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HOLOMORPHIC FUNCTIONS ON ALMOST COMPLEX MANIFOLDS

  • Han, Chong-Kyu ;
  • Kim, Hye-Seon
  • Received : 2011.02.08
  • Published : 2012.03.01

Abstract

Given an almost complex structure ($\mathbb^{C}^m$, J), $m\geq2$, that is defined by setting $\theta^{\alpha}=dz^{\alpha}+a_{\beta}^{\alpha}d\bar{z}^{\beta}$, ${\alpha}=1,\ldots$,m, to be (1, 0)-forms, we find conditions on ($a_{\beta}^{\alpha}$) for the existence of holomorphic functions an classify the almost complex structures by type ($\nu$,q). Then we determine types for several examples in $\mathbb{C}^2$ and $\mathbb{C}^3$ including the natural almost complex structure on $S^6$.

Keywords

almost complex manifolds;J-holomorphic functions;Nijenhuis tensor;Newlander-Nirenberg theorem

References

  1. H. Ahn and C. K. Han, Local geometry of Levi-forms associated with the existence of complex manifolds and the minimality of generic CR manifolds, J. Geom. Anal., to appear.
  2. S. Berhanu, P. Cordaro, and J. Hounie, An Introduction to Involutive Structures, Cambridge U. Press, 2008.
  3. R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior Differential Systems, Springer-Verlag, New York, 1991.
  4. E. Cartan, Les systemes differentiels exterieurs et leurs applications geometriques, Herman, Paris, 1945.
  5. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, John Wiley and Sons, New York, 1962.
  6. A. Frolicher, Zur Differentialgeometrie der komplexen Strukturen, Math. Ann. 129 (1955), 50-95. https://doi.org/10.1007/BF01362360
  7. T. Fukami and S. Ishihara, Almost Hermitian structure on $S^{6}$, Tohoku Math. J. (2) 7 (1955), 151-156.
  8. R. B. Gardner, Invariants of Pfaffian systems, Trans. Amer. Math. Soc. 126 (1967), 514-533. https://doi.org/10.1090/S0002-9947-1967-0211352-5
  9. C. K. Han and K. H. Lee, Integrable submanifolds in almost complex manifolds, J. Geom. Anal. 20 (2010), no. 1, 177-192. https://doi.org/10.1007/s12220-009-9099-2
  10. H. Kim and K. H. Lee, Complete prolongation for innitesimal automorphisms on almost complex manifolds, Math. Z. 264 (2010), no. 4, 913-925. https://doi.org/10.1007/s00209-009-0496-9
  11. B. S. Krugilov, Nijenhuis tensors and obstructions to constructing pseudoholomorphic mappings, Mathematical Notes 63 (1998), 476-493. https://doi.org/10.1007/BF02311251
  12. O. Muskarov, Almost complex manifolds without almost holomorphic functions, C. R. Acad. Bulgare Sci. 34 (1981), no. 9, 1225-1228.
  13. O. Muskarov, Existence of holomorphic functions on almost complex manifolds, Math. Z. 192 (1986), no. 2, 283-295. https://doi.org/10.1007/BF01179429
  14. A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391-404. https://doi.org/10.2307/1970051
  15. F. Treves, Approximation and representation of functions and distributions annihilated by a system of complex vector fields, Ecole Polytechnique, Centre de Mathematiques, Palaiseau, 1981.

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  1. Partial integrability of almost complex structures and the existence of solutions for quasilinear Cauchy–Riemann equations vol.265, pp.1, 2013, https://doi.org/10.2140/pjm.2013.265.59
  2. Invariant submanifolds for systems of vector fields of constant rank vol.59, pp.7, 2016, https://doi.org/10.1007/s11425-016-5139-0