@IanGershonTeixeira That is correct. Here it's important that by "homotopic" we mean as a map, not as a diffeomorphism (i.e., the intermediate maps are allowed to not be invertible). Note that once you have a homotopy, you can always perturb it to be smooth.

@IgorBelegradek Hmm, I couldn't find it there. Is it in one of the linked papers?

@IanGershonTeixeira Just to be clear, the result I wrote only works for Serre fibrations, which are not just the "up to homeomorphism" but also the "up to homotopy" version of bundles (I edited my preamble to make that clearer). I'm not sure whether the "up to homeomorphism" result is true, perhaps someone else can comment on this.

Right. Tom Goodwillie's answer came while I was writing mine, and I added a disclaimer.

@JasonStarr Thank you! I had trouble finding the local cohomology identification (the pdf I have isn't easily searchable). But I'm pretty sure you're right. One argument to see this is to interpret the Serre dualizing complex using Verdier duality of holonomic D modules, which can be done in the analytic topology.

Thank you! Somehow I convinced myself that you need extra conditions for the right adjoint to exist, but of course you are right that it works for general proper maps. In the singular case (in characteristic 0), is the solid structure an invariant of the variety, or does it depend on a compactification?