Complex algebraic morphisms admit compatible finite smooth stratifications

We record a partition construction. Let $Z$ be a reduced separated complex algebraic variety of finite type and let $\mathcal{C}$ be a finite list of closed subsets of $Z$. There is a finite partition of $Z$ into irreducible smooth locally closed strata such that every member of $\mathcal{C}$ is a union of strata and every stratum is open dense in its own irreducible closure. If $Z=\mathrm{\varnothing }$ take the empty partition. Otherwise let $Z_1,\dots ,Z_m$ be the irreducible components of $Z$, finitely many by [F1]. Remove from $Z_i$ its nonsmooth locus, the other components $Z_j$, and every $C\cap Z_i$ with $C\in \mathcal{C}$ and $C⊉Z_i$. Each removed set is closed, and each is proper in the irreducible $Z_i$; by [F3] and [F2] the nonsmooth locus has smaller dimension, and by [F2] the other removed sets do too, so the result $U_i$ is a nonempty dense open subset of $Z_i$ and is contained in or disjoint from every $C\in \mathcal{C}$. The $U_i$ are pairwise disjoint and open in $Z$, because their complement is the finite union of the removed closed sets and the sets $Z_j$ with $j\mathrm{\ne }i$. The remainder $Z^{\mathrm{\prime }}:=Z\setminus {\bigcup }_i{U}_i$ is closed and meets each $Z_i$ in a proper closed subset, so $\dim Z^{\mathrm{\prime }}<\dim Z$ by [F2]. Partition $Z^{\mathrm{\prime }}$ by the same construction with $\mathcal{C}$ replaced by the restrictions of all members of $\mathcal{C}$ and of all $Z_i$ to $Z^{\mathrm{\prime }}$, and iterate. The dimension of the remainder strictly decreases at every round, so the recursion terminates; because each round respects the restricted lists, every member of $\mathcal{C}$ is a union of strata and every stratum is open dense in its irreducible closure. The closure of a stratum emitted in a recursion round is a component of that round's closed remainder; it meets no earlier stratum and no other stratum emitted in that round, and the added component restrictions make it a union of subsequent strata. Thus this partition also has the frontier property.

Now construct the target strata in rounds on a closed residual subvariety $R\subseteq Y$, starting with $R=Y$; at each stage keep finite lists of prescribed closed subsets of $R$ and of $f^{-1}(R)$, initially the given lists and later those lists enlarged by the restrictions of the closures of all previously emitted strata. Apply step 1.1 to $f^{-1}(R)$ with the maintained source list, obtaining a finite partition into irreducible smooth locally closed strata $Z_{\alpha }$. Fix an irreducible component $R_i$ of $R$ and remove from it its nonsmooth locus, the other components of $R$, and every maintained target prescribed closed subset not containing $R_i$; by [F1] this leaves a nonempty dense open subset $V_i^0\subseteq R_i$. For each $\alpha$ the image $f(Z_{\alpha })$ is irreducible and lies in a component of $R$, so its closure in $R$ is an irreducible closed subset of some component. If that closure is not $R_i$, then its intersection with $R_i$ is a proper closed subset, and we remove it from $V_i^0$; if the closure equals $R_i$, then $f{\mathrm{\mid }}_{Z_{\alpha }}$ is dominant, $Z_{\alpha }$ is smooth and irreducible, and [F4] supplies a dense open subset of $R_i$ over which $Z_{\alpha }\to R_i$ is smooth, with nonempty fibres over the closed points after shrinking to lie in the image. There are finitely many $\alpha$, so intersecting $V_i^0$ with all the supplied dense opens contained in the images produces a nonempty dense open $V_i\subseteq R_i$ such that every $Z_{\alpha }$ whose image closure is not $R_i$ has no point over $V_i$, while every $Z_{\alpha }$ whose image closure is $R_i$ maps smoothly and onto $V_i$.

Emit the target strata $V_i$ of this round and, for every $\alpha$ whose image closure $\overline{f(Z_{\alpha })}$ in $R$ equals $R_i$, the source stratum $S_{\alpha ,i}:=Z_{\alpha }\cap f^{-1}(V_i)$, which is nonempty. It is open in $Z_{\alpha }$ because $V_i$ is open in $R_i$ and $Z_{\alpha }\subseteq f^{-1}(R_i)$, hence it is an irreducible smooth locally closed subvariety of $X$, and it maps smoothly and onto $V_i$ as the restriction of the map of step 2.1. Every source point over $V_i$ lies in exactly one such stratum, because the $Z_{\alpha }$ partition $f^{-1}(R)$ and their intersections with $f^{-1}(V_i)$ are exactly the emitted pieces; those with other image closures have empty intersection by step 2.1. Within the round, closures behave as in step 1.1: the closure of $S_{\alpha ,i}$ meets $f^{-1}(V_i)$ in the union of the strata $S_{\beta ,i}$ with $Z_{\beta }\subseteq \overline{Z_{\alpha }}$, since $f^{-1}(V_i)$ is open in the closed set $f^{-1}(R_i)$.

Put $R^{\mathrm{\prime }}:=R\setminus {\bigcup }_i{V}_i$; it is closed and has $\dim R^{\mathrm{\prime }}<\dim R$ by [F2], since each component of $R$ lost a dense open subset. Enlarge both lists by the closures of the emitted strata, restrict to $R^{\mathrm{\prime }}$ and $f^{-1}(R^{\mathrm{\prime }})$, and repeat steps 2.1 and 3.1. The target dimension strictly decreases each round, so after finitely many rounds the emitted strata partition $Y$ and $X$. Every emitted stratum $S$ is open dense in its irreducible closure: a target stratum $V_i$ is open dense in $R_i$, while a source stratum $S_{\alpha ,i}$ is a nonempty open subset of $Z_{\alpha }$ and has the same irreducible closure. For source strata the within-round frontier property follows from steps 1.1 and 3.1; closures lie in $f^{-1}(R_i)$ and meet no source stratum over another $V_j$. Target closures likewise meet no other target stratum of the round. Each closure lies in the current closed residual and hence meets no earlier stratum. Finally the maintained lists contain these closures in every later round, forcing every later stratum to be contained in or disjoint from each closure. Consequently $\bar{S}$ is a union of strata, and every stratum $P\subseteq \bar{S}\setminus S$ has $\dim P<\dim S$: indeed $\bar{P}$ is an irreducible closed subset of $\bar{S}$ with $\dim \bar{P}\le \dim S$, equality would give $\bar{P}=\bar{S}$, and then $P$ and $S$, being dense opens of the irreducible $\bar{P}=\bar{S}$, would meet, contradicting disjointness of strata. Thus every stratum closure is a union of strata of no larger dimension and its boundary has strictly smaller dimension. Every prescribed subset is a union of strata by the invariance maintained in steps 1.1 and 2.1, and the empty source contributes no source strata.

Finally let $S\to T$ be one emitted restriction onto a target stratum. Zero-dimensional smooth strata are locally single points: they are zero-dimensional Noetherian varieties, hence finite sets of closed points, so their analytic charts and maps to a zero-dimensional target require no implicit-function or constant-rank theorem. A standard smooth presentation with no equations already gives open coordinate charts. In the remaining positive-dimensional cases, the stratum $S$ is a smooth complex variety by construction, so by [F3] it is locally standard smooth over $\mathbb{C}$; the defining equations of a standard smooth presentation near a point have an invertible Jacobian minor, and [F5] turns them into holomorphic charts of $S$ near that point, with holomorphic transition maps. Similarly $T$ carries holomorphic charts. The map $S\to T$ is a morphism of classical varieties, so in these charts its components are restrictions of regular functions on affine charts, hence locally polynomial quotients with nonvanishing denominators in the ambient coordinates and therefore holomorphic in the chart parameters. At every point the differential of $S\to T$ is surjective by the smoothness of the restriction and the criterion of [F5]; since surjectivity onto a target of dimension $n$ is the nonvanishing of some $n\times n$ minor, the rank stays full on a neighbourhood, and the holomorphic constant rank theorem exhibits the map near each point as a linear projection in suitable holomorphic coordinates. Hence the restriction is a submersion of complex manifolds, in particular a submersion of the underlying real smooth manifolds. No properness or topological triviality is used, and AC is inherited exactly through the cited suppliers.

Each emitted source and target piece is nonempty and irreducible, so [F6] proves its analytic connectedness. Its analytic closure is its algebraic closure by [F6]; therefore the algebraic frontier bookkeeping of step 4.1 is the analytic frontier condition as well. Combined with step 5.1 this produces finite connected smooth analytic strata with the required frontier and submersion restrictions, while preserving all algebraic compatibility and dimension conclusions.