Spectral Sequences for Almost Complex Manifolds

Author

Qian Chen, The Graduate Center, City University of New YorkFollow

Date of Degree

9-2020

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Scott O. Wilson

Committee Members

Luis Fernandez

Bianca Santoro

Subject Categories

Algebra | Algebraic Geometry | Analysis | Geometry and Topology | Harmonic Analysis and Representation | Other Mathematics

Keywords

Almost complex manifolds, Complex manifolds, Cohomology groups, Spectral Sequences, Nijenhuis Tensor, Almost complex structure, Kahler Manifolds, Vector valued forms, Bott Chern cohomology, Six sphere, Kadaira Thurston manifold, Iwasawa manifold

Abstract

In recent work, two new cohomologies were introduced for almost complex manifolds: the so-called J-cohomology and N-cohomology [CKT17]. For the case of integrable (complex) structures, the former cohomology was already considered in [DGMS75], and the latter agrees with de Rham cohomology. In this dissertation, using ideas from [CW18], we introduce spectral sequences for these two cohomologies, showing the two cohomologies have natural bigradings. We show the spectral sequence for the J-cohomology converges at the second page whenever the almost complex structure is integrable, and explain how both fit in a natural diagram involving Bott-Chern cohomology and the Frolicher spectral sequence. Using explicit formulas that we derive for the pages, as well as topology in some cases, we deduce several properties of the groups and the natural maps in various degrees. As applications, we study the Kodaira-Thurston and Iwasawa manifolds, as well as a hypothetical complex structure of the six-sphere.

Recommended Citation

Chen, Qian, "Spectral Sequences for Almost Complex Manifolds" (2020). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/4003