Related papers: Stability Conditions for Multigraded Rings

In this paper, I present some sufficient conditions for projective hypersurfaces to be GIT (semi-)stable. These conditions will be presented in terms of dimension and degree of the hypersurfaces, dimension of the singular locus and…

We prove that the atom spectrum, which is a topological space associated to an arbitrary abelian category introduced by Kanda, of the category of finitely presented graded modules over a graded ring $R$ is given as a union of the…

Let $G$ be a reductive affine algebraic group, and let $X$ be an affine algebraic $G$-variety. We establish a (poly)stability criterion for points $x\in X$ in terms of intrinsically defined closed subgroups $H_{x}$ of $G$, and relate it…

We construct a geometric model for the root category $\mathcal{D}^b(Q)/[2]$ of any Dynkin diagram $Q$, which is an $h_Q$-gon $\mathbf{V}_Q$ with cores, where $h_Q$ is the Coxeter number and $\mathcal{D}^b(Q)$ is the bounded derived category…

Let $G$ be a connected, simply connected, simple, complex, linear algebraic group. Let $P$ be an arbitrary parabolic subgroup of $G$. Let $X=G/P$ be the $G$-homogeneous projective space attached to this situation. Let $d\in H_2(X)$ be a…

We introduce two extensions of the space of Bridgeland stability conditions of a triangulated category. First we consider lax stability conditions where semistable objects are allowed to have mass zero but still have a phase. The…

The moduli space $\cM_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\bcM_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding…

We study the GIT compactification $\mathbb{P}(\mathrm{Sym}^3\mathbb{C}^7)//\mathrm{SL}(7)$ of the moduli space of cubic fivefolds $X\subset\mathbb{P}^6$ and give an explicit description of its strictly semistable boundary. We construct…

We find stability conditions ([Do], [Br]) on some derived categories of differential graded modules over a graded algebra studied in [RZ], [KS]. This category arises in both derived Fukaya categories and derived categories of coherent…

The present paper focuses on the study of t-stabilities on a triangulated category in the sense of Gorodentsev, Kuleshov and Rudakov. We give an equivalent description for the finest t-stability on a piecewise hereditary triangulated…

We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant…

We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of $G$-bundles on a smooth projective curve for a reductive algebraic group…

The unitary $ \mathrm{U}(d)$-equivalence relation between elements of the space $\mathfrak{P}_+\,$ of mixed states of $d$-dimensional quantum system defines the orbit space $ \mathfrak{P}_+/ \mathrm{U}(d)\,$ and provides its description in…

We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two…

For a discrete metric space (or more generally a large scale space) $X$ and an action of a group $G$ on $X$ by coarse equivalences, we define a type of coarse quotient space $X_G$, which agrees up to coarse equivalence with the orbit space…

Let $\mathcal{X}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack $\mathcal{K}_{g,n}(\mathcal{X},d)$ of twisted stable maps is a quotient stack and can…

Let $R$ be a 3-dimensional complete local Gorenstein isolated singularity. For a basic maximal modifying $R$-module $M$, we construct a wall-and-chamber structure, denoted by ${\sf Cone}(M)$ and called the mutation cone of $M$, in the real…

Let X be a projective surface, let \sigma be an automorphism of X, and let L be a \sigma-ample invertible sheaf on X. We study the properties of a family of subrings, parameterized by geometric data, of the twisted homogeneous coordinate…

Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomlov-Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric…

We develop a framework relating semiorthogonal decompositions of a triangulated category $\mathcal{C}$ to paths in its space of stability conditions. We prove that when $\mathcal{C}$ is the homotopy category of a smooth and proper…