Volume 2016, Issue 711

We introduce a new family of superalgebras which should be considered as a super version of the Khovanov–Lauda–Rouquier algebras. Let I be the set of vertices of a Dynkin diagram with a decomposition I=I even ⊔I odd $I=I_{{\mathrm {even}}}\sqcup I_{{\mathrm {odd}}}$. To this data, we associate a family of graded superalgebras R n , the quiver Hecke superalgebras . When I odd =∅$I_{{\mathrm {odd}}}=\emptyset $, these algebras are nothing but the usual Khovanov–Lauda–Rouquier algebras. We then define another family of graded superalgebras RC n , the quiver Hecke–Clifford superalgebras, and show that the superalgebras R n and RC n are weakly Morita superequivalent to each other. Moreover, we prove that the affine Hecke–Clifford superalgebras, as well as their degenerate version, the affine Sergeev superalgebras, are isomorphic to quiver Hecke–Clifford superalgebras RC n after a completion.

For a proper semistable curve X over a DVR of mixed characteristics and perfect residue field we prove the “invariant cycles theorem” with trivial coefficients, i.e. that the group of elements of the first de Rham cohomology group of the generic fiber of X annihilated by the monodromy operator coincides with the first rigid cohomology group of its special fiber. This is done using an explicit description of the monodromy operator on the de Rham cohomology of the generic fiber of X with coefficients a convergent F -isocrystal (see [Représentations p -adiques de groupes p -adiques III: Méthodes globales et géométriques, Astérisque 331, 179–254]) and the proof exhibits an interesting interplay between this cohomology group and the combinatorics of the graph of the reduction of X . The result was proved in a different way in the case the DVR has finite residue field in [Duke Math. J. 97 (1999), no. 1, 155–169]. We also study the case where the coefficients are unipotent convergent F -isocrystals on the special fiber of X (without log-structure): we show that the invariant cycles theorem does not hold in general in this setting and give a sufficient condition for non-exactness.

Let X be an abelian scheme over a base variety S , with endomorphism algebra D . We prove that the relative Chow motive R(X/S)$R(X/S)$ has a canonical decomposition as a direct sum of motives R(ξ)$R^{(\xi )}$ where ξ runs over an explicitly determined finite set of irreducible representations of the group Dop,*$D^{\textup {op},*}$, such that R(ξ)$R^{(\xi )}$, seen as a functor from Chow motives to Dop,*$D^{\textup {op},*}$-representations, is ξ-isotypic. Our decomposition refines the motivic decomposition of Deninger and Murre, as well as Beauville's decomposition of the Chow group. The second main result is that we construct a canonical generalized motivic Lefschetz decomposition. Inspired by work of Looijenga and Lunts in [Invent. Math. 129 (1997), 361–412], the role of 𝔰𝔩2$\mathfrak {sl}_2$ in the classical theory is here replaced by a larger Lie algebra, generated by all Lefschetz and Lambda operators associated to non-degenerate line bundles. We construct an action of this larger Lie algebra on R(X/S)$R(X/S)$ and deduce from this a generalized Lefschetz decomposition. We also give a precise structure result for the Lefschetz components that arise. As an application of our techniques, we provide a positive answer to a question of Claire Voisin concerning the analogue for abelian varieties of a conjecture of Beauville.

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C 1,1 open sets D in ℝ d , of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided Dirichlet heat kernel estimates hold for all t > 0. Integrating the heat kernel estimates with respect to the time variable t , we obtain sharp two-sided estimates for the Green functions, in bounded C 1,1 open sets, of such subordinate Brownian motions with Gaussian components.

In this paper, we first show an interpretation of the Kähler–Ricci flow on a manifold X as an exact elliptic equation of Einstein type on a manifold M of which X is one of the (Kähler) symplectic reductions via a (non-trivial) torus action. There are plenty of such manifolds (e.g. any line bundle on X will do). Such an equation is called V -soliton equation, which can be regarded as a generalization of Kähler–Einstein equations or Kähler–Ricci solitons. As in the case of Kähler–Einstein metrics, we can also reduce the V -soliton equation to a scalar equation on Kähler potentials, which is of Monge–Ampère type. We then prove some preliminary results towards establishing existence of solutions for such a scalar equation on a compact Kähler manifold M . One of our motivations is to apply the interpretation to studying finite time singularities of the Kähler–Ricci flow.

We show the semi-continuity property of minimal log discrepancies for varieties which have a crepant resolution in the category of Deligne–Mumford stacks. Using this property, we also prove the ideal-adic semi-continuity problem for toric pairs.

The Haagerup property for locally compact groups is generalised to the context of locally compact quantum groups, with several equivalent characterisations in terms of the unitary representations and positive-definite functions established. In particular it is shown that a locally compact quantum group 𝔾 has the Haagerup property if and only if its mixing representations are dense in the space of all unitary representations. For discrete 𝔾 we characterise the Haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group 𝔾^$\hat{\mathbb {G}}$; by the existence of a real proper cocycle on 𝔾, and further, if 𝔾 is also unimodular we show that the Haagerup property is a von Neumann property of 𝔾. This extends results of Akemann, Walter, Bekka, Cherix, Valette, and Jolissaint to the quantum setting and provides a connection to the recent work of Brannan. We use these characterisations to show that the Haagerup property is preserved under free products of discrete quantum groups.

The spt-function spt( n ) was introduced by Andrews as the weighted counting of partitions of n with respect to the number of occurrences of the smallest part. Andrews, Garvan and Liang defined the spt-crank of an S -partition which leads to combinatorial interpretations of the congruences of spt( n ) modulo 5 and 7. Let N S ( m , n ) denote the net number of S -partitions of n with spt-crank m . Andrews, Garvan and Liang showed that N S ( m , n ) is nonnegative for all integers m and positive integers n , and they asked the question of finding a combinatorial interpretation of N S ( m , n ). In this paper, we introduce the structure of doubly marked partitions and define the spt-crank of a doubly marked partition. We show that the number N S ( m , n ) can be interpreted as the number of doubly marked partitions of n with spt-crank m . Moreover, we establish a bijection between marked partitions of n and doubly marked partitions of n . A marked partition is defined by Andrews, Dyson and Rhoades as a partition with exactly one of the smallest parts marked. They consider it a challenge to find a definition of the spt-crank of a marked partition so that the set of marked partitions of 5n$5n$ + 4 and 7 n + 5 can be divided into five and seven equinumerous classes. The definition of spt-crank for doubly marked partitions and the bijection between the marked partitions and doubly marked partitions in principle lead to a solution to the problem of Andrews, Dyson and Rhoades.