| Title:
|
Characteristic forms of complex Cartan geometries III: $G$-structures (English) |
| Author:
|
McKay, Benjamin |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2025 |
| Pages:
|
133-150 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Characteristic class relations in Dolbeault cohomology follow from the existence of a holomorphic geometric structure (for example, holomorphic conformal structures, holomorphic Engel distributions, holomorphic projective connections, and holomorphic foliations). These relations can be calculated directly from the representation theory of the structure group, without selecting any metric or connection or having any knowledge of the Dolbeault cohomology groups of the manifold. This paper improves on its predecessor [17] by allowing infinite type geometric structures. (English) |
| Keyword:
|
complex projective manifold |
| Keyword:
|
G-structure |
| MSC:
|
53A55 |
| MSC:
|
53B21 |
| MSC:
|
53C56 |
| DOI:
|
10.5817/AM2025-4-133 |
| . |
| Date available:
|
2025-12-19T12:50:38Z |
| Last updated:
|
2025-12-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153200 |
| . |
| Reference:
|
[1] : The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 9.2.0), 2021, https://www.sagemath.org. |
| Reference:
|
[2] Atiyah, M.F.: Complex analytic connections in fibre bundles.Trans. Amer. Math. Soc. 85 (1957), 181–207. 10.1090/S0002-9947-1957-0086359-5 |
| Reference:
|
[3] Baum, P., Bott, R.: Singularities of holomorphic foliations.J. Differential Geom. 7 (1972), 279–342. 10.4310/jdg/1214431158 |
| Reference:
|
[4] Beauville, A.: Complex manifolds with split tangent bundle.Complex analysis and algebraic geometry, de Gruyter, Berlin, 2000, pp. 61–70. |
| Reference:
|
[5] Biswas, I.: On connections on principal bundles.Arab. J. Math. Sci. 23 (1) (2017), 32–43. |
| Reference:
|
[6] Bryant, R., Griffiths, Ph., Hsu, L.: Toward a geometry of differential equations.Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995. |
| Reference:
|
[7] Bryant, R.L., Chern, Shiing-Shen, Gardner, R.B., Goldschmidt, H.L., Griffiths, P.A.: Exterior differential systems.Springer-Verlag, New York, 1991. |
| Reference:
|
[8] Bryant, R.L., Hsu, L.: Rigidity of integral curves of rank $2$ distributions.Invent. Math. 114 (2) (1993), 435–461. 10.1007/BF01232676 |
| Reference:
|
[9] Chern, Shiing Shen, Simons, J.: Characteristic forms and geometric invariants.Ann. of Math. (2) 99 (1974), 48–69. 10.2307/1971013 |
| Reference:
|
[10] Gardner, R.B.: The method of equivalence and its applications.CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. Zbl 0694.53027 |
| Reference:
|
[11] Griffiths, Ph.A.: On certain homogeneous complex manifolds.Proc. Nat. Acad. Sci. USA 48 (1962), 780–783. 10.1073/pnas.48.5.780 |
| Reference:
|
[12] Hilgert, J., Nee, K.-H.: Structure and geometry of Lie groups.Springer, New York, 2012. |
| Reference:
|
[13] Hochschild, G.P.: Basic theory of algebraic groups and Lie algebras.Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1981. |
| Reference:
|
[14] Ivey, T.A., Landsberg, J.M.: Cartan for beginners.Graduate Studies in Mathematics, vol. 175, American Mathematical Society, Providence, RI, 2016. |
| Reference:
|
[15] Kamber, F.W., Tondeur, Ph.: Foliated bundles and characteristic classes.Lecture Notes in Mathematics, vol. 493, Springer-Verlag, Berlin-New York, 1975. |
| Reference:
|
[16] Knapp, A.W.: Lie groups beyond an introduction.Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 2002. |
| Reference:
|
[17] McKay, B.: Characteristic forms of complex cartan geometries ii.arXiv:2206.04495 [math.DG]. |
| Reference:
|
[18] McKay, B.: Characteristic forms of complex Cartan geometries.Adv. Geom. 11 (1) (2011), 139–168. 10.1515/advgeom.2010.044 |
| Reference:
|
[19] Mumford, D.: Hirzebruch’s proportionality theorem in the noncompact case.Invent. Math. 42 (1977), 239–272. 10.1007/BF01389790 |
| Reference:
|
[20] Presas, F., Solá Conde, L.E.: Holomorphic Engel structures.Rev. Mat. Comput. 27 (1) (2014), 327–344. 10.1007/s13163-013-0129-z |
| Reference:
|
[21] Sternberg, S.: Lectures on differential geometry.Chelsea Publishing Co., New York, 1983, second ed. With an appendix by Sternberg and Victor W. Guillemin. |
| Reference:
|
[22] Varadarajan, V.S.: Lie groups, Lie algebras, and their representations.Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984, Reprint of the 1974 edition. |
| . |
