non-singular variety and CW complex structure

Question

We know non-singular projective variety is a manifold, then it's natural to ask when a non-singular projective(affine) variety can be given a CW-structure? Is there any reference to give a proof?

Answer 1

If $X \subset \Bbb{CP}^n$ is a non-singular algebraic variety then on any affine chart $\text{U} \subset \Bbb{CP}^n$, the subvariety $\text{U} \cap X$ is cut out of $\text{U}$, which we identify with $\Bbb C^n$, using a system of polynomial equations $f_1 = \cdots = f_m = 0$ where they jointly constitute a map $F =(f_1, \cdots, f_m) : \Bbb C^n \to \Bbb C^m$ such that $0$ is a regular value of $F$. By inverse function theorem, this implies $X$ is locally a smooth submanifold of $\Bbb{CP}^n$, hence globally so as well.

In particular, $X$ is a compact smooth manifold. Such things always admit triangulation: Consider a Morse function $f : X \to \Bbb R$ and we can ensure that the critical values are arranged $c_0 < c_1 < \cdots < c_m$ in the increasing order of index $0 = \lambda_0 < \lambda_1 < \cdots < \lambda_m = \dim X$. By Morse theory, $f^{-1}[c_0, c_k + \varepsilon]$ is obtained from $f^{-1}[c_0, c_k - \varepsilon]$ by attatching a $\lambda_k$-handle ($D^{\lambda_k} \times D^{2n - \lambda_k})$ to the boundary $f^{-1}(c_k - \varepsilon)$ of $f^{-1}[c_0, c_k - \varepsilon]$. Write $M_k = f^{-1}[c_0, c_k - \varepsilon]$, where $\varepsilon$ is sufficiently small so that there is a unique critical value, namely $c_k$, between $c_k - \varepsilon$ and $c_k + \varepsilon$ for all $k$. Then $X$ has a filtration $M_1 \subset M_2 \subset \cdots \subset M_{m - 1} \subset M$ such that each $M_{k+1}$ is iteratively obtained from $M_k$ upto homeomorphism using the handle-attatching recipie above. This gives a handlebody decomposition of $X$, and by collapsing each handle to it's core, we get a CW-decomposition of $X$. See, for example, Milnor "Morse theory".

If $X \subset \Bbb C^n$ is an affine variety, similar arguments as above would work. In this case there's a very natural Morse function $L_p : X \to \Bbb R$ given by $L_p(\mathbf{x}) = \|\mathbf{x} - p\|^2$, that is the squared-distance from $p$. For generic choice of $p$, this can be shown to be a Morse function (roughly speaking, the critical points of $L_p$ occur where the secant joining $\mathbf{x}$ and $p$ points normally to $X$, and it is a degenerate critical point if there is some tangent direction in $T_\mathbf{x} X$, the normals keep intersecting $p$ even when shifted along that direction, so that $p$ is a focal point of $M$ near $\mathbf{x}$ with a certain multiplicity determined by the number of independent tangent directions along which this happens. Fortunately, focal points to $X$ form a measure zero subset of $\Bbb C^n$)

As an addendum, the "nonsingular" hypothesis should not be really necessary. If $X \subset \Bbb{CP}^n$ is an arbitrary projective algebraic variety, then notice that the subset $X_{sing} \subset X$ consisting of the singular points of $X$ form a subvariety. This can be checked locally on an affine chart: say $A \subset \Bbb C^n$ is an affine variety cut out by $f_1, \cdots, f_m$. Then $A_{sing} \subset A$ consists of the points where rank of the Jacobian $J(f_1, \cdots, f_m)$ is strictly less than $m$. This is the algebraic condition that $\det J = 0$, which therefore cuts out $A_{sing} \subset A$ as an algebraic subvariety. However this singular subvariety might also be singular! So we can iteratively build a filtration $X_0 \subset X_1 \subset \cdots \subset X_{k-1} \subset X_k = X$ where $X_{i - 1}$ is the singular subvariety of $X_i$, so $X_i \setminus X_{i-1}$ is a smooth quasiprojective variety.

I haven't checked, but this should be a Whitney stratification of $X$. I intuit this as follows: it's a corollary of Thom's first isotopy lemma that Whitney stratified spaces are locally conical, that is, if $L_1, L_2$ are two strata such that $L_1 \subset \overline{L_2}$ then any $p \in L_1$ has a neighborhood $U$ in $\overline{L_2}$ such that there is a stratum-preserving homeomorphism $U \to \Bbb R^k \times \text{Cone}(S)$ where $S$ is a stratified set of appropriate dimension. But in the case of algebraic varieties, we already have cone-like neighborhoods to singular points given by the Zariski tangent cone! EDIT: This statement is not true, as pointed out correctly in the comments. It's instructive to look at a kind of example described by Stephen, given by $X = \{(x, y, z) \in \Bbb C^3 : y^2 = x^3 + z^2x^2\}$. The singular set is given by $X_0 = \{x = y = 0\}$ as can be checked by computing when the Jacobian of $y^2 - x^3 - z^2x^2$ vanishes. However $(X_0, X \setminus X_0)$ does not satisfy the Whitney condition (b), as can be checked by looking at the real locus: the limiting tangent planes along any appropriate sequence of points $p_n \to (0, 0, 0)$ converges to the plane $z = 0$, whereas the secants joining $p_n$ with $(0, 0, 1/n)$ converge to a line in $x = y = 0$, which is orthogonal to the limiting tangent plane! The problem, however, is resolved by taking $X_1 = \{(0, 0, 0)\}$ and defining a refined stratification $(X, X_0 \setminus X_1, X_1)$ which turns out to be a Whitney stratification. It's in fact a theorem of Whitney that any complex projective variety can be stratified and his idea seems to be precisely to come up with a finer stratification than the singular stratification where the condition (b) works. I haven't read the proof, but I plan to in the near future.

Goresky has proved here that any Thom-Mather stratified space admits a triangulation. Whitney stratified spaces always admit controlled tubular neighborhoods (See, for example, Mather's "Notes on Topological Stability"), so are in particular triangulable.