We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma _{\infty }$ , where $\Gamma $ denotes the value group of K. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

We give a geometric interpretation of sheaf cohomology for higher degrees $n\geq 1$ in terms of torsors on the member of degree $d=n-1$ in hypercoverings of type $r=n-2$, endowed with an additional datum, the so-called rigidification. This generalizes the fact that cohomology in degree one is the group of isomorphism classes of torsors, where the rigidification becomes vacuous, and that cohomology in degree two can be expressed in terms of bundle gerbes, where the rigidification becomes an associativity constraint.

Let $X$ be a topological space. We consider certain generalized configuration spaces of points on $X$, obtained from the cartesian product $X^{n}$ by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on $X$. Suppose that $X$ is a ‘nice’ topological space, $R$ is any commutative ring, $H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$ is the zero map, and that $H_{c}^{\bullet }(X,R)$ is a projective $R$-module. We prove that the compact support cohomology of any generalized configuration space of points on $X$ depends only on the graded $R$-module $H_{c}^{\bullet }(X,R)$. This generalizes a theorem of Arabia.

We use microlocal sheaf theory to show that knots can only have Legendrian isotopic conormal tori if they themselves are isotopic or mirror images.

Let M be a Hamiltonian K-space with proper moment map $\mu$. The symplectic quotient $X=\mu^{-1}(0)/K$ is a singular stratified space with a symplectic structure on the strata. In this paper we generalise the Kirwan map, which maps the K equivariant cohomology of $\mu^{-1}(0)$ to the middle perversity intersection cohomology of X, to this symplectic setting.

The key technical results which allow us to do this are Meinrenken's and Sjamaar's partial desingularisation of singular symplectic quotients and a decomposition theorem, proved in Section 2 of this paper, exhibiting the intersection cohomology of a ‘symplectic blowup’ of the singular quotient X along a maximal depth stratum as a direct sum of terms including the intersection cohomology of X.