In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott–Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.

Let $\Sigma $ be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro-$\Sigma $ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of $\Sigma $ are invertible. The present paper focuses on the topic of comparison between the theory developed in earlier papers concerning pro-$\Sigma $ fundamental groups and various discrete versions of this theory. We begin by developing a theory concerning certain combinatorial analogues of the section conjecture and Grothendieck conjecture. This portion of the theory is purely combinatorial and essentially follows from a result concerning the existence of fixed points of actions of finite groups on finite graphs (satisfying certain conditions). We then examine various applications of this purely combinatorial theory to scheme theory. Next, we verify various results in the theory of discrete fundamental groups of hyperbolic topological surfaces to the effect that various properties of (discrete) subgroups of such groups hold if and only if analogous properties hold for the closures of these subgroups in the profinite completions of the discrete fundamental groups under consideration. These results make possible a fairly straightforward translation, into discrete versions, of pro-$\Sigma $ results obtained in previous papers by the authors. Finally, we discuss a construction that was considered previously by M. Boggi in the discrete case from the point of view of the present paper.

We invoke the Bernstein–Gel$'$fand–Gel$'$fand (BGG) correspondence to study subcomplexes of free resolutions given by two well-known complexes, the Koszul and the Eagon–Northcott. This approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon–Northcott case.

We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete self-injective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.

Let $\mathbf {D}$ be a bounded homogeneous domain in ${\mathbb {C}}^n$. In this note, we give a characterization of the Stein domains in $\mathbf {D}$ which are invariant under a maximal unipotent subgroup N of $Aut(\mathbf {D})$. We also exhibit an N-invariant potential of the Bergman metric of $\mathbf {D}$, expressed in a Lie theoretical fashion. These results extend the ones previously obtained by the authors in the symmetric case.

We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.

Using the technique of Gabor analysis, we characterize the boundedness of $e^{i\Delta }: W^{p_1,q_1}_m\rightarrow W^{p_2,q_2}$ with modulation and translation operators, where and m is a v-moderate weight. The sharp exponents for the boundedness are also characterized in the case of power weight.

Sextic double solids, double covers of $\mathbb P^3$ branched along a sextic surface, are the lowest degree Gorenstein terminal Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are $\mathbb Q$-factorial with ordinary double points, are known to be birationally rigid. In this paper, we study sextic double solids with an isolated compound $A_n$ singularity. We prove a sharp bound $n \leq 8$, describe models for each n explicitly, and prove that sextic double solids with $n> 3$ are birationally nonrigid.