Generic complex algebraic Whitney regularity

Put $d=\dim S$ and let $G$ be the graph of the map $(x,y)\mapsto (T_{x}S,[x-y])$ from $S\times Y$ to $\mathrm{Gr}(d,n)\times {\mathbb{P}}^{n-1}$; it is algebraic by [F5], because on chart domains the tangent plane is the kernel of the Jacobian matrix of local equations, whose basis entries are regular ratios on each fixed nonzero-minor chart and the secant coordinates are regular since $x\mathrm{\ne }y$. Let $Z$ be the reduced closure of $G$ in $\bar{S}\times Y\times \mathrm{Gr}(d,n)\times {\mathbb{P}}^{n-1}$. The graph $G$ is isomorphic to $S\times Y$, hence irreducible by [F6], and $\bar{S}$, $Y$, $\mathrm{Gr}(d,n)$ and ${\mathbb{P}}^{n-1}$ are irreducible; the ambient product is therefore irreducible by [F6], and its closed subset $Z$, the closure of the irreducible $G$, is irreducible as well. Let $B=Z\setminus G$, a proper closed subset, and let $g:Z\to Y$ be the projection; it is dominant because it is onto on $G$.

Apply [F2] to $g$ to obtain a dense open subset of $Y$ over which the fibres $Z_y$ are nonempty and pure of dimension $r=\dim Z-\dim Y$. This number is positive, because each fibre contains $G_y=\{(x,y,T_{x}S,[x-y]):x\in S\}\cong S$ of dimension $\dim S\ge 1$. Each irreducible component $B_i$ of the proper closed subset $B$ has $\dim B_i<\dim Z$ by [F4]; if $B_i\to Y$ is dominant, [F2] gives a further dense open of $Y$ over which its fibres have pure dimension $\dim B_i-\dim Y\le r-1$, and if it is not dominant we remove its image closure. Intersecting the finitely many dense opens leaves a dense open $Y^0\subseteq Y$ on which $Z_y$ is pure $r$-dimensional while $B_y$ has dimension at most $r-1$; hence no irreducible component of $Z_y$ is contained in $B_y$, and for every component $V$ of $Z_y$ the set $V\cap G_y=V\setminus B_y$ is a nonempty dense open subset of the irreducible $V$.

Fix $y\in Y^0$ and an incidence point $z_0=(y,y,T,[u])\in Z_y$ with $u\mathrm{\ne }0$. Let $V$ be an irreducible component of $Z_y$ containing $z_0$ and let $\gamma$ be the holomorphic arc supplied by [F3] for the variety $V$, the nonempty Zariski open $V\setminus B_y$ and the point $z_0$. Since $\gamma$ maps into $Z_y$ and $Z_y\subseteq g^{-1}(y)$, its second coordinate is the constant $y$; write $\gamma (s)=(x(s),y,T(s),[u(s)])$ with $\gamma (0)=z_0$ and $\gamma (s)\in G$ for $s\mathrm{\ne }0$. Then $T(s)=T_{x(s)}S$ and $[u(s)]=[x(s)-y]$ for $s\mathrm{\ne }0$, so $x(s)\mathrm{\ne }y$ for $s\mathrm{\ne }0$. Choosing a projective chart containing $[u]$, the normalized representative of the arc is holomorphic, and since $x(s)-y$ is not identically zero some coordinate has finite positive vanishing order; with $m\ge 1$ the minimum of these orders, $x(s)-y=s^{m}v(s)$ with $v$ holomorphic and $v(0)\mathrm{\ne }0$, and $[v(0)]=[u]$.

Let $p_0$ be a complex covector annihilating the complex plane $T$. In a Grassmannian matrix chart around $T$ write $T(s)$ as the graph of a holomorphic matrix $A(s)$ with $A(0)=A_0$, possible for small $s$ because $T(s)\to T$; every annihilator of the graph of $A(s)$ has the form $q(s)=(-A(s{)}^{t}b,b)$ for $b\in ({\mathbb{C}}^{n-d}{)}^{\ast }$, and $b$ can be chosen with $(-A_0^{t}b,b)=p_0$ because $p_0$ annihilates $T$. The resulting $q(s)$ is holomorphic and annihilates each $T(s)$. Since $x(s)\in S$ for $s\mathrm{\ne }0$, its derivative lies in $T_{x(s)}S=T(s)$ by [F5], so $q(s)(x^{\mathrm{\prime }}(s))=0$ for $s\mathrm{\ne }0$. Differentiating $x(s)-y=s^{m}v(s)$ gives $x^{\mathrm{\prime }}(s)=s^{m-1}(mv(s)+s{v}^{\mathrm{\prime }}(s))$, and dividing by $s^{m-1}$ and letting $s\to 0$ yields $p_0(mv(0))=0$, hence $p_0(v(0))=0$. Every annihilator of $T$ therefore kills $v(0)$, so $v(0)\in T$ and the limiting complex secant line $[u]$ lies in $T$.

Now let $x_j\in S$ and $y_j\in Y$ be sequences converging to the same point $y\in Y^0$, with tangent planes $T_{x_j}S$ converging to a plane $T$ and real secant lines through $x_j,y_j$ converging to a real line $\mathrm{\ell }$. After choosing their signs, the secant directions normalized to unit vectors converge to a unit vector $w$ spanning $\mathrm{\ell }$, and the complex projective points $[x_j-y_j]=[w_j]$ converge to $[w]$, the complex line spanned by $\mathrm{\ell }$; moreover $(x_j,y_j,T_{x_j}S,[x_j-y_j])\in G\subseteq Z$ converges to $(y,y,T,[w])\in Z_y$ because $Z$ is closed. Applying steps 3.1 and 4.1 to this incidence point shows that its limiting complex secant line lies in $T$, so $w\in T$ and $\mathrm{\ell }=\mathbb{R}w\subseteq T$. This is Whitney (b) at $y$ in the full moving-point form.

For Whitney (a) at $y$, let $x_j\in S$ converge to $y$ with tangent planes converging to $T$, and let $w\in T_{y}Y$ be a nonzero real tangent vector (the zero vector already belongs to $T$). In a holomorphic chart of the smooth manifold $Y$ near $y$ choose points $y_j\in Y$ of the form $y+{ϵ}_{j}w+o({ϵ}_j)$ with ${ϵ}_j>0$, ${ϵ}_j\to 0$ and $\mathrm{\parallel }{x}_j-y\mathrm{\parallel }\mathrm{/}{ϵ}_j\to 0$, for instance ${ϵ}_j=\sqrt{\mathrm{\parallel }{x}_j-y\mathrm{\parallel }}$ and a straight segment in the chart. Then $\mathrm{\parallel }{x}_j-y_j\mathrm{\parallel }\ge {ϵ}_j\mathrm{\parallel }w\mathrm{\parallel }\mathrm{/}2$ for large $j$, and the secant directions from $y_j$ to $x_j$ converge to the real line spanned by $w$. Since the tangent planes $T_{x_j}S\to T$, Whitney (b) from step 5.1 places that real line in $T$, so $w\in T$. Every real tangent vector of $T_{y}Y$ arises in this way, so $T_{y}Y\subseteq T$, proving Whitney (a).

Set $E=Y\setminus Y^0$; it is a proper closed algebraic subset by step 2.1, and steps 5.1 and 6.1 show that $(S,Y)$ satisfies (a) and (b) at every point of $Y\setminus E$. For finitely many prescribed incident upper strata $S_1,\dots ,S_m$ over the same irreducible $Y$, apply the construction to each and take the union of the finitely many proper closed exceptional sets; since $Y$ is irreducible, that finite union is still proper, by [F4]. For a quasi-projective ambient manifold, use the finitely many affine ambient charts $U_{\alpha }$ that meet $Y$; their intersections with the irreducible $Y$ are nonempty opens. Inside each chart apply the affine result to every pair of irreducible components $(S_i,Y_j)$ of $(S\cap U_{\alpha },Y\cap U_{\alpha })$ with $Y_j\subseteq \overline{S_i}$ and $\dim Y_j<\dim S_i$, and for each point of an irreducible component of $Y\cap U_{\alpha }$ one of these pairs accounts for every sequence in $S$ converging to it after passing to a subsequence in a component. Each chart exceptional set is closed in $Y\cap U_{\alpha }$ and has smaller dimension than $Y$, so its closure in $Y$ is proper by [F4]. The union of these finitely many closures and the proper closed complements $Y\setminus U_{\alpha }$ is still proper because $Y$ is irreducible. Outside this union every chart argument applies, giving the required closed $E⊊Y$.