From Taubes currents to almost K?hler forms

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• 作者：Weiyi Zhang (1)
  
• 关键词：53D35 ; 58A25 ; 53C15
• 刊名：Mathematische Annalen
• 出版年：2013
• 出版时间：July 2013
• 年：2013
• 卷：356
• 期：3
• 页码：969-978
• 全文大小：170KB
• 参考文献：1. Bott, R., Tu, L.W.: Differential forms in algebraic topology. Graduate texts in mathematics, 82. Springer, New York-Berlin, pp xiv+331 (1982)
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  3. Draghici, T., Li, T.J., Zhang, W.: Symplectic forms and cohomology decomposition of almost complex 4-manifolds. Int. Math. Res. Not. IMRN 1, 1-7 (2012)
  4. Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307-47 (1985) CrossRef
  5. Li, T.J., Zhang, W.: Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds. Comm. Anal. Geom. 17(4), 651-83 (2009)
  6. Li, T.J., Zhang, W.: Almost K?hler forms on rational $4$ -manifolds. arXiv:1210.2377 (2012)
  7. Peternell, T.: Algebraicity criteria for compact complex manifolds. Math. Ann. 275(4), 653-72 (1986) CrossRef
  8. Taubes, C.H.: Tamed to compatible: symplectic forms via moduli space integration. J. Symplectic Geom. 9, 161-50 (2011)
  9. Tosatti, V., Weinkove, B.: The Calabi-Yau equation, symplectic forms and almost complex structures. In: Geometry and analysis, Vol. I, pp. 475-93, Adv. Lect. Math. (ALM) 17, International Press, Somerville (2011)
  
• 作者单位：Weiyi Zhang (1)
  
  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA
  
• ISSN：1432-1807

文摘

The Taubes current is introduced by Taubes in (J Symplectic Geom 9:161-50, 2011) as an intermediate step to construct almost K?hler forms. In this paper, we prove that an almost complex structures being almost K?hler structure in dimension four is the equivalent of the existence of Taubes currents. Precisely, we show that Taubes currents could be regularized to almost K?hler forms up to small perturbations of cohomology classes on any 4-dimensional almost complex manifold $(M, J)$ . A similar result is established for higher dimensions under the assumption of almost K?hler. An application to Donaldson’s “tamed to compatible-question is provided.