zbMATH — the first resource for mathematics

Unfolding CR singularities. (English) Zbl 1194.32016
Mem. Am. Math. Soc. 962, vii, 90 p. (2010).
This monograph treats the local equivalence of real submanifolds of complex manifolds under biholomorphic transformations. The real dimension $$m$$ of the submanifold $$M$$ under consideration is assumed to be at most equal to the complex dimension $$n$$ of the ambient complex manifold. The simplest example of this situation are surfaces in $$\mathbb{C}^2$$ having at some point a complex tangent, which was studied in [J. K. Moser and S. M. Webster, Acta Math. 150, 255–296 (1983; Zbl 0519.32015)]. The points of $$M$$ to be considered are those carrying CR singularities. The general idea is to find first normal forms, and secondly to investigate how different types of singularities fit into parametrized families of maps (unfoldings). The classification of unfoldings reduces again to normal forms for defining expressions under an appropriate group of transformations.
The starting point for surfaces $$M$$ in $$\mathbb{C}^2$$ is the quadric normal form of E. Bishop [Duke Math. J. 32, 1–21 (1965; Zbl 0154.08501)], and some subsequent refinements. Deformation brings the issue of the stability of CR singularities and of properties that are stable under perturbations. One approach uses a Grassmann variety construction to define a general position notion and give an expected codimension for the locus of CR singularities.
To study deformations of $$M$$ depending on $$k$$ real parameters, an $$(m+k)$$ dimensional real submanifold $$\hat{M}$$ of $$\mathbb{C}^{n+k}$$, containing $$M$$, is introduced. The classification amounts to find normal forms for $$\hat{M}$$ under a group of holomorphic transformations that keeps into account the difference between the coordinates of the original $$\mathbb{C}^n$$ and the added parameters.
The situation of $$n=m$$ and $$n>m$$ are qualitatively different and treated separately. In §{6} it is proved (Main Theorem) that, when $$M$$ is a real analytic submanifold of $$\mathbb{C}^n$$ of real dimension $$m=\frac{2}{3}(n+1)<n$$, then there is a coordinate transformation making a non trivial unfolding $$\hat{M}$$ of $$M$$ real algebraic.
The proofs involve linear approximation and rapid convergence, to solve a system of nonlinear functional equations.

MSC:
 32S30 Deformations of complex singularities; vanishing cycles 58K35 Catastrophe theory 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 32V40 Real submanifolds in complex manifolds
Citations:
Zbl 0519.32015; Zbl 0154.08501
PovRay; Maple
Full Text:
References:
 [1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. · Zbl 0395.30001 [2] Dynamical systems. VI, Encyclopaedia of Mathematical Sciences, vol. 6, Springer-Verlag, Berlin, 1993. Singularity theory. I; A translation of Current problems in mathematics. Fundamental directions, Vol. 6 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988 [ MR1088738 (91h:58010a)]; Translation by A. Iacob; Translation edited by V. I. Arnol$$^{\prime}$$d. [3] Thomas Banchoff and Frank Farris, Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space, Pacific J. Math. 161 (1993), no. 1, 1-24. · Zbl 0815.57024 [4] M. S. Baouendi, P. Ebenfelt, and Linda Preiss Rothschild, Local geometric properties of real submanifolds in complex space, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 309-336 (electronic). · Zbl 0955.32027 [5] V. K. Beloshapka, The normal form of the germ of a four-dimensional real manifold in $${\mathbf C}^5$$ at an $${\mathbf RC}$$-singular point in general position, Mat. Zametki 61 (1997), no. 6, 931-934 (Russian); English transl., Math. Notes 61 (1997), no. 5-6, 777-779. · Zbl 0917.32015 [6] Gautam Bharali, Surfaces with degenerate CR singularities that are locally polynomially convex, Michigan Math. J. 53 (2005), no. 2, 429-445. · Zbl 1087.32009 [7] Errett Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1-21. · Zbl 0154.08501 [8] James Callahan, Singularities and plane maps. II. Sketching catastrophes, Amer. Math. Monthly 84 (1977), no. 10, 765-803. · Zbl 0389.58006 [9] A. COFFMAN, Enumeration and Normal Forms of Singularities in Cauchy-Riemann Structures, dissertation, University of Chicago, 1997. [10] Adam Coffman, CR singular immersions of complex projective spaces, Beiträge Algebra Geom. 43 (2002), no. 2, 451-477. · Zbl 1029.32020 [11] Adam Coffman, Real congruence of complex matrix pencils and complex projections of real Veronese varieties, Linear Algebra Appl. 370 (2003), 41-83. · Zbl 1049.14042 [12] Adam Coffman, Analytic normal form for CR singular surfaces in $$\mathbb C^3$$, Houston J. Math. 30 (2004), no. 4, 969-996. · Zbl 1074.32013 [13] Adam Coffman, CR singularities of real threefolds in $$\mathbb C^4$$, Adv. Geom. 6 (2006), no. 1, 109-137. · Zbl 1120.32024 [14] Adam Coffman, Analytic stability of the CR cross-cap, Pacific J. Math. 226 (2006), no. 2, 221-258. · Zbl 1123.32018 [15] Andrzej Derdzinski and Tadeusz Januszkiewicz, Immersions of surfaces in Spin$$^c$$-manifolds with a generic positive spinor, Ann. Global Anal. Geom. 26 (2004), no. 2, 175-199. , Erratum: “Immersions of surfaces in $${\mathrm Spin}^c$$-manifolds with a generic positive spinor” [Ann. Global Anal. Geom. 26 (2004), 175-199; MR2071487] by A. Derdzinski and T. Januszkiewicz, Ann. Global Anal. Geom. 26 (2004), no. 3, 319+25. · Zbl 1067.53037 [16] A. V. Domrin, On the number of RC-singular points of a four-dimensional real submanifold in a five-dimensional complex manifold, Mat. Zametki 57 (1995), no. 2, 240-245, 318 (Russian, with Russian summary); English transl., Math. Notes 57 (1995), no. 1-2, 167-170. · Zbl 0853.57028 [17] A. V. Domrin, A description of characteristic classes of real submanifolds in complex manifolds in terms of RC-singularities, Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), no. 5, 19-40 (Russian, with Russian summary); English transl., Izv. Math. 59 (1995), no. 5, 899-918. · Zbl 0877.57011 [18] Franc Forstnerič, Complex tangents of real surfaces in complex surfaces, Duke Math. J. 67 (1992), no. 2, 353-376. · Zbl 0761.53032 [19] Thomas Garrity, Global structures on CR manifolds via Nash blow-ups, Michigan Math. J. 48 (2000), 281-294. Dedicated to William Fulton on the occasion of his 60th birthday. · Zbl 0995.32023 [20] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. · Zbl 0294.58004 [21] Gary A. Harris, Lowest order invariants for real-analytic surfaces in $${\mathbf C}^2$$, Trans. Amer. Math. Soc. 288 (1985), no. 1, 413-422. · Zbl 0574.32029 [22] Gary A. Harris, Real $$k$$-flat surfaces in $${\mathbf C}^2$$, Houston J. Math. 14 (1988), no. 4, 501-506. · Zbl 0694.32011 [23] Reese Harvey and H. Blaine Lawson Jr., Geometric residue theorems, Amer. J. Math. 117 (1995), no. 4, 829-873. · Zbl 0851.58036 [24] Xiao Jun Huang and Steven G. Krantz, On a problem of Moser, Duke Math. J. 78 (1995), no. 1, 213-228. · Zbl 0846.32010 [25] S. IVASHKOVICH and V. SHEVCHISHIN, Complex Curves in Almost-Complex Manifolds and Meromorphic Hulls, arXiv:math.CV/9912046. · Zbl 0930.32017 [26] Hon Fei Lai, Characteristic classes of real manifolds immersed in complex manifolds, Trans. Amer. Math. Soc. 172 (1972), 1-33. · Zbl 0222.32002 [27] Yung Chen Lu, Singularity theory and an introduction to catastrophe theory, Springer-Verlag, New York, 1976. With an introduction by Peter Hilton; Universitext. · Zbl 0354.58008 [28] MAPLE 9.50, WATERLOO MAPLE INC., 2004. WWW.MAPLESOFT.COM [29] Jean Martinet, Singularities of smooth functions and maps, London Mathematical Society Lecture Note Series, vol. 58, Cambridge University Press, Cambridge-New York, 1982. Translated from the French by Carl P. Simon. · Zbl 0522.58006 [30] Jürgen Moser, Analytic surfaces in $${\mathbf C}^2$$ and their local hull of holomorphy, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 397-410. · Zbl 0585.32007 [31] Jürgen K. Moser and Sidney M. Webster, Normal forms for real surfaces in $${\mathbf C}^{2}$$ near complex tangents and hyperbolic surface transformations, Acta Math. 150 (1983), no. 3-4, 255-296. · Zbl 0519.32015 [32] Tim Poston and Ian Stewart, Catastrophe theory and its applications, Pitman, London-San Francisco, Calif.-Melbourne: distributed by Fearon-Pitman Publishers, Inc., Belmont, Calif., 1978. With an appendix by D. R. Olsen, S. R. Carter and A. Rockwood; Surveys and Reference Works in Mathematics, No. 2. · Zbl 0382.58006 [33] POVRAY POV-RAY FOR WINDOWS VERSION 3.6.1C.ICL8.WIN32, PERSISTENCE OF VISION RAYTRACER PTY. LTD., 2006. WWW.POVRAY.ORG [34] Alejandro Sanabria-García, Polynomial hulls of smooth discs: a survey, Irish Math. Soc. Bull. 45 (2000), 135-153. · Zbl 1065.32005 [35] Marko Slapar, Real surfaces in elliptic surfaces, Internat. J. Math. 16 (2005), no. 4, 357-363. · Zbl 1072.32015 [36] C. T. C. Wall, Classification and stability of singularities of smooth maps, Singularity theory (Trieste, 1991) World Sci. Publ., River Edge, NJ, 1995, pp. 920-952. · Zbl 0991.58507 [37] S. M. Webster, Real submanifolds of $${\mathbf C}^n$$ and their complexifications, Topics in several complex variables (Mexico, 1983) Res. Notes in Math., vol. 112, Pitman, Boston, MA, 1985, pp. 69-79. [38] S. M. Webster, The Euler and Pontrjagin numbers of an $$n$$-manifold in $${\mathbf C}^n$$, Comment. Math. Helv. 60 (1985), no. 2, 193-216. · Zbl 0566.32015 [39] S. M. Webster, On the relation between Chern and Pontrjagin numbers, Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984) Contemp. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1986, pp. 135-143. [40] R. O. Wells Jr., Compact real submanifolds of a complex manifold with nondegenerate holomorphic tangent bundles, Math. Ann. 179 (1969), 123-129. · Zbl 0167.21604 [41] Hassler Whitney, The general type of singularity of a set of $$2n-1$$ smooth functions of $$n$$ variables, Duke Math. J. 10 (1943), 161-172. · Zbl 0061.37207 [42] Hassler Whitney, The singularities of a smooth $$n$$-manifold in $$(2n-1)$$-space, Ann. of Math. (2) 45 (1944), 247-293. · Zbl 0063.08238
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.