Abstract
Based on recent work of T. Draghici, T.-J. Li, and W. Zhang, we further investigate properties of the dimension \(h_J^-\) of the \(J\)-anti-invariant cohomology subgroup \(H_J^-\) of a closed almost Hermitian 4-manifold \((M,g,J,F)\) using metric compatible almost complex structures. We prove that \(h_J^-=0\) for generic almost complex structures \(J\) on \(M\).
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Acknowledgments
The authors would like to thank J. Draghici and T.-J. Li for stimulating email discussions and T.-J. Li and K.-C. Chen for their help in sending reference papers. The authors are grateful to the referees for their valuable comments and suggestions. Especially, the referees suggest a more direct argument about the openness part. The second author would like to thank East China Normal University and Qing Zhou for hosting his visit in the fall semester in 2011. Supported by NSFC (China) Grants 11071208, 11371309 (Wang), 11271276 (Zhang), 11101352 (Zhu), Fund of Jiangsu University of Technology Grant KYY13005 (Zhu), Qing Lan Project (Zhu) and the Postgraduate Innovation Project of Jiangsu Province (NO.CXZZ13\(_{-}\)0888) (Tan).
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Communicated by Jiaping Wang.
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Tan, Q., Wang, H., Zhang, Y. et al. On Cohomology of Almost Complex 4-Manifolds. J Geom Anal 25, 1431–1443 (2015). https://doi.org/10.1007/s12220-014-9477-2
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DOI: https://doi.org/10.1007/s12220-014-9477-2
Keywords
- Almost Hermitian 4-manifold
- \(J\)-anti-invariant cohomology
- Metric compatible almost complex structure