Affine algebraic Whitney strata admit distance exhaustions with finitely many stratified Morse points

For a smooth stratum $S$ of complex dimension $s$, its algebraic conormal bundle $C_S\subset T^{\ast }{\mathbb{C}}^N$ has complex dimension $N$: on a Jacobian chart the independent differentials of local equations span the conormal fibres, of rank $N-s$, giving a locally trivial algebraic bundle. On each irreducible component the closure of $C_S$ has dimension $N$ and its boundary has dimension at most $N-1$, since a proper closed subset of an irreducible variety of dimension $N$ has dimension at most $N-1$. Let $D$ be the finite union of these conormal boundaries. If a covector $\lambda$ kills a limiting tangent plane $Q=\lim T_{p_m}S$ at a point outside $S$, then $\lambda$ lies in that boundary: the orthogonal projections of $\lambda$ onto the annihilators of $T_{p_m}S$ are conormal covectors at the points $p_m$ and converge to the projection of $\lambda$ onto $Q^{\perp }$, which is $\lambda$ because $\lambda$ kills $Q$. Thus every bad limiting-tangent covector lies in the complex algebraic set $D$ of complex dimension at most $N-1$.

Consider the real critical incidence $I_S=\{(a,z):z\in S,d({\varphi }_a\mathrm{\mid }S{)}_z=0\}$. Differentiation in $a$ surjects onto $T_z^{\ast }S$, since every real covector on $T_{z}S$ extends to one on the ambient Euclidean space; concretely $I_S$ is the real normal bundle via $(z,v)\mapsto (a=z+v,z)$ with $v\perp T_{z}S$, and the smooth Gram-matrix charts of that normal bundle exhibit it as a smooth real manifold of dimension $2s+(2N-2s)=2N$. In local stratum coordinates its tangent equation is $H_{z}v+B_z\delta a=0$, where $H_z$ is the Hessian of ${\varphi }_a\mathrm{\mid }S$ and $B_z$ is the surjective parameter derivative. The differential of $(a,z)\mapsto a$ from $I_S$ to ${\mathbb{C}}^N$ has kernel $\{(0,v):H_{z}v=0\}$; since source and target have equal real dimension, it is surjective exactly where $H_z$ is invertible. Hence [F1] makes the parameters with a degenerate stratum critical point null, and a finite union over the strata preserves nullity.

Identify real ambient covectors with complex covectors by $\lambda \mapsto \mathrm{Re}\lambda$. The map $(a,z)\mapsto (z,d{\varphi }_a(z))$ from ${\mathbb{C}}^N\times {\mathbb{C}}^N$ to the real cotangent bundle is a real smooth diffeomorphism, because $d{\varphi }_a(z)$ is twice the real covector corresponding to $\overline{z-a}$. Therefore the inverse image of each smooth algebraic stratum of $D$ has real dimension at most $2N-2$, and its projection to the $a$-coordinate is a smooth map whose source dimension is smaller than its target dimension $2N$, so every point is critical and [F1] makes the image null; a finite algebraic stratification of $D$ is obtained by successively taking each component's smooth locus and its lower-dimensional singular complement, which terminates because the dimension strictly decreases, and the finite union of the null images is null. Outside this null set, $d{\varphi }_a$ kills no forbidden limiting tangent space.

Finiteness of the critical set uses algebraic geometry rather than a finiteness claim for real analytic sets. Complexify the squared distance to $F_{u,v}(z,w)={\sum }_{k=1}^N(z_k-u_k)(w_k-v_k)$ on $S\times \bar{S}$, where $\bar{S}$ is obtained by conjugating the equations of $S$. Its complex critical incidence over the parameter space ${\mathbb{C}}^N\times {\mathbb{C}}^N$ is the algebraic incidence cut out by the two conormal conditions $w-v\perp T_{z}S$ and $z-u\perp T_w\bar{S}$; it is isomorphic to the algebraic conormal bundle of $S\times \bar{S}$, of complex dimension $2N$, equal to the parameter dimension, because the two gradient coefficients $w-v$ and $z-u$ are exactly the conormal vectors that uniquely determine $(u,v)$ for given $(z,w)$. If an irreducible component projects dominantly, [F2] gives a nonempty open of parameters over which its fibres are zero-dimensional and hence finite by [F2]; if not, one avoids its proper closed image closure. The set of parameters $a$ with $(a,\bar{a})$ in this nonempty complex Zariski open is Euclidean open and dense. Indeed, a nonzero polynomial vanishing on its proper algebraic complement restricts to a nonzero polynomial in the real coordinates $(x,y)$ under the invertible substitution $(a,\bar{a})=(x+iy,x-iy)$; such a polynomial cannot vanish on any nonempty real open box. Thus the inverse image of the complement has empty interior. This inverse image involves complex conjugation, so no complex Zariski openness of the good real-parameter set is asserted. Therefore the prescribed nonempty real open subset meets it in a nonempty real open set, which cannot be contained in the null exceptional sets of steps 1.2 and 2.1; choose such a real parameter $a$. Its real stratum critical points lie in the finite complex fibre over $(a,\bar{a})$ just constructed: at $(z,\bar{z})$ the real diagonal tangent vectors $(t,\bar{t})$ span $T_{z}S\times T_{\bar{z}}\bar{S}$ over $\mathbb{C}$, and the complex differential of $F$ on them is the real differential of ${\varphi }_a$, so vanishing on the real form implies vanishing on the whole complex tangent.

Choose such an $a$. The Euclidean square distance is bounded below and proper on the closed subset $Z$, and its ambient Levi form is the positive definite Euclidean Hermitian form, so its restriction to every positive-dimensional complex stratum is strongly plurisubharmonic and the sublevel sets exhaust $Z$; on the singular variety the strong plurisubharmonicity statement is the stated inline verification for the concrete function $\mathrm{\parallel }z-a{\mathrm{\parallel }}^2$, whose ambient complex Hessian is the identity. Steps 1.2, 2.1 and 3.1 give exactly the nondegeneracy, limiting-tangent and finiteness conclusions: on a zero-dimensional stratum the Hessian condition is vacuous because its tangent space is zero, and the ambient differential condition there is the step 2.1 exclusion, while the stratum contributes only its finitely many points, and the empty variety has no critical points. The argument works verbatim on any closed affine complex algebraic embedding of $Z$, since only the Euclidean structure of the ambient affine space is used. No stratified product theorem or complex-link identification is used to establish this exhaustion.