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CR singularities of real fourfolds in $$\mathbb C^{3}$$. (English) Zbl 1233.32027
The author studies embedded real $$4$$-manifolds into complex $$3$$-space. Points where the tangent space of the $$4$$-manifold is a complex $$2$$-plane are called CR-singular. The main result of the paper is the classification of the quadratic part of the CR-singularities up to local holomorphic coordinate changes. A table of normal forms is presented. It is also shown how the quadratic part determines the intersection index, an enumerative invariant occurring in global formulas.

##### MSC:
 32V40 Real submanifolds in complex manifolds 15A21 Canonical forms, reductions, classification 32S05 Local complex singularities 32S20 Global theory of complex singularities; cohomological properties
##### Keywords:
CR singularities; 4-manifolds; normal form
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##### References:
 [1] B. Aebischer, M. Borer, M. Kälin, C. Leuenberger and H. Reimann, Symplectic geometry , PiM, vol. 124, Birkhäuser, Basel, 1994. · Zbl 0932.53002 [2] T. Banchoff and F. Farris, Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space , Pacific J. Math. 161 (1993), 1–24. · Zbl 0815.57024 [3] E. Bishop, Differentiable manifolds in complex Euclidean space , Duke Math. J. 32 (1965), 1–21. · Zbl 0154.08501 [4] C. Bohr, Immersions of surfaces in almost-complex 4-manifolds , Proc. Amer. Math. Soc. 130 (2002), 1523–1532. JSTOR: · Zbl 0992.57011 [5] D. Chakrabarti and R. Shafikov, Holomorphic extension of CR functions from quadratic cones , Math. Ann. 341 (2008), 543–573; and Erratum , Math. Ann. 345 (2009), 491–492. · Zbl 1153.32023 [6] A. Coffman, Enumeration and normal forms of singularities in Cauchy–Riemann structures , dissertation, University of Chicago, 1997. [7] A. Coffman, CR singular immersions of complex projective spaces , Beiträge zur Algebra und Geometrie 43 (2002), 451–477. · Zbl 1029.32020 [8] A. Coffman, Unfolding CR singularities , to appear in Mem. Amer. Math. Soc. 205 (2010). · Zbl 1194.32016 [9] P. Dolbeault, On Levi-flat hypersurfaces with given boundary in $$\co^n$$ , Sci. China Ser. A 51 (2008), 541–552. · Zbl 1151.32013 [10] P. Dolbeault, G. Tomassini and D. Zaitsev, On boundaries of Levi-flat hypersurfaces in $$\co^n$$ , C. R. Math. Acad. Sci. Paris 341 (2005), 343–348. · Zbl 1085.32019 [11] P. Dolbeault, G. Tomassini and D. Zaitsev, On Levi-flat hypersurfaces with prescribed boundary , preprint; available at arXiv :0904.0481. · Zbl 1214.32010 [12] A. V. Domrin, A description of characteristic classes of real submanifolds in complex manifolds via RC-singularities , Izv. Ross. Akad. Nauk Ser. Mat. (English transl.) 59 (1995), 899–918. · Zbl 0877.57011 [13] Ju. B. Ermolaev, The simultaneous reduction of symmetric and Hermitian forms , Izv. Vysš. Učebn. Zaved. Matematika 21 (1961), 10–23. [14] F. Forstnerič, Complex tangents of real surfaces in complex surfaces , Duke Math. J. 67 (1992), 353–376. · Zbl 0761.53032 [15] T. Garrity, Global structures on CR manifolds via Nash blow-ups , W. Fulton Birthday Volume, Michigan Math. J. 48 (2000), 281–294. · Zbl 0995.32023 [16] X. Gong, Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics , Indiana Univ. Math. J. 53 (2004), 83–95. · Zbl 1060.32020 [17] P. Griffiths and J. Harris, Principles of algebraic geometry , Wiley, New York, 1978. · Zbl 0408.14001 [18] F. R. Harvey and H. B. Lawson, Jr., A theory of characteristic currents associated with a singular connection , Astérisque 213 (1993). · Zbl 0804.53037 [19] F. R. Harvey and H. B. Lawson, Jr., Geometric residue theorems , Amer. J. Math. 117 (1995), 829–873. JSTOR: · Zbl 0851.58036 [20] R. Herbert, Multiple points of immersed manifolds , Mem. Amer. Math. Soc. 34 (1981). [21] M. Hirsch, Immersions of manifolds , Trans. Amer. Math. Soc. 93 (1959), 242–276. JSTOR: · Zbl 0113.17202 [22] R. Horn and C. Johnson, Matrix analysis , Cambridge University Press, Cambridge, 1985. · Zbl 0576.15001 [23] R. Horn and V. Sergeichuk, Canonical forms for complex matrix congruence and $$^*$$congruence , Linear Algebra Appl. 416 (2006), 1010–1032. · Zbl 1098.15004 [24] L.-K. Hua, On the theory of automorphic functions of a matrix variable I—geometrical basis , Amer. J. Math. 66 (1944), 470–488. JSTOR: · Zbl 0063.02919 [25] X. Huang and S. Krantz, On a problem of Moser , Duke Math. J. 78 (1995), 213–228. · Zbl 0846.32010 [26] X. Huang and W. Yin, A codimension two CR singular submanifold that is formally equivalent to a symmetric quadric , Int. Math. Res. Notices 2009 (2009), 2789–2828. · Zbl 1182.32014 [27] G. E. Izotov, Simultaneous reduction of a quadratic and a Hermitian form , Izv. Vysš. Učebn. Zaved. Matematika 1957 (1957), 143–159. · Zbl 0091.01903 [28] H. Jacobowitz and P. Landweber, Manifolds admitting generic immersions into $$\co^N$$ , Asian J. Math. 11 (2007), 151–165. · Zbl 1132.57027 [29] H.-F. Lai, Characteristic classes of real manifolds immersed in complex manifolds , Trans. Amer. Math. Soc. 172 (1972), 1–33. · Zbl 0247.32005 [30] P. Lancaster and L. Rodman, Canonical forms for Hermitian matrix pairs under strict equivalence and congruence , SIAM Review 47 (2005), 407–443. · Zbl 1087.15014 [31] B. H. Li and F. Peterson, Immersions of $$n$$-manifolds into $$(2n-2)$$-manifolds , Proc. Amer. Math. Soc. 97 (1986), 531–538. JSTOR: · Zbl 0606.57017 [32] Maple 12, Waterloo Maple Inc., 2008., www.maplesoft.com. · Zbl 1160.68689 [33] J. Milnor, On the immersion of $$n$$-manifolds in $$(n+1)$$-space , Comment. Math. Helv. 30 (1956), 275–284. · Zbl 0070.40202 [34] J. Moser and S. Webster, Normal forms for real surfaces in $$\co^2$$ near complex tangents and hyperbolic surface transformations , Acta Math. 150 (1983), 255–296. · Zbl 0519.32015 [35] E. Thomas, Submersions and immersions with codimension one or two , Proc. Amer. Math. Soc. 19 (1968), 859–863. JSTOR: · Zbl 0169.26102 [36] J. Thorpe, Elementary topics in differential geometry , UTM, Springer, New York, 1979. · Zbl 0404.53001 [37] S. Webster, On the relation between Chern and Pontrjagin numbers , Contemp. Math. 49 (1986), 135–143. · Zbl 0587.57010 [38] R. Wells, Jr., Compact real submanifolds of a complex manifold with nondegenerate holomorphic tangent bundles , Math. Ann. 179 (1969), 123–129. · Zbl 0167.21604
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