Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 108, 19 pages      arXiv:2405.13614      https://doi.org/10.3842/SIGMA.2024.108

On Relative Tractor Bundles

Andreas Čap, Zhangwen Guo and Michał Andrzej Wasilewicz
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Received June 03, 2024, in final form November 18, 2024; Published online November 29, 2024

Abstract
This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated to the geometry in question. In many situations of interest, it is known that this sequence provides a resolution of a sheaf that can locally be realized as a pullback from a local leaf space of a foliation that is naturally available in this situation. An explicit description of the latter sheaf was only available under much more restrictive assumptions. For any geometry which admits relative tractor bundles, we construct a large family of such bundles for which we obtain a simple, explicit description of the resolved sheaves under weak assumptions on the torsion of the geometry. In particular, we discuss the cases of Legendrean contact structures and of generalized path geometries, which are among the most important examples for which the relative BGG machinery is available. In both cases, we show that essentially all relative tractor bundles are obtained by our construction and our description of the resolved sheaves applies whenever the BGG sequence is a resolution.

Key words: relative BGG-machinery; relative BGG resolution; relative tractor bundle; parabolic geometries; Legendrean contact structure; generalized path geometry.

pdf (426 kb)   tex (32 kb)  

References

  1. Baston R.J., Almost Hermitian symmetric manifolds. II. Differential invariants, Duke Math. J. 63 (1991), 113-138.
  2. Baston R.J., Eastwood M.G., The Penrose transform. Its interaction with representation theory, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1989.
  3. Bernšteǐn I.N., Gel'fand I.M., Gel'fand S.I., Differential operators on the base affine space and a study of $\mathfrak{g}$-modules, in Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted Press, New York, 1975, 21-64.
  4. Calderbank D.M.J., Diemer T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math. 537 (2001), 67-103, arXiv:math.DG/0001158.
  5. Čap A., Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582 (2005), 143-172, arXiv:math.DG/0102097.
  6. Čap A., Slovák J., Weyl structures for parabolic geometries, Math. Scand. 93 (2003), 53-90, arXiv:math.DG/0001166.
  7. Čap A., Slovák J., Parabolic geometries. I. Background and general theory, Math. Surveys Monogr., Vol. 154, American Mathematical Society, Providence, RI, 2009.
  8. Čap A., Slovák J., Souček V., Bernstein-Gelfand-Gelfand sequences, Ann. of Math. 154 (2001), 97-113, arXiv:math.DG/0001164.
  9. Čap A., Souček V., Relative BGG sequences: I. Algebra, J. Algebra 463 (2016), 188-210, arXiv:1510.03331.
  10. Čap A., Souček V., Relative BGG sequences; II. BGG machinery and invariant operators, Adv. Math. 320 (2017), 1009-1062, arXiv:1510.03986.
  11. Doubrov B., Medvedev A., The D., Homogeneous Levi non-degenerate hypersurfaces in $\mathbb{C}^3$, Math. Z. 297 (2021), 669-709, arXiv:1711.02389.
  12. Doubrov B., Merker J., The D., Classification of simply-transitive Levi non-degenerate hypersurfaces in $\mathbb{C}^3$, Int. Math. Res. Not. 2022 (2022), 15421-15473, arXiv:2010.06334.
  13. Eastwood M., A complex from linear elasticity, Rend. Circ. Mat. Palermo (2) Suppl. 63 (2000), 23-29.
  14. Fels M.E., The equivalence problem for systems of second-order ordinary differential equations, Proc. London Math. Soc. 71 (1995), 221-240.
  15. Guo Z., Generalized path geometries and almost Grassmannian structures, Ph.D. Thesis, University of Vienna, 2024, in preparation.
  16. Kostant B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961), 329-387.
  17. Lepowsky J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra 49 (1977), 496-511.
  18. Ma T., Flood K.J., Matveev V.S., Zádník V., Canonical curves and Kropina metrics in Lagrangian contact geometry, Nonlinearity 37 (2024), 015007, 36 pages.
  19. Michor P.W., Topics in differential geometry, Grad. Stud. Math., Vol. 93, American Mathematical Society, Providence, RI, 2008.
  20. Takeuchi M., Lagrangean contact structures on projective cotangent bundles, Osaka J. Math. 31 (1994), 837-860.
  21. Wasilewicz M.A., Elementary relative tractor calculus for Legendrean contact structures, Arch. Math. (Brno) 58 (2022), 339-347, arXiv:2205.05326.
  22. Wasilewicz M.A., Geometry of Legendrean distributions, Ph.D. Thesis, University of Vienna, 2024, in preparation.

Previous article  Next article  Contents of Volume 20 (2024)