Bridgeland Stability Conditions on Fano Threefolds

Abstract

We show the existence of Bridgeland stability conditions on all Fano threefolds, by proving a modified version of a conjecture by Bayer, Toda, and the second author. The key technical ingredient is a strong Bogomolov inequality, proved recently by Chunyi Li. Additionally, we prove the original conjecture for some toric threefolds by using the toric Frobenius morphism. Comment: 24 pages, 1 figure. Fifth version: Official version of the journal

Zero Rank Asymptotic Bridgeland Stability

“…In [3] Bayer-Macrì-Toda conjectured a way to generate Bridgeland stability conditions for threefolds, which later was proven to hold for a few cases of threefolds, see [2,4,11,10]. The idea was a generalization to the threefold case of the approach given in [5] by using the stability notion for surfaces as a weaker notion of stability for threefolds and apply a new tilting to the already tilted category.…”

Bridgeland wall-crossing for some Simpson moduli spaces on $\mathbb{P}^1 \times \mathbb{P}^1$

“…This field has grown quickly in the past decade, even though the very existence of such stability conditions for a specific variety X is already something that needs a lot of effort to be proven. For curves all is known (see [Mac07]), while for surfaces a great deal of general machinery has been developed for the study of the structure of Stab(X) (see for example [Bri08], or [MS17] for a survey); the most recent results in this direction have been achieved in [BMS16], [Li16] and [BMSZ17], with the construction of Bridgeland stability conditions for abelian and Fano threefolds, the ultimate goal being the non-emptiness of the stability manifold for Calabi-Yau threefolds.…”

Stability conditions on threefolds with nef tangent bundles

“…[20]), some toric threefolds (cf. [8]), product threefolds of projective spaces and Abelian varieties (cf. [19]), and quintic threefolds (cf.…”

“…The failure of the conjecture is related to the existence of a kind of negative effective divisors on a threefold ( [28], see Lemma 2.10). The modification of the conjecture is discussed in the paper [8], and they prove that the modified version of the BG type inequality holds when X is a Fano threefold of arbitrary Picard rank.…”

“…In this paper, we treat the remaining cases, i.e. ( 6) - (8) in Theorem 1.3. Note that P 1 × A, P 2 × C, and P 1 × P 1 × C are treated in the author's previous paper [19], which are the special cases of ( 6) - (8) in Theorem 1.3.…”