On the Chow ring of the classifying space BSO(2N, C).

摘要

This thesis computes the Chow ring of the classifying space BSO2n,C completely for n≤3 and partially for all n, confirming conjectures of Totaro and Pandharipande, and gives the additional theorem that this Chow ring is not generated by Chern classes of any representations of SO2n,C , which was not previously conjectured. This thesis shows that, for n≤3 , the Chow ring is a polynomial ring in the Chern classes of the standard representation and a class yn in codimension n (defined by Edidin and Graham) which maps to 2n-1 times the Euler class in cohomology modulo the expected relations (the odd Chern classes are 2-torsion, the class yn kills odd Chern classes, and the relation y2n=22n-2c2n which corresponds to the fact that c2=pn in cohomology). For n > 3, the Chow ring is shown to be at least the ring above, but there may be more generators in codimensions higher than n. The immediate corollary is that for n≤3 this Chow ring injects into cohomology.; This Chow ring is not generated by Chern classes of any representations, because the representation ring for SO2n,C is generated by exterior powers of the standard representation and D+n , the space of self dual n forms: the nth Chern class of D+n is 2n-1n-1!c (modulo the Chern classes of the standard representation) in cohomology.; The proof uses a fibration of BSO2n,C over BGl2n,C with fiber Gl2n,C/ SO2n,C , along with a theorem of Totaro, to find generators of the Chow ring of BSO2n,C as a module over the Chow ring of BGl2n,C by computing the Chow ring of the quotient. This symmetric space is a spherical variety, and we use a theorem of Fulton, MacPherson, Sottile, and Sturmfels along with an examination of the double cover Gl2n,C/ O2n,C →Gl2n ,C/SO 2n,C and Schubert calculus to compute this Chow ring for n≤3 .