Polynomial Bridgeland stability conditions and the large volume limit

By Arend Bayer. 2007

91 citations recorded by the source corpus.

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 wallcrossing in our family of polynomial stability conditions. Similarly, we show that the relation between stable pairs and invariants of one-dimensional torsion sheaves (proven recently by the same authors) is a wall-crossing formula. 14F05, 18E30; 14J32, 14D20, 14N35

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b2c65b1fae85b879bf2be547147df58015532efa

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Citations in the corpus

Inducing stability conditions (2007, background): “The reader is encouraged to consult [26, 40, 41, 2 , 49, 50] for motivation and applications.”
Bridgeland’s stabilities on abelian surfaces (2012, unknown): “Y ) ∈G/b{±1}. Then the action of θ(ΦE X→Y DX) on Sym2(Z,n) is the same as the action of Φ E X→Y DX. Lemma 5.3 ([14, Lemma 6.18]). If θ(ΦE X→Y ) = a b c d , then θ(ΦE Y →X) = ± d b c a , θ(ΦE ∨[2] Y →X) = ± d −b −c a , θ(ΦE ∨[2] X→Y ) = ± a −b −c d . 5.2. The arithmetic group G and numerical ”
Stability conditions and extremal contractions (2012, methodology): “By combining it with the techniques developed in [3], [6], we establish the BG inequality for σ0-semistable objects in Bf∗ω.”
Curve counting via stable pairs in the derived category (2007, background): “on the other side of the wall. No worse conﬁgurations occur since the K-theory classes of IZ, IZ′ and Op are all primitive and distinct. So the mth term in (3.5) is the contribution 12Recently, Bayer [4] and Toda [52] have deﬁned variants of Bridgeland’s axioms for a stability condition within which”
Hilbert schemes and stable pairs: GIT and derived category wall crossings (2009, methodology): “of In(X,β) on one side of a codimension one wall and the stable pairs of Pn(X,β) on the other side of the wall. In fact this can be justiﬁed using the variants of stability conditions deﬁned by Bayer [Ba] and Toda [To1], but we shall just use that (and the GIT stabilities of the ﬁrst half of this pa”
Stable pairs and BPS invariants (2007, background): “2.3. Wall-crossing. Arend Bayer [ 1 ] and Yukinoba Toda [33] have made the beautiful observation that (2.4) should be seen as a wallcrossing formula.”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “Unlike other constructions (as in [5, 78]), the space of Bridgeland stability conditions admits a well-behaved wall and chamber structure: the moduli space of stable objects with given invariants remains unchanged unless the stability condition crosses a wall.”
The space of stability conditions on the local projective plane (2009, background): “ central charge is approximately given by Z(E) = − R P2 ch(E)e −ah, where his the class of a line in P2; the limit stability condition as z → 0 can be described as a polynomial stability condition of [Bay09]. The point z = ∞ is the orbifold point: the heart of the bounded t-structure is A1 ∼= Coh0[C”
Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, background): “[34],) E → LHilbn(X,β), satisfying that E ∼= E[1].”
Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities (2011, background): “conditions rather than slope-stability. We will compare the two constructions in Section 5.1. 4.1. Polynomial stability conditions. The notion of polynomial stability condition has been introduced in [Bay09]. We refer to loc. cit. for all basic deﬁnitions. We will repeatedlyconstruct polynomialstabi”
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008, background): “that t>0 and t<xfor any x∈R>0). Some of our considerations below make sense in this situation. In the framework of stability conditions on triangulated categories such structures appeared in [1], [73]. 2.2 Reformulation of the stability data In what follows we will consider various cones in”
The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds (2014, background): “6 (a)Alternatively, the statements of this appendix could be proved via the relation of tilt-stability to a certain polynomial stability condition (in the sense of [4]); see Sects.”