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.

References
[1]
A. Aeppli, On the cohomology structure of Stein manifolds, Proc. Conf. Complex Analysis (Minneapolis, Minn., 1964), Springer, Berlin, 1965, 58-70. MR0221536
[2]
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).
[3]
D. Angella, Cohomological aspects of non-Kähler manifolds, Ph.D. Thesis, Università di Pisa, arXiv:1302.0524 [math.DG], 2013.
[4]
D. Angella, On the Bott-Chern and Aeppli cohomology, arXiv:1507.07112 [math.CV], 2015.
[5]
D. Angella and A. Tomassini, On Bott-Chern cohomology and formality, J. Geom. Phys. 93 (2015), 52-61. MR3340173
[6]
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
[7]
B. Bigolin, Gruppi di Aeppli, Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 259-287. MR0245836
[8]
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
[9]
J. R. Brown, Properties of a hypothetical exotic complex structure on \(\mathbb{C} P^3\), Math. Bohem. 132 (2007), 59-74. MR2311754
[10]
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
[11]
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).
[12]
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
[13]
A. McHugh, Narrowing cohomology on complex \(S^6\), Eur. J. Pure Appl. Math. 10 (2017), 440-454. MR3647611
[14]
I. Nakamura, Complex parallelisable manifolds and their small deformations, J. Differential Geometry 10 (1975), 85-112. MR0393580
[15]
D. Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bull. Soc. Math. France 143 (2015), 763-800. MR3450501
[16]
M. Schweitzer, Autour de la cohomologie de Bott-Chern, Prépublication de l'Institut Fourier no. 703 (2007); arXiv:0709.3528v1 [math.AG], 2007.
[17]
J. Stelzig, On the structure of double complexes, arXiv:1812.00865v1 [math.RT], 2018.
[18]
A. Teleman, The pseudo-effective cone of a non-Kählerian surface and applications, Math. Ann. 335 (2006), 965-989. MR2232025
[19]
V. Tosatti, 483-1 Algebraic Geometry, Solution of Homework 6, Northwestern University, https://sites.math.northwestern.edu/ tosatti/hw6_ag.pdf
[20]
L.-S. Tseng and S.-T. Yau, Non-Kähler Calabi-Yau manifolds, Proc. Sympos. Pure Math., 85, Amer. Math. Soc., Providence, RI, 2012. MR2985333
[21]
L. Ugarte, Hodge numbers of a hypothetical complex structure on the six sphere, Geom. Dedicata 81 (2000), 173-179. MR1772200
[22]
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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