Intersection cycle in a product of Grassmannians

Question

Let $G(k,n)$ denote the Grassmiannian of $k$-planes in $\mathbb C^n$. Let's define $$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$ These are an analytic subvarieties in $G(k,n) \times G(l,n)$. I would like to know if something is known about their homology class. It would be perfect if it was possible to express the Poincare duals of these homology classes in terms of the universal Chern classes (Chern classes of tautological bundles on the two factor Grassmannians, pulled back to the product). I am most interested in the case $k=l$, $n=2k$. The question seems to be closely related to Schubert calculus.

Answer 1

Let $V$ be the $n$-dimensional space such that $\Lambda_i \subset V$. Then the condition $\dim(\Lambda_1 \cap \Lambda_2) \ge j$ is equivalent to the condition $$ \mathrm{rank}(\Lambda_1 \hookrightarrow V \to V/\Lambda_2) \le k - j. $$ This means that $I_j$ is a degeneracy locus for the morphism $$ \mathcal{U}_1 \hookrightarrow V \otimes \mathcal{O} \to \mathcal{Q}_2 $$ on $\mathrm{Gr}(k,V) \times \mathrm{Gr}(l,V)$ from the pullback of the tautological subbundle of the first factor to the tautological quotient bundle of the second factor. Therefore, its class is computed by Porteous formula in terms of Chern classes of $\mathcal{U}_1$ and $\mathcal{Q}_2$.