Bridgeland Stability on Threefolds - Some Wall Crossings
Following up on the construction of Bridgeland stability condition on by Macrì, we compute first examples of wall crossing behaviour. In particular, for Hilbert schemes of curves such as twisted cubics or complete intersections of the same degree, we show that there are two chambers in the stability manifold where the moduli space is given by a smooth projective irreducible variety respectively the Hilbert scheme. In the case of twisted cubics, we compute all walls and moduli spaces on a path between those two chambers. In between slope stability and Bridgeland stability there is the notion of tilt stability that is defined similarly to Bridgeland stability on surfaces. We develop tools to use computations in tilt stability to compute wall crossings in Bridgeland stability.
Key words and phrases:Bridgeland stability conditions, Derived categories, Threefolds, Hilbert Schemes of Curves
2010 Mathematics Subject Classification:14F05 (Primary); 14J30, 18E30 (Secondary)
The introduction of stability condition on triangulated categories by Bridgeland in [Bri07] has revolutionized the study of moduli spaces of sheaves on smooth projective surfaces. We introduce techniques that worked on surfaces into the realm of threefolds. As an application we deal with moduli spaces of sheaves on . It turns out that for certain Chern characters there is a chamber in the stability manifold where the corresponding moduli space is smooth, projective and irreducible. The following theorem applies in particular to complete intersections of the same degree or twisted cubics.
Theorem 1.1 (See also Theorem 7.1).
Let where are integers with and are positive integers. Assume that is a primitive vector. There is a path that satisfies the following properties.
At the beginning of the path the semistable objects are exactly slope stable coherent sheaves with .
Before the last wall on the moduli space is smooth, irreducible and projective.
At the end of the path there are no semistable objects, i.e. the moduli space is empty.
As an example we compute all walls on the path of the last Theorem in the case of twisted cubics.
Theorem 1.2 (See also Theorem 7.2).
Let where is a twisted cubic curve. There is a path such that the moduli spaces for in its image outside of walls are given in the following order.
The empty space .
A smooth projective variety that contains ideal sheaves of twisted cubic curves as an open subset.
A space with two components . The space is a blow up of in a smooth locus. The exceptional locus parametrizes plane singular cubic curves with a spatial embedded point at a singularity. The second component is a -bundle over . An open subset in parametrizes plane cubic curves together with a potentially but not necessarily embedded point that is not scheme theoretically contained in the plane.
The Hilbert scheme of curves with . It is given as where is a blow up of in a smooth locus. The exceptional locus parametrizes plane cubic curves together with a point scheme theoretically contained in the plane.
The Hilbert scheme of twisted cubics has been heavily studied. In [PS85] it was shown that it has two smooth irreducible components of dimension and intersecting transversally in a locus of dimension . In [EPS87] it was shown that the closure of the space of twisted cubics in this Hilbert scheme is the blow up of another smooth projective variety in a smooth locus. This matches exactly the description we obtain using stability.
The literature on Hilbert schemes on projective space from a more classical point of view is vast. It turns out that the geometry of these spaces can be quite badly behaved. For example Mumford observed that there is an irreducible component in the Hilbert scheme on containing smooth curves that is generically non reduced in [Mum62]. However, Hartshorne proved that Hilbert schemes in projective space are at least connected in [Har66].
Bridgeland’s original work was motivated by Calabi-Yau threefolds and related questions in physics. A fundamental issue in the theory of stability conditions on threefolds is the actual construction of Bridgeland stability conditions. A conjectural way has been proposed in [BMT14] and has been proven for in [MacE14], for the smooth quadric threefold in [Sch14] and for abelian threefolds in both [MP13a, MP13b] and [BMS14]. In order to do so the notion of tilt stability has been introduced in [BMT14] as an intermediate notion between classical slope stability and Bridgeland stability on a smooth projective threefold over . The construction is analogous to Bridgeland stability on surfaces. The heart is a certain abelian category of two term complexes while the central charge is given by
where is ample, , and is the twisted Chern character. More details on the construction of stability is given in Section 3. Many techniques that worked in the case of surfaces still apply to tilt stability. Bayer, Macrì and Toda propose that doing another tilt will lead to a Bridgeland stability condition with central charge
where . The following theorem connects Bridgeland stability with the simpler notion of tilt stability. It is one of the key ingredients for the two theorems above.
Theorem 1.3 (See also Theorem 6.1).
Let be the Chern character of an object in such that is primitive. Then there are two paths such that all moduli spaces of tilt stable objects outside of walls occur as moduli spaces of Bridgeland stable objects along either or .
Notice that the Theorem does not preclude the existence of further chambers along those paths. In many cases, for example for twisted cubics as above, there are different exact sequences defining identical walls in tilt stability because the defining objects only differ in the third Chern character. However, by definition, changes in cannot be detected via tilt stability. In Bridgeland stability those identical walls often move apart and give rise to further chambers.
The computations in tilt stability in this article are very similar in nature to many computations about stability of sheaves on surfaces in [ABCH13, BM14, CHW14, LZ13, MM13, Nue14, Woo13, YY14]. Despite the tremendous success in the surface case, the threefold case has barely been explored. Beyond the issue of constructing Bridgeland stability condition there are further problems that have made progress difficult.
1.2. Further Questions
For surfaces, or more generally, tilt stability parametrized by the upper half-plane, there is at most one unique vertical wall, while all other walls are nested inside two piles of non intersecting semicircles. This structure is rather simple. However, in the case of Bridgeland stability on threefolds walls are given by real degree 4 equation. Already in the case of twisted cubics we can observe that they intersect in Theorem 7.2.
Given a path in the stability manifold and a class is there a numerical criterion that determines all the walls on with respect to ? If not, can we at least numerically restrict the amount of potential walls on in an effective way?
We are only able to answer this question for the two paths described in Theorem 6.1. The general situation seems to be more intricate. If we want to study stability in any meaningful way beyond tilt stability, we need at least partial answers to this question.
Another serious problem is the construction of reasonably behaved moduli spaces of Bridgeland semistable objects. A recent result by Piyaratne and Toda is a major step towards this.
Theorem 1.5 ([Pt15]).
Let be a smooth projective threefold such that the conjectural construction of Bridgeland stability from [BMT14] works. Then any moduli space of semistable objects for such a Bridgeland stability condition is a universally closed algebraic stack of finite type over .
If there are no strictly semistable objects, the moduli space becomes a proper algebraic space of finite type over . For certain applications such as birational geometry we would like our moduli spaces to be projective.
Assume is a Bridgeland stability condition and . Is the moduli space of -stable objects with class quasi-projective?
1.3. Organization of the Article
In Section 2 we recall the notion of a very weak stability condition from [BMS14] and [PT15]. All our examples of stability conditions fall under this notion. Section 3 describes the construction of both tilt stability and Bridgeland stability and establishes some basic properties. In particular, we remark which techniques for Bridgeland stability on surfaces work without issues in tilt stability. In Section 4 we deal with stability of line bundles or powers of line bundles on by connecting these questions to moduli of quiver representations. Section 5 deals with computing specific examples in for tilt stability. Moreover, we discuss how many of those calculations can be handled by computer calculations. In Section 6 we prove our main comparison theorem between Bridgeland stability and tilt stability. Finally, in Section 7 we use this connection to finish the computations necessary to establish the two main theorems.
|smooth projective variety over ,|
|fixed ample divisor on ,|
|,||ideal sheaf of a closed subscheme ,|
|bounded derived category of coherent|
|sheaves on ,|
|,||Chern character of an object ,|
|for an ample divisor on ,|
|for an ample divisor on ,|
|the numerical Grothendieck group of ,|
I would like to thank David Anderson, Arend Bayer, Patricio Gallardo, César Lozano Huerta and Emanuele Macrì for insightful discussions and comments on this article. I especially thank my advisor Emanuele Macrì for carefully reading preliminary versions of this article. Most of this work was done at the Ohio State University whose mathematics department was extraordinarily accommodating after my advisor moved. In particular, Thomas Kerler and Roman Nitze helped me a lot with handling the situation. Lastly, I would like to thank Northeastern University at which the finals details of this work were finished for their hospitality. The research was partially supported by NSF grants DMS-1160466 and DMS-1523496 (PI Emanuele Macrì) and a presidential fellowship of the Ohio State University.
2. Very Weak Stability Conditions and the Support Property
All forms of stability occurring in this article are encompassed by the notion of a very weak stability condition introduced in Appendix B of [BMS14]. It will allow us to treat different forms of stability uniformly. We will recall this notion more closely to how it was defined in [PT15].
A heart of a bounded t-structure on is a full additive subcategory such that
for integers and , the vanishing holds,
for all there are integers and a collection of triangles
The heart of a bounded t-structure is automatically abelian. A proof of this fact and a full introduction to the theory of t-structures can be found in [BBD82]. The standard example of a heart of a bounded t-structure on is given by . While it is generally not true that it is still an intuitive way to partially comprehend this notion.
Definition 2.2 ([Bri07]).
A slicing of is a collection of subcategories for all such that
if and , then ,
for all there are and a collection of triangles
For this filtration of an element we write and . Moreover, for we call the phase of .
The last property is called the Harder-Narasimhan filtration. By setting to be the extension closure of the subcategories one gets the heart of a bounded t-structure from a slicing. In both cases of a slicing and the heart of a bounded t-structure it is not particularly difficult to show that the Harder-Narasimhan filtration is unique.
Let be a homomorphism where is a finite rank lattice. Fix to be an ample divisor on . Then will usually be one of the homomorphisms defined by
for some .
Definition 2.3 ([Pt15]).
A very weak pre-stability condition on is a pair where is a slicing of and is a homomorphism such that any non zero satisfies
This definition is short and good for abstract argumentation, but it is not very practical for defining concrete examples. As before, the heart of a bounded t-structure can be defined by . The usual way to define a very weak pre-stability condition is to instead define the heart of a bounded t-structure and a central charge such that maps to the upper half plane plus the non positive real line . The subcategory for consists of all semistable objects such that
More precisely, we can define a slope function by
where dividing by is interpreted as . Then an object is called (semi-)stable if for all monomorphisms in we have . More generally, an element is called (semi-)stable if there is such that is (semi-)stable. A semistable but not stable object is called strictly semistable. Moreover, one needs to show that Harder-Narasimhan filtrations exist inside with respect to the slope function to actually get a very weak pre-stability condition. We interchangeably use and to denote the same very weak pre-stability condition.
An important tool is the support property. It was introduced in [KS08] for Bridgeland stability conditions, but can be adapted without much trouble to very weak stability conditions (see [PT15, Section 2]). We also recommend [BMS14, Appendix A] for a nicely written treatment of this notion. Without loss of generality we can assume that implies . If not we replace by a suitable quotient.
A very weak pre-stability condition satisfies the support property if there is a bilinear form on such that
all semistable objects satisfy the inequality and
all non zero vectors with satisfy .
A very weak pre-stability condition satisfying the support property is called a very weak stability condition.
By abuse of notation we will write instead of for . We will also use the notation .
Let be the set of very weak stability conditions on with respect to . This set can be given a topology as the coarsest topology such that the maps , and for any are continuous.
Lemma 2.5 ([Bms14][Section 8, Lemma A.7 & Proposition A.8]).
Assume that has signature and is a path connected open subset of such that all satisfy the support property with respect to .
If with is -stable for some then it is -stable for all unless it is destabilized by an object with .
Let be a ray in starting at the origin. Then
is a convex cone for any very weak stability condition .
Moreover, any vector with generates an extremal ray of .
Only the situation of an actual stability condition is handled in [BMS14]. In that situation there are no objects in the heart with . However, exactly the same arguments go through in the case of a very weak stability condition.
A numerical wall inside (or a subspace of it) with respect to an element is a proper non trivial solution set of an equation for a vector .
A subset of a numerical wall is called an actual wall if for each point of the subset there is an an exact sequence of semistable objects in where and numerically defines the wall.
Walls in the space of very weak stability conditions satisfy certain numerical restrictions with respect to .
Let be a very weak stability condition satisfying the support property with respect to (it is actually enough for to be negative semi-definite on ).
Let be semistable objects. If , then .
Assume there is an actual wall defined by an exact sequence . Then .
We start with the first statement. If or , then . If not, there is such that . Therefore, we get
The inequalities and lead to . For the second statement we have
Since all four terms are positive, the claim follows. ∎
Since has to be only negative semi-definite on for the Lemma to apply, it is sometimes possible to define on a bigger lattice than . For example, we will define a very weak stability condition factoring through , but apply the Lemma for where everything is still well defined later on.
The most well known example of a very weak stability condition is slope stability. We will slightly generalize it for notational purposes. Let be a fixed ample divisor on . Moreover, pick a real number . Then the twisted Chern character is defined to be . In more detail, one has
In this case . The central charge is given by
The heart of a bounded t-structure in this case is simply . The existence of Harder-Narasimhan filtration was first proven for curves in [HN74], but holds in general. Finally the support property is satisfied for . We will denote the corresponding slope function by
Note that the modification by does not change stability itself but just shifts the value of the slope.
3. Constructions and Basic Properties
3.1. Tilt Stability
In [BMT14] the notion of tilt stability has been introduced as an auxiliary notion in between classical slope stability and Bridgeland stability on threefolds. We will recall its construction and prove a few properties. From now on let .
The process of tilting is used to obtain a new heart of a bounded t-structure. For more information on the general theory of tilting we refer to [HRS96]. A torsion pair is defined by
A new heart of a bounded t-structure is defined as the extension closure . In this case . Let be a positive real number. The central charge is given by
The corresponding slope function is
Note that in regard to [BMT14] this slope has been modified by switching with . We prefer this point of view for aesthetical reasons because it will make the walls semicircles and not just ellipses. Every object in has a Harder-Narasimhan filtration due to [BMT14, Lemma 3.2.4]. The support property is directly linked to the Bogomolov inequality. This inequality was first proven for slope semistable sheaves in [Bog78]. We define the bilinear form by .
Theorem 3.1 (Bogomolov Inequality for Tilt Stability, [Bmt14, Corollary 7.3.2]).
Any -semistable object satisfies
As a consequence satisfies the support property with respect to . On smooth projective surfaces this is already enough to get a Bridgeland stability condition (see [Bri08, AB13]). On threefolds this notion is not able to properly handle geometry that occurs in codimension three as we will see.
Proposition 3.2 ([Bms14, Appendix B]).
The function defined by is continuous. Moreover, walls with respect to a class in the image of this map are locally finite.
Numerical walls in tilt stability satisfy Bertram’s Nested Wall Theorem. For surfaces it was proven in [MacA14].
Theorem 3.3 (Structure Theorem for Walls in Tilt Stability).
Fix a vector . All numerical walls in the following statements are with respect to .
Numerical walls in tilt stability are of the form
for , and . In particular, they are either semicircles with center on the -axis or vertical rays.
If two numerical walls given by and intersect for any and then , and are linearly dependent. In particular, the two walls are completely identical.
The curve is given by the hyperbola
Moreover, this hyperbola intersect all semicircles at their top point.
If there is exactly one vertical numerical wall given by . If there is no vertical wall.
If a numerical wall has a single point at which it is an actual wall, then all of it is an actual wall.
Part (1) and (3) are straightforward but lengthy computations only relying on the numerical data.
A wall can also be described as two vectors mapping to the same line under the homomorphism . This homomorphism maps surjectively onto . Therefore, at most two linearly independent vectors can be mapped onto the same line. That proves (2).
In order to prove (4), observe that a vertical wall occurs when holds. By the above formula for this implies
in case . A direct computation shows that the equation simplifies to . If and , then . This implies that the two slopes are the same for all or no . If , then all objects with this Chern character are automatically semistable and there are no walls at all.
Let be an exact sequence of tilt semistable objects in that defines an actual wall. If there is a point on the numerical wall at which this sequence does not define a wall anymore, then either , or have to destabilize at another point along the numerical wall in between the two points. But that would mean two numerical walls intersect in contradiction to (2). ∎
A generalized Bogomolov inequality involving third Chern characters for tilt semistable objects with has been conjectured in [BMT14]. In [BMS14] it was shown that the conjecture is equivalent to the following more general inequality that drops the hypothesis .
Conjecture 3.4 (BMT Inequality).
Any -semistable object satisfies
By using the definition of and expanding the expression one can find depending on such that the inequality becomes
This means the solution set is given by the complement of a semi-disc with center on the -axis or a quadrant to one side of a vertical line. The conjecture is known for [MacE14], the smooth quadric threefold [Sch14] and all abelian threefolds [BMS14, MP13a, MP13b].
Another question that comes up in concrete situations is the question whether a given tilt semistable object is a sheaf. For a fixed let
Lemma 3.5 ([Bmt14, Lemma 7.2.1 and 7.2.2]).
An object that is -semistable for all is given by one of three possibilities.
is a pure sheaf supported in dimension greater than or equal to two that is slope semistable.
is a sheaf supported in dimension less than or equal to one.
is a torsion free slope semistable sheaf and is supported in dimension less than or equal to one. Moreover, if then for all sheaves of dimension less than or equal to one.
An object with is -semistable if and only if it is given by one of the three types above.
Notice that part of the second statement follows directly from the first as follows. Any subobject of in must have or . In the second case the corresponding quotient satisfies . Therefore, in both cases either the quotient or the subobject have infinite slope. This means there is no wall that could destabilize for any . This type of argument will be used several times in the next sections. Using the same proof as in the surface case in [Bri08, Proposition 14.1] leads to the following lemma.
Assume is a slope stable sheaf and . Then is -stable for all .
3.2. Bridgeland Stability
We will recall the definition of a Bridgeland stability condition from [Bri07] and show how they can be conjecturally constructed on threefolds based on the BMT-inequality as described in [BMT14]. It is known that the inequality holds on due to [MacE14] and we will apply it in a later section to study concrete examples of moduli spaces of complexes in this case.
A Bridgeland (pre-)stability condition on the category is a very weak (pre-)stability condition such that for all semistable objects . By we denote the subspace of Bridgeland stability conditions in .
If is the corresponding heart, then we could have equivalently defined a Bridgeland stability condition by the property for all non zero . Note that in this situation choosing the heart to be instead of for any is arbitrary and any other choice works just as well. In some very special cases it is possible to choose such that the corresponding heart is equivalent to the category of representations of a quiver with relations. This will be particularly useful in the case of .
Theorem 3.8 ([Bri07, Section 7]).
The map from to is a local homeomorphism. In particular, is a complex manifold.
In order to have any hope of actually computing wall-crossing behaviour it is necessary for walls in Bridgeland stability to be somewhat reasonably behaved. The following result due to [Bri08, Section 9] is a major step towards that.
Walls in Bridgeland stability are locally finite, i.e. for a fixed vector there are only finitely many walls in any compact subset of .
An important question is how moduli spaces change set theoretically at walls. In case the destabilizing subobject and quotient are both stable this has a satisfactory answer due to [BM11, Lemma 5.9]. Note that this proof does not work in the case of very weak stability conditions due to the lack of unique factors in the Jordan-Hölder filtration.
Let such that there are stable object with . Then there is an open neighborhood around where non trivial extensions are stable for all such that .
Since stability is an open property there is an open neighborhood of in which both and are stable. The category is of finite length with simple objects corresponding to stable objects. In fact is a Jordan-Hölder filtration. By shrinking if necessary we know that if is unstable at a point in , there is a sequence that becomes a Jordan-Hölder filtration at . Since the Jordan-Hölder filtration has unique factors and is a non trivial extension, we get and . Therefore, there is no destabilizing sequence if . ∎
It turns out that while constructing very weak stability conditions is not very difficult, constructing Bridgeland stability conditions is in general a wide open problem. Note that for any smooth projective variety of dimension bigger than or equal to two, there is no Bridgeland stability condition factoring through the Chern character for due to [Tod09, Lemma 2.7].
Tilt stability is no Bridgeland stability as can be seen by the fact that skyscraper sheaves are mapped to the origin. In [BMT14] it was conjectured that one has to tilt again as follows in order to construct a Bridgeland stability condition on a threefold. Let
and set . For any they define
In this case the bilinear form is given by
for some . Notice that for this comes directly from the quadratic form in the BMT-inequality.
If the BMT inequality holds, then is a Bridgeland stability condition for all . The support property is satisfied with respect to .
Note that as a consequence the BMT inequality holds for all -stable objects. In [BMS14, Proposition 8.10] it is shown that this implies a continuity result just as in the case of tilt stability.
The function defined by is continuous.
In the case of tilt stability we have seen that the limiting stability for is closely related with slope stability. The first step in connecting Bridgeland stability with tilt stability is a similar result. For an object we denote the cohomology with respect to the heart by . It is defined by the property that is a factor in the Harder-Narasimhan filtration of .
Lemma 3.13 ([Bms14, Lemma 8.9]).
If is -semistable for all , then one of the following two conditions holds.
is a -semistable object.
is -semistable and is a sheaf supported in dimension .
4. Stability on
In the case of more can be proven than in the general case. In this section the connection to stability of quiver representations will be recalled and a stability result about line bundles will be proven. It was already shown in [BMT14] that a line bundle is tilt stable if . This condition always holds in Picard rank . However, we need a slightly more refined result that holds in the special case of .
Let for integers with . Then or a shift of it is the unique tilt semistable and Bridgeland semistable object with Chern character for any and . Moreover, in the case the line bundle is stable.
For the proof we will need a connection between Bridgeland stability and quiver representations. We will recall exceptional collections after [Bon90].
An object is called an exceptional object if for all and .
A sequence of exceptional objects is a full exceptional collection if for all and and , i.e., is generated from by shifts and extensions.
A full exceptional collection is called strong if additionally for all and .
Theorem 4.3 ([Bon90]).
Let be a strong full exceptional collection on , and be the category of right -modules of finite rank. Then the functor
is an exact equivalence. Under this identification the correspond to the indecomposable projective -modules.
In particular, the category becomes the heart of a bounded t-structure on with this identification. In the case of this heart can be connected to some stability conditions.
Theorem 4.4 ([MacE14]).
If and then
for some and the Bridgeland stability condition for small enough . Moreover, is the category for some finite dimensional algebra coming from an exceptional collection as in Theorem 4.3. The four objects generating correspond to the simple representations.
Proof of Proposition 4.1.
By using the autoequivalence given by tensoring with , we can reduce to the case . Then .
We start by proving the statement in Bridgeland stability for and . By Theorem 4.4 the object corresponds to a simple representation at this point. Then any object in the quiver category with corresponds to a representation of the form . The statement follows in this case, since there is a unique such representation and it is semistable.
Next, we will extend this to all , . Notice that . By Lemma 2.5 the object is Bridgeland stable for all , . Let be -semistable with . By Lemma 2.5, the class spans an extremal ray of the cone . In particular, that means all its Jordan-Hölder factors are scalar multiples of . If , then is primitive in the lattice. Therefore, is actually stable and then is also stable for and , i.e. is or a shift of it. Assume . Since there are no stable objects with class at and , Lemma 2.5 implies that is strictly semistable. Therefore, the case implies that all the Jordan-Hölder factors are .
The next step is to show semistability of in tilt stability. For this, we just need deal with . We have . By Lemma 2.5 we know that is tilt stable everywhere or nowhere unless it is destabilized by an object supported in dimension . In that case is a wall. However, that cannot happen since there are no morphism from or to for any skyscraper sheaf. Since is primitive, semistability of is equivalent to stability. For and we know that is semistable due to Lemma 3.5.
Now we will show that any tilt semistable object with has to be for , . We have . Therefore, is in the category . The Bridgeland slope is independently of . This means is Bridgeland semistable and by the previous argument .
We will use and Lemma 2.5 similarly as in the Bridgeland stability case to extend it to all of tilt stability. We start with the case . Let be a tilt semistable object with . By using Lemma 2.5, the class spans an extremal ray of the cone . In particular, that means all its stable factors have Chern character . The BMT inequality shows . But since all the stable factor add up to this means . Therefore, we reduced to the case . In this case Lemma 2.5 does the job as before.
If , the situation is more involved, since skyscraper sheaves can be stable factors. All stable factor have Chern characters of the form or . In this case . Let be such a stable factor with Chern character . By openness of stability is stable in a whole neighborhood that includes points with and . The BMT-inequality in both cases together implies . But then follows from the fact that Chern characters are additive. Again we reduced to the case . By openness of stability and the result for we are done with this case. The case can now be handled in the same way as by using Lemma 2.5 again. ∎
In the case of tilt stability there is an even stronger statement. If , we do not need to fix to get the same conclusion.
Let for integers with . Then is the unique tilt semistable object with for any and .
The semistability of has already been shown in Proposition 4.1. As in the previous proof, we can use tensoring by to reduce to the case . This means .
Let be a tilt stable object for some and with . The BMT-inequality implies . Since , we can use Lemma 2.5 to get that is tilt stable for all . If is also stable for , then using the BMT-inequality for implies . Assume becomes strictly semistable at . By Lemma 2.5 the class spans an extremal ray of the cone . That means all stable factors must have Chern characters of the form for some . If then using the BMT-inequality for both and implies . If , then . However, all the third Chern characters add up to the non positive number . This is only possible if and no stable factor has . By Proposition 4.1 this means and since is stable this is only possible if .
Let be a strictly tilt semistable object for some and with . Since , we can use Lemma 2.5 again to get that all stable factors have