Abstract
The Taubes current is introduced by Taubes in (J Symplectic Geom 9:161–250, 2011) as an intermediate step to construct almost Kähler forms. In this paper, we prove that an almost complex structures being almost Kähler structure in dimension four is the equivalent of the existence of Taubes currents. Precisely, we show that Taubes currents could be regularized to almost Kähler 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 Kähler. An application to Donaldson’s “tamed to compatible” question is provided.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Unless otherwise stated, all the manifolds are assumed to be closed, smooth and oriented.
References
Bott, R., Tu, L.W.: Differential forms in algebraic topology. Graduate texts in mathematics, 82. Springer, New York-Berlin, pp xiv+331 (1982)
Donaldson, S.K.: Two-forms on four-manifolds and elliptic equations. Inspired by S. S. Chern, 153–172, Nankai Tracts Math., 11, World Sci. Publ., Hackensack (2006)
Draghici, T., Li, T.J., Zhang, W.: Symplectic forms and cohomology decomposition of almost complex 4-manifolds. Int. Math. Res. Not. IMRN 1, 1–17 (2012)
Gromov, M.: Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985)
Li, T.J., Zhang, W.: Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds. Comm. Anal. Geom. 17(4), 651–683 (2009)
Li, T.J., Zhang, W.: Almost Kähler forms on rational \(4\)-manifolds. arXiv:1210.2377 (2012)
Peternell, T.: Algebraicity criteria for compact complex manifolds. Math. Ann. 275(4), 653–672 (1986)
Taubes, C.H.: Tamed to compatible: symplectic forms via moduli space integration. J. Symplectic Geom. 9, 161–250 (2011)
Tosatti, V., Weinkove, B.: The Calabi-Yau equation, symplectic forms and almost complex structures. In: Geometry and analysis, Vol. I, pp. 475–493, Adv. Lect. Math. (ALM) 17, International Press, Somerville (2011)
Acknowledgments
The author would like to express the deepest gratitude to his advisor Tian-Jun Li for his constant encouragement, collaborations and especially his interest in the current work. He appreciates the collaboration with Daniel Angella and Adriano Tomassini. He is in debt to Tedi Draghici for his interest, collaborations, and helpful discussions. He also would like to thank Clifford Taubes for discussions and encouragement. Last but not least, the author is grateful to Valentino Tosatti for his careful reading of an early draft and many helpful comments. He thanks the referee for comments. This work is partially supported by the AMS-Simons Travel Grants.