Homotopy Types of Schemes in Algebraic Geometry

ag.algebraic-geometryat.algebraic-topologycv.complex-variableshomotopy-theoryschemes

Let $X$ be a scheme over $\mathbb{C}$.

  1. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?

  2. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the weak homotopy type of a finite CW-complex, (i.e. when is it a finite space)?

By "when", I mean what adjectives do I have to add to make this true, e.g. finite type, separated, smooth…

  1. I'm also interested in question 1.) for when $X$ is affine.

If you happen to know a reference also, that would be fantastic. Thanks!

P.S. I'm aware of this mathoverflow question:

How to prove that a projective variety is a finite CW complex?

However, it addresses only the case of varieties, unless I am missing something..

Best Answer

Any scheme which is separated of finite type, has at least a triangulation, hence is, in particular, a CW-complex. In fact, by a theorem of Lojasiewicz, this is true for any semi-algebraic set (one can even get this for subanalytic sets, by a result of Hironaka, in Triangulation of algebraic sets, Proc. Amer. Math. Soc. Inst. Algebra Geom. Arcata(1974)); however, the case of (possibly singular) algebraic varieties goes back to the early times of Algebraic Topology: e.g. these papers of van der Waerden and of Lefschetz and Whitehead).

If you only are interested in weak homotopy types, it follows from Lurie's proper base change theorem that considering complex points satisfies proper (hyper)descent (this is Prop. 3.21 in this paper of A. Blanc, which is now published in Compositio Math.). Using Hironaka's resolution of singularities theorem, this implies that, for any scheme of finite type $X$, the space $X(\mathbf{C})$ is a finite homotopy colimit of spaces of the form $Y(\mathbf{C})$ with $Y$ affine and smooth (using Mayer-Vietoris-like homotopy pushouts associated to blow-ups and to coverings by Zariski open subschemes). A smooth affine algebraic variety has the homotopy type of a finite CW-complex: this follows from Morse theory, as can be seen from (the proof of) Theorem 7.2, page 39 in Milnor's book Morse Theory.

From all this, we get that a sufficient condition for $X(\mathbf{C})$ to have the weak homotopy type of a finite CW-complex is to be of finite type, while a sufficient condition to get the homotopy type of a finite CW-complex is to be separated of finite type. A sufficient condition to get an actual finite triangulated space is to be proper.

If we drop the assumption that the scheme is of finite type, I don't see how we can control/define what happens unless we work with pro-homotopy types of some sort.

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