Some Examples of Tilt-Stable Objects on Threefolds

Abstract

We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarisation ω gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.

Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds

“…A conjectural construction of Bridgeland stability conditions for projective threefolds was introduced in [4] and the problem is reduced to proving an inequality, which the authors call a Bogomolov-Gieseker (B-G for short) type inequality, holds for certain tilt stable objects. This inequality has been shown to hold for three dimensional projective space (see [4] and [12]) and smooth quadric threefold (see [17]), and some progress has been made for more general threefolds (see [19] and [9]). However, there is no known example of a stability condition on a projective Calabi-Yau threefold and this case is especially significant because of the interest from Mathematical Physics and also in connection with Donaldson-Thomas invariants.…”

Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds II

“…Here they introduced the notion of tilt stability for objects in an abelian subcategory of the derived category which is a tilt of coherent sheaves. These have now been studied extensively: [Tod2,LM,BMT,Mac2,MP,Sch]. This conjectural construction has boiled down to the requirement that certain tilt stable objects satisfy a socalled (weak) Bogomolov-Gieseker (B-G for short) type inequality.…”

Stability conditions on Kuznetsov components of Gushel–Mukai threefolds and Serre functor

“…Moreover, $$\mathcal {G}$$ is a slope semistable torsion‐free sheaf. Since $$\mathop {\mathrm{Hom}}\nolimits (\mathcal {G}, \mathcal {O}_X[2])=\mathop {\mathrm{Hom}}\nolimits (\mathcal {O}_X(H),\mathcal {G}[-1])=0$$ by stability, [33, Theorem 3.14] implies that $$\mathcal {G}$$ is a vector bundle with $$\mathop {\mathrm{ch}}\nolimits (\mathcal {G})=\mathop {\mathrm{ch}}\nolimits (\mathcal {U}_X)$$. It follows that $$\mathcal {G}\cong \mathcal {U}_X$$.…”

“…Moreover,  is a slope semistable torsion-free sheaf. Since Hom(,  𝑋[2]) = Hom( 𝑋 (𝐻), [−1]) = 0 by stability,[33, Theorem 3.14] implies that  is a vector bundle with ch() = ch( 𝑋 ). It follows that  ≅  𝑋 .Step We end by showing the statement of the lemma, arguing as in Step 3 of the proof of Lemma 3.7.…”