We generalise the construction of Rouquier complexes to setting singular Soergel bimodules by taking minimal restriction complexes. show that they retain many properties ordinary complexes: are delta-split, satisfy a vanishing formula and, when Soergel's conjecture holds perverse. As an application, we use establish Hodge theory for bimodules.

Abstract We study two kinds of categorical traces (monoidal) dg categories, with particular interest in categories Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, category bimodules arbitrary types. Secondly, introduce notion horizontal trace a monoidal and type $A$. As an application obtain annular Khovanov–Rozansky link invariant actio...

Abstract We present a categorical point of view on dynamical quantum groups in terms categories Harish-Chandra bimodules. prove Tannaka duality theorems for forgetful functors into the monoidal category bimodules slight modification notion bialgebroid. Moreover, we show that standard $F(G)$ and $F_q(G)$ are related to parabolic restriction classical Finally, exhibit natural Weyl symmetry functo...

We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphis...

Let X be a log Q-Fano variety, i.e, there exists an effective Q-divisor D such that (X,D) is Kawamata log terminal (klt) and −(KX + D) is nef and big. By a result of Miyaoka-Mori [15], X is uniruled. The conjecture ([10], [16]) predicts that X is rationally connected. In this paper, apply the theory of weak (semi) positivity of (log) relative dualizing sheaves f∗(KX/Y + ∆) (which has been devel...

Let $X$ be a sufficiently nice scheme. We survey some recent progress on dualizing complexes. It turns out that a complex in $kinj X$ is dualizing if and only if tensor product with it induces an equivalence of categories from Murfet's new category $kmpr X$ to the category $kinj X$. In these terms, it becomes interesting to wonder how to glue such equivalences.

A necessary and sufficient condition is given for semi-ampleness of a nef and big line bundle in positive characteristic. One application is to the geometry of the universal stable curve over Mg , specifically, the semi-ampleness of the relative dualizing sheaf, in positive characteristic. An example is given which shows this, and the semi-ampleness criterion, fail in characteristic zero. A sec...

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. We obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results...