Extension Phenomena for Holomorphic Geometric Structures
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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The most commonly encountered types of complex analytic $G$-structures and Cartan geometries cannot have singularities of complex codimension 2 or more.
Keywords: Hartogs extension; Cartan geometry; parabolic geometry; $G$-structure.
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Benjamin McKay. Extension Phenomena for Holomorphic Geometric Structures. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a57/

[1] Adachi K., Suzuki M., Yoshida M., “Continuation of holomorphic mappings, with values in a complex Lie group”, Pacific J. Math., 47 (1973), 1–4 | MR | Zbl

[2] Aeppli A., “On the cohomology structure of Stein manifolds”, Proc. Conf. Complex Analysis (Minneapolis, Minn., 1964), Springer, Berlin, 1965, 58–70 | MR

[3] Alekseevsky D. V., Michor P. W., “Differential geometry of Cartan connections”, Publ. Math. Debrecen, 47 (1995), 349–375 ; math.DG/9412232 | MR | Zbl

[4] Atiyah M. F., “Complex analytic connections in fibre bundles”, Trans. Amer. Math. Soc., 85 (1957), 181–207 | DOI | MR | Zbl

[5] Beloshapka V., Ezhov V., Schmalz G., “Canonical Cartan connection and holomorphic invariants on Engel CR manifolds”, Russ. J. Math. Phys., 14 (2007), 121–133 ; math.CV/0508084 | DOI | MR | Zbl

[6] Borel A., “Élie Cartan, Hermann Weyl et les connexions projectives”, Essays on Geometry and Related Topics, Vols. 1, 2, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001, 43–58 | MR | Zbl

[7] Bryant R. L., “Metrics with exceptional holonomy”, Ann. of Math. (2), 126 (1987), 525–576 | DOI | MR | Zbl

[8] Bryant R. L., “An introduction to Lie groups and symplectic geometry”, Geometry and Quantum Field Theory (Park City, UT, 1991), IAS/Park City Math. Ser., 1, Amer. Math. Soc., Providence, RI, 1995, 5–181 | MR | Zbl

[9] Bryant R. L., Griffiths P. A., Hsu L., “Toward a geometry of differential equations”, Geometry, Topology, Physics, Conf. Proc. Lecture Notes Geom. Topology, 4, Internat. Press, Cambridge, MA, 1995, 1–76 | MR | Zbl

[10] Buchdahl N. P., Harris A., “Holomorphic connections and extension of complex vector bundles”, Math. Nachr., 204 (1999), 29–39 | MR | Zbl

[11] Čap A., “Two constructions with parabolic geometries”, Rend. Circ. Mat. Palermo (2) Suppl., 79 (2006), 11–37 ; math.DG0504389 | MR | Zbl

[12] Čap A., Schichl H., “Parabolic geometries and canonical Cartan connections”, Hokkaido Math. J., 29 (2000), 453–505 | MR | Zbl

[13] Clelland J. N., “Geometry of conservation laws for a class of parabolic partial differential equations”, Selecta Math. (N.S.), 3 (1997), 1–77 | DOI | MR | Zbl

[14] Doubrov B., “Generalized Wilcynski invariants for nonlinear ordinary differential equations”, Symmetries and Overdetermined Systems of Partial Differential Equations (July 17 – August 4, 2006, Minneapolis, MN, USA), IMA Vol. Math. Appl., 144, Springer, New York, NY, 2008, 25–40 ; math.DG/0702251 | MR | Zbl

[15] Dunajski M., Tod P., “Paraconformal geometry of $n$th-order ODEs, and exotic holonomy in dimension four”, J. Geom. Phys., 56 (2006), 1790–1809 ; math.DG/0502524 | DOI | MR | Zbl

[16] Ehlers K., Koiller J., Montgomery R., Rios P. M., “Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization”, The Breadth of Symplectic and Poisson geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 75–120 ; math-ph/0408005 | MR | Zbl

[17] Fox D. J. F., “Contact projective structures”, Indiana Univ. Math. J., 54 (2005), 1547–1598 ; math.DG/0402332 | DOI | MR | Zbl

[18] Gallo D., Kapovich M., Marden A., “The monodromy groups of Schwarzian equations on closed Riemann surfaces”, Ann. of Math. (2), 151 (2000), 625–704 ; math.CV/9511213 | DOI | MR | Zbl

[19] Gardner R. B., The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989 | MR | Zbl

[20] Godliński M., Nurowski P., $GL(2,R)$ geometry of ODEs, arXiv:0710.0297

[21] Gunning R. C., On uniformization of complex manifolds: the role of connections, Mathematical Notes, 22, Princeton University Press, Princeton, N.J., 1978 | MR

[22] Hartogs F., “Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten”, Math. Ann., 62:1 (1906), 1–88 | DOI | MR | Zbl

[23] Hong J., “Fano manifolds with geometric structures modeled after homogeneous contact manifolds”, Internat. J. Math., 11 (2000), 1203–1230 | MR | Zbl

[24] Hörmander L., An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, 7, North-Holland Publishing Co., Amsterdam, 1990 | MR | Zbl

[25] Huckleberry A. T., “The classification of homogeneous surfaces”, Exposition. Math., 4 (1986), 289–334 | MR | Zbl

[26] Hwang J.-M., Mok N., “Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation”, Invent. Math., 131 (1998), 393–418 ; math.AG/9604227 | DOI | MR | Zbl

[27] Hwang J.-M., Mok N., “Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation”, Invent. Math., 160 (2005), 591–645 | DOI | MR | Zbl

[28] Ivashkovich S. M., “Extension of locally biholomorphic mappings to a product of complex manifolds”, Izv. Akad. Nauk SSSR Ser. Mat., 49 (1985), 884–890 (in Russian) | MR | Zbl

[29] Ivashkovich S. M., “The Hartogs phenomenon for holomorphically convex Kähler manifolds”, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 866–873 (in Russian) | MR

[30] Ivashkovich S. M., “The Hartogs-type extension theorem for meromorphic maps into compact Kähler manifolds”, Invent. Math., 109 (1992), 47–54 | DOI | MR | Zbl

[31] Ivashkovich S. M., Extra extension properties of equidimensional holomorphic mappings: results and open questions, arXiv:0810.4588 | MR

[32] Ivey T. A., Landsberg J. M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, 61, American Mathematical Society, Providence, RI, 2003 | MR | Zbl

[33] Kajiwara J., Sakai E., “Generalization of Levi–Oka's theorem concerning meromorphic functions”, Nagoya Math. J., 29 (1967), 75–84 | MR | Zbl

[34] Kazarian M., Montgomery R., Shapiro B., “Characteristic classes for the degenerations of two-plane fields in four dimensions”, Pacific J. Math., 179 (1997), 355–370 ; dg-ga/9704001 | DOI | MR | Zbl

[35] Knapp A. W., Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002 | MR | Zbl

[36] Krantz S. G., “The Hartogs extension phenomenon redux”, Complex Var. Elliptic Equ., 53 (2008), 343–353 | DOI | MR | Zbl

[37] Landsberg J. M., Secant varieties, shadows and the universal Lie algebra: perspectives on the geometry of rational homogeneous varieties, From 2004 lectures at Harvard University, to appear

[38] Matsushima Y., Morimoto A., “Sur certains espaces fibrés holomorphes sur une variété de Stein”, Bull. Soc. Math. France, 88 (1960), 137–155 | MR | Zbl

[39] McKay B., Rational curves and parabolic geometries, math.DG/0603276

[40] McKay B., Complete Cartan connections, arxiv:0802.1473

[41] McKay B., Holomorphic parabolic geometries and Calabi–Yau manifolds, November 2008, unpublished.

[42] Merker J., Porten E., “A Morse-theoretical proof of the Hartogs extension theorem”, J. Geom. Anal., 17 (2007), 513–546 ; math.CV/0610985 | MR | Zbl

[43] Mok N., “On Fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents”, Trans. Amer. Math. Soc., 354 (2002), 2639–2658 | DOI | MR | Zbl

[44] Molzon R., Mortensen K. P., “The Schwarzian derivative for maps between manifolds with complex projective connections”, Trans. Amer. Math. Soc., 348 (1996), 3015–3036 | DOI | MR | Zbl

[45] Mostow G. D., “The extensibility of local Lie groups of transformations and groups on surfaces”, Ann. of Math. (2), 52 (1950), 606–636 | DOI | MR | Zbl

[46] Ochiai T., “Geometry associated with semisimple flat homogeneous spaces”, Trans. Amer. Math. Soc., 152 (1970), 159–193 | DOI | MR | Zbl

[47] Okonek C., Schneider M., Spindler H., Vector bundles on complex projective spaces, Progress in Mathematics, 3, Birkhäuser, Boston, Mass., 1980 | MR | Zbl

[48] Olver P. J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[49] Procesi C., Lie groups. An approach through invariants and representations, Universitext, Springer, New York, 2007 | MR | Zbl

[50] Remmert R., “Holomorphe und meromorphe Abbildungen komplexer Räume”, Math. Ann., 133 (1957), 328–370 | DOI | MR | Zbl

[51] Sato H., Yoshikawa A. Y., “Third order ordinary differential equations and Legendre connections”, J. Math. Soc. Japan, 50 (1998), 993–1013 | DOI | MR | Zbl

[52] Serre J.-P., “Prolongement de faisceaux analytiques cohérents”, Ann. Inst. Fourier (Grenoble), 16 (1966), 363–374 | MR | Zbl

[53] Serre J.-P., Complex semisimple Lie algebras, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001 | MR

[54] Sharpe R. W., Differential geometry. Cartan's generalization of Klein's Erlangen program, With a foreword by S. S. Chern, Graduate Texts in Mathematics, 166, Springer-Verlag, New York, 1997 | MR | Zbl

[55] Sharpe R. W., “An introduction to Cartan geometries”, Proceedings of the 21st Winter School “Geometry and Physics” (Srní, 2001), Rend. Circ. Mat. Palermo (2) Suppl., 69, 2002, 61–75 | MR | Zbl

[56] Shevchishin V. V., “The Thullen type extension theorem for holomorphic vector bundles with $L^2$-bounds on curvature”, Math. Ann., 305 (1996), 461–491 | DOI | MR | Zbl

[57] Siu Y. T., Techniques of extension of analytic objects, Lecture Notes in Pure and Applied Mathematics, 8, Marcel Dekker, Inc., New York, 1974 | MR | Zbl

[58] Sternberg S., Lectures on differential geometry, 2nd ed., Chelsea Publishing Co., New York, 1983 | MR | Zbl

[59] Vogel T., “Existence of Engel structures”, Ann. of Math. (2), 169 (2009), 79–137 ; math.GT/0411217 | DOI | MR | Zbl

[60] Wang H.-C., “Closed manifolds with homogeneous complex structure”, Amer. J. Math., 76 (1954), 1–32 | DOI | MR | Zbl

[61] Wehler J., “Versal deformations for Hopf surfaces”, J. Reine Ang. Math., 328 (1981), 22–32 | MR | Zbl

[62] Funct. Anal. Appl., 24:2 (1990), 150–152 | DOI | MR | Zbl