Symplectic forms and cohomology decomposition of almost complex 4-manifolds

2008/12/31 by Tedi Draghici, Tian-Jun Li, Weiyi Zhang · 1 citation
Mathematics · #math.DG #math.SG

paper · pdf

published as IMRN 2010, no. 1, 1-17 · v2. This is the published version of some of the results of v1; Other parts of v1 have been considerably extended and included in arXiv:1104.2511

arxiv created 2011/04/14 · arxiv updated 2011/04/15

Abstract

For any compact almost complex manifold (M,J), the last two authors defined two subgroups H_J⁺(M), H_J^-(M) of the degree 2 real de Rham cohomology group H²(M, ℝ) in arXiv:0708.2520. These are the sets of cohomology classes which can be represented by J-invariant, respectively, J-anti-invariant real 2-forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of H²(M, ℝ). This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini. 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.

Citations

• Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds
• On the J-anti-invariant cohomology of almost complex 4-manifolds

Cited by

• Symplectically aspherical Kähler manifolds, scalar curvature, and the fundamental group

Related

• On the J-anti-invariant cohomology of almost complex 4-manifolds
• On cohomology of almost complex 4-manifolds
• On non-pure forms on almost complex manifolds
• A note on the deformations of almost complex structures on compact four-manifolds
• On L₂-cohomology of almost Hermitian manifolds