Abstract
We present here some conjectures on the diagonalizability of uniform principal bundles on rational homogeneous spaces, that are natural extensions of classical theorems on uniform vector bundles on the projective space, and study the validity of these conjectures in the case of classical groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ballico, E.: Uniform vector bundles on quadrics. Ann. Univ. Ferrara Sez. VII (N.S.) 27, 135–146 (1982)
Ballico, E.: Uniform vector bundles of rank \((n+1)\) on \({{ P}}_{n}\). Tsukuba J. Math. 7(2), 215–226 (1983)
Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 4. Springer, Berlin (1984)
Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. Math. 57, 115–207 (1953)
Drezet, J.-M.: Exemples de fibres uniformes non homogènes sur \({{\bf P}}_{n-}\). C. R. Acad. Sci. Paris Sér. A B 291(2), A125–A128 (1980)
Eisenbud, D., Harris, J.: 3264 and All That—A Second Course in Algebraic Geometry. Cambridge University Press, Cambridge (2016)
Elencwajg, G.: Des fibrés uniformes non homogènes. Math. Ann. 239(2), 185–192 (1979)
Elencwajg, G., Hirschowitz, A., Schneider, M.: Les fibrés uniformes de rang au plus \(n\) sur \({{ P}}_{n}({{ C}})\) sont ceux qu’on croit. In: Vector Bundles and Differential Equations. Progress in Mathematics, vol. 7, Birkhäuser, Boston, pp. 37–63 (1980)
Ellia, Ph.: Sur les fibrés uniformes de rang \((n+1)\) sur \({{\bf P }}^{n}\). Mém. Soc. Math. France (N.S.) 7, 1–60 (1982)
Fulton, W.: Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J. 65(3), 381–420 (1992)
Fulton, W.: Intersection Theory, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 2. Springer, Berlin (1998)
Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)
Fulton, W., MacPherson, R.D., Sottile, F., Sturmfels, B.: Intersection theory on spherical varieties. J. Algebraic Geom. 4(1), 181–193 (1995)
Grothendieck, A.: Sur la classification des fibrés holomorphes sur la sphére de Riemann. Amer. J. Math. 79, 121–138 (1957)
Guyot, M.: Caractérisation par l’uniformité des fibrés universels sur la grassmanienne. Math. Ann. 270(1), 47–62 (1985)
Hiller, H.: Geometry of Coxeter Groups. Research Notes in Mathematics, vol. 54. Pitman, Boston (1982)
Humphreys, J.E.: Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21. Springer, New York (1975)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9. Springer, New York (1978)
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
Landsberg, J.M., Manivel, L.: On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78(1), 65–100 (2003)
Lazarsfeld, R.: Some applications of the theory of positive vector bundles. In: Greco, S., Strano, R. (eds.) Complete Intersections. Lecture Notes in Mathematics, vol. 1092, pp. 29–61. Springer, Berlin (1984)
Muñoz, R., Occhetta, G., Solá Conde, L.E.: Rank two Fano bundles on \({\mathbb{G}}(1,4)\). J. Pure Appl. Algebra 216(10), 2269–2273 (2012)
Muñoz, R., Occhetta, G., Solá Conde, L.E.: Uniform vector bundles on Fano manifolds and applications. J. Reine Angew. Math. 664, 141–162 (2012)
Muñoz, R., Occhetta, G., Solá Conde, L.E.: On uniform flag bundles on Fano manifolds (2016). arXiv:1610.05930. To appear in Kyoto Journal of Mathematics
Muñoz, R., Occhetta, G., Solá Conde, L.E., Watanabe, K.: Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle. Math. Ann. 361(3), 583–609 (2015)
Occhetta, G., Solá Conde, L.E., Wiśniewski, J.A.: Flag bundles on Fano manifolds. J. Math. Pures Appl. 106(4), 651–669 (2016)
Okonek, Chr., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces. Progress in Mathematics, vol. 3. Birkhäuser, Boston (1980)
Pan, X.: Triviality and split of vector bundles on rationally connected varieties. Math. Res. Lett. 22(2), 529–547 (2015)
Sato, E.: Uniform vector bundles on a projective space. J. Math. Soc. Japan 28(1), 123–132 (1976)
Serre, J.-P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier Grenoble 6, 1–42 (1955–1956)
Tango, H.: On morphisms from projective space \(P^{n}\) to the Grassmann variety \({Gr}{(n,d)}\). J. Math. Kyoto Univ. 16(1), 201–207 (1976)
Van de Ven, A.: On uniform vector bundles. Math. Ann. 195, 245–248 (1972)
Wiśniewski, J.A.: Uniform vector bundles on Fano manifolds and an algebraic proof of Hwang–Mok characterization of Grassmannians. In: Bauer, I., et al. (eds.) Complex Geometry, pp. 329–340. Springer, Berlin (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Roberto Muñoz and Luis E. Solá Conde are supported by the Spanish Government Project MTM2015-65968-P, Gianluca Occhetta and Luis E. Solá Conde are supported by PRIN Project 2015EYPTSB_002: “Geometria delle varietà algebriche”.
Rights and permissions
About this article
Cite this article
Muñoz, R., Occhetta, G. & Solá Conde, L.E. Splitting conjectures for uniform flag bundles. European Journal of Mathematics 6, 430–452 (2020). https://doi.org/10.1007/s40879-019-00343-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-019-00343-6