scholarly journals A short proof of the deformation property of Bridgeland stability conditions

2019 ◽  
Vol 375 (3-4) ◽  
pp. 1597-1613
Author(s):  
Arend Bayer
Keyword(s):  
Stability Condition ◽  
Deformation Property ◽  
Short Proof ◽  
Central Charge ◽  
Direct Proof ◽  
Small Deformation ◽  
Stability Conditions ◽  
Effective Version ◽  
Bridgeland Stability Conditions ◽  
The Stability

Abstract The key result in the theory of Bridgeland stability conditions is the property that they form a complex manifold. This comes from the fact that given any small deformation of the central charge, there is a unique way to correspondingly deform the stability condition. We give a short direct proof of an effective version of this deformation property.

scholarly journals Systolic inequalities for K3 surfaces via stability conditions

2021 ◽  
Author(s):  
Yu-Wei Fan
Keyword(s):  
Stability Condition ◽  
Symplectic Geometry ◽  
K3 Surfaces ◽  
Stability Conditions ◽  
Triangulated Categories ◽  
K3 Surface ◽  
Bridgeland Stability Conditions ◽  
Systolic Inequality

AbstractWe introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface X, there exists a constant C depending only on the rank and discriminant of NS(X), such that $$\begin{aligned} \mathrm {sys}(\sigma )^2\le C\cdot \mathrm {vol}(\sigma ) \end{aligned}$$ sys ( σ ) 2 ≤ C · vol ( σ ) holds for any stability condition on $$\mathcal {D}^b\mathrm {Coh}(X)$$ D b Coh ( X ) . This is an algebro-geometric generalization of a classical systolic inequality on two-tori. We also discuss applications of this inequality in symplectic geometry.

scholarly journals Stability of Multi-Dimensional Birth-and-Death Processes with State-Dependent 0-Homogeneous Jumps

2014 ◽  
Vol 46 (01) ◽  
pp. 59-75 ◽  
Author(s):  
Matthieu Jonckheere ◽  
Seva Shneer
Keyword(s):  
Stochastic Systems ◽  
Direct Proof ◽  
Geometric Interpretation ◽  
Stability Conditions ◽  
Birth And Death Processes ◽  
Piecewise Constant ◽  
State Dependent ◽  
The Stability ◽  
Deterministic Dynamical System ◽  
Simple Geometric Interpretation

We study the conditions for positive recurrence and transience of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. First, using an associated deterministic dynamical system, we provide a generic method to construct a Lyapunov function when the drift is a smooth function on ℝN. This approach gives an elementary and direct proof of ergodicity. We also provide instability conditions. Our main contribution consists of showing how discontinuous drifts change the nature of the stability conditions and of providing generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piecewise constant drifts in dimension two.

scholarly journals Semi-continuity of stability for sheaves and variation of Gieseker moduli spaces

2019 ◽  
Vol 2019 (749) ◽  
pp. 227-265 ◽  
Author(s):  
Daniel Greb ◽  
Julius Ross ◽  
Matei Toma
Keyword(s):  
Finite Number ◽  
Moduli Spaces ◽  
Stability Condition ◽  
Invariant Theory ◽  
Geometric Invariant Theory ◽  
Stability Conditions ◽  
Geometric Invariant ◽  
Moduli Spaces Of Sheaves ◽  
Stable Sheaves ◽  
The Stability

Abstract We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that generalises the classical notion of Gieseker-stability to allow for several polarisations. As such we are able to prove that on smooth threefolds certain moduli spaces of Gieseker-stable sheaves are related by a finite number of Thaddeus-flips (that is flips arising for Variation of Geometric Invariant Theory) whose intermediate spaces are themselves moduli spaces of sheaves.

scholarly journals Contractibility of the stability manifold for silting-discrete algebras

2018 ◽  
Vol 30 (5) ◽  
pp. 1255-1263 ◽  
Author(s):  
David Pauksztello ◽  
Manuel Saorín ◽  
Alexandra Zvonareva
Keyword(s):  
Derived Category ◽  
Stability Conditions ◽  
Bridgeland Stability Conditions ◽  
The Stability ◽  
Stability Manifold

AbstractWe show that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. has a length heart with finitely many simple objects. As a corollary, we obtain that the space of Bridgeland stability conditions for a silting-discrete algebra is contractible.

scholarly journals PREFERENTIAL MATING IN SYMMETRIC MULTILOCUS SYSTEMS: LIMITS FOR MULTIALLELISM AND FOR MANY LOCI

Genetics ◽  
1982 ◽  
Vol 100 (1) ◽  
pp. 149-158
Author(s):  
J Raper
Keyword(s):  
Sexual Selection ◽  
Frequency Dependence ◽  
Assortative Mating ◽  
Stability Condition ◽  
Local Stability ◽  
Stability Conditions ◽  
Mating Pattern ◽  
Viability Selection ◽  
The Stability ◽  
The One

ABSTRACT Models in which general forms of preferential mating have been superimposed on the framework of the symmetric heterozygosity selection regime have been examined previously with respect to the existence and local stability of a central polymorphic equilibrium. The results are now extended to produce the limiting form of the stability conditions in two cases: First, where the number of alleles per locus is assumed to be very large; second, where the number of loci affecting the character is very large. It is argued that some type of frequency dependence in the mating pattern must be included, and a particular case is examined in detail. It is shown that multiallelism is ambiguous in its effect on stability, while an increasing number of loci, at least under zero linkage, leads to a simple stability condition which is analogous to the one-locus heterosis principle. Assortative mating appears to be more likely to produce a stable central polymorphism under high levels of allelism than is sexual selection, but is relatively very much weaker than sexual or viability selection if the number of loci involved is large.

scholarly journals Donaldson–Thomas invariants of abelian threefolds and Bridgeland stability conditions

2021 ◽  
Vol 31 (1) ◽  
pp. 13-73
Author(s):  
Georg Oberdieck ◽  
Dulip Piyaratne ◽  
Yukinobu Toda
Keyword(s):  
Discriminant Function ◽  
Stability Conditions ◽  
Full Support ◽  
Wall Crossing ◽  
Rank One ◽  
Bridgeland Stability Conditions ◽  
Support Property ◽  
The Stability ◽  
Definition Of ◽  
Stability Manifold

We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author. For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given.

scholarly journals Stability Conditions for Affine Type A

2019 ◽  
Vol 23 (5) ◽  
pp. 2079-2111 ◽  
Author(s):  
P. J. Apruzzese ◽  
Kiyoshi Igusa
Keyword(s):  
Representation Theory ◽  
Linear Stability ◽  
Stability Condition ◽  
Central Charge ◽  
Type A ◽  
Maximal Length ◽  
Stability Conditions ◽  
Background Material ◽  
Affine Type

Abstract We construct maximal green sequences of maximal length for any affine quiver of type A. We determine which sets of modules (equivalently c-vectors) can occur in such sequences and, among these, which are given by a linear stability condition (also called a central charge). There is always at least one such maximal set which is linear. The proofs use representation theory and three kinds of diagrams shown in Fig. 1. Background material is reviewed with details presented in two separate papers Igusa (2017a, b).

scholarly journals Stability of Multi-Dimensional Birth-and-Death Processes with State-Dependent 0-Homogeneous Jumps

2014 ◽  
Vol 46 (1) ◽  
pp. 59-75 ◽  
Author(s):  
Matthieu Jonckheere ◽  
Seva Shneer
Keyword(s):  
Stochastic Systems ◽  
Direct Proof ◽  
Geometric Interpretation ◽  
Stability Conditions ◽  
Birth And Death Processes ◽  
Piecewise Constant ◽  
State Dependent ◽  
The Stability ◽  
Deterministic Dynamical System ◽  
Simple Geometric Interpretation

We study the conditions for positive recurrence and transience of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. First, using an associated deterministic dynamical system, we provide a generic method to construct a Lyapunov function when the drift is a smooth function on ℝN. This approach gives an elementary and direct proof of ergodicity. We also provide instability conditions. Our main contribution consists of showing how discontinuous drifts change the nature of the stability conditions and of providing generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piecewise constant drifts in dimension two.

STABILITY CONDITIONS FOR GATED M/G/∞ QUEUES

2004 ◽  
Vol 18 (1) ◽  
pp. 103-110
Author(s):  
Dimitra Pinotsi ◽  
Michael A. Zazanis
Keyword(s):  
Stability Condition ◽  
Service Time ◽  
Time Distribution ◽  
Stability Conditions ◽  
Service Time Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient ◽  
First Moment

The question of stability for the M/G/∞ queue with gated service is investigated using a Foster–Lyapunov drift criterion. The necessary and sufficient condition for positive recurrence is shown to be the finiteness of the first moment of the service time distribution, thus weakening the stability condition given in Browne et al. [3].

scholarly journals Characters of irrelevant deformations

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Shouvik Datta ◽  
Yunfeng Jiang
Keyword(s):  
Deformation Parameter ◽  
Central Charge ◽  
Scaling Limit ◽  
Small Deformation ◽  
State Dependent ◽  
Dependent Change ◽  
Left And Right ◽  
Double Scaling Limit

Abstract We analyse the $$ T\overline{T} $$ T T ¯ deformation of 2d CFTs in a special double-scaling limit, of large central charge and small deformation parameter. In particular, we derive closed formulae for the deformation of the product of left and right moving CFT characters on the torus. It is shown that the 1/c contribution takes the same form as that of a CFT, but with rescalings of the modular parameter reflecting a state-dependent change of coordinates. We also extend the analysis for more general deformations that involve $$ T\overline{T} $$ T T ¯ , $$ J\overline{T} $$ J T ¯ and $$ T\overline{J} $$ T J ¯ simultaneously. We comment on the implications of our results for holographic proposals of irrelevant deformations.

Sign in / Sign up

Export Citation Format

Share Document