Bridgeland Stability Conditions on Fano Threefolds

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.


Introduction
In the recent article [Li15], Chunyi Li proved the existence of Bridgeland stability conditions on Fano threefolds of Picard number 1. The purpose of this paper is to extend Li's method to any Fano threefold, when we choose the polarization to be the anti-canonical divisor.
Tilt stability was introduced in [BMT14] as an intermediate notion to define Bridgeland stability on threefolds. Its definition mimics Bridgeland's original construction in the surface case [Bri08,AB13]. Let X be a smooth projective complex threefold, and let H be a polarization on X. Also, let α, β ∈ R, α > 0. The usual notion of slope stability (with respect to H) induces a family of torsion pairs on the category of coherent sheaves Coh(X), parameterized by β, roughly obtained by truncating Harder-Narasimhan filtrations at the slope β. We denote the corresponding family of tilted abelian categories by Coh β (X). Tilt stability is the (weak) stability condition defined on Coh β (X) with respect to the slope ν α,β := H · ch β 2 − α 2 2 H 3 · ch β 0 H 2 · ch β 1 , where, as usual, dividing by 0 equals to +∞. Here ch β denotes the twisted Chern character ch ·e −βH . We will review all of this in Section 2.. The intuition is that tilt stability on threefolds should play the role of usual slope stability of sheaves on surfaces. By keeping this idea in mind, we should look for a generalization of the classical Bogomolov inequality for slope stable sheaves on surfaces to tilt stability. This is exactly the content of our main theorem. An analogous result, with a similar proof, appears independently in [Piy16, Theorem 1.3].
When the Picard rank of X is 1, Theorem 1.1 was proved in [Li15] with Γ = 0 (the cases of P 3 and of the quadric threefold were proved earlier, with a completely different proof, in [Mac14b] and [Sch14]). The idea of the proof of Theorem 1.1 follows along the same lines as in [Li15], by using an Euler characteristic estimate. This will be carried out in Section 4.. The key result is a slight generalization of [Li15,Proposition 3.2] to the higher Picard rank case. This is the content of Theorem 3.1, whose proof will take the whole Section 3..
There are examples in which the class Γ is indeed 0, even if the Picard number is greater than 1 (see Theorem 1.2). More generally, it would be interesting to know what the optimal class Γ is. For example, all known counter-examples ( [Sch16,Mar16]) to the original conjecture from [BMT14] on Fano threefolds do satisfy a stronger inequality for some choice of Γ with Γ · H = 0 (see also Section 6.). It is interesting to observe that when such a strong version of the conjecture holds, then all possible applications of such an inequality either to the birational geometry of threefolds or to counting invariants (pointed out in a sequence of papers; e.g., [BBMT14,Tod12,Tod14]) would hold unchanged. Finally, we remark that the original conjecture is also proved for abelian threefolds (and theirétale quotients) in [MP15,MP16,BMS16].
In the case of P 1 × P 2 (and partly in the case of the blow-up of P 3 in a line) we are able to prove the original conjecture with Γ = 0 by using the toric Frobenius morphism. The proof in these two cases will then follow a similar "limit" argument of Euler characteristics as in [BMS16].
Theorem 1.2. Let X be a P 2 -bundle over P 1 . Let H be an ample divisor such that, for all effective divisors D on X, we have H ·D 2 ≥ 0. Then, for any ν α,β -stable object E ∈ Coh β (X) with ν α,β (E) = 0, we have The assumption on the polarization H is quite strong. It holds for all polarizations on P 1 × P 2 . On the blow-up of P 3 in a line, it holds for some polarizations, but not for the anti-canonical bundle. In fact, we will prove Theorem 1.2 under a slightly more general technical assumption. The precise statement is Theorem 5.1. In particular, it will hold for all polarizations H on P 1 × P 1 × P 1 as well. Notation X smooth projective threefold over C H fixed ample divisor on X D b (X) bounded derived category of coherent sheaves on X ch(E) Chern character of an object

Background on tilt stability
In [BMT14] the notion of tilt stability was introduced as an auxiliary notion in between slope stability and Bridgeland stability on threefolds. Let X be a smooth projective threefold over the complex numbers and H be a fixed ample divisor on X. The classical slope for a coherent sheaf E ∈ Coh(X) is defined as where division by zero is interpreted as +∞. As usual a coherent sheaf E is called slope-(semi)stable if for any non trivial proper subsheaf F ⊂ E the inequality µ(F ) < (≤)µ(E/F ) holds. Let β be an arbitrary real number. Then the twisted Chern character ch β is defined to be e −βH ·ch. Explicitly: The process of tilting is used to construct a new heart of a bounded t-structure. For more information on the general theory of tilting we refer to [HRS96] and [BVdB03]. A torsion pair is defined by The heart of a bounded t-structure is given as the extension closure Coh β (X) := F β [1], T β . Let α > 0 be a positive real number. The tilt slope is defined as As before, an object E ∈ Coh β (X) is called tilt-(semi)stable (or ν α,β -(semi)stable) if for any non trivial proper subobject F ⊂ E the inequality ν α,β (F ) < (≤)ν α,β (E/F ) holds.
2 Z be the image of the map H · ch ≤2 . Notice that ν α,β factors through H · ch ≤2 . Varying (α, β) changes the set of stable objects. A numerical wall in tilt stability with respect to a class v ∈ Λ is a non trivial proper subset W of the upper half plane given by an equation of the form ν α,β (v) = ν α,β (w) for another class w ∈ Λ. A subset S of a numerical wall W is called an actual wall if the set of semistable objects with class v changes at S. The structure of walls in tilt stability is rather simple. Part (1) -(5) is usually called Bertram's Nested Wall Theorem and appeared first in [Mac14a], while part (6), (7), and (8) are in Lemma 2.7 and Appendix A of [BMS16]. The last part of (8) about reflexivity is to be found in [LM16, Proposition 3.1].
Theorem 2.2. (Structure Theorem for Tilt Stability) Let v ∈ Λ be a fixed class. All numerical walls in the following statements are with respect to v.
1. Numerical walls in tilt stability are either semicircles with center on the β-axis or rays parallel to the α-axis.
2. If two numerical walls given by classes w, u ∈ Λ intersect, then v, w and u are linearly dependent.
In particular, the two walls are completely identical.
3. The curve ν α,β (v) = 0 is given by a hyperbola. Moreover, this hyperbola intersects all semicircular walls at their top point.

4.
If v 0 = 0, there is exactly one numerical vertical wall given by β = v 1 /v 0 . If v 0 = 0, there is no actual vertical wall.
5. If a numerical wall has a single point at which it is an actual wall, then all of it is an actual wall.
7. If ∆ H (E) = 0 for a tilt semistable object E, then E can only be destabilized at the unique numerical vertical wall.
8. If E is a tilt stable object for fixed β ∈ R and all α 0, then E is one of the following.
• If ch 0 (E) < 0, then H 0 (E) is a torsion sheaf supported in dimension smaller than or equal to 1 and H −1 (E) is a reflexive slope semistable sheaf with positive rank.
A generalized Bogomolov type inequality involving third Chern characters for tilt semistable objects with ν α,β = 0 has been conjectured in [BMT14]. Its main goal was the construction of Bridgeland stability conditions on arbitrary threefolds. For any ν α,β -stable object E ∈ Coh β (X) with ν α,β (E) = 0 the inequality holds.
The conjecture has been proved for P 3 in [Mac14b] and for the smooth quadric hypersurface Q ⊂ P 4 in [Sch14]. All other Fano threefolds of Picard rank one were handled in [Li15]. Finally, it is known to hold for abelian threefolds with independent proofs in [BMS16] and [MP15,MP16]. It turns out that the conjecture fails on the blow up of P 3 in a single point as shown in [Sch16]. In this article we give an affirmative answer to the following natural follow up question in case X is a Fano threefold and the polarization H is given by the anticanonical divisor.
Question 2.4. Is there a cycle Γ ∈ A 1 (X) R depending at most on H such that Γ · H ≥ 0 and for any ν α,β -stable object E with ν α,β (E) = 0, we have The condition Γ · H ≥ 0 is crucial for the reduction to the case α = 0. In order to state it precisely, we first need the notion of β-stability.
Definition 2.5. For any object E ∈ Coh β (X), we define Moreover, we say that E is β-(semi)stable, if it is (semi)stable in a neighborhood of (0, β(E)).
Proposition 2.6. ([BMT14, Proposition 5.1.3]) Assume E ∈ Coh β (X) is ν α,β -semistable with ν α,β (E) = ∞. Then there is a ν α,−β -semistable objectẼ ∈ Coh −β (X) and a sheaf T supported in dimension 0 together with a trianglẽ The following lemma was first proved for the original conjecture in [BMS16], but it works out in our case as well without any change in the proof.
Proof. By Proposition 2.6 we can use the derived dual to reduce to the case ch 0 (E) ≥ 0. Tensoring with lines bundles O(aH) for a ∈ Z makes it possible to further reduce to β(E) ∈ [0, 1). A straightforward computation shows that is decreasing along the hyperbola ν α,β (E) = 0 as α decreases, because Γ · H ≥ 0 and ch 0 (E) ≥ 0. By using Theorem 2.2, (6), we can then proceed by induction on ∆ H (E) to show that it is enough to prove the inequality for β-stable objects.
Otherwise, E is destabilized along a wall between (α, β) and (0, β(E)). Let F 1 , . . . , F n be the stable factors of E along this wall. By induction, the inequality holds for F 1 , . . . , F n and so it does for E by linearity of the Chern character.
As in [BMS16], we get a quadratic inequality for any tilt semistable object. This still implies the support property for Bridgeland stability conditions, as in [BMS16, Section 8].
Proposition 2.8. Assume that Question 2.4 has an affirmative answer. Then any ν α,β -stable object E satisfies As a consequence, (α, β) lies on a unique numerical semicircular wall W with respect to v. One computes that there are x, y ∈ R, δ ∈ R ≥0 such that Moreover, the equality Q α,β (E) = 0 holds if and only if In particular, the equation Q α,β (E) ≥ 0 defines either the complement of a semi-disc with center on the β-axis or a quadrant to one side of a vertical line. Moreover, Q α,β (E) = 0 is a numerical wall with respect to v. Since numerical walls do not intersect, the inequality holds on either any or no point of W . Computing it at the top point of W concludes the proof.

Li's Bogomolov inequality
Let X be a Fano threefold of index i X . Consider tilt stability with respect to the polarization Theorem 3.1. Let E be a tilt stable object other than a shift of O X or an ideal sheaf of points. If additionally ch 0 (E) = 0 and 0 ≤ β − (E) ≤ β + (E) < 1, then the inequality Remark 3.2. Note, that when X has index 1 or 2 and its Picard rank is at least 2, then H 3 ≥ 2, and That turns out to be the only case in which we use this theorem in the following sections.
The idea of the proof is as follows. Assuming there exists an object E contradicting the theorem, we will show that E can be chosen such that either E or E[1] is tilt stable for all α > 0 and β ∈ R. The conditions on β ± (E) will then allow to prove ext 2 (E, E) = 0. A contradiction can be obtained from estimating the Euler characteristic χ(E, E) ≤ 1. We will fill the details in a series of lemmas.
then E is a shift of O X or an ideal sheaf of points. That means β(E) = 0. In particular, Theorem 3.1 holds for rank ±1 objects.

Proof. The Hodge Index Theorem implies
with equality if and only if ch 1 (E) is numerically equivalent to a multiple of H. Part (7) of the Structure Theorem for Tilt Stability shows that E is stable for α 0, and therefore, part (8) applies. In any of the two cases, E satisfies the classical Bogomolov inequality and we get H · (ch 1 (E) 2 − 2 ch 2 (E)) = 0. In particular, ch 1 (E) is numerically equivalent to a multiple of H.
If ch 0 (E) = −1, then H −1 (E) is reflexive of rank one, i.e., a line bundle. Since line bundles have no extensions with skyscraper sheaves, we must have H 0 (E) = 0. The hypotheses on β ± (E) directly Assume ch 0 (E) = 1. Then E ⊗ O(− ch 1 (E)) is an ideal sheaf of a subscheme of dimension smaller than or equal to one. Its second Chern character equals Since H is ample and it intersects this curve class as zero, we must have that E ⊗ O(− ch 1 (E)) is an ideal sheaf of a finite number of points. Again, the hypotheses on β ± (E) directly imply the claim.
In particular, let E be a tilt stable object with ch 0 (E) > 0 that destabilizes at a semicircular wall W . Then there is a stable factor F in the Jordan-Hölder filtration of E at W satisfying µ(F ) < µ(E) and ch 0 (F ) > 0. If instead ch 0 (E) < 0, then we get µ(F ) > µ(E) and ch 0 (F ) < 0.
Proof. We will only do the case ch 0 (E) > 0, and leave ch 0 (E) < 0 to the reader. The proof is by induction on the number of stable factors n in a Jordan-Hölder filtration of E. If n = 1, then we can simply choose E = F .
Assume the statement holds for some n ∈ N and that E has a Jordan-Hölder filtration of length n + 1. Let F → E be a stable subobject in such a Jordan-Hölder filtration of E, and let G be the quotient F/E. Because of ch 0 (E) > 0, we can divide the argument into the following three cases.
• Assume that ch 0 (F ) ≥ ch 0 (E) > 0 holds. That means we have This implies , but G is only semistable. By the inductive hypothesis G has a stable factor with the desired properties.
) lie in the upper half plane and add up to v = (−H 2 · ch 1 (E), H 3 · ch 0 (E)). It follows that exactly one of the slopes µ(F ) and µ(G) is larger than or equal to µ(E). If µ(F ) ≤ µ(E), we have proved the statement. If µ(G) ≤ µ(E), then we can use the inductive hypothesis on G to obtain an object with the desired properties.
For the last part observe that since the wall is not vertical, we get µ(F ) = µ(E) for all stable factors F . Lemma 3.6. Assume there is an object E contradicting Theorem 3.1 such that ∆ H (E) ≥ 0 is minimal among all such objects. Then E can only be destabilized at the unique numerical vertical wall.
Proof. By Lemma 3.3, such an object E must satisfy | ch 0 (E)| ≥ 2. We will give the proof in the case ch 0 (E) ≥ 2. The negative rank case is almost identical. 1 Assume E is destabilized at some semicircular wall W . We will show that there is a stable factor F of E at W that also contradicts Theorem 3.1. Then, part (6) of the Structure Theorem for Walls in Tilt Stability (Theorem 2.2) creates a contradiction to the minimality of ∆ H (E).
By Lemma 3.5 there is a stable factor F of E such that µ(F ) < µ(E) and ch 0 (F ) > 0. Since the ranks of both E and F are positive, the wall W is to the left of the vertical numerical walls of both E and F . In particular, the hyperbolas ν α,β (E) = 0 and ν α,β (F ) = 0 are both decreasing. Let (α 0 , β 0 ) be the top point on the semicircle W . By Lemma 3.4 the derivative of α by β at (α 0 , β 0 ) along the hyperbola ν α,β (E) = 0 is smaller than along the hyperbola ν α,β (F ) = 0 (see Figure 1). This proves holds. From here we see that F also contradicts the theorem. Lemma 3.7. Assume that E is a tilt stable object for some α > 0 and β ∈ R with ∆ H (E) = 0, β(E) = 0, and | ch 0 (E)| ≥ 2. Then there is an object E with ∆ H (E ) = 0, β(E ) = 0, and ch 0 (E ) ≥ 2 such that for all α > 0 and β ∈ R either E or E [1] is tilt stable.
Proof. By hypothesis we know H · ch ≤2 (E) = (r, 0, 0), for r = ch 0 (E). Theorem 2.6 implies that we can use the derived dual to reduce to r ≥ 2. By Theorem 2.2 (7), E can only destabilize at the vertical wall β = 0. Moreover, by part (8) of the same Theorem, E is a torsion free slope semistable sheaf.
Let E[1] G be a stable quotient in a Jordan-Hölder filtration of E[1] at the numerical vertical wall. By Theorem 2.2 (6), we have H · ch ≤2 (G) = (r , 0, 0), for r = ch 0 (G) ≤ 0. Since E is a sheaf, G cannot be a torsion sheaf, i.e., r = 0. If r ≤ −2, then we can choose E = G[−1]. If r = −1, then by Lemma 3.3, G ∼ = O X [1]. Consider the map E → O X . Since ∆ H (E) = 0, we must have that both kernel and image are also slope semistable sheaves with ∆ H = 0 and slope zero. In particular, they all have the same tilt slope independent of α and β. Since r ≥ 2, both have also non-zero ranks, and therefore, E is strictly tilt semistable, a contradiction.
Lemma 3.8. Assume there is an object E contradicting Theorem 3.1. Then E can be chosen such that for all α > 0 and β ∈ R either E or E[1] is tilt stable and ch 0 (E) ≥ 2.
Proof. Assume E contradicts the theorem such that ∆ H (E ) ≥ 0 is minimal among all such objects. By Lemma 3.3 the inequality | ch 0 (E)| ≥ 2 is automatic. If E or E [1] is tilt stable for all (α, β), we can set E = E and be done. If E or E [−1] is tilt stable for all (α, β), we can set E = E [−1] and be done. Otherwise, we can use Lemma 3.6 to show that E destabilizes at the vertical wall. Let F 1 , . . . , F n be all the stable factors of E that satisfy H · ch ≤2 (F i ) = 0 for i = 1, . . . , n.
The shape of the wall implies either β = µ(F i ) = µ(E ) or (H 3 · ch 0 (F i ), H 2 · ch 1 (F i )) = (0, 0). By definition of Coh β (X) the rank of an object at the vertical wall can never be positive, and rank 0 objects have to be torsion sheaves. Therefore, the vectors v i = (−H·ch β 2 (F i ), −H 3 ·ch β 0 (F i )) lie either in the upper half plane or on the negative real line. They also add up to v = (−H ·ch β 2 (E ), −H 3 ·ch β 0 (E )). This is only possible if at least one of the slopes is smaller than or equal to the slope If µ(E) = 0, then the condition β − (E) ≥ 0 implies ∆ H (E) = 0 and β(E) = 0, and we are done by Lemma 3.7. In all other cases, F i cannot be O X [1]. By part (6) of Theorem 2.2, ∆ H (F i ) is minimal again. In particular, the object E = F i [−1] also contradicts the theorem.
Proof of Theorem 3.1. Assume the theorem does not hold, and choose E contradicting it as in Lemma 3.8. Then ch 0 (E) ≥ 2. Stability of E implies hom(E, E) = 1. By assumption the inequality holds. Due to the fact that E(−i X H)[1] and E are both stable at β(E), we get ext 2 (E, E) = hom(E, E(−i X H)[1]) = 0. The Hirzebruch-Riemann-Roch Theorem together with the Hodge-Index Theorem implies By rearranging the terms one gets the contradiction

The main theorem
Let X be a Fano threefold of index i X with polarization H = − K X i X and consider tilt-stability with respect to it. In this section we prove Theorem 1.1. Chunyi Li already showed that the theorem holds with Γ = 0 for Picard rank 1 in [Li15]. Therefore, we only have to deal with the case where the Picard rank is at least two.
The idea of the proof is similar to Li's approach. As reviewed in Lemma 2.7, we will assume we have an object E which is β-stable with 0 ≤ β(E) < 1 and ch 0 (E) ≥ 0. Then stability is used to bound the Euler characteristic χ(E(−H)). Finally, Theorem 3.1 and the Hirzebruch-Riemann-Roch Theorem allow us to deduce the bound on the Chern character. To simplify notation, we will often write β(E) = β.

4.A. The index two case
First of all, recall that there are only three Fano threefolds with index two and large Picard number.
Theorem 4.1. ([Fuj82, Theorem 1]) A Fano threefold X of index two and Picard rank greater than or equal to two is given by either 1. the blow up of P 3 in a point, or 2. P 1 × P 1 × P 1 , or 3. the projective bundle P(T P 2 ), where T P 2 is the tangent bundle of P 2 .
In this section, we will prove Theorem 1.1 in the latter two cases. For P 1 × P 1 × P 1 we will actually prove the statement for all polarizations in Section 5. with a different method. The following lemma holds for all three cases. In particular, Another application of the Hirzebruch-Riemann-Roch Theorem leads to Finally, the last equality in the lemma is a straightforward computation using H · ch β 2 (E) = 0.
Throughout the rest of this section we will assume that X is either P 1 × P 1 × P 1 or P(T P 2 ). In fact, we will choose Γ = 0 and prove the original Conjecture 2.3. This does not work for the blow up of P 3 in a point due to the counterexample in [Sch16]. We will handle this case later individually. Lemma 4.3. Let X be either P 1 × P 1 × P 1 or P(T P 2 ). The equalities ch 2 (T X ) = 0 and H 3 = 6 hold.
Proof. These are standard calculations based on Chow-Künneth formulas for the first case and the Euler sequence on P 2 for the second case.
We are now in position to prove Theorem 1.1 for both P 1 ×P 1 ×P 1 and P(T P 2 ) with Γ = 0. Assume we have an object E which is β-stable with 0 ≤ β(E) < 1 and ch 0 (E) ≥ 0. Then Lemma 4.2 and Lemma 4.3 imply

4.B. Exceptional case of index two: the blow-up of P 3 in a point
Let X be the blow-up of P 3 in a point. In this case, X has index 2, and H 3 = 7. Denoting the exceptional divisor of the blow-up by e and the pull-back of the hyperplane section of P 3 to X by h, we have H = 2h − e. Moreover, e 3 = h 3 = 1, and h · e = 0 as a cycle in A 1 (X). Finally, a calculation of ch 2 (T X ) yields ch 2 (T X ) = 2h 2 + 2e 2 .

4.C. The index one case
In this section we prove Theorem 1.1 when X is a Fano threefold of index 1. Let H = −K X and consider tilt stability with respect to it. The approach is exactly as in the index two case, but we have to distinguish two cases according to whether 0 < β(E) < 1 or β(E) = 0. We have to define a 1-cycle Γ. For 0 ≤ β ≤ 1 and C ∈ R we define l X,C (β) := f X,C (β) H 3 + g X (β). We have g X (0) = − 1 H 3 < 0 and g X (1) = 1 H 3 > 0. Therefore, we can define β 0 to be the largest zero of g X (β) in the interval [0, 1]. Hence, g X (β) ≥ 0, for all β ∈ [β 0 , 1].
Remark 4.5. If H 3 ≤ 48, one gets β 0 = 1 2 . We refer to [IP99, Table 12.3-6] for the fact that there are only seven types of Fano threefolds of degree strictly greater than 48: 1. the blow up of P 3 in a point, also given as P(O P 2 ⊕ O P 2 (1)), which has degree d = 56, Picard number ρ = 2, and index i = 2, 2. the product P 1 × P 2 , which has degree d = 54, Picard number ρ = 2, and index i = 1, 3. the blow up of P 3 along a line, also given as P(O ⊕2 P 2 ⊕ O P 2 (1)), which has degree d = 54, Picard number ρ = 2, and index i = 1, 4. the projective bundle P(O P 2 ⊕ O P 2 (2)), which has degree d = 62, Picard number ρ = 2, and index i = 1, 5. the double blow up of P 3 first in a point and then in a line contained in the exceptional divisor, which has degree d = 50, Picard number ρ = 3, and index i = 1, 6. the double blow up of P 3 first in a point x and then in the strict transform of a line through x, which has degree d = 50, Picard number ρ = 3, and index i = 1, 7. the projective bundle P(O P 1 ×P 1 ⊕ O P 1 ×P 1 (1, 1)), which has degree d = 52, Picard number ρ = 3, and index i = 1.
Lemma 4.6. Let E be a β-stable object with 0 < β(E) < 1. Then In particular, Proof. This proof is the same as for Lemma 4.2.
We are now in position to prove Theorem 1.1 when E is a β-stable object with 0 < β(E) < 1 and ch 0 (E) ≥ 0. In this situation Lemma 4.6 implies where the second expression is only valid for ch 0 (E) = 0 and follows from the definition of β. As in the index two case, we will show that the right hand side is non-positive. Since both H 2 · ch β 1 (E) ≥ 0 and ch β 0 (E) ≥ 0 we are done if β ≥ β 0 , or ch β 0 (E) = 0. If 0 < β < β 0 and ch β 0 (E) = 0 we will split the two cases β + (E) ≥ 1 and β + (E) < 1.
If β + (E) ≥ 1, we get If β + (E) < 1, we can use Theorem 3.1 to obtain We are left to deal with the case of a β-stable object E with ch 0 (E) ≥ 0 and β(E) = 0. If ∆ H (E) = 0 then, by Theorem 3.1, E is a shift of O X or an ideal of points. In those cases, Theorem 1.1 is obvious.
Proof. Pick a β-stable object E such that H 2 · ch 1 (E) is minimal with the properties ch 0 (E) ≥ 0, β(E) = 0, and ch 3 (E) − Γ · ch 1 (E) > 0. Then ∆ H (E) = 0, and equivalently H 2 · ch 1 (E) = 0. If E could be chosen with ch 0 (E) arbitrarily large, the conditions of Theorem 3.1 would be fulfilled. This would imply even in the limit ch 0 (E) → ∞, a contradiction. Therefore, we can additionally choose E such that ch 0 (E) is maximal for objects with these properties.
If Ext 1 (E, O X ) = 0, we can get a non trivial triangle We want to show that E is in Coh 0 (X), i.e., that the morphism E → O X [1] is surjective. Let T be the cokernel of this map in Coh 0 (X). Since O X [1] is semistable, we get ν 0,α (T ) = ∞. Since it is even stable, this means T a sheaf supported on points (this is because those are the only objects that are mapped to the origin by Z 0,α ). But then Hom(O X [1], T ) = 0, a contradiction unless T = 0. We will show that E is also β-stable to get a contradiction to the maximality of ch 0 (E). Let be the Harder-Narasimhan filtration of E for β = 0 and α 1. The fact that E is β-stable and that E → E is injective in Coh 0 (X) implies Taking the limit α → 0 implies H · ch 2 (E i ) = 0 for all i. For every semistable factor E i /E i−1 we choose a Jordan Hölder filtration and call the stable factors of all those filtrations F 1 , . . . , F m . Then H · ch 2 (F i ) = 0 for all i = 1, . . . , m. The values ch 3 (F i ) − Γ · ch 1 (F i ) add up to ch 3 (E ) − Γ · ch 1 (E ) = ch 3 (E) − Γ · ch 1 (E) > 0. Choose j such that ch 3 (F j ) > Γ · ch 1 (F j ). By definition of Coh 0 (X) we have H 2 · ch 1 (F i ) ≤ H 2 · ch 1 (E ) = H 2 · ch 1 (E) for all i = 1, . . . , m.
Assume H 2 · ch 1 (F j ) < H 2 · ch 1 (E). If ch 0 (F j ) ≥ 0, then this contradicts the minimality of E. If ch 0 (F j ) < 0, then we can use the derived dual via Proposition 2.6 to reduce to the positive rank case. Hence, we must have H 2 · ch 1 (F j ) = H 2 · ch 1 (E) and H 2 · ch 1 (F i ) = 0 for i = j. Since H 2 · ch 1 (E) = 0, the existence of the morphism F 1 → E shows that j = 1. But the slopes of the factors in the Harder-Narasimhan filtration are strictly decreasing. Therefore, we must have m = 1 and E is indeed β-stable.
We can now finish the proof of Theorem 1.1 in the case of a β-stable object E with ch 0 (E) ≥ 0 and β(E) = 0. Assume there is E as in Lemma 4.7 contradicting the theorem, i.e.,

Toric threefolds
In this section, we use a variant of the method in [BMS16] to prove Conjecture 2.3 in some toric (not necessarily Fano) cases, with respect to certain polarizations. The results presented here arose from discussions together with Arend Bayer and Paolo Stellari.
Theorem 5.1. Let X be a smooth projective complex toric threefold. Let H be an ample divisor such that, for all effective divisors D on X, we have H · D 2 ≥ 0, and H · D 2 = 0 implies that D is an extremal ray of the effective cone. Then, for any ν α,β -stable object E ∈ Coh β (X) with ν α,β (E) = 0, we have Since the statement of Theorem 5.1 is independent of scaling H, we will assume throughout this section that H is primitive. The extra condition on the extremality of divisors with H · D 2 = 0 is probably not necessary. It is trivially satisfied by P 2 -bundles over P 1 . For us, it will simplify the proof, since it directly implies (see e.g., [CLS11, Lemma 15.1.8]) that a primitive such D is the class of an irreducible torus-invariant divisor. Also, since X is a threefold, there cannot be more than 3 such irreducible torus-invariant divisors with the same class. Indeed, if D is not movable, then there is only one irreducible divisor with that class. If D is movable, then since it is extremal in the effective cone, it cannot be big and so it induces a rational morphism on a lower dimensional toric variety of Picard rank 1, namely P 1 or P 2 .
The assumptions are satisfied for any polarization on P 1 × P 2 and on P 1 × P 1 × P 1 . In the blow-up of P 3 in a line, if we denote the pull-back of O P 3 (1) by h and the pull-back of O P 1 (1) by f , then the assumption is satisfied by any polarization of the form ah + bf , with a, b > 0 and a ≤ b. The class of the anticanonical bundle is 3h + f , and so it is not covered by the above result.
In order to prove Theorem 5.1, we use the toric Frobenius and the formula from Theorem 5.2 below as follows. First, as in the previous sections, we can work only with β-stable objects. In the case where β(E) = 0, assuming ch 3 (E) > 0, the Euler characteristic χ(O X , m * E) grows as a polynomial in m of degree 3. On the other hand, by using adjointness, Theorem 5.2, and stability of line bundles, we can bound both hom(O X , m * E) and ext 2 (O X , m * E) and show that χ(O X , m * E) has to grow at most as a polynomial in m 2 , thus giving a contradiction to ch 3 (E) > 0. The basic idea is then to reduce to this case. When β(E) is an integer, this is easy to do by tensoring with line bundles. When it is a rational number, we pull-back again via the toric Frobenius morphism to simplify denominators. Finally, the irrational case can be dealt with by elementary Dirichlet approximation.

5.A. Toric Frobenius morphism
Let X be a smooth projective toric threefold. We denote the irreducible torus-invariant divisors by D ρ (corresponding to the rays ρ of the fan associated to X). The canonical line bundle is then For an integer m ∈ N, we denote the toric Frobenius morphism by m : X → X.
It is defined via multiplication by m on the lattice and is a finite flat map of degree m 3 . The main property we will need is the following result from [Tho00] on direct images of line bundles (see also [Ach15] for a short proof).
Theorem 5.2. Let D ∈ Pic(X) be a line bundle on X. Then where • the line bundles L j are given by all the possible integral divisors in the formula where a ρ varies in between 0, . . . , m − 1, • the multiplicity η j counts the number of a ρ 's giving the same line bundle L j .
Example 5.3. Let X = P 2 × P 1 and let us denote the pull-backs of the hyperplane classes from P 2 and P 1 by h and f respectively, as before. There are five torus-invariant irreducible divisors, three of them lie in the class h and two of them in the class f . It follows that where r 1 , r 3 grow like m 2 and r 2 , r 4 grow like m 3 .
Example 5.4. Notice that Frobenius pull-back in general does not preserve β-stability. Indeed, in the case of the blow-up of P 3 in a point, we can consider O(h). It is not too hard to check that O(h) is β-stable, while for any m ≥ 2, the pull-back m * O(h) = O(mh) is not β-semistable (see [Sch16] and Section 6. for more details).

5.B. Proof of Theorem 5.1: the integral case
We use the statement in [BMS16, Conjecture 5.3] (or Lemma 2.7 in the present note). Let E be a β-stable object. In this section, we assume that β(E) ∈ Z. By tensoring by multiples of O X (H), we can assume that β(E) = 0. We want to show ch 3 (E) ≤ 0. Assume the contrary ch 3 (E) > 0. By using the Hirzebruch-Riemann-Roch Theorem we can compute If m * E were β-semistable, we could easily get a contradiction. Unfortunately, in general, it is not true, as remarked in Example 5.4. But we can still use the push-forward m * and Theorem 5.2. Indeed, by adjointness, we have The last inequality follows since, given a line bundle L on X, we have hom(L, E[i]) = 0 if i < −1 or i > 3. We want to prove that the right hand side has order ≤ m 2 . In the notation of Theorem 5.2, First of all, since H is ample and L j is effective, we have and H 2 · L j = 0 if and only if L j = O X . Hence, L j ∈ Coh β=0 (X), when L j = O X , and as remarked before, by [BMS16], they are all ν α,0 -stable, for all α > 0. If H · L 2 j > 0, then and so Hom(L j , E) = 0. If H · L 2 j = 0, then by our assumption a ρ = 0 for all but at most 3 rays ρ. Hence, η j has order at most m 2 : there are at most three non-trivial coefficients a ρ , with the property 0 ≤ a ρ < m, and related by a linear equation defining L j in Theorem 5.2. Summing up, we have To bound the ext 2 , by Serre duality, we have Since Since 0 ≤ a ρ < m, then m − a ρ > 0, for all ρ. Therefore, H 2 · L j < 0 and H · (L j ) 2 > 0. Hence, L j [1] ∈ Coh β=0 (X) and lim giving the required contradiction.

5.C. Proof of Theorem 5.1: the rational case
In this section, we assume that β(E) ∈ Q \ Z. We write β(E) = p q , with p and q coprime, q > 0. We want to show that ch p/q 3 (E) ≤ 0. As before, by the Hirzebruch-Riemann-Roch Theorem, we have and, by adjointness, and 0 ≤ a ρ < mq. Therefore, is an effective divisor and cannot be O X because p/q is not an integer. It follows that H 2 ·ch As in the β(E) = 0 case, the equality holds if and only if the corresponding η j has order at most m 2 , since q is constant.
and so hom(L j , E) = 0. As in the integral case, this shows that The vanishing of ext 2 follows as in the integral case, by using Serre duality and stability.
By the Dirichlet approximation theorem, there exists a sequence β n = pn qn n∈N of rational numbers such that β(E) − p n q n < 1 q 2 n < for all n, where q n → +∞ as n → +∞. We compute, for n 0, The last inequality follows since, by definition, ch β 3 (E) has a local minimum at β = β(E). As in the previous cases, we assume for a contradiction that ch β(E) 3 (E) > 0, and we want to bound χ (q n * O X (−p n H)) ∨ , E ≤ hom (q n * O X (−p n H)) ∨ , E + ext 2 (q n * O X (−p n H)) ∨ , E for n 0. By Theorem 5.2, and 0 ≤ a (n) ρ < q n . Notice that, since L (n) j is an integral divisor, pn qn and a (n) ρ qn are both universally bounded with respect to n, there is only a finite number of isomorphism classes of L (n) j for all n. We have that ch pn/qn 1 (L (n) is an effective divisor, and cannot be O X for n 0. Since ρ a (n) ρ D ρ is an integral divisor, we have Now, for n 0. Therefore L j , E) = 0 by stability. Since q n → ∞, we can have f n,j ≤ 0 only if the ratio 2H 2 · a (n) is not bounded from above for n 0, since H · a (n) ρ D ρ 2 ≥ 0 by assumption. But for a constant K > 0 which is independent on n and j, and H · a (n) ρ D ρ 2 = ρ,τ a (n) ρ a (n) τ (H · D ρ · D τ ).
Fix ρ 0 such that a (n) ρ 0 = 0. Then unless a (n) τ = 0 for all τ for which H · D ρ 0 · D τ = 0. That is, we can have a (n) τ = 0 possibly only for τ such that H · D ρ 0 · D τ = 0. By our assumption, the latter equality means that D τ has the same divisor class as D ρ 0 , H · D 2 ρ 0 = 0. Indeed, we can show first that H · D ρ 0 · D τ = 0 implies that H · D 2 τ ≤ 0. Assume not and set λ = H 2 ·Dρ 0 H 2 ·Dτ . The Hodge Index Theorem says 0 ≥ H · (D ρ 0 − λD τ ) 2 = λ 2 H · D 2 τ > 0, a contradiction. By the assumption in the theorem this implies H · D 2 τ = 0. Thus, we get H · (D τ + D ρ 0 ) 2 = 0. By the extremality assumption, we then have that D τ is in the same class as D ρ 0 . It follows that As before, the vanishing of ext 2 follows as in the integral case, by using Serre duality and stability. This shows that ch β(E) 3 (E) ≤ 0 also in this case, and therefore completes the proof of Theorem 5.1.
6. Details about the blow-up of P 3 in a point Theorem 1.1 implies the existence of Bridgeland stability conditions on all Fano threefolds. However, it would be interesting to know what the optimal class Γ is. A condition that is coherent with the case of Picard rank 1 would be Γ · H = 0.
In this section we will study the blow-up of P 3 in a point more carefully. We use the notations for divisors on X which were introduced in Section 4.B.. In [Sch16], it was shown that the line bundle O(h) does not satisfy Theorem 1.1 with Γ = 0. In [Mar16], it was shown that the structure sheaf O e of the exceptional divisor also does not satisfy Theorem 1.1 with Γ = 0. We will do the following computation.
Proof. Recall that H = 2h − e, so by twisting a suitable choice of O(H), we can assume that the line bundle is of the form O(mh) for an integer m.