Geodesic
Orthogonal complex structures in $\mathbb{R}^4$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 73 (2018) no. 1, pp. 91-159
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Orthogonal complex structures in domains in $\mathbb{R}^4$ are studied using methods of multidimensional complex analysis. New results on removable singularities of such structures are established. The simplest multivalued orthogonal complex structures are investigated. A classification of quadrics in $\mathbb{CP}_3$ with respect to the action of the conformal group is given, and the discriminant sets of the twistor projections of model quadrics are described. Bibliography: 39 titles.
Keywords: twistor bundles, removable singularities
Mots-clés : complex structures, conformal maps, discriminant sets. 
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E. M. Chirka. Orthogonal complex structures in $\mathbb{R}^4$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 73 (2018) no. 1, pp. 91-159. http://geodesic.mathdoc.fr/item/RM_2018_73_1_a2/

[1] L. V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, 10, D. Van Nostrand Co., Inc., Toronto, ON–New York–London, 1966, v+146 pp. | MR | MR | Zbl | Zbl

[2] J. Armstrong, M. Povero, S. Salamon, “Twistor lines on cubic surfaces”, Rend. Semin. Mat. Univ. Politec. Torino, 71:3-4 (2013), 317–338 | MR | Zbl

[3] J. Armstrong, S. Salamon, “Twistor topology of the Fermat cubic”, SIGMA, 10 (2014), 061, 12 pp. | DOI | MR | Zbl

[4] M. F. Atiyah, Geometry of Yang–Mills fields, Scuola Norm. Super. Pisa, Pisa, 1979, 99 pp. | MR | Zbl

[5] M. F. Atiyah, N. J. Hitchin, I. M. Singer, “Self-duality in four-dimensional Riemannian geometry”, Proc. Roy. Soc. London Ser. A, 362:1711 (1978), 425–461 | DOI | MR | Zbl

[6] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. (3), 10, Springer-Verlag, Berlin, 1987, xii+510 pp. | DOI | MR | MR | Zbl | Zbl

[7] E. Bishop, “Conditions for the analyticity of certain sets”, Michigan Math. J., 11 (1964), 289–304 | DOI | MR | Zbl

[8] L. Borisov, S. Salamon, J. Viaclovsky, “Twistor geometry and warped product orthogonal complex structures”, Duke Math. J., 156:1 (2011), 125–166 | DOI | MR | Zbl

[9] D. Burns, P. de Bartolomeis, “Applications harmoniques stables dans $\mathbf{P}^n$”, Ann. Sci. École Norm. Sup. (4), 21:2 (1988), 159–177 | DOI | MR | Zbl

[10] E. M. Chirka, Complex analytic sets, Math. Appl. (Soviet Ser.), 46, Kluwer Acad. Publ., Dordrecht, 1989, xx+372 pp. | DOI | MR | MR | Zbl | Zbl

[11] E. M. Chirka, “Holomorphic motions and uniformization of holomorphic families of Riemann surfaces”, Russian Math. Surveys, 67:6 (2012), 1091–1165 | DOI | DOI | MR | Zbl

[12] E. M. Chirka, “On the $\bar\partial$-problem with $L^2$-estimates on a Riemann surface”, Proc. Steklov Inst. Math., 290:1 (2015), 264–276 | DOI | DOI | MR | Zbl

[13] E. M. Chirka, “Removable singularities of holomorphic functions”, Sb. Math., 207:9 (2016), 1335–1343 | DOI | DOI | MR | Zbl

[14] E. M. Chirka, “On removable singularities of complex analytic sets”, Sb. Math., 208:7 (2017), 1073–1086 | DOI | DOI | MR

[15] W.-L. Chow, “On compact complex analytic varieties”, Amer. J. Math., 71:4 (1949), 893–914 | DOI | MR | Zbl

[16] J. Garnett, Analytic capacity and measure, Lecture Notes in Math., 297, Springer-Verlag, Berlin–New York, 1972, iv+138 pp. | DOI | MR | Zbl

[17] G. Gentili, S. Salamon, C. Stoppato, “Twistor transforms of quaternionic functions and orthogonal complex structures”, J. Eur. Math. Soc. (JEMS), 16:11 (2014), 2323–2353 | DOI | MR | Zbl

[18] R. Harvey, “Holomorphic chains and their boundaries”, Several complex variables (Williams College, Williamstown, MA, 1975), Proc. Sympos. Pure Math., 30, part 1, Amer. Math. Soc., Providence, RI, 1977, 309–382 | MR | Zbl

[19] F. R. Harvey, H. B. Lawson, Jr., “On boundaries of complex analytic varieties. I”, Ann. of Math. (2), 102:2 (1975), 223–290 | DOI | MR | Zbl

[20] L. Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, NJ–Toronto, ON–London, 1966, x+208 pp. | MR | MR | Zbl | Zbl

[21] N. H. Kuiper, “On conformally-flat manifolds in the large”, Ann. of Math. (2), 50:4 (1949), 916–924 | DOI | MR | Zbl

[22] C. LeBrun, “Orthogonal complex structures on $S^6$”, Proc. Amer. Math. Soc., 101:1 (1987), 136–138 | DOI | MR | Zbl

[23] A. Nijenhuis, W. B. Woolf, “Some integration problems in almost-complex and complex manifolds”, Ann. of Math. (2), 77:3 (1963), 424–489 | DOI | MR | Zbl

[24] W. F. Osgood, Lehrbuch der Funktionen Theorie, v. II, Chelsea Publishing Co., New York, 1965, xiv+686 pp. | MR | Zbl

[25] M. Pontecorvo, “Uniformization of conformally flat Hermitian surfaces”, Differential Geom. Appl., 2:3 (1992), 295–305 | DOI | MR | Zbl

[26] M. Pontecorvo, “On twistor spaces of anti-self-dual Hermitian surfaces”, Trans. Amer. Math. Soc., 331:2 (1992), 653–661 | DOI | MR | Zbl

[27] R. Remmert, K. Stein, “Über die wesentlichen Singularitäten analytischer Mengen”, Math. Ann., 126 (1953), 263–306 | DOI | MR | Zbl

[28] Yu. G. Reshetnyak, “Liouville's theorem on conformal mappings for minimal regularity assumptions”, Sib. Math. J., 8:4 (1967), 631–634 | DOI | MR | Zbl

[29] Yu. G. Reshetnyak, Space mappings with bounded distortion, Transl. Math. Monogr., 73, Amer. Math. Soc., Providence, RI, 1989, xvi+362 pp. | MR | MR | Zbl | Zbl

[30] H. Rossi, “Continuation of subvarieties of projective varieties”, Amer. J. Math., 91:2 (1969), 565–575 | DOI | MR | Zbl

[31] W. Rothstein, “Zur Theorie der analytischen Mengen”, Math. Ann., 174 (1967), 8–32 | DOI | MR | Zbl

[32] S. Salamon, J. Viaclovsky, “Orthogonal complex structures on domains in $\mathbb{R}^4$”, Math. Ann., 343 (2009), 853–899 | DOI | MR | Zbl

[33] B. V. Shabat, Introduction to complex analysis. Part II. Functions of several variables, Transl. Math. Monogr., 110, Amer. Math. Soc., Providence, RI, 1992, x+371 pp. | MR | MR | Zbl | Zbl

[34] B. Shiffman, “On the removal of singularities of analytic sets”, Michigan Math. J., 15 (1968), 111–120 | DOI | MR | Zbl

[35] M. Spivak, A comprehensive introduction to differential geometry, v. 4, 2nd ed., Publish or Perish, Inc., Wilmington, DE, 1979, viii+561 pp. | MR | Zbl

[36] N. Steenrod, The topology of fibre bundles, Princeton Univ. Series, 4, Princeton Univ. Press, Princeton, NJ, 1951, viii+224 pp. | MR | Zbl

[37] E. L. Stout, Polynomial convexity, Progr. Math., 261, Birkhäuser Boston, Inc., Boston, MA, 2007, xii+439 pp. | MR | Zbl

[38] J. C. Wood, “Harmonic morphisms and Hermitian structures on Einstein 4-manifolds”, Internat. J. Math., 3:3 (1992), 415–439 | DOI | MR | Zbl

[39] J. C. Wood, “Harmonic maps and harmonic morphisms”, Differentsialnaya geometriya, gruppy Li i mekhanika. 15–1, Zap. nauch. sem. POMI, 234, POMI, SPb., 1996, 190–200 ; J. Math. Sci. (N. Y.), 94:2 (1999), 1263–1269 | MR | Zbl | DOI