. MR 3340173, DOI 10.1016/j.geomphys.2015.03.004
Raoul Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965), 71–112. MR 185607, DOI 10.1007/BF02391818
Jean-Luc Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114. MR 950556
G. R. Cavalcanti, New aspects of the ddc-lemma, Oxford University D. Phil thesis, arXiv:math/0501406.
Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 382702, DOI 10.1007/BF01389853
Alfred Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 641–644. MR 73262, DOI 10.1073/pnas.41.9.641
Nigel Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308. MR 2013140, DOI 10.1093/qjmath/54.3.281
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
D.-M. Lu, J. H. Palmieri, Q.-S. Wu, and J. J. Zhang, $A$-infinity structure on Ext-algebras, J. Pure Appl. Algebra 213 (2009), no. 11, 2017–2037. MR 2533303, DOI 10.1016/j.jpaa.2009.02.006
Joseph Neisendorfer and Laurence Taylor, Dolbeault homotopy theory, Trans. Amer. Math. Soc. 245 (1978), 183–210. MR 511405, DOI 10.1090/S0002-9947-1978-0511405-5
D. Popovici, Volume and Self-Intersection of Differences of Two Nef Classes, arXiv:1505.03457.
M. Schweitzer, Autour de la cohomologie de Bott-Chern, Prépublication de l’Institut Fourier no. 703 (2007), arXiv:0709.3528.
Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078
N. Tardini and A. Tomassini, On geometric Bott-Chern formality and deformations, to appear in Annali di Matematica Pura ed Applicata, DOI 10.1007/s10231-016-0575-6. arXiv:1502.03706.
Valentino Tosatti and Ben Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), no. 4, 1187–1195. MR 2669712, DOI 10.1090/S0894-0347-2010-00673-X
Li-Sheng Tseng and Shing-Tung Yau, Cohomology and Hodge theory on symplectic manifolds: I, J. Differential Geom. 91 (2012), no. 3, 383–416. MR 2981843
Li-Sheng Tseng and Shing-Tung Yau, Cohomology and Hodge theory on symplectic manifolds: II, J. Differential Geom. 91 (2012), no. 3, 417–443. MR 2981844
J. Varouchas, Propriétés cohomologiques d’une classe de variétés analytiques complexes compactes, Séminaire d’analyse P. Lelong-P. Dolbeault-H. Skoda, années 1983/1984, Lecture Notes in Math., vol. 1198, Springer, Berlin, 1986, pp. 233–243 (French). MR 874775, DOI 10.1007/BFb0077057