We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the litterature. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles.
We present a notion of $Δ$-stability and stability filtration in arbitrary categories which is equivalent to the existence of Harder-Narasimhan (HN) sequences on objects. Indeed it is equivalent to the existence of a zero morphism, a partial order on objects, and a collection of some universal sequences. In additive categories embedded in an ambient triangulated category, we could obtain a numerical polynomial or central charge of objects by calculating the Euler characteristic of slope sequences and objects, inducing a partial order and HN sequences. In the case of normal projective surfaces …
Stability conditions are a mathematical way to understand $Π$-stability for D-branes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditions, and giving some pointers for future research.
Given a stability condition on a smooth projective variety $X$, we construct a family of stability conditions on $X\times C$, where $C$ is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macrì and weak stability conditions on the Abramovich-Polishchuk heart of a bounded t-structure in $D(X\times C)$.
We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpson-stability, and large volume limits of Bridgeland stability conditions. We show that the PT/DT-correspondence relating stable pairs to Donaldson-Thomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wall-crossing in our family of polynomial stability conditions. Similarly, we show that the relation be…
We shall study stability conditions and Fourier-Mukai transforms on an elliptic surface. In particular we shall explain duality of elliptic surfaces by Fourier-Mukai transforms.
We study some examples of Bridgeland-Douglas stability conditions on triangulated categories. From one side we give a complete description of the stability manifolds for smooth projective curves of positive genus. From the other side we study stability conditions on triangulated categories generated by an exceptional collection. In the case of the projective line this leads to the connectedness and simply-connectedness of the stability manifold.
We give some remarks on our papers with Minamide and Yanagida on Bridgeland stability conditions. We also give a remark on stability conditions on Enriques surfaces, and give another proof of the projectivity of the coarse moduli spaces of semi-stable objects, which were obtained by Nuer.
In this paper, we prove BG-type inequality conjecture for threefolds in the title. In particular, there exist Bridgeland stability conditions on these threefolds.
In this paper, we prove the Bogomolov-Gieseker type inequality conjecture for threefolds with nef tangent bundles. As a corollary, there exist Bridgeland stability conditions on these threefolds.
We show that, under mild conditions, the space of numerical Bridgeland stability conditions Stab(T) on a triangulated category T is complete. In particular the metric on a full component of Stab(T) for which the central charges factor through a finite rank quotient of the Grothendieck group K(T) is complete. As an example, we compute the metric on the space of numerical stability conditions on a smooth complex projective curve of genus greater than one, and show that in this case the quotient Stab(T)/C by the natural action of the complex numbers is isometric to the upper half plane equipped w…
The space of Bridgeland stability conditions on the bounded derived category of coherent sheaves on P2 has a principle connected component Stab^†(P2). We show that Stab^†(P2) is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for Stab^†(P2) and show that Stab^†(P2) is contractible.
In this paper we describe a simply connected component of the complex manifold of stability conditions on the bounded derived category of a generic complex torus of any dimension. A torus is called generic if there are no nontrivial integral (p,p)-classes. We give an explicit description of all stability conditions in that connected component.
We introduce $q$-stability conditions $(σ,s)$ on Calabi-Yau-$\mathbb{X}$ categories $\mathcal{D}_\mathbb{X}$, where $σ$ is a stability condition on $\mathcal{D}_\mathbb{X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname{QStab}_s\mathcal{D}_\mathbb{X}$ is a complex manifold of dimension $n$ for fixed $s$, where $n$ is the rank of the Grotendieck group of $\mathcal{D}_\mathbb{X}$ over $\mathbb{Z}[q^{\pm 1}]$. When $s=N$ is an integer, we show that the $q$-stability conditions can be identified with the stability conditions on $\mathcal{D}_N$, provid…
In this paper, we study the space of stability conditions on triangulated categories generated by an exceptional collection. We give an exact description of subspace of stability conditions generated by length 4 complete exceptional collection.
We construct a topological embedding of the maximal connected component of Bridgeland stability conditions of a (twisted) Abelian surface into the distinguished connected component of the stability manifold of the associated (twisted) Kummer surface. We use methods developed for orbifold conformal field theories.
On a triangulated category $\mathbf D$ equipped with a semiorthogonal decomposition $\mathbf D=\langle{\mathbf D_{1}},{\mathbf D_{2}}\rangle$, Collins and Polishchuk develop a gluing construction of stability condition on $\mathbf D$. The gluing construction gives a stability condition on $\mathbf D$ from these on $\mathbf D_{1}$ and $\mathbf D_{2}$. We study a deformation of gluing stability conditions on for a nice semiorthogonal decomposition. As a consequence, we construct a continuous family of stability conditions by showing a deformation property introduced by Bridgeland's original pape…
Calabi-Yau Fermat varieties are obtained from moduli spaces of Lagrangian connect sums of graded Lagrangian vanishing cycles on stability conditions on Fukaya-Seidel categories. These graded Lagrangian vanishing cycles are stable representations of quivers on their mirror stability conditions.
