Stable-flag incidence spaces admit the oriented proper correspondence calculus

Incidence as an associated bundle. A zero-dimensional $V_v$ has the unique empty flag, so its flag variety and group factor are points. Fix a graded reference flag. Its stabilizer $P_{\nu }={\prod }_v{P}_v$ is the product of the upper triangular subgroups of the individual $\mathrm{G}\mathrm{L}(V_v)$, and by the transitive action of [F3] its orbit is $B_{\nu }={\prod }_v\mathrm{Flag}(V_v)$, a product of smooth projective flag varieties, so $B_{\nu }=G\mathrm{/}{P}_{\nu }$ is smooth projective. On an ordinary flag chart choose ordered basis vectors compatible with the flags; the stability condition $x_h(F_v^a)\subseteq F_w^a$ for an arrow $h:v\to w$ says that the matrix coefficient of $x_h$ from source position $a$ to target position $b$ vanishes whenever $b>a$, positions being read in the full word $\nu$. Over the chart the fiber of $X_{\nu }\to B_{\nu }$ is therefore the fixed linear subspace $S_{\nu }\subset E_V$ of tuples whose entries vanish in the forbidden positions, of dimension the sum over arrows of the number of allowed pairs; the stabilizer $P_{\nu }$ preserves $S_{\nu }$ and the chart trivializations identify $X_{\nu }=G{\times }_{P_{\nu }}{S}_{\nu }$ as a complex algebraic vector bundle over $B_{\nu }=G\mathrm{/}{P}_{\nu }$. Multiple arrows contribute separate summands, and arrows in both directions impose conditions on separate matrices.

Smoothness, projectivity and properness. By step 1.1 the total space $X_{\nu }$ is the total space of a vector bundle over the smooth variety $B_{\nu }$, hence is smooth of pure complex dimension $d_{\nu }={\sum }_v\dim \mathrm{Flag}(V_v)+\dim S_{\nu }$. The incidence conditions are polynomial in the chart coordinates, so $X_{\nu }$ is closed in $E_V\times B_{\nu }$, and ${\pi }_{\nu }$ is the composite of this closed immersion with the product projection. Since $B_{\nu }$ is a closed subvariety of a product of projective spaces by [F3], the product projection $E_V\times B_{\nu }\to E_V$ is a closed subvariety of a trivial projective-space bundle over the affine space $E_V$, so it is projective, and passing to the closed subvariety $X_{\nu }$ keeps it projective; by [F5] a projective morphism over the affine base is proper. The middle-forgetting map $X_{\nu }{\times }_{E_V}{X}_{\eta }{\times }_{E_V}{X}_{\theta }\to X_{\nu }{\times }_{E_V}{X}_{\theta }$ is the base change of ${\pi }_{\eta }$ along the projection $X_{\nu }{\times }_{E_V}{X}_{\theta }\to E_V$, hence proper by the base-change stability in [F5], although neither its fibers nor its total space is asserted smooth.

Orientations and equivariance. Each $X_{\nu }$ is smooth by step 2.1, with its complex orientation and diagonal normal bundle $T{X}_{\nu }$. The supported Thom operation of [F1] therefore gives the refined middle-diagonal operation on all fiber products, and step 2.1 supplies the proper traces. These are $G$-equivariant constructions, descending to the Stiefel models with the stabilization comparisons of Cartesian simplicial sheaves model actual equivariant derived Hom. Thus [F2] applies with shift $d_{\nu }$. On a Stiefel quotient the graded flags are nested subspaces of the fixed frame spaces, hence a closed incidence locus in a product of Grassmannians, projective by the same incidence equations and projective-product construction as step 2.1. The incidence approximation is the associated vector bundle of step 1.1 over that projective base. Fiber scaling retracts its total to the zero section, so [F3] applied to the projective base gives CW type for the total. The base has a metric from its finite-polyhedral model. A finite trivializing cover of the compact base admits a partition of unity supported inside its charts; multiplying the local vector coordinates by these functions embeds the total bundle into the product of the base with a finite-dimensional vector space, giving the total a metric as well.

Zero sections and relative orientations. The complex bundles on the metric approximations of step 3.1 have finite trivializing covers. For a finite trivializing cover $(U_i)$ of a metric space $X$, put $r_i(x)=\mathrm{dist}(x,X\setminus U_i)$, using $r_i=1$ when $U_i=X$, and $R(x)={\max }_i{r}_i(x)>0$. The continuous functions ${\psi }_i=\max (0,r_i-R\mathrm{/}2)$ have positive sum and support contained in $\{r_i\ge R\mathrm{/}2\}\subset U_i$; normalizing their sum gives a subordinate finite numeration. Hence the Thom/Euler clause of [F1] applies. Zero-section self-intersection is cap product with the Euler class. Pulling back along a CW-model homotopy equivalence and using naturality identifies it with the top Chern class by [F4], componentwise; extend the integral identity to $k$. A smooth vector-bundle or flag-bundle projection of relative complex dimension $r$ locally has the product duality formula $f^{!}=f^{\ast }[2r]$. Complex transition maps preserve the fiber orientation, so these formulas glue and give the oriented Borel–Moore pullback of degree $2r$. Projection formula and the trace/base-change mates of [F1] give compatibility with proper pushforward and refined Cartesian pullback, without asserting ordinary purity on singular fiber products. These are the calculus operations, rather than numerical crossing formulas. AC is inherited through [A1].