Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019
Andrew McHugh^{[a]}
BottChernAeppli, Dolbeault, and Frölicher on compact complex \(3\)folds
Pages: 2562
Received: 20 November 2018
Accepted in revised form: 31 Januay 2019
Mathematics Subject Classification (2010): Dolbeault cohomology, Hodge numbers, Aeppli cohomology,
BottChern cohomology, Frölicher spectral sequence.
Keywords: 32Q99, 53C55, 32C35.
Authors address:
[a]: Pennsylvania State UniversityHarrisburg, 777 W. Harrisburg Pike, Middletown, PA 17057 USA.
Full Text (PDF)
Abstract:
We give the complete BottChernAeppli cohomology for compact complex \(3\)folds in terms of Dolbeault,
Frölicher, a bidegree de Rhamlike 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 phuriharmonic functions.)
We then work out the complete BottChernAeppli cohomology in some examples.
We give the BottChernAeppli cohomology for a hypothetical complex structure on \(S^6\) in terms of Dolbeault and Frölicher.
We also give the BottChernAeppli cohomology on a CalabiEckmann \(3\)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 BottChernAeppli 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, 5870.
MR0221536
 [2]

M. Albanese (MathOverflow http://mathoverflow.net/users/21564/michaelalbanese),
Are there compact complex manifolds with nonconstant pluriclosed functions?,
http://mathoverflow.net/q/263003 (version: 20170224).
 [3]

D. Angella, Cohomological aspects of nonKähler manifolds,
Ph.D. Thesis, Università di Pisa, arXiv:1302.0524 [math.DG], 2013.
 [4]

D. Angella, On the BottChern and Aeppli cohomology,
arXiv:1507.07112 [math.CV], 2015.
 [5]

D. Angella and A. Tomassini, On BottChern cohomology and formality,
J. Geom. Phys. 93 (2015), 5261.
MR3340173
 [6]

D. Angella, G. Dloussky and A. Tomassini,
On BottChern cohomology of compact complex surfaces,
Ann. Mat. Pura Appl. (4) 195 (2016), 199217.
MR3453598
 [7]

B. Bigolin, Gruppi di Aeppli,
Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 259287.
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), 71112.
MR0185607
 [9]

J. R. Brown,
Properties of a hypothetical exotic complex structure on \(\mathbb{C} P^3\),
Math. Bohem. 132 (2007), 5974.
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 BottChern/Aeppli cohomology determined by the de Rham and Dolbeault cohomologies?,
https://mathoverflow.net/q/245702 (version: 20160803).
 [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., 251255.
MR1456264
 [13]

A. McHugh,
Narrowing cohomology on complex \(S^6\),
Eur. J. Pure Appl. Math. 10 (2017), 440454.
MR3647611
 [14]

I. Nakamura,
Complex parallelisable manifolds and their small deformations,
J. Differential Geometry 10 (1975), 85112.
MR0393580
 [15]

D. Popovici,
Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds,
Bull. Soc. Math. France 143 (2015), 763800.
MR3450501
 [16]

M. Schweitzer, Autour de la cohomologie de BottChern,
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 pseudoeffective cone of a nonKählerian surface and applications,
Math. Ann. 335 (2006), 965989.
MR2232025
 [19]

V. Tosatti, 4831 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,
NonKähler CalabiYau 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), 173179.
MR1772200
 [22]

R. O. Wells, Jr., Differential analysis on complex manifolds,
Second ed., Graduate Texts in Mathematics, 65, SpringerVerlag, New YorkBerlin, 1980.
MR0608414
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