[proofplan] We realize the special Schubert variety as a rank-drop locus for the bundle map from the universal subbundle $S$ to a fixed quotient [vector space](/page/Vector%20Space) of $\mathbb C^n$. The chosen index $F_{n-k+1-j}$ makes the expected complex codimension equal to $j$, and the Thom-Porteous degeneracy-locus formula identifies the Poincare dual of this locus with the $j$-th Chern class of the virtual bundle $(\mathbb C^n/F_{n-k+1-j})-S$, which is the same Chern class as $Q$ by the universal exact sequence. Finally, Whitney multiplicativity applied to the universal exact sequence gives $c(S)c(Q)=1$, and substituting the Schubert identifications gives the inverse formula for $c(S)$. [/proofplan]

[step:Express the Schubert condition as a rank-drop condition] Fix $j\in\{1,\dots,n-k\}$ and define \begin{align*} a:=n-k+1-j. \end{align*} Then $1\le a\le n-k$. Let \begin{align*} \rho_a:\mathbb C^n\longrightarrow \mathbb C^n/F_a \end{align*} be the quotient map. Pulling $\rho_a$ back over $\operatorname{Gr}(k,n)$ and restricting it to the universal subbundle gives a morphism of complex vector bundles \begin{align*} \varphi_a:S\longrightarrow (\mathbb C^n/F_a)\times \operatorname{Gr}(k,n), \end{align*} whose fiber over $V\in\operatorname{Gr}(k,n)$ is the [linear map](/page/Linear%20Map) \begin{align*} (\varphi_a)_V:V\longrightarrow \mathbb C^n/F_a \end{align*} defined by $(\varphi_a)_V(v)=v+F_a$ for every $v\in V$. The kernel of $(\varphi_a)_V$ is $V\cap F_a$. Therefore \begin{align*} \dim_{\mathbb C}(V\cap F_a)\ge 1 \end{align*} if and only if $(\varphi_a)_V$ is not injective, which is equivalent to \begin{align*} \operatorname{rank}(\varphi_a)_V\le k-1. \end{align*} Thus \begin{align*} \Sigma_j(F_\bullet)=D_{k-1}(\varphi_a), \end{align*} where $D_{k-1}(\varphi_a)$ denotes the degeneracy locus on which $\varphi_a$ has rank at most $k-1$. [/step]

[step:Check the expected codimension and apply Thom-Porteous]The bundle $S$ has rank $k$, and the product bundle $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$ has rank \begin{align*} n-a=n-(n-k+1-j)=k-1+j. \end{align*} For the rank condition $\operatorname{rank}(\varphi_a)\le k-1$, the expected complex codimension is \begin{align*} (k-(k-1))((k-1+j)-(k-1))=j. \end{align*} The locus $D_{k-1}(\varphi_a)$ is precisely the special Schubert variety indexed by the one-row partition $(j)$ for the quotient-bundle convention determined in the statement. The Schubert cell dimension formula gives complex codimension $j$ for this one-row Schubert variety, so the degeneracy locus has the expected codimension. In the standard Schubert determinantal description, the local defining equations for this one-row Schubert variety are exactly the maximal minors of the map obtained from $\varphi_a$ after choosing local frames for $S$ and for $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$. These determinantal charts are reduced for Schubert varieties in the Grassmannian, hence the determinantal scheme structure induced by $\varphi_a$ agrees with the reduced complex subvariety structure on $\Sigma_j(F_\bullet)$. Therefore its cycle class is the integral fundamental class used to define $\sigma_j$. Since $S$ and $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$ are complex vector bundles over the smooth complex Grassmannian, and $D_{k-1}(\varphi_a)$ is the corresponding complex algebraic degeneracy locus of the expected codimension with this natural integral fundamental class, the integral Thom-Porteous degeneracy-locus formula applies. By the integral Thom-Porteous degeneracy-locus formula for the bundle map \begin{align*} \varphi_a:S\longrightarrow (\mathbb C^n/F_a)\times\operatorname{Gr}(k,n), \end{align*} the Poincare dual of the integral fundamental class of $D_{k-1}(\varphi_a)$ is \begin{align*} c_j\bigl((\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)-S\bigr). \end{align*} Since $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$ is a product complex vector bundle with constant fiber $\mathbb C^n/F_a$, its total Chern class is $1$. Hence \begin{align*} c\bigl((\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)-S\bigr)=c(S)^{-1}. \end{align*}[/step]

[guided]Fix $j\in\{1,\dots,n-k\}$ and define $a:=n-k+1-j$, so $1\le a\le n-k$. Let $F_a$ be the $a$-dimensional member of the fixed complete flag, and let $\rho_a:\mathbb C^n\to \mathbb C^n/F_a$ be the quotient map. Pulling $\rho_a$ back to $\operatorname{Gr}(k,n)$ and restricting to the universal subbundle defines the complex vector bundle morphism \begin{align*} \varphi_a:S\longrightarrow (\mathbb C^n/F_a)\times\operatorname{Gr}(k,n). \end{align*} The point of introducing $\varphi_a$ is that the Schubert condition is exactly a failure of injectivity. For each $V\in\operatorname{Gr}(k,n)$, the fiber map \begin{align*} (\varphi_a)_V:V\longrightarrow \mathbb C^n/F_a \end{align*} is defined by $(\varphi_a)_V(v)=v+F_a$ for every $v\in V$. Its kernel is precisely the set of vectors of $V$ that already lie in $F_a$, namely \begin{align*} \ker(\varphi_a)_V=V\cap F_a. \end{align*} Thus the condition $\dim_{\mathbb C}(V\cap F_a)\ge 1$ says exactly that $(\varphi_a)_V$ has nonzero kernel. Since the domain has dimension $k$, this is equivalent to \begin{align*} \operatorname{rank}(\varphi_a)_V\le k-1. \end{align*} We next verify the codimension required by the Thom-Porteous formula. The domain bundle $S$ has rank $k$. The target bundle is the product bundle with fiber $\mathbb C^n/F_a$, whose rank is \begin{align*} \dim_{\mathbb C}(\mathbb C^n/F_a)=n-a=n-(n-k+1-j)=k-1+j. \end{align*} For a map from a rank-$k$ bundle to a rank-$(k-1+j)$ bundle, the degeneracy locus where the rank is at most $k-1$ has expected complex codimension \begin{align*} (k-(k-1))((k-1+j)-(k-1))=j. \end{align*} This is the standard Schubert variety of one-row shape $(j)$ in the convention where special Schubert classes are represented by the quotient-bundle degeneracy condition. The Schubert cell dimension formula says that this one-row Schubert variety has complex codimension $j$, matching the expected codimension computed above. Locally on the Grassmannian, choose frames for $S$ and for $(\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)$. In those frames, the rank condition on $\varphi_a$ is imposed by the maximal minors of the representing matrix. These are exactly the usual determinantal equations for this one-row Schubert variety, and the corresponding Schubert determinantal charts are reduced. Thus the determinantal scheme defined by the rank condition on $\varphi_a$ is the reduced Schubert variety $\Sigma_j(F_\bullet)$, so the cycle class appearing in Thom-Porteous is the integral fundamental class used in the definition of $\sigma_j$. The bundles involved are complex vector bundles over the smooth complex Grassmannian, and the degeneracy locus is a complex algebraic Schubert variety of the expected codimension equipped with this natural integral fundamental class. These are precisely the geometric hypotheses needed to apply the integral Thom-Porteous degeneracy-locus formula. The integral Thom-Porteous degeneracy-locus formula now applies to the morphism \begin{align*} \varphi_a:S\longrightarrow (\mathbb C^n/F_a)\times\operatorname{Gr}(k,n), \end{align*} because the degeneracy locus has the expected codimension $j$. It gives that the Poincare dual of the integral fundamental class of this codimension-$j$ locus is the $j$-th Chern class of the virtual bundle \begin{align*} (\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)-S. \end{align*} The target bundle is a product bundle with constant fiber $\mathbb C^n/F_a$, so its total Chern class is $1$, and therefore the total Chern class of this virtual bundle is \begin{align*} c\bigl((\mathbb C^n/F_a)\times\operatorname{Gr}(k,n)-S\bigr)=c(S)^{-1}. \end{align*}[/guided]

[step:Identify the resulting virtual Chern class with $c_j(Q)$] The universal exact sequence \begin{align*} 0\longrightarrow S\longrightarrow \mathbb C^n\times\operatorname{Gr}(k,n)\longrightarrow Q\longrightarrow 0 \end{align*} and the Chern relation for tautological and quotient bundles [citetheorem:9802] give \begin{align*} c(S)c(Q)=1. \end{align*} Hence \begin{align*} c(Q)=c(S)^{-1}. \end{align*} Combining this identity with the Thom-Porteous computation from the previous step yields \begin{align*} [\Sigma_j(F_\bullet)]^\vee=c_j(Q), \end{align*} where $[\Sigma_j(F_\bullet)]^\vee\in H^{2j}(\operatorname{Gr}(k,n);\mathbb Z)$ denotes the Poincare dual of the integral fundamental class of $\Sigma_j(F_\bullet)$. By the definition of $\sigma_j$, this says \begin{align*} \sigma_j=c_j(Q). \end{align*} [/step]

[step:Invert the total quotient Chern class] Since $Q$ has rank $n-k$, its total Chern class is \begin{align*} c(Q)=1+c_1(Q)+\cdots+c_{n-k}(Q). \end{align*} Using the identifications $c_j(Q)=\sigma_j$ for $1\le j\le n-k$, we obtain \begin{align*} c(Q)=1+\sigma_1+\cdots+\sigma_{n-k}. \end{align*} The identity $c(S)c(Q)=1$ in $H^*(\operatorname{Gr}(k,n);\mathbb Z)$ therefore gives \begin{align*} c(S)=c(Q)^{-1}=\bigl(1+\sigma_1+\cdots+\sigma_{n-k}\bigr)^{-1}. \end{align*} This proves both the Schubert compatibility statement and the asserted formula for the total Chern class of $S$. [/step]