Odd Type Generalized Complex Structures on 4-Manifolds
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We prove that a compact smooth 4-manifold admits generalized complex structures of odd type if and only if it has a transversely holomorphic 2-foliation. We also obtain an equivalent description of generalized complex structures in terms of almost bihermitian structures.
We would like to thank Professor Tian-Jun Li for the support, encouragement, and many useful discussions, especially for pointing out Corollary 1.3. Also, we like to thank Professors Tedi Draghici, Weiwei Wu for the discussions and Professor Marco Gualtieri for helpful communications. The authors thank the anonymous referees whose comments and suggestions helped improve the paper. The first-named author is partially supported by Zhejiang Provincial Natural Science Foundation (Grant No. LY19A010017). The second-named author is partially supported by National Natural Science Foundation of China (Grant No. 11801516).
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