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Quantitative and qualitative cohomological properties for non-Kähler manifolds


Authors: Daniele Angella and Nicoletta Tardini
Journal: Proc. Amer. Math. Soc. 145 (2017), 273-285
MSC (2010): Primary 32Q99, 32C35
DOI: https://doi.org/10.1090/proc/13209
Published electronically: July 12, 2016
MathSciNet review: 3565379
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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.


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Daniele Angella
Affiliation: Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy
Email: daniele.angella@gmail.com, daniele.angella@unifi.it

Nicoletta Tardini
Affiliation: Dipartimento di Matematica, Università di Pisa, largo Bruno Pontecorvo 5, 56127 Pisa, Italy
Email: tardini@mail.dm.unipi.it

DOI: https://doi.org/10.1090/proc/13209
Keywords: Complex manifold, non-K\"ahler geometry, Bott-Chern cohomology, Aeppli cohomology, $\partial\overline\partial$-Lemma
Received by editor(s): December 19, 2015
Received by editor(s) in revised form: March 18, 2016
Published electronically: July 12, 2016
Additional Notes: During the preparation of this work, the first author was also granted by a Junior Visiting Position at Centro di Ricerca “Ennio de Giorgi” in Pisa. The first author was supported by the Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, by the Project FIRB “Geometria Differenziale e Teoria Geometrica delle Funzioni”, by SNS GR14 grant “Geometry of non-Kähler manifolds”, by SIR2014 project RBSI14DYEB “Analytic aspects in complex and hypercomplex geometry”, and by GNSAGA of INdAM. The second author was supported by GNSAGA of INdAM
Dedicated: Dedicated to the memory of Professor Pierre Dolbeault
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society