Related papers: A comparison between compactly supported rigid and…

We first describe a canonical mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a D-module on the…

We prove a finiteness theorem and a comparison theorem in the theory of \'etale cohomology of rigid analytic varieties. By a result of Huber, for a quasi-compact separated morphism of rigid analytic varieties with target being of dimension…

We study the $p$-adic analogue of the $\ell$-adic hypergeometric sheaves for reductive groups, called the hypergeometric $\mathscr{D}^{\dagger}(\infty)$-modules. They are overholonomic objects in the derived category of arithmetic…

We construct the $\Lambda$-adic crystalline and Dieudonn\'e analogues of Hida's ordinary $\Lambda$-adic \'etale cohomology, and employ integral $p$-adic Hodge theory to prove $\Lambda$-adic comparison isomorphisms between these cohomologies…

A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over…

Let $M= G/\Gamma$ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space ${\cal C} ({\frak g})$ of invariant complex structures on $M$, the…

In the first part we deepen the six-functor theory of (holonomic) logarithmic D-modules, in particular with respect to duality and pushforward along projective morphisms. Then, inspired by work of Ogus, we define a logarithmic analogue of…

We construct a fully-faithful functor of $\infty$-categories from complexes of D-cap modules with Fr\'echet cohomology to quasi-coherent sheaves on an analytic stack. We prove various descent results for $\infty$-categories of D-cap modules…

Contraherent cosheaves are module objects over algebraic varieties defined by gluing using the colocalization functors. Contraherent cosheaves are designed to be used for globalizing contramodules and contraderived categories for the…

L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of a locally symmetric space. We define the micro-support of an L-module; it is a set of irreducible modules for the Levi…

We introduce a category of 'rigid spaces with overconvergent structure sheaf' which we call dagger spaces --- this is the correct category in which de Rham cohomology in rigid analysis should be studied. We compare it with the (usual)…

Let R be a regular ring essentially of finite type over a perfect field k. An R-module M is called a unit R[F]-module if it comes equipped with an isomorphism F*M-->M where F denotes the Frobenius map on Spec R, and F* is the associated…

We compute the cohomology with compact supports of a Picard modular surface as a virtual module over the product of the appropriate Galois group and the appropriate Hecke algebra. We use the method developed by Ihara, Langlands, and…

We compute formal invariants associated with the cohomology sheaves of the direct image of holonomic D-modules of exponential type. We also prove that every formal C[[t]]<\partial_t>-modules is isomorphic, after a ramification, to a germ of…

Following earlier work with Cris Negron on the cohomology of Drinfeld doubles $D(\mathbb G_{(r)})$, we develop a "geometric theory" of support varieties for "extended Drinfeld doubles" $\tilde D(\mathbb G_{(r)})$ of Frobenius kernels…

We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify…

We study the behaviour of D-cap-modules on rigid analytic varieties under pushforward along a proper morphism. We prove a D-cap-module analogue of Kiehl's Proper Mapping Theorem, considering the derived sheaf-theoretic pushforward from…

We state a conjecture relating de Rham cohomology of a smooth rigid analytic variety to its compactly supported pro-\'etale cohomology. We prove the conjecture in the cases where the variety is a Stein curve of dimension one or a Stein…

Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the…

This chapter sets out preliminaries for the duality theory in later chapters. An underlying idea is that local cohomology functors are higher derived functors of colocalizations (a.k.a.~coreflections). Predominantly well-known facts about…