Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019

Andrew McHugh[a]

Bott-Chern-Aeppli, Dolbeault, and Frölicher on compact complex \(3\)-folds

Pages: 25-62
Received: 20 November 2018
Accepted in revised form: 31 Januay 2019
Mathematics Subject Classification (2010): Dolbeault cohomology, Hodge numbers, Aeppli cohomology, Bott-Chern cohomology, Frölicher spectral sequence.
Keywords: 32Q99, 53C55, 32C35.
Authors address:
[a]: Pennsylvania State University-Harrisburg, 777 W. Harrisburg Pike, Middletown, PA 17057 USA.

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Abstract: We give the complete Bott-Chern-Aeppli cohomology for compact complex \(3\)-folds in terms of Dolbeault, Frölicher, a bi-degree de Rham-like type of cohomology, \(K^{p,q}\), defined as

\( K^{p,q}=\displaystyle\frac{ker( \partial ) \cap ker( {\bar{\partial}}) }{im( \partial )\cap ker( {\bar{\partial}} )+im( {\bar{\partial}})\cap ker( \partial )}\)

and \({\check{H}}^1({\mathcal{PH}})\). (Here \(\mathcal{PH}\) is the sheaf of phuri-harmonic functions.) We then work out the complete Bott-Chern-Aeppli cohomology in some examples. We give the Bott-Chern-Aeppli cohomology for a hypothetical complex structure on \(S^6\) in terms of Dolbeault and Frölicher. We also give the Bott-Chern-Aeppli cohomology on a Calabi-Eckmann \(3\)-fold concurring with the calculations of Angella and Tomassini [5]. Finally, we show agreement of our results with the calculation by Angella [3] of the Bott-Chern-Aeppli cohomology for small Kuranishi deformations of the Iwasawa manifold.

A. Aeppli, On the cohomology structure of Stein manifolds, Proc. Conf. Complex Analysis (Minneapolis, Minn., 1964), Springer, Berlin, 1965, 58-70. MR0221536
M. Albanese (MathOverflow http://mathoverflow.net/users/21564/michael-albanese), Are there compact complex manifolds with non-constant pluriclosed functions?, http://mathoverflow.net/q/263003 (version: 2017-02-24).
D. Angella, Cohomological aspects of non-Kähler manifolds, Ph.D. Thesis, Università di Pisa, arXiv:1302.0524 [math.DG], 2013.
D. Angella, On the Bott-Chern and Aeppli cohomology, arXiv:1507.07112 [math.CV], 2015.
D. Angella and A. Tomassini, On Bott-Chern cohomology and formality, J. Geom. Phys. 93 (2015), 52-61. MR3340173
D. Angella, G. Dloussky and A. Tomassini, On Bott-Chern cohomology of compact complex surfaces, Ann. Mat. Pura Appl. (4) 195 (2016), 199-217. MR3453598
B. Bigolin, Gruppi di Aeppli, Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 259-287. MR0245836
R. Bott and S. S. Chern, Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math. 114 (1965), 71-112. MR0185607
J. R. Brown, Properties of a hypothetical exotic complex structure on \(\mathbb{C} P^3\), Math. Bohem. 132 (2007), 59-74. MR2311754
J. Cirici, A Short course on the interactions of rational homotopy theory and complex (algebraic) geometry, Notes for the summer school on "Rational Homotopy Theory and its Interactions", (July 2016, Rabat, Morocco). Article
Daniele (MathOverflow http://mathoverflow.net/users/29341/daniele), Is the Bott-Chern/Aeppli cohomology determined by the de Rham and Dolbeault cohomologies?, https://mathoverflow.net/q/245702 (version: 2016-08-03).
A. Gray, A property of a hypothetical complex structure on the six sphere, Boll. Un. Mat. Ital. B (7) 11 (1997), no. 2, suppl., 251-255. MR1456264
A. McHugh, Narrowing cohomology on complex \(S^6\), Eur. J. Pure Appl. Math. 10 (2017), 440-454. MR3647611
I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Differential Geometry 10 (1975), 85-112. MR0393580
D. Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bull. Soc. Math. France 143 (2015), 763-800. MR3450501
M. Schweitzer, Autour de la cohomologie de Bott-Chern, Prépublication de l'Institut Fourier no. 703 (2007); arXiv:0709.3528v1 [math.AG], 2007.
J. Stelzig, On the structure of double complexes, arXiv:1812.00865v1 [math.RT], 2018.
A. Teleman, The pseudo-effective cone of a non-Kählerian surface and applications, Math. Ann. 335 (2006), 965-989. MR2232025
V. Tosatti, 483-1 Algebraic Geometry, Solution of Homework 6, Northwestern University, https://sites.math.northwestern.edu/ tosatti/hw6_ag.pdf
L.-S. Tseng and S.-T. Yau, Non-Kähler Calabi-Yau manifolds, Proc. Sympos. Pure Math., 85, Amer. Math. Soc., Providence, RI, 2012. MR2985333
L. Ugarte, Hodge numbers of a hypothetical complex structure on the six sphere, Geom. Dedicata 81 (2000), 173-179. MR1772200
R. O. Wells, Jr., Differential analysis on complex manifolds, Second ed., Graduate Texts in Mathematics, 65, Springer-Verlag, New York-Berlin, 1980. MR0608414

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