I've the following problem:

I know how to calculate Chern classes of the tautological bundle over the Grassmannian $G=G(2,4)$ using the Schubert calculus. If I am right, the Chern character should be $$1-\sigma_1+\sigma_{1,1}$$ where $\sigma_{i_1,i_2}$ indicates the Schubert cycle corresponding to the Schubert variety $\Sigma_{i_1,i_2}=\{\Lambda\in G\mid\dim(V_{2-i_j+j}\cap\Lambda)\geq j\;\forall j\}$ (here $\{V_j\}\subset V$ is a flag in the 4-dimensional vector space $V$). Now these classes are element in the Chow ring, but I want to work with classes in the integral cohomology group. How can I do this translation?

Thank you!

  • 3
    $\begingroup$ The natural map $CH(G)\rightarrow H^*(G,\mathbb{Z})$ is a ring isomorphism. $\endgroup$ – abx Jul 15 '14 at 14:28
  • $\begingroup$ I know that a such map exists, but I don't how it works. So in my specific example, what is the image of $-\sigma_1$? And that of $\sigma_{1,1}$? I never did such a calculation.. $\endgroup$ – User3773 Jul 17 '14 at 9:25

The cohomology ring of $G(k,n)$ is isomorphic to $$\frac{\mathbb{Z}[c_1(T),...,c_k(T),c_1(Q),..., c_{n−k} (Q)]}{(c(T)c(Q) = 1)},$$ where $T$ and $Q$ are respectively the tautological and the quotient bundle. The chern classes of the tautological and the quotient bundle are given in terms of Schubert cycles by the following formulas:

  • $c_i(T) = (−1)^i\sigma_{1,...,1}$,
  • $c_i(Q) = \sigma_i$.
  • 1
    $\begingroup$ In the displayed formula, you should have a $\mathbb{Z}$ where you have a "coefficient field" $k$. $\endgroup$ – Jason Starr Jul 15 '14 at 17:15
  • $\begingroup$ You are right. I corrected it. $\endgroup$ – F_L Jul 15 '14 at 19:16

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