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.

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. NotationWe denote by CAT the 2-"category" 1 of categories, by (S)MCAT the 2-"category" of (symmetric) multicategories, and by Cat the 2-category of small categories. We consider a partially ordered set (poset) X as a small category by considering the relation x ≤ y to be equivalent to the existence of a unique morphism x → y. We denote the positive integers (resp. non-negative integers) by N (resp. N 0 ). The ordered sets {0, . . . , n} ⊂ N 0 considered as a small category are denoted by ∆ n . We denote by Mor(D) (resp. Iso(D)) the class of morphisms (resp. isomorphisms) in a category D. The final category (which consists of only one object and its identity) is denoted by ⋅ or ∆ 0 . The same notation is also used for the final multi-category, i.e. that with one object and precisely one n-ary morphism for any n. Our conventions about multicategories and fibered (multi-)categories are summarized in appendix A.

We develop the theory of (op)fibrations of 2-multicategories and use it to define abstract sixfunctor-formalisms. We also give axioms for Wirthmüller and Grothendieck formalisms (whereFinally, 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.

No abstract

No abstract