Gluing Stability Conditions

By John Collins, Alexander Polishchuk. 2009

55 citations recorded by the source corpus.

Stability conditions Definition. A stability condition σ is given by a pair (Z, P ), where Z : K0(D) → C is a homomorphism from the Grothendieck group K0(D) of D, and P is a slicing. Such a slicing is given by a collection of subcategories P (φ) of semistable objects of phase φ for each φ ∈ R, where Hom(P (φ1), P (φ2)) = 0 for φ1 > φ2, and P (φ)[1] = P (φ + 1). For an object E ∈ P (φ) we will use the notation φ(E) = φ. Similarly to the case of vector bundles, for each object E of D there should exist a Harder-Narasimhan filtration (HN-filtration), i.e., a collection of exact triangles building E from the semistable factors E1, . . . , En (called the HN-factors of E), where φ(E1) > . . . > φ(En) (E1 → E is an analog of the subbundle of maximal phase, etc.).

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5c97fceda6ddd7fe310e6171cfc14eba7f3761d8

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Citations in the corpus

Dynamical systems and categories (2013, background): “ Ext-exceptional collection11 in D. Then for any 0 i<j nthe extension closure A ij of E i;E i+1;:::;E j is a heart of a bounded t-structure in T ij = hE i;E i+1;:::;E jiˆD (see [39, Lemma 3.14],[17]), hence we have a corresponding family HA ij ˆStab(T ij). In this setting all stability condition”
Derived categories of cyclic covers and their branch divisors (2014, result): “µn = hjk+1∗Db(Z),jk+2∗Db(Z),...,jn−1∗Db(Z), f∗ 0D b(Y ),j ∗Db(Z),j 1Db(Z),...,jk− Db(Z)i. (4.8) This decomposition is well-known to the experts. The case n = 2 was proved by Collins and Polishchuk in [6]. The general case follows from a result of Ishii and Ueda [14, Theorem 1.6], which gives a semio”
Stability conditions on Kuznetsov components (2017, result): “Using the results in [CP10], one can show that the stability condition σKu(X) just constructed extends to a Bridgeland stability condition on the whole cubic fourfold X.”