An effective restriction theorem via wall-crossing and Mercat's conjecture

We prove an effective restriction theorem for stable vector bundles $E$ on a smooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors $D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we show that Mercat's conjecture in any rank greater than $2$ fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgeland stability conditions which we also use to reprove Camere's result on slope stability of the tangent bundle of $\mathbb{P}^n$ restricted to a K3 surface.


Introduction
Inspired by the construction of Bridgeland stability conditions on K3 surfaces [Bri08] Toda, Bayer, Macrì and Stellari [BMT14,BMS16] studied weak stability conditions on any smooth complex projective variety. In this paper, we use wall-crossing with respect to these weak stability conditions to prove an effective restriction theorem that expresses sufficient conditions on a slope-stable reflexive sheaf such that its restriction to a hypersurface remains stable. Restriction theorems provide us with the possibility of studying higher dimensional varieties via the geometry of their subvarieties. That is why they have been long-studied via different approaches; see [HL10, Chapter 7] for a survey.
Let X be a smooth complex projective variety of dimension n ≥ 2 with an ample divisor H. For a µ-stable coherent sheaf E of positive rank on X, we define Theorem 1.1. Let E be a µ-stable reflexive sheaf on X of rank rk > 0. The restricted sheaf E| D for any irreducible divisor 1 D ∈ |kH| is µ-semistable on D if Moreover, E| D is µ-stable on D if the inequalities in (3) are both strict.
1 D can be singular. The µ-slope of a coherent sheaf and the notion of µ-(semi)stability are defined in section 2. Note that if k ≥ 2δ(E), the conditions in (3) are equivalent to When rk > 1, we always have δ(E) ≥ 1 H n rk(rk −1) . If we substitute this lower bound, we obtain one of Langer's restriction theorems [Lan04], see Corollary 4.4 and Remark 4.5 for more details.
Clifford indices. The Clifford index Cliff(C) of a smooth curve C is the second important invariant of C after the genus g, which carries the information of special line bundles C. Lange and Newstead [LN10] proposed a generalisation of Cliff(C) to higher rank Clifford index Cliff r (C) which depends on rank r-semistable vector bundles on C.
Take a vector bundle E of rank r and degree d on C. The Clifford index of E is defined as Cliff(E) = 1 r (d − 2(h 0 (E) − r)). We say E contributes to the rank r-Clifford index of C if E is µ-semistable with degree d ≤ r(g − 1) and h 0 (C, E) ≥ 2r. Then the rank r-Clifford index of C is defined as the quantity Cliff r (C) := min Cliff(E) : E contributes to the rank r-Clifford index of C .
Note that Cliff 1 (C) = Cliff(C) is the classical Clifford index of C. In terms of these new invariants, Mercat's conjecture [Mer02] can be expressed as M r (C) : Cliff r (C) = Cliff(C) which says higher rank Clifford indices are equal to the rank one Clifford index.
Curves over K3 surfaces have played an important role in the Brill-Noether theory of vector bundles on curves. In [BF18], the conjecture M 2 (C) was proved for any smooth curve C ∈ |H| on a K3 surface S when Pic(S) = Z.H. Using this, M 2 (C) was shown for generic curves of every genus. However, the restriction of Lazarsfeld-Mukai bundles on S to the curve C (see Section 5 for a definition) have led to counterexamples to M 3 (C) [FO12] and M 4 (C) [AFO16]. As a consequence of Theorem 1.2, we generalise these results to higher ranks and show M r (C) fails for r ≥ 3 and any smooth curve C ∈ |H|.
Theorem 1.2. Let (S, H) be a smooth polarised K3 surface such that H 2 divides H.D for all curve classes D on S, e.g. Pic(S) = Z.H. Take a µ-stable vector bundle E on S with Chern character ch(E) = (r, H, ch 2 ) such that r ≥ 2 and ch 2 ≥ 0. Then for any smooth curve C ∈ |H| of genus g, the restricted bundle E| C contributes to the rank r-Clifford index of C. If r ≥ 4, or, r = 3 and g − ch 2 < 4 + 3 2 ⌊ g−1 2 ⌋, then In particular, M r (C) does not hold if either (i) r ≥ 4 and g ≥ r 2 , or (ii) r = 3 and g = 7, 9 or g ≥ 11.
Slope stability of the restriction of tangent bundle of P n to a K3 surface. Let S be a smooth projective K3 surface and let L be an ample line bundle generated by its global sections on S. Then there is a well-defined morphism In the final part of the paper, we apply wall-crossing with respect to Bridgeland stability conditions on K3 surfaces to reprove Camere's result on slope-stability of φ * L T P n . The restriction of the Euler exact sequence to the K3 surface S and tensoring with L * gives the short exact sequence Theorem 1] Let S be a smooth projective K3 surface over C, and let L be a globally generated ample line bundle on S. Then the kernel M L of the evaluation map on the global sections of L Method of the proof of Theorem 1.1. There is an abelian subcategory A ⊂ D(X) of the bounded derived category of coherent sheaves on X which includes E and E(−kH)[1] for k ∈ N. Thus for any irreducible divisor D ∈ |kH|, the restricted sheaf E| D lies in an exact sequence in A. The slope-stability of the reflexive sheaf E implies that there is a weak stability condition σ on A such that both E and E(−kH)[1] are stable with respect to σ. If k is large enough, we may apply wall-crossing techniques to show that we can deform the weak stability condition σ, while keeping E and E(−kH)[1] stable, to reach a weak stability condition σ ′ where E and E(−kH)[1] have the same slope (phase). Thus their extension E| D is σ ′ -semistable of the same slope. Then a general argument immediately implies that E| D is slope-stable. The main advantage of this method is the possibility to strengthen effective restriction theorems for special sheaves E as soon as we have more control over the position of the walls for E and E(−kH)[1]; see for instance Proposition 4.6.
Related work. Theorem 1.1 appeared in an earlier version of this paper [Fey16], where it was stated for K3 surfaces only. However the proof works for any variety, as the current version of the paper makes explicit. The result has now found applications in [Li19,Kos20] to construct Bridgeland stability conditions on Calabi-Yau threefolds, and in [Kop20] to investigate the stability of the restriction of stable vector bundles on a surface to any integral curve, not necessarily multiples of H.
In joint work with Chunyi Li [FL18], we employed the techniques in the current paper and [Fey19] to compute rank ≥ 2 Clifford indices of curves over K3 surfaces.
Acknowledgements. I would like to thank Arend Bayer for many useful discussions. I am grateful for comments by Gavril Farkas, Chunyi Li, Angela Ortega and Richard Thomas. The author was supported by the ERC starting grant (PI: Arend Bayer) WallXBirGeom 337039 and EPSRC postdoctoral fellowship EP/T018658/1.

Two-dimensional slice of weak stability conditions
In this section, we recall the notion of (weak) Bridgeland stability conditions on the bounded derived category of coherent sheaves. The main references are [BMS16,BMT14].
Let X be a projective scheme over C and let H be the class of an ample divisor on X. Recall that the Hilbert polynomial P (E, m) of a coherent sheaf E is defined via the Euler characteristic m → χ(E ⊗ O X (mH)). It can be uniquely written as . If E is a coherent sheaf of dimension d = dim X, we define the rank and degree of E as Then for any coherent sheaf E, the slope is defined as if α dim(X) = 0, +∞ oth.
From now on, we assume X is a smooth projective complex variety of dimension n ≥ 2. For any coherent sheaf E on X, the Hirzebruch-Riemann-Roch formula shows that deg(E) = ch 1 (E).H n−1 .
We denote the bounded derived category of coherent sheaves on X by D(X) and its Grothendieck group by K(X). For any b ∈ R, consider the full abelian subcategory of complexes Here µ ± H (E) for a coherent sheaf E is the maximum (minimum) slope in the Harder-Narasimhan filtration of E with respect to µ-stability. By [Bri08, Lemma 6.1], Coh b (X) is the heart of a bounded t-structure on D(X).
For any w > b 2 2 , we define the function Z b,w : K(X) → C such that for any [E] ∈ K(X), Proposition 12.2] 2 . Thus for objects in Coh b (X), we have the slope function Note that ν b,w -stability defines a Harder-Narasimhan (HN) filtration on Coh b (X), and so a weak stability condition on D(X) [BMS16, Proposition 12.2].
In Figure 1 we plot the (b, w)-plane simultaneously with the image of the projection map 2 To have a weak stability function whose imaginary part is non-negative and the real part is linear, we replace Z a,B=bH in [BMS16, Proposition 12.2] by Also, the image Π(E) of ν b,w -semistable objects E with ch 0 (E) = 0 is always outside U due to the classical Bogomolov-Gieseker-type inequality for the H-discriminant, Objects of D(X) give the space of weak stability conditions a wall and chamber structure, explained in [FT20, Proposition 4.1] for instance. Proposition 2.3 (Wall and chamber structure). Fix an object E ∈ D(X) such that the vector ch 0 (E), ch 1 (E).H 2 , ch 2 (E).H is non-zero. There exists a set of lines {ℓ i } i∈I in R 2 such that the segments ℓ i ∩ U (called " walls") are locally finite and satisfy (a) If ch 0 (E) = 0 then all lines ℓ i pass through Π(E).
in one of the two chambers adjacent to the wall ℓ i .
The line segments ℓ i ∩ U are walls for E.

large volume limit
As before X is a smooth projective complex variety of dimension n ≥ 2 and H is an ample divisor on X. In this section, we first prove the following general statement which holds for any coherent sheaf.
Theorem 3.1. Given a coherent sheaf E of rank bigger than one. There is no wall for E crossing the vertical line b = b 0 at a point inside U whenever Proof. Suppose for a contradiction that there is a wall ℓ for E which intersects the vertical at the intersection points of the wall ℓ with ∂U , see Figure 3. The inequality (13) gives One can easily check Step 1. First assume r ≥ 3H n or ∆ H (E) ≥ 3, then (15) gives be a destabilising sequence for E which makes it unstable in one side of the wall ℓ with We consider two different cases: for (b, w) ∈ U above the wall ℓ E , i.e. our destabilising sequence destabilises E below ℓ. Then r 1 = 0 and Suppose b 2 < b < b 1 , then (17) implies r 1 > 0 and so The limit of (17) when , The last inequality comes from the point that b * r and b * r 1 are integral multiples of 1 ch 0 (E)−1 . Therefore, by (19) we get which is not possible by our assumption (14).
Case ii. Suppose our destabilising sequence makes E unstable above the line ℓ E , then Then limit of (17 Hence applying the same argument as in (20) and (21) reach a contradiction.
Step 2. Finally, we show that if r = 2H n and ∆ H (E) = 1, or 2, there is no wall for E in U , so the claim follows. Suppose there is a wall with the destabilising sequence (16). The same argument as in [BMS16, Corollary 3.10] gives which is not possible. Similarly, if ∆ H (E) = 2, one of the following cases happens: (a) ∆ H (E i ) = 0 for i = 1, 2. Since H n |r i , we get H n |c 2 i , but which is not again possible. Similar argument applies to the case ∆ H (E 1 ) = 0 and By combing the ideas in the proof of Theorem 3.1 and the notion of safe area introduced in [FT21], we get the following.
Proposition 3.2. Take a µ-stable coherent sheaf E of positive rank, and let ℓ be a line passing through Π(E) which intersects ∂U at two points with and Then there is a unique line through Π(E) which is tangent to ∂U at a point to the left of Π(E). If we move this line upwards while making it pass through Π(E), its intersection point with ∂U move further apart until we find a unique ℓ E for which the b-values b * 2 < b * 1 of the two points of ℓ E ∩ ∂U satisfy We show there is no wall for E above the line ℓ E to the left of Π(E). Since the line ℓ in the Proposition lies above ℓ E , the claim follows.
b, H n−1 ch 1 b * 1 <b 1 < µ(E). Let E 1 ֒→ E ։ E 2 be a destabilising sequence for E which make it unstable belowl. We first show that both E 1 and E 2 are coherent sheaves. Taking cohomology from the destabilising sequence gives the long exact sequence of coherent sheaves This immediately gives E 1 is a sheaf. Suppose H −1 (E 2 ) is of rank r ′ = 0. As we move b ↑b 1 or b ↓b 2 alongl, E 1 and E 2 remain in the heart Coh b (X) by Proposition 2.3. This implies µ(H −1 (E 2 )) ≤b 2 and µ(H 0 (E i )) ≥b 1 for i = 1, 2. Combining this with (24) gives which is not possible by (23), so r ′ = 0. We know H −1 (E 2 ) is a torsion-free sheaf by the definition (8) of Coh b (X), thus it is zero and E 2 is a sheaf. Hence E 1 is a subsheaf of E. The walll intersects the vertical line b = b * 1 at a point (b * , w) inside U where E 1 and E have the same ν b * 1 ,w -slope, see Figure 4. Thus . The second inequality comes from µ-stability of E. Note that the quotient sheaf E 2 cannot be supported in co-dimension at least 2, otherwise its ν b,w -slope is +∞ and it cannot destabilise E, so µ(E 1 ) = µ(E). Therefore The next step is to analyse points (b, w) ∈ U where the shift of a slope-stable reflexive sheaf is stable. The following Lemma is well-known, we add it for completeness.
If ch 0 (H 0 (E 1 )) = 0, we get µ(H 0 (E 1 )) = b 0 which is not possible by the definition (8) of Coh b 0 (X). Therefore, E 1 is a sheaf supported in co-dimension at least 2. Taking cohomology from (25) gives the short exact sequence of coherent sheaves Since E is a reflexive sheaf, [Har80, Proposition 1.1] implies that E lies in the short exact sequence where G is a locally free sheaf and G ′ is torsion-free. Since H 0 (E 1 ) is supported in codimension at least 2, we have This implies Ext 1 (H 0 (E 1 ), E) = 0, thus (27) gives . Taking cohomology from (28) gives a short exact sequence of coherent sheaves Thus they all have the same µ-slope which is not possible by µ-stability of E.
By applying a similar argument to the proof of Proposition 3.2, we get the following.
Proposition 3.4. Take a µ-stable reflexive sheaf E of positive rank, and let ℓ be a line passing through Π(E) which intersects ∂U at two points with b-values b 2 < b 1 such that and To prove the claim we only need to show there is no wall for E[1] above the line ℓ E[1] to the right of Π(E), see Figure 5. Suppose there is such a walll above ℓ E[1] whose intersection point with ∂U has b-valuesb 2 <b 1 satisfyingb 2 < b * 2 and b * 1 <b 1 . Let be a destabilising sequence which makes E[1] unstable belowl. Taking cohomology from the destabilising sequence gives the long exact sequence of coherent sheaves We first show that H 0 (E 1 ) is of rank zero. Suppose it is of rank r ′ = 0, then which is not possible by (29). Thus H 0 (E 1 ) is of rank zero and the exact sequence (30) gives Sincel crosses the vertical line b = b * 2 at a point inside U , we gain Combining this with (31) gives  However, we gave here independent proof. We finish this section by mentioning another way to control the position of walls.
Proof. Assume otherwise, so when we move along the vertical line b = b 0 , E gets strictly ] which is not possible by our assumption.

slope stability of the restricted sheaf
In this section, we prove our main result Theorem 1.1 which provides sufficient conditions for a slope-stable reflexive sheaf whose restriction to a hypersurface is slope semistable. Firstly, we examine µ-stability of a sheaf supported on a hypersurface via the Chern character of its push-forward. Proof. For any coherent sheaf Q on D, by adjuction, we get Since X is a smooth projective variety, Hirzebruch-Riemann-Roch Formula gives Since Q is supported on an irreducible divisor D ∈ |kH|, ch 1 (i * Q) is a multiple of H, so Therefore, if α n−1 (Q) = 0, the slope of Q is Hence, for a quotient sheaf F ։ F ′ on D, we have µ(F ) < (≤) µ(F ′ ) if and only if (32) holds.
The next proposition describes how we can use the 2-dimensional slice U of weak stability conditions to prove the µ-stability of the restricted sheaf.
Proposition 4.2. Let E be a reflexive sheaf on X such that for a point (b, w) ∈ U , E and Then the restricted sheaf E| D for any irreducible divisor D ∈ |kH| is µ-semistable on D if it has no quotient sheaf E| D ։ F whose point Z b,w (F ) lies inside the triangle △opp ′ or on its sides except op ′ where p = Z b,w (E) and p ′ = Z b,w (E| D ).
Re[Z b,w ] = − ch 2 H n−2 + wH n ch 0 Figure 6. The triangle △opp ′ Proof. Since E and E(−kH)[1] are in the heart Coh b (X), the exact triangle implies that E| D ∈ Coh b (X). Let F + ֒→ E| D be the subobject of E| D of maximum ν b,w -slope in the HN filtration of A similar argument also implies Then there is a sequence of coherent sheaves . By definition of the heart Coh b (X), any torsion sheaf Q is in the heart and its ν b,w -slope is equal to ch 2 (Q).H n−2 ch 1 (Q)H n−1 if ch 1 (Q)H n−1 = 0, otherwise it is +∞. Thus (37) is an exact sequence in Coh b (X) and by Lemma 4.1, This implies Z b,w (F ) lies to the right of op ′ , see Figure 6. Moreover, we have . Combining this with (35) and (36) shows that The first inequality implies Z b,w (F ) cannot be to the right of op, and the second one shows Z b,w (F ) cannot be above p ′ p. Note that the slope of the line segment p ′ p corresponds to ν b,w (E(−kH)[1]). Hence Z b,w (F ) lies inside △opp ′ or on the line segments op or pp ′ .
Proposition 4.2 in particular, implies the following result.
Corollary 4.3. Let E be a slope-stable reflexive sheaf such that E and E(−kH)[1] are ν b,w -stable of the same slope, then E| D is µ-stable for any irreducible divisor D ∈ |kH|.
Proof. By proposition 4.2 E| D is µ-semistable. Suppose it is strictly µ-semistable and E| D ։ F is a proper quotient sheaf with µ(E| D ) = µ(F ). We know F is also a quotient object of E| D in Coh b (X) with the same ν b,w -slope as E| D . Thus E| D is strictly ν b,wsemistable.
Since E is ν b,w -stable, the composition E ֒→ E| D ։ F in Coh b (X) must either be zero or injective. It cannot be zero because this would give a surjection E(−kH)[1] ։ F in Coh b (X), contradicting the ν b,w -stability of E(−kH)[1]. Thus it is injective, then its cokernel C in Coh b (X) lies in a commutative diagram Since E and F are ν b,w -semistable of the same slope, C is also ν b,w -semistable. Hence the right hand surjection again contradicts the ν b,w -stability of E(−kH)[1].
Proof of Theorem 1.1. Let ℓ be the line passing through Π(E) and Π(E(−kH)) for k > 0, and let r = H n ch 0 (E). If By Proposition 3.2 and 3.4, E and E(−kH)[1] are ν b,w -stable for any (b, w) ∈ ℓ ∩ U if the following three conditions are satisfied: Figure 7. Stability of the restricted bundle The condition (a) is equivalent to and thus Note that (40) and so condition (a) implies (39). Also (b) and (c) are equivalent to Hence, if we have strict inequalities in (3), then E and E(−kH)[1] are ν b,w -stable of the same slope for (b, w) ∈ ℓ ∩ U , so E| D is µ-stable by Corollary 4.3. Now suppose we have equality in (a), (b), or (c) which means equality in one of the inequalities in (3). The computation in (40) shows that still (39) holds, so ℓ ∩ U = ∅. Then the structure of walls and Proposition 3.2 and 3.4 imply that E and E(−kH)[1] are ν b,w -semistable for (b, w) ∈ ℓ ∩ U . Hence E| D is µ-semistable on D by Proposition 4.2.
As a consequence of Theorem 1.1, we obtain a variant of Langer's restriction Theorem [Lan04, Theorem 5.2].
Corollary 4.4. Let E be a µ-stable reflexive sheaf of rank rk > 1. The restricted sheaf E| D for any irreducible divisor D ∈ |kH| is µ-stable if Proof. We always have .
Hence if (43) holds, then the second inequality in (3) is satisfied. Also, if rk ≥ 3 or ∆ H (E) ≥ 3, then Remark 4.5. [Lan04, Theorem 5.2] covers more general cases than Corollary 4.4, it allows X to be a smooth projective variety over an arbitrary algebraically closed field, and assumes E to be a µ-stable torsion free sheaf with respect to some nef divisors.
In some special cases, we have more control over the position of the walls for E and E(−kH)[1], so we gain stronger effective restriction results. For instance, suppose (X, H) is a smooth polarised complex projective variety of dimension n ≥ 2 such that Let E be a µ-stable reflexive sheaf with gcd H n ch 0 (E), ch 1 (E)H n−1 = H n . We define Proposition 4.6. Let E be a µ-stable reflexive sheaf as above with rank rk > 1. The restricted sheaf E| D for any irreducible divisor D ∈ |kH| is µ-(semi)stable on D if .
Proof. First of all, since δ ≤ 1 2 , (46) is equivalent to As in the proof of Theorem 1.1, let ℓ be the line passing through Π(E) and Π(E(−kH)). First assume the inequality in (46) is strict, so Thus (39) which is equivalent to (47).

Clifford indices of curves over K3 surfaces
In this section, we assume (S, H) is a smooth polarized K3 surface over C such that H 2 divides H.D for all curve classes D on S.
Let ι : C ֒→ S be any smooth curve in the linear system |H| of genus g. It is proved in [Laz86] and [Bay18, Theorem 1.1] that there exists a globally generated degree d line bundle A on C with r global sections if and only if Take such a line bundle A on C. Then the kernel of the evaluation of sections of ι * A is a vector bundle of rank r. The Lazarsfeld-Mukai bundle E C,A associated to the pair (C, A) is the dual of F C,A and it is of character ch(E C,A ) = r, H, g − 1 − d .
The bundle E C,A has been appeared, for example, in Lazarsfeld's proof of Brill-Noether Petri Theorem [Laz86], in the Mukai's classification of prime Fano manifolds of coindex 3, or in Voisin's proof of Green's canonical syzygy conjecture [Voi05]; see [Apr13] for a survey of applications.
The µ-slope of any destabilising subsheaf of F C,A must be less than zero because of the exact sequence (52), thus F C,A and so E C,A are µ-stable. Hence (51) implies that for any ch 2 ∈ Z with there is a stable vector bundle E C,A of Chern character v = (r, H, ch 2 ) for any curve C ∈ |H|.
The restriction of Lazarsfeld-Mukai bundle E C,A to curves on the K3 surface S has led to counterexamples to M 3 (C) [FO12] and M 4 (C) [AFO16]. By applying Theorem 1.1, we extend these results to higher ranks.
The last inequality follows from hom(O S , E(−H)) = 0. So E| C contributes to the rank r-Clifford index of C. By (51), we obtain Cliff(C) = g−1
because w > b 2 2 − 1 H 2 by our assumption. Thus we only need to check positivity of the real part for spherical bundles as mentioned.
The next step is to control the position of the projection of spherical bundles. Since in Coh b 0 (X) implies that M L [1] is also ν b 0 ,w 0 -semistable.
We claim M L [1] is ν b 0 ,w + -stable where w 0 < w + ≪ w 0 + 1. Otherwise, the structure of locally finite set of walls described in Proposition 2.3 shows that there is a destabilising sequence E 1 ֒→ M L [1] ։ E 2 such that the E i have the same ν b 0 ,w 0 -slope as M L [1] and they make M L [1] unstable above ℓ, i.e. (58) for any w > w 0 . Hence, rank of E 1 is negative. We know ν b 0 ,w 0 -stable factors of E 1 and If k > 1, we claim there is no wall for M L [1] in the region V (55) and it is ν b,wstable for any (b, w) ∈ V . Suppose there is a wall ℓ ′ in V with a destabilising sequence E 1 ֒→ M L [1] ։ E 2 . We know this wall ends at Π(M L ). For any (b, w) ∈ ℓ ′ ∩ V , we have Z b,w (E i ) = 0 and |Z b,w (E 1 )| + |Z b,w (E 2 )| = |Z b,w (M L [1])| which implies |Z b,w (E i )| < |Z b,w (M L )|. If we move (b, w) along the wall ℓ ′ towards Π(M L ), then |Z b,w (M L [1])| → 0, thus Therefore ch 0 (E i ) = 0 and Π(E i ) = Π(M L ), so the E i have the same ν b,w -slope as M L [1] for any (b, w) ∈ V and cannot make a wall. Thus, in particular, M L [1] is ν b,w -stable for any (b, w) ∈ V with b = µ(M L ), so it is µ-semistable [BMS16, Lemma 2.7]. If M L is strictly µ-semistable, there is an exact sequence F 1 ֒→ M L ։ F 2 of µ-semistable coherent sheaves of the same slope. Thus the F i [1] have the same ν b=µ(M L ),w -slope as M L [1] and they are in the heart Coh µ(M L ) (X). Thus M L [1] is strictly ν b=µ(M L ),w -semistable which is not possible by the above argument.