An Algebraic Computation of the Height of the First Chern Class of the Canonical K-Plane Bundle over the Complex Grassmannian CG_(n,k)

Vimala Ramani

doi:10.32628/IJSRST241161123

Authors

Vimala Ramani Department of Mathematics, Anna University, Chennai, Tamil Nadu, India Author

DOI:

https://doi.org/10.32628/IJSRST241161123

Keywords:

Complex Grassmann manifold, Chern classes, height, complex flag manifold

Abstract

We prove algebraically that the height of the first Chern class of the canonical k-plane bundle over the complex Grassmannian CG_(n,k) is k(n-k) in H^* ( CG_(n,k);Q) for k=2,3,4, and any positive integer n≥2k. We use the technique of R. E. Stong, of embedding the rational cohomology ring of the complex Grassmannian CG_(n,k) into the rational cohomology ring of the complete complex flag manifold F_C (⏟(1,…,1)┬n) .

Downloads

Download data is not yet available.

References

Borel A.(1953). Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. of Math. (2). 1953; 57 (2):115-207. https://people.math.rochester.edu/faculty/doug/otherpapers/Borel-Sur.pdf DOI: https://doi.org/10.2307/1969728

Korbas ̌ J, Lo ̈rinc J. The Z_2-cohomology cup-length of real flag manifolds. Fund. Math. 2003; 178(2) : 143-158. https://eudml.org/doc/283181 DOI: https://doi.org/10.4064/fm178-2-4

Ramani V, Sankaran P. On degrees of maps between Grassmannians. Proc. Indian Acad. Sci. (Math. Sci.) 1997;107(1):13-19. https://doi.org/10.1007/BF02840469 DOI: https://doi.org/10.1007/BF02840469

Ramani V. On the topological complexity of Grassmann manifolds. Math. Slovaca. 2020;70(5): 1197-1210. https://doi.org/10.1515/ms-2017-0425 DOI: https://doi.org/10.1515/ms-2017-0425

Ramani V. The rational zero-divisor cup-length of oriented partial flag manifolds. Math. Slovaca. 2023; 73 (5):1325-1357. https://doi.org/10.1515/ms-2023-0097 DOI: https://doi.org/10.1515/ms-2023-0097

Ramani V. An application of Gro ̈bner basis. International Journal of Scientific Research in Science and Technology. 2024; 11 (6): 455-460. https://doi.org/10.32628/IJSRST24114309 DOI: https://doi.org/10.32628/IJSRST24114309

Stong R E. Cup-products in Grassmannians. Topology Appl. 1982; 13: 03-113. https://doi.org/10.1016/0166-8641(82)90012-8 DOI: https://doi.org/10.1016/0166-8641(82)90012-8