Motives of complex-analytic spaces

Question

In any setting where we have a notion of space and a notion of cohomology theory, we in principle could ask "what are motives in this setting?". In some settings the question can be interesting (i.e. algebraic varieties and Weil cohomology theories), in some settings not so much (i.e. topological spaces and cohomology theories satisfying all of the Eilbenberg--Steenrod axioms).

For complex-analytic spaces, I am not completely sure how should we define a good cohomology theory. First question: what are some ways to define this notion? Presumably rational Betti cohomology and de Rham cohomology should be examples of good cohomology theories. Second: can we study motives in this setting? I would guess that the answer turns out to be boring, but it would be nice if there is a 3-page paper where this is proved.

Answer 1

See theorem 1.8 of (Joseph Ayoub. Note sur les opérations de Grothendieck et la réalisation de Betti. J. Inst. Math. Jussieu.)

You can set up a theory of analytic motives, and the resulting category is equivalent to the derived category of vector spaces (over your field of coefficients, typically $\mathbb{Q}$).

In https://arxiv.org/pdf/1810.04968v1.pdf the analogue for p-adic analytic geometry is proven by Bambozzi and Vezzani. (Maybe also relevant: https://arxiv.org/pdf/1708.04284.pdf)