Home » Mathematics » Algebraic geometry » Addendum to last post: the total Chern classes of tautological bundles on G(k,n)

# Addendum to last post: the total Chern classes of tautological bundles on G(k,n)

I realised after finishing the last post that it’s simpler than I had thought to compute the total Chern classes of the tautological bundles appearing in the short exact sequence

$0 \to \mathcal{S} \to \mathbb{C}^n \to \mathcal{Q} \to 0$

on the Grassmannian $G(k,n)$. I’m keeping it in a separate post, though, both because the last post is already very long and because I haven’t yet defined the Schubert varieties that appear in the answer.

In particular, it’s easy to compute $c(\mathcal{Q})$, as follows. First, $\mathcal{Q}$ is globally generated, and all its global sections come from the trivial bundle (including global sections). So consider choosing $i$ general global sections, that is, choosing $v_1, \ldots, v_i \in \mathbb{C}^n$, a “general, constant choice” of vectors from our ambient vector space. To compute the Chern class, we need to know: for which subspaces $V \in G(k,n)$ do our sections become linearly dependent in $\mathbb{C}^n / V$? (Recall that $\mathbb{C}^n/V$ is the fiber of $\mathcal{Q}$ over $V$.)

Well, they are linearly dependent in $C^n/V$ if and only if $V$ intersects the subspace $\langle v_1, \ldots, v_i \rangle$ nontrivially. But the set of $V$ intersecting a given $i$-dimensional subspace nontrivially is precisely the Schubert variety $X_{n-k+1-i}$, or, more prosaically, the Schubert variety corresponding to a one-row partition of the appropriate length.

So, in fact, the total Chern class of $\mathcal{Q}$ is the sum $1 + [X_1] + [X_2] + \ldots + [X_{n-k}]$, the sum of all the one-row partitions fitting in a $k \times (n-k)$ box.

Repeating this logic for $\mathcal{S}^*$ (thinking of it as the tautological quotient bundle on the dual Grassmannian) shows that $\mathcal{S}^*$ is the sum $1 + [X_1] + [X_{1,1}] + \cdots + [X_{1,\ldots,1}]$, the sum of all the single-vertical-column partitions. To turn this into the total Chern class of $\mathcal{S}$, just change the plus signs to alternating signs.