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Volume 30, Issue 2
A Note on Donaldson's "Tamed to Compatible" Question

Qiang Tan & Haifeng Xu

DOI: 10.13447/j.1674-5647.2014.02.08

Commun. Math. Res., 30 (2014), pp. 179-182.

Published online: 2021-05

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  • Abstract

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson's "tamed to compatible" question (Draghici T, Zhang W. A note on exact forms on almost complex manifolds. arXiv: 1111. 7287v1 [math. SG]. Submitted on 30 Nov. 2011). That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure? They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson's question in dimension 4. In this note, we give another simpler proof of their result.

  • Keywords

compact almost complex 4-manifold, $ω$-tame almost complex structure, $ω$-compatible almost complex structure.

  • AMS Subject Headings

53D35, 53D45, 58D29

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-30-179, author = {Tan , Qiang and Xu , Haifeng}, title = {A Note on Donaldson's "Tamed to Compatible" Question}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {30}, number = {2}, pages = {179--182}, abstract = {

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson's "tamed to compatible" question (Draghici T, Zhang W. A note on exact forms on almost complex manifolds. arXiv: 1111. 7287v1 [math. SG]. Submitted on 30 Nov. 2011). That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure? They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson's question in dimension 4. In this note, we give another simpler proof of their result.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2014.02.08}, url = {http://global-sci.org/intro/article_detail/cmr/18980.html} }
TY - JOUR T1 - A Note on Donaldson's "Tamed to Compatible" Question AU - Tan , Qiang AU - Xu , Haifeng JO - Communications in Mathematical Research VL - 2 SP - 179 EP - 182 PY - 2021 DA - 2021/05 SN - 30 DO - http://doi.org/10.13447/j.1674-5647.2014.02.08 UR - https://global-sci.org/intro/article_detail/cmr/18980.html KW - compact almost complex 4-manifold, $ω$-tame almost complex structure, $ω$-compatible almost complex structure. AB -

Recently, Tedi Draghici and Weiyi Zhang studied Donaldson's "tamed to compatible" question (Draghici T, Zhang W. A note on exact forms on almost complex manifolds. arXiv: 1111. 7287v1 [math. SG]. Submitted on 30 Nov. 2011). That is, for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form, is there a symplectic form compatible with this almost complex structure? They got several equivalent forms of this problem by studying the space of exact forms on such a manifold. With these equivalent forms, they proved a result which can be thought as a further partial answer to Donaldson's question in dimension 4. In this note, we give another simpler proof of their result.

Tan , Qiang and Xu , Haifeng. (2021). A Note on Donaldson's "Tamed to Compatible" Question. Communications in Mathematical Research . 30 (2). 179-182. doi:10.13447/j.1674-5647.2014.02.08
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