arXiv subjects
Publications and source records attributed to Fritz Hörmann.
For complex tori with algebraic normalized period matrix, in particular for Abelian varieties of CM-type, the map from the second cohomology group of the analytic sheaf of the second Milnor K-groups to the rational cohomology classes of Hodge type (2,2) is surjective (up to denominators).
We discuss Lurie's (derived) bar and cobar constructions, the classical ones for simplicial groups and sets (due to Eilenberg-MacLane and Kan), and the classical ones for differential graded (co)algebras (due to Eilenberg-MacLane and Adams) and their relations, putting them into an abstract framework which makes sense much more generally for any cofibration of infinity-operads. Along these lines we give new and rather conceptual existence proofs of Lurie's adjunction (where bar is left adjoint) and the classical adjunction (where bar is right adjoint). We also recover various classical comparison maps, e.g. the Szczarba and Hess-Tonks maps comparing Adams cobar with Kan's loop group.
Starting from simple and necessary axioms on a (derivator enhanced) four-functor-formalism, we construct derivator six-functor-formalisms using compactifications. This works, for instance, for the stable homotopy categories of Morel-Voevodsky-Ayoub, and also for the classical setting of unbounded complexes of sheaves of Abelian groups on `nice' topological spaces. The formalism of derivator six-functor-formalisms elegantly encodes all isomorphisms between compositions of the six functors (and their compatibilities) and moreover it gives coherent enhancements over diagrams of correspondences. Such a formalism allows to extend six-functor-formalisms to stacks using (co)homological descent.
Several possible presentations for the homotopy theory of (non-hypercomplete) $\infty$-stacks on a classical site S are discussed. In particular, it is shown that an elegant combinatorial description in terms of diagrams in S exists, similar to Cisinski's presentation, based on work of Quillen, Thomason and Grothendieck, of usual homotopy theory by small categories and their smallest (basic) localizer. As an application it is shown that any (local) fibered (a.k.a. algebraic) derivator over S with stable fibers extends to $\infty$-stacks in a well-defined way under mild assumptions.
In this article, it is shown that derivator six-functor-formalisms on any (classical) site canonically extend to higher geometric stacks as defined by Toën-Vezzosi under some natural locality conditions. As an application, it is shown that the six-functor-formalisms of Morel-Voevodsky-Ayoub extend to higher (Nisnevich-)Artin stacks locally of finite type over some fixed base scheme.
We give an elementary description of the space of formal periods of a mixed motive. This allows for a simplified reformulation of the period conjectures of Grothendieck and Kontsevich-Zagier. Furthermore, we develop a machinery which in principle allows to determine the space of formal periods for an arbitrary mixed motive explicitly.
A theory of a derivator version of six-functor-formalisms is developed, using an extension of the notion of fibered multiderivator due to the author. Using the language of (op)fibrations of 2-multicategories this has (like a usual fibered multiderivator) a very neat definition. This definition not only encodes all compatibilities among the six functors but also their interplay with homotopy Kan extensions. One could say: a nine-functor-formalism. This is essential for dealing with (co)descent questions. Finally, it is shown that every fibered multiderivator (for example encoding any kind of derived four-functor formalism $(f_*, f^*, \otimes, \mathcal{HOM})$ occurring in nature) satisfying base-change and projection formula formally gives rise to such derivator six-functor-formalism in which $f_!=f_*$, i.e. a derivator Grothendieck context.
A general method for lifting weak factorization systems in a category S to model category structures on simplicial objects in S is described, analogously to the lifting of cotorsion pairs in Abelian categories to model category structures on chain complexes. This generalizes Quillen's original treatment of (projective) model category structures on simplicial objects. As a new application we show that there is always a model category structure on simplicial objects in the coproduct completion of any (even large) category S with finite limits, which - if S is small - defines the same homotopy theory as any global model category of simplicial pre-sheaves on S. If S is, in addition, extensive, a similar model category structure exists on simplicial objects in S itself. In any case the associated left derivator exists on all diagrams despite the lack of colimits in the original category.
We show that weak equivalences in a (cofibrantly generated) left Bousfield localization of the projective model category of simplicial presheaves can be characterized by a local lifting property if and only if the localization is exact.
We axiomatize the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalized automorphic sheaf is proposed which includes sheaves of (meromorphic) sections of automorphic vector bundles with prescribed vanishing and pole orders along strata in the compactification, and their quotients. These include, for instance, sheaves of Jacobi forms and weakly holomorphic modular forms. Using this machinery we give a short and purely algebraic proof of the proportionality theorem of Hirzebruch and Mumford.
We show that the theory of derivators (or, more generally, of fibered multiderivators) on all small categories is equivalent to this theory on partially ordered sets, in the following sense: Every derivator (more generally, every fibered multiderivator) defined on partially ordered sets has an enlargement to all small categories that is unique up to equivalence of derivators. Furthermore, extending a theorem of Cisinski, we show that every bifibration of multi-model categories (basically a collection of model categories, and Quillen adjunctions in several variables between them) gives rise to a left and right fibered multiderivator on all small categories.
We develop the theory of (op)fibrations of 2-multicategories and use it to define abstract six-functor-formalisms. We also give axioms for Wirthmüller and Grothendieck formalisms (where either $f^!=f^*$ or $f_!=f_*$) or intermediate formalisms where we have e.g. a natural morphism $f_! \rightarrow f_*$. Finally, it is shown that a fibered multiderivator (in particular, a closed monoidal derivator) can be interpreted as a six-functor-formalism on diagrams (small categories). This gives, among other things, a considerable simplification of the axioms and of the proofs of basic properties, and clarifies the relation between the internal and external monoidal products in a (closed) monoidal derivator. Our main motivation is the development of a theory of derivator versions of six-functor-formalisms.
For a regular noetherian scheme $X$ with a divisor with strict normal crossings $D$ we prove that coherent sheaves satisfy descent w.r.t. the 'covering' consisting of the open parts in the various completions of $X$ along the components of $D$ and their intersections.
The theory of derivators enhances and simplifies the theory of triangulated categories. In this article a notion of fibered (multi-)derivator is developed, which similarly enhances fibrations of (monoidal) triangulated categories. We present a theory of cohomological as well as homological descent in this language. The main motivation is a descent theory for Grothendieck's six operations.
We apply the theory of Borcherds products to calculate arithmetic volumes (heights) of Shimura varieties of orthogonal type up to contributions from very bad primes. The approach is analogous to the well-known computation of their geometric volume by induction, using special cycles. A functorial theory of integral models of toroidal compactifications of those varieties and a theory of arithmetic Chern classes of integral automorphic vector bundles with singular metrics are used. We obtain some evidence in the direction of Kudla's conjectures on relations of heights of special cycles on these varieties to special derivatives of Eisenstein series.
We investigate recursive properties of certain p-adic Whittaker functions (of which representation densities of quadratic forms are special values). The proven relations can be used to compute them explicitly in arbitrary dimensions, provided that enough information about the orbits under the orthogonal group acting on the representations is available. These relations have implications for the first and second special derivatives of the Euler product over all p of these Whittaker functions. These Euler products appear as the main part of the Fourier coefficients of Eisenstein series associated with the Weil representation. In case of signature (m-2,2), we interpret these implications in terms of the theory of Borcherds' products on orthogonal Shimura varieties. This gives some evidence for Kudla's conjectures in higher dimensions.