Smooth and noncharacteristic complex pullbacks are perverse-exact

If $P$ is in the upper perverse aisle, a support of ${\mathcal{H}}^{q+d}(P)$ has dimension at most $-(q+d)$. Its inverse image under a smooth relative-$d$ map has dimension at most $-(q+d)+d=-q$. Since ordinary pullback is exact for field-module sheaves, this is exactly the support condition for $a^{\ast }P[d]$, so $a^{\ast }[d]$ preserves the upper aisle. Its relative disk bundle is complex oriented, so the smooth exceptional pullback formula in [F2] gives $a^{!}=a^{\ast }[2d]$. Then $D(a^{\ast }P[d])=a^{!}DP[-d]=a^{\ast }DP[d]$, so the upper-aisle calculation for $DP$ also proves preservation of the lower aisle.

Suppose $P$ is in the upper perverse aisle and constructible for $\mathfrak{S}$. The nonzero stalk loci of its cohomology sheaves are unions of strata, each of dimension bounded by the corresponding support dimension. For a transverse embedding $i$, a stratum $S$ of dimension $s$ meets $Z$ either trivially or in dimension $s-c$, so the support of ${\mathcal{H}}^{q-c}(P)$ meets $Z$ in dimension at most $-(q-c)-c=-q$; this is the upper perverse condition for $i^{\ast }P[-c]$. The exact shift uses transversality on every stratum supporting that cohomology sheaf.

In a local controlled product $W\times D^{2c}$ from [F1], shrink $W$ to a finite conical local model of its induced stratification and triangulate it compatibly by [F3]; the disk also has a finite deleted-boundary model. The pullback of $P$ to this product has bounded finite-rank locally constant cohomology on each simplex times the disk. Such a piece is contractible, so its cohomology sheaves are constant by local-system transport along paths and contracting homotopies. Apply [F3] with base a point and this contractible product as fiber: pullback from the point is fully faithful, and its essential image is closed under finite extensions because connecting morphisms lift. Ordinary truncation triangles therefore show that the complex on each piece is pulled back from a finite coefficient complex. Filter $W$ by closed skeleta. Pullback along $\mathrm{pr}:W\times D^{2c}\to W$ commutes with open extension by zero and closed pushforward: their natural comparisons are isomorphisms on stalks, including zero stalks off the relevant product subsets. On each skeleton [F3] makes this pullback fully faithful; lifting the connecting morphisms in the open/closed triangles and taking their cones proves inductively that $P{\mathrm{\mid }}_{W\times D^{2c}}\cong {\mathrm{pr}}^{\ast }B$, with $B$ its zero-slice restriction. Support localization [F2], computed in the finite product models [F3], now identifies its exceptional zero-slice restriction with $B{\otimes }_k{C}^{\ast }(D^{2c},D^{2c}\setminus \{0\};k)$. The relative normal complex has its sole cohomology $k$ in degree $2c$, with the complex-oriented generator, giving $i^{!}P\cong i^{\ast }P[-2c]$. For $c=0$ this is the identity calculation. These are the natural support comparisons with the oriented normal class; transitions preserve the orientation of the complex normal bundle, so they agree on overlaps and glue.

Verdier duality preserves constructibility for $\mathfrak{S}$, by the stratumwise finite conical coefficient models in [F3] and duality in [F2]. Dualizing gives $D(i^{\ast }P[-c])=i^{!}DP[c]=i^{\ast }DP[-c]$; apply step 1.2 to $DP$ and use [F2] to obtain lower perverse exactness on that subcategory. Together with steps 1.1 and 1.3 this proves both exactness assertions and both oriented exceptional-pullback formulas, including the cases of relative dimension or codimension zero.