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.

neverrealizable in this manner, since the map would extend through the (simply connected) $\mathbb{P}^{1}$ $\endgroup$ – EBz Mar 15 '19 at 12:591more comment