HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS

We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.

COMPACT ORBITS OF PARABOLIC SUBGROUPS

We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U with Lie algebra $\mathfrak {u}$ extends holomorphically to an action of the complexified group $U^{\mathbb {C}}$ and that the U-action on Z is Hamiltonian. If $G\subset U^{\mathbb {C}}$ is compatible, there exists a gradient map $\mu _{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$ . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map $\mu _{\mathfrak p}$ .

PRINCIPAL RADICAL SYSTEMS, LEFSCHETZ PROPERTIES, AND PERFECTION OF SPECHT IDEALS OF TWO-ROWED PARTITIONS

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

FUJITA DECOMPOSITION AND MASSEY PRODUCT FOR FIBERED VARIETIES

Let $f\colon X\to B$ be a semistable fibration where X is a smooth variety of dimension $n\geq 2$ and B is a smooth curve. We give the structure theorem for the local system of the relative $1$ -forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega _{X/B}$ . We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if $B=\mathbb {P}^1$ .

CONICS IN SEXTIC -SURFACES IN

We prove that the maximal number of conics in a smooth sextic $K3$ -surface $X\subset \mathbb {P}^4$ is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible.

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $\mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ of the loop algebra $\widehat {\mathfrak {gl}}_{m|n}$ of ${\mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${\widehat {{\mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ is to determine the action of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $\widehat {\mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ and ${\widehat {{\mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $\mathbb Z$ -form for ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.

GENERALIZED GREEN FUNCTIONS AND UNIPOTENT CLASSES FOR FINITE REDUCTIVE GROUPS, III

Lusztig’s algorithm of computing generalized Green functions of reductive groups involves an ambiguity on certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.

ON THE INSTABILITY OF PERIODIC WAVES FOR DISPERSIVE EQUATIONS—REVISITED

The purpose of this paper is to present an extension of the results in [8]. We establish a more general proof for the moving kernel formula to prove the spectral stability of periodic traveling wave solutions for the regularized Benjamin–Bona–Mahony type equations. As applications of our analysis, we show the spectral instability for the quintic Benjamin–Bona–Mahony equation and the spectral (orbital) stability for the regularized Benjamin–Ono equation.

ENERGY CONCENTRATION PROPERTIES OF A p-GINZBURG–LANDAU MODEL

This paper is concerned with the p-Ginzburg–Landau (p-GL) type model with $p\neq 2$ . First, we obtain global energy estimates and energy concentration properties by the singularity analysis. Next, we give a decay rate of $1-|u_\varepsilon |$ in the domain away from the singularities when $\varepsilon \to 0$ , where $u_\varepsilon $ is a minimizer of p-GL functional with $p \in (1,2)$ . Finally, we obtain a Liouville theorem for the finite energy solutions of the p-GL equation on $\mathbb {R}^2$ .