On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

Generalized heredity in ℬ-free systems

Let [Formula: see text] be a primitive set, [Formula: see text], [Formula: see text], and denote by [Formula: see text] the orbit closure of [Formula: see text] under the shift. We complement results on heredity of [Formula: see text] from [Dymek et al., [Formula: see text]-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489] in two directions: In the proximal case we prove that a certain subshift [Formula: see text], which coincides with [Formula: see text] when [Formula: see text] is taut, is always hereditary. (In particular there is no need for the stronger assumption that the set [Formula: see text] has light tails, as in [Dymek et al., [Formula: see text]-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489].) We also generalize the concept of heredity to include the non-proximal (and hence non-hereditary) case by proving that [Formula: see text] is always “hereditary above its unique minimal (Toeplitz) subsystem”. Finally, we characterize this Toeplitz subsystem as being a set [Formula: see text], where [Formula: see text] for a set [Formula: see text] that can be derived from [Formula: see text], and draw some further conclusions from this characterization. Throughout results from [Kasjan et al., Dynamics of [Formula: see text]-free sets: A view through the window, Int. Math. Res. Not. 2019 (2019) 2690–2734] are heavily used.

Poincaré polynomials of generic torus orbit closures in Schubert varieties

The closure of a generic torus orbit in the flag variety G / B G/B of type A A is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in G / B G/B . When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

Pointwise Topological Stability

We decompose the topological stability (in the sense of P. Walters) into the corresponding notion for points. Indeed, we define a topologically stable point of a homeomorphism f as a point x such that for any C0-perturbation g of f there is a continuous semiconjugation defined on the g-orbit closure of x which tends to the identity as g tends to f. We obtain some properties of the topologically stable points, including preservation under conjugacy, vanishing for minimal homeomorphisms on compact manifolds, the fact that topologically stable chain recurrent points belong to the periodic point closure, and that the chain recurrent set coincides with the closure of the periodic points when all points are topologically stable. Next, we show that the topologically stable points of an expansive homeomorphism of a compact manifold are precisely the shadowable ones. Moreover, an expansive homeomorphism of a compact manifold is topologically stable if and only if every point is topologically stable. Afterwards, we prove that a pointwise recurrent homeomorphism of a compact manifold has no topologically stable points. Finally, we prove that every chain transitive homeomorphism with a topologically stable point of a compact manifold has the pseudo-orbit tracing property. Therefore, a chain transitive expansive homeomorphism of a compact manifold is topologically stable if and only if it has a topologically stable point.

NILPOTENT SUBSPACES AND NILPOTENT ORBITS

Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$. For a nilpotent $G$-orbit ${\mathcal{O}}\subset \mathfrak{g}$, let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$. In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$, then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_{2}$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$, or (2) is the only $B$-stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$.