2018, 25: 87-95. doi: 10.3934/era.2018.25.009

On the embeddings of the Riemann sphere with nonnegative normal bundles

R. Pantilie, Institutul de Matematicǎ "Simion Stoilow" al Academiei Române, C.P. 1-764, 014700, Bucureşti, România

Received  May 30, 2018 Revised  November 23, 2018 Published  January 2019

Fund Project: This work is supported by a grant of the Ministery of Research and Innovation CNCS-UEFISCDI, project no. PN-Ⅲ-P4-ID-PCE-2016-0019, within PNCDI Ⅲ.

We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere with nonnegative normal bundles.

Citation: Radu Pantilie. On the embeddings of the Riemann sphere with nonnegative normal bundles. Electronic Research Announcements, 2018, 25: 87-95. doi: 10.3934/era.2018.25.009
References:
[1]

M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85 (1957), 181-207.  doi: 10.1090/S0002-9947-1957-0086359-5.  Google Scholar

[2]

M. F. AtiyahN. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362 (1978), 425-461.  doi: 10.1098/rspa.1978.0143.  Google Scholar

[3]

T. N. Bailey and M. G. Eastwood, Complex paraconformal manifolds – their differential geometry and twistor theory, Forum Math., 3 (1991), 61-103.  doi: 10.1515/form.1991.3.61.  Google Scholar

[4]

E. Bonan, Sur les $G$-structures de type quaternionien, Cahiers Topologie Géom. Différentielle, 9 (1967), 389-461.   Google Scholar

[5]

R. L. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory, in Complex Geometry and Lie Theory (Sundance, UT, 1989), Proc. Sympos. Pure Math., 53, Amer. Math. Soc., Providence, RI, 1991, 33–88. doi: 10.1090/pspum/053/1141197.  Google Scholar

[6]

J. B. Carrell, A remark on the Grothendieck residue map, Proc. Amer. Math. Soc., 70 (1978), 43-48.  doi: 10.1090/S0002-9939-1978-0492408-1.  Google Scholar

[7]

Q.-S. Chi and L. J. Schwachhöfer, Exotic holonomy on moduli spaces of rational curves, Differential Geom. Appl., 8 (1998), 105-134.  doi: 10.1016/S0926-2245(97)00019-3.  Google Scholar

[8]

W. Fulton and J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9.  Google Scholar

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H. Grauert and R. Remmert, Coherent Analytic Sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 265, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69582-7.  Google Scholar

[10]

M. Gualtieri, Generalized complex geometry, Ann. of Math. (2), 174 (2011), 75–123. doi: 10.4007/annals.2011.174.1.3.  Google Scholar

[11]

N. J. Hitchin, Complex manifolds and Einstein's equations, in Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Math., 970, Springer, Berlin-New York, 1982, 73–99.  Google Scholar

[12]

P. Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, in Projective Varieties With Unexpected Properties, Walter de Gruyter, Berlin, 2005,317–335.  Google Scholar

[13]

D. Joyce, Compact hypercomplex and quaternionic manifolds, J. Differential Geom., 35 (1992), 743-761.  doi: 10.4310/jdg/1214448266.  Google Scholar

[14]

K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. (2), 75 (1962), 146–162. doi: 10.2307/1970424.  Google Scholar

[15]

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Ⅰ, Ⅱ, Ann. of Math. (2), 67 (1958), 328–466. doi: 10.2307/1970009.  Google Scholar

[16]

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Ⅲ. Stability theorems for complex structures, Ann. of Math. (2), 71 (1960), 43–76. doi: 10.2307/1969879.  Google Scholar

[17]

C. R. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math., 118 (1994), 109-132.  doi: 10.1007/BF01231528.  Google Scholar

[18]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147.  doi: 10.1112/blms/27.2.97.  Google Scholar

[19]

S. MarchiafavaL. Ornea and R. Pantilie, Twistor Theory for CR quaternionic manifolds and related structures, Monatsh. Math., 167 (2012), 531-545.  doi: 10.1007/s00605-011-0326-0.  Google Scholar

[20]

S. Marchiafava and R. Pantilie, Twistor theory for co-CR quaternionic manifolds and related structures, Israel J. Math., 195 (2013), 347-371.  doi: 10.1007/s11856-013-0001-3.  Google Scholar

[21]

S. Merkulov and L. Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections, Ann. of Math. (2), 150 (1999), 77–149. doi: 10.2307/121098.  Google Scholar

[22]

J. Morrow and H. Rossi, Submanifolds of $\mathbb{P}^N$ with splitting normal bundle sequence are linear, Math. Ann., 234 (1978), 253-261.  doi: 10.1007/BF01420647.  Google Scholar

[23]

M. S. Narasimhan, Deformations of complex structures and holomorphic vector bundles, in Complex Analysis, Proc. Summer School (Trieste, 1980) (ed. J. Eells), Lecture Notes in Math., Springer, 1982,196–209. doi: 10.1007/BFb0061878.  Google Scholar

[24]

R. Pantilie, On the classification of the real vector subspaces of a quaternionic vector space, Proc. Edinb. Math. Soc. (2), 56 (2013), 615–622. doi: 10.1017/S0013091513000011.  Google Scholar

[25]

R. Pantilie, On the twistor space of a (co-)CR quaternionic manifold, New York J. Math., 20 (2014), 959-971.   Google Scholar

[26]

R. Pantilie, On the integrability of co-CR quaternionic structures, New York J. Math., 22 (2016), 1-20.   Google Scholar

[27]

D. Quillen, Quaternionic algebra and sheaves on the Riemann sphere, Quart. J. Math. Oxford Ser. (2), 49 (1998), 163–198. doi: 10.1093/qmathj/49.2.163.  Google Scholar

[28]

S. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4), 19 (1986), 31–55. doi: 10.24033/asens.1503.  Google Scholar

show all references

References:
[1]

M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., 85 (1957), 181-207.  doi: 10.1090/S0002-9947-1957-0086359-5.  Google Scholar

[2]

M. F. AtiyahN. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, 362 (1978), 425-461.  doi: 10.1098/rspa.1978.0143.  Google Scholar

[3]

T. N. Bailey and M. G. Eastwood, Complex paraconformal manifolds – their differential geometry and twistor theory, Forum Math., 3 (1991), 61-103.  doi: 10.1515/form.1991.3.61.  Google Scholar

[4]

E. Bonan, Sur les $G$-structures de type quaternionien, Cahiers Topologie Géom. Différentielle, 9 (1967), 389-461.   Google Scholar

[5]

R. L. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory, in Complex Geometry and Lie Theory (Sundance, UT, 1989), Proc. Sympos. Pure Math., 53, Amer. Math. Soc., Providence, RI, 1991, 33–88. doi: 10.1090/pspum/053/1141197.  Google Scholar

[6]

J. B. Carrell, A remark on the Grothendieck residue map, Proc. Amer. Math. Soc., 70 (1978), 43-48.  doi: 10.1090/S0002-9939-1978-0492408-1.  Google Scholar

[7]

Q.-S. Chi and L. J. Schwachhöfer, Exotic holonomy on moduli spaces of rational curves, Differential Geom. Appl., 8 (1998), 105-134.  doi: 10.1016/S0926-2245(97)00019-3.  Google Scholar

[8]

W. Fulton and J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129, Readings in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9.  Google Scholar

[9]

H. Grauert and R. Remmert, Coherent Analytic Sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 265, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69582-7.  Google Scholar

[10]

M. Gualtieri, Generalized complex geometry, Ann. of Math. (2), 174 (2011), 75–123. doi: 10.4007/annals.2011.174.1.3.  Google Scholar

[11]

N. J. Hitchin, Complex manifolds and Einstein's equations, in Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Math., 970, Springer, Berlin-New York, 1982, 73–99.  Google Scholar

[12]

P. Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, in Projective Varieties With Unexpected Properties, Walter de Gruyter, Berlin, 2005,317–335.  Google Scholar

[13]

D. Joyce, Compact hypercomplex and quaternionic manifolds, J. Differential Geom., 35 (1992), 743-761.  doi: 10.4310/jdg/1214448266.  Google Scholar

[14]

K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. (2), 75 (1962), 146–162. doi: 10.2307/1970424.  Google Scholar

[15]

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Ⅰ, Ⅱ, Ann. of Math. (2), 67 (1958), 328–466. doi: 10.2307/1970009.  Google Scholar

[16]

K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. Ⅲ. Stability theorems for complex structures, Ann. of Math. (2), 71 (1960), 43–76. doi: 10.2307/1969879.  Google Scholar

[17]

C. R. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math., 118 (1994), 109-132.  doi: 10.1007/BF01231528.  Google Scholar

[18]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147.  doi: 10.1112/blms/27.2.97.  Google Scholar

[19]

S. MarchiafavaL. Ornea and R. Pantilie, Twistor Theory for CR quaternionic manifolds and related structures, Monatsh. Math., 167 (2012), 531-545.  doi: 10.1007/s00605-011-0326-0.  Google Scholar

[20]

S. Marchiafava and R. Pantilie, Twistor theory for co-CR quaternionic manifolds and related structures, Israel J. Math., 195 (2013), 347-371.  doi: 10.1007/s11856-013-0001-3.  Google Scholar

[21]

S. Merkulov and L. Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections, Ann. of Math. (2), 150 (1999), 77–149. doi: 10.2307/121098.  Google Scholar

[22]

J. Morrow and H. Rossi, Submanifolds of $\mathbb{P}^N$ with splitting normal bundle sequence are linear, Math. Ann., 234 (1978), 253-261.  doi: 10.1007/BF01420647.  Google Scholar

[23]

M. S. Narasimhan, Deformations of complex structures and holomorphic vector bundles, in Complex Analysis, Proc. Summer School (Trieste, 1980) (ed. J. Eells), Lecture Notes in Math., Springer, 1982,196–209. doi: 10.1007/BFb0061878.  Google Scholar

[24]

R. Pantilie, On the classification of the real vector subspaces of a quaternionic vector space, Proc. Edinb. Math. Soc. (2), 56 (2013), 615–622. doi: 10.1017/S0013091513000011.  Google Scholar

[25]

R. Pantilie, On the twistor space of a (co-)CR quaternionic manifold, New York J. Math., 20 (2014), 959-971.   Google Scholar

[26]

R. Pantilie, On the integrability of co-CR quaternionic structures, New York J. Math., 22 (2016), 1-20.   Google Scholar

[27]

D. Quillen, Quaternionic algebra and sheaves on the Riemann sphere, Quart. J. Math. Oxford Ser. (2), 49 (1998), 163–198. doi: 10.1093/qmathj/49.2.163.  Google Scholar

[28]

S. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4), 19 (1986), 31–55. doi: 10.24033/asens.1503.  Google Scholar

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