2019-03-14 23:52:23 8 Comments

My question is simple, but I don't expect there are any simple answers.

Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points. Suppose we are given spaces $X'\simeq X(\mathbb{C})$ and $Y'\simeq Y(\mathbb{C})$, where $\simeq$ denotes weak equivalence of spaces.

What is known about necessary or sufficient conditions for a map $f':X'\to Y'$ to be homotopy equivalent to a map coming from a morphism $f:X\to Y$ of schemes?

I am specifically interested in showing that a particular map of spaces is *not* homotopy equivalent to a map coming from a morphism of schemes.

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