The space of stability conditions on the local projective plane

Abstract

ABSTRACT. We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply-connected. We determine the group of autoequivalences preserving this connected component.Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with level structure, with the morphism given by solutions of Picard-Fuchs equations. This result is motivated by the notion of Π-stability and by mirror symmetry.

Stable pair invariants on Calabi–Yau threefolds containing ℙ2

“…Therefore We now investigate the limiting point lim t→+0 σ t of the above weak stability conditions. The following lemma shows that the one parameter family in Lemma 4.13 connects the 'large volume limit point' with the 'orbifold point', a similar picture obtained by Bayer-Macri [BM11] for the space of Bridgeland stability conditions on local P 2 (cf. Figure 1 in Subsection 1.4).…”

“…The proof of Theorem 1.2 follows from wall-crossing argument in the space of weak stability conditions, as in the author's previous papers [Tod10a], [Tod13], [Tod12]. In order to explain the argument, we first recall Bayer-Macri's description of the space Stab(ω P 2 ) of Bridgeland stability conditions on D b Coh(ω P 2 ) in [BM11]. They showed that the double quotient stack of Stab(ω P 2 ) by the actions of Aut D b Coh(ω P 2 ) and the additive group C contains the parameter space of the mirror family of ω P 2 .…”

The space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds

“…This notion is equivalent to σ being "full" in the sense of [14], see [10,Proposition 12.4]. The definition is quite natural: it implies that if W is in an -neighborhood of Z with respect to the operator norm on Hom( R , C) induced by and the standard norm on C, then W (E) is in a disc of radius C |Z (E)| around Z (E) for all semistable objects E; in particular, we can bound the difference of the arguments of the complex numbers Z (E) and W (E).…”

Lectures on Bridgeland Stability

“…The main result is the following (see [Bri08,AB13,BM11]). We will choose Λ = K num (X) and v as the Chern character map as in Example 5.7.…”

“…Arcara and Bertram realized that the construction can be generalized to any surface by using the Bogomolov Inequality in [AB13]. The proof of the support property is in [BM11,BMT14,BMS14]. As a consequence, these stability conditions vary continuously in ω and B.…”