Voevodsky’s “geometric” criterion for 6-functor formalisms with applications to the stable motivic homotopy theory of complex analytic stacks

In this talk I will present some enhancements and generalizations of a criterion for six-functor formalisms first sketched by Voevodsky in 2001. This principle was then implemented by Ayoub in order to show that the stable motivic homotopy theory of quasi-projective schemes has the structure of a six-functor formalism, although it has later been generalized by works of Cisinski, Déglise, Hoyois, Khan, and Ravi, leading to a six-functor formalism of genuine stable motivic homotopy theory on qcqs derived algebraic stacks with separated diagonals and nice stabilizers.

In our framework, we produce six-functor formalism using the cohomological behaviour of smooth maps, closed immersions, and smooth proper maps (where the relevant cohomological property is expressed by a version of Atiyah duality). This is related to recent results of Dauser-Kuijper and Cnossen-Lenz-Linskens, which enhances work of Mann following Liu-Zheng on the construction of six-functor formalisms using the cohomological behaviour of étale maps and proper maps. Our general results are then used to produce a six-functor formalism of complex analytic stable motivic homotopy theory, as well as equivariant analytification functors that are compatible with the six operations.