Symplectic forms and cohomology decomposition of almost complex four-manifolds

Tedi Draghici, Tian Jun Li, Weiyi Zhang

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Abstract

For any compact almost complex manifold (M, J), the last two authors [8] defined two subgroups H+J(M), H-J(M) of the degree 2 real de Rham cohomology group. These are the sets of cohomology classes which can be represented by J-invariant, respectively, J-antiinvariant real 2-forms. In this paper, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of. This is a specifically four-dimensional result, as it follows from a recent work of Fino and Tomassini [6]. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.

Original languageEnglish (US)
Pages (from-to)1-17
Number of pages17
JournalInternational Mathematics Research Notices
Volume2010
Issue number1
DOIs
StatePublished - 2010

Bibliographical note

Funding Information:
We appreciate V. Apostolov for his very useful comments, R. Hind and T. Perutz for their interest, A. Fino and A. Tomassini for sending us their paper [6], and National Science Foundation for the partial support. We also thank the referees for their careful reading of the manuscript and useful remarks.

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