A short proof of the deformation property of Bridgeland stability conditions

The key result in the theory of Bridgeland stability conditions is the property that they form a complex manifold. This comes from the fact that given any small deformation of the central charge, there is a unique way to correspondingly deform the stability condition. We give a short direct proof of a strong version of this deformation property.

Definition 1.1 ([Bri07], [KS08]). Let Q : Λ R → R be a quadratic form. We say that a pre-stability condition (Z, P) satisfies the support property with respect to Q if (a) the kernel Ker Z ⊂ Λ R of the central charge is negative definite with respect to Q, and (b) for any semistable object E, i.e. E ∈ P(φ) for some φ ∈ R, we have Q(v(E)) ≥ 0.
If such Q exists, we call σ a stability condition. Let Stab Λ (D) denote the topological space (see Section 2) of stability conditions. It comes with a canonical map Z : Stab Λ (D) → Hom(Λ, C) given by Z(Z, P) = Z. We will prove: Theorem 1.2. Let Q be a quadratic form on Λ ⊗ R, and assume that the stability condition σ = (Z, P) satisfies the support property with respect to Q. Then: (a) There is an open neighboorhood σ ∈ U σ ⊂ Stab Λ (D) such that the restriction Z : U σ → Hom(Λ, C) is a covering of the set of Z ′ such that Q is negative definite on Ker Z ′ . (b) All σ ∈ U σ satisfy the support property with respect to Q.
In other words, Stab Λ (D) is a manifold, and any path Z t ∈ Hom(Λ, C) for t ∈ [0, 1] with Z 0 = Z and Ker Z t negative definite for all t ∈ [0, 1] lifts uniquely to a continuous path σ t = (Z t , P t ) in the space of stability conditions starting at σ 0 = σ.
Part (a) is an effective variant of [Bri07, Theorem 1.2] (which says that there is some neighbourhood of Z 0 in which paths can be lifted uniquely). The entire result first appeared as [BMS16,Proposition A.5] with an indirect proof based on reduction to Bridgeland's previous result.
Remarks. The support property can be a deep and interesting property in itself: a quadratic Bogomolov-Gieseker type inequality for Chern classes of semi-stable objects which, by Theorem 1.2, is preserved under wall-crossing.
Theorem 1.2 was crucial in [BMS16] in order to describe an entire component of the space of stability conditions on abelian threefolds, and on some Calabi-Yau threefolds. It also greatly simplifies the construction of stability conditions on surfaces (or of tilt-stability on higher-dimensional varieties [BMT14]). In this case, the quadratic form Q is the classical Bogomolov-Gieseker inequality, and Theorem 1.2 gives an open subset of stability conditions that otherwise has to be glued together from many small pieces (see e.g. [BM11, Section 4]).
Theorem 1.2 of [Bri07] also allows for components of the space of stability conditions modelled on a linear subspace L ⊂ Hom(Λ, C). When L is defined over Q, we can recover that statement by replacing Λ with Λ/ Ker L. (See [MP14] for examples where this is not satisfied; however, to achieve well-behaved wall-crossing one always has to assume that L is defined over Q.) Proof idea. Our proof is based on two ideas. First, we reduce to the case where the imaginary part of Z is constant; then we only have to deform stability in a fixed abelian category. Secondly, we use the elementary convex geometry of the Harder-Narasimhan polygon, see Section 3.
This avoids the need for quasi-abelian categories, of ǫ or of 1 8 . It also avoids some of the more technical arguments of [Bri07, Section 7]. We still need a few arguments similar to ones in [Bri07]; we have reproduced most of them, except for the proofs of Proposition 2.6 and Lemma 2.9.
Application. Assume that D is a 2-Calabi-Yau category, i.e. for all E, F ∈ D we have a bifunctorial isomorphism Hom(E, F ) = Hom(F, E[2]) ∨ . Let Λ be the numerical K-group of D, and assume that Λ is finitely generated. Then there is a surjection v : K(D) → Λ, and Λ admits a non-degenerate bilinear form ( , ), called Mukai-pairing, with Let P 0 (D) ⊂ Hom(Λ, C) be the set of central charges Z such that Ker Z is negative definite with respect to the Mukai pairing, and such that Ker Z contains no root δ ∈ Λ, (δ, δ) = −2.
The proof, given in Section 7, is fairly similar to the case of K3 surfaces [Bri08,Proposition 8.3]. The point of including it here is to show that in terms of the support property via quadratic forms, and equipped with Theorem 1.2, the proof becomes natural and short. This result was also proved previously for preprojective algebras of quivers in [Tho08,Bri09b,Ike14]. In each of these cases, there is in fact a connected component of Stab(D) that is a covering of a connected of P 0 (X); such deeper statements rely crucially on non-emptiness of moduli spaces of stable objects.
Acknowledgements. I would like to thank Emanuele Macrì and Paolo Stellari; as indicated above, Theorem 1.2 first appeared with a different proof in our joint work [BMS16]. I presented a clumsier version of the arguments in this article at my lectures at the Hausdorff school on derived categories in Bonn, April 2016; I am grateful to the organisers for the opportunity, and the participants for their feedback. My work was supported by the ERC starting grant WallXBirGeom 337039.

REVIEW: DEFINITIONS AND BASIC PROPERTIES
Throughout, D will be a triangulated category, equipped with a group homomorphism v : K(D) → Λ from its K-group to an abelian group Λ ∼ = Z m .
Definitions. We first recall the main definitions from [Bri07].
The objects of P(φ) are called semistable of phase φ; its simple objects are called stable. The sequence of maps in (c) is called the HN filtration of E.

Definition 2.2.
A pre-stability condition on D is a pair σ = (Z, P) where P is a slicing, and Z : Λ → C is a group homomorphism, that satisfy the following condition: for all 0 = E ∈ P(φ), we have Z(v(E)) ∈ R >0 · e iπφ .
We will abuse notation and write Z(E) instead of Z(v(E)).
Basic properties. Let GL + 2 (R) denote the group of real 2 × 2-matrices with positive determinant, and let GL + 2 (R) be its universal cover. Since GL + 2 (R) acts on S 1 , its universal cover acts on the universal cover R → S 1 given explicitly by φ → e iπφ . Forg ∈ GL + 2 (R) we will write g for the corresponding element of GL + 2 (R), andg.φ for the given action on R.
Proposition 2.3. There is a natural action of GL + 2 (R) on the set of pre-stability conditions given byg.(Z, P) = (Z ′ , P ′ ) with The heart of a bounded t-structure is a full subcategory A ⊂ D such that . It is automatically an abelian subcategory; and stability conditions on D can be constructed from slope-stability in A.

Definition 2.4. A stability function Z on an abelian category
Definition 2.5. We say that a stability function Z on an abelian category A satisfies the HN property if every object E ∈ A admits a Harder-Narasimhan (HN) filtration: Proposition 2.6 ([Bri07, Proposition 5.3]). To give a pre-stability condition on D is equivalent to giving a heart A of a bounded t-structure, and a stability function Z on A with the HN property.
Here we tacitly assume that the stability function Z on A also factors via Given (Z, A), the slicing is determined by setting P(φ) to be the Z-semistable objects E ∈ A of phase φ for φ ∈ (0, 1]. Conversely, given (Z, P), the heart A is the smallest extension-closed subcategory of D containing P(φ) for φ ∈ (0, 1].
Definition 2.7. A stability condition σ is a pre-stability condition that satisfies the support property in the sense of Definition 1.1 with respect to some quadratic form Q on Λ ⊗ R.

Topology and local injectivity.
There is a generalised metric, and thus a topology, on the set of slicings Slice(D) given as follows. Given two slicings P, Q, we write φ ± (E) and ψ ± (E) for the largest and smallest phase in the associated HN filtration of an object E for P and Q, respectively. Then we define the distance of P and Q by We recall that this distance can be computed by considering P-semistable objects alone: where the latter is defined by Proof. The inequality d(P, Q) ≥ d ′ (P, Q) is immediate. For the converse, consider E ∈ D, and let A i be one of its HN factors with respect to P.
The topology on Stab Λ (D) (and similarly on the set of pre-stability conditions) is the finest topology such that both maps  In other words, the intersection of HN Z (E) with the closed half-plane to the left of the line through 0 and Z(E) is the polygon with vertices z 0 , z 1 , . . . , z m . Our proof of Theorem 1.2 is based on the following well-known statement; we provide a proof for completeness:

PROOF OF THE DEFORMATION PROPERTY
Throughout Section 4 and 5, we will make the following assumption: Assumption 4.1. The quadratic form Q has signature (2, rk Λ − 2).

Lemma 4.2.
Up to the action of GL + 2 (R) on Stab Λ (D), we may assume that we are in the following situation. There is a norm · on Ker Z such that if p : Λ R → Ker Z denotes the orthogonal projection with respect to Q, then Proof. Let K the kernel of Z, and let K ⊥ denote its orthogonal complement. Then Q is negative definite on K; let · be the norm associated to −Q. As Z| K ⊥ is injective, Assumption 4.1 can only hold if Q is positive definite on K ⊥ , and if we have an isomorphism of real vector spaces Using the GL + 2 (R)-action, we may assume this to be an isometry. Then the the claim follows. Remark 4.3. In different context, namely for the Mukai quadratic form instead of Q, the analogous normalisation is used extensively in [Bri08].
Consider the subset in Hom(Λ, C) of central charges whose kernel is negative definite with respect to Q; let P Z (Q) be its connected component containing Z.

Lemma 4.4. Assume we are in the situation of Lemma 4.2. Up to the action of GL
Ker Z → C is a linear map with operator norm satisfying u < 1.
Proof. As in the previous Lemma, let K ⊥ be the orthogonal complement of Ker Z. The restriction of Z ′ to K ⊥ is an isomorphism for any Z ′ ∈ P Z (Q). Hence for any path Z(t) in P Z (Q) starting at Z there is a corresponding path γ(t) ∈ GL + 2 (R) such that γ(t) • Z(t) is constant. So we may assume that Z ′ and Z agree when restricted to K ⊥ . Let u be the restriction of Z ′ to Ker Z, and the claim follows.
Lemma 4.5. In order to prove Theorem 1.2, it is enough to show the following: given any stability conditions σ 0 = (Z 0 , P 0 ), and any path of central charges of the form t → Z t = Z + t · u • p for t ∈ [0, 1], where u : Ker Z → R is a linear map to the real numbers with u < 1, there exists a continuous lift t → σ t to the space of stability conditions; moreover, all σ t satisfy the support property with respect to the same quadratic form Q.
Proof. Due to Corollary 2.10, it is enough to prove the existence of a lift for any given path, and moreover we can freely replace any path in P Z (Q) by a homotopic one. Observe that due the GL + 2 (R)-action such a result would equally hold when u is purely imaginary. Now write Z 1 = Z + u • p and u = ℜu + iℑu. Since ℜu ≤ u , we first obtain a path from σ 0 = σ to a stability condition σ 1 = (Z 1 , P 1 ) with Z 1 = Z + ℜu • p. By part (b) of Theorem 1.2, we can apply the result again starting at σ 1 to construct the desired stability condition with central charge Z + u • p = Z 1 + iℑu • p.
Our next key observation is that when u is real, we may (and in fact, have to) leave the heart A := P(0, 1] unchanged. Hence we will apply Proposition 2.6 and prove that (A, Z t ) produces a stability condition for all t ∈ [0, 1]. Clearly we just need to prove the case t = 1.
Lemma 4.6. Let Z, u be as in Lemma 4.5. Then Z 1 = Z + u • p is a stability function on A.
Proof. Consider E ∈ A; if ℑZ(E) = ℑZ 1 (E) > 0, there is nothing to prove. Otherwise, E is semistable with Z(E) ∈ R <0 and thus p(E) ≤ −Z(E). From u < 1 we conclude Next, we want to prove that (A, Z 1 ) satisfies the HN property. We will use Proposition 3.3 and Corollary 3.6.
Let us define the mass m Z (E) of E with respect to Z as the length of the boundary of HN Z (E) on the left between 0 and Z(E).
Combined with the triangle inequality, this gives The following Lemma needs no proof:  Proof. This follows from the previous Lemma, convexity and a picture, see fig. 4. Indeed, choose x > ℜZ(A), ℜZ(E); let a = x+iℑZ(A) and e = x+iℑZ(E). Let γ A be the path that follows by boundary of HN Z (A) from 0 to Z(A), and then continues horizontally to a; similarly γ E follows the boundary of HN Z (E) and then continues to e. Their lengths are given as On the other hand, convexity and Lemma 4.8 imply |γ A | ≤ |γ E |; for example, if γ I denotes the intermediate path that follows the boundary of HN Z (E) up to height ℑZ(A) and then goes horizontally to a, we clearly have |γ A | ≤ |γ I | ≤ |γ E |. Z 0 (E) Proof. Given any such A, we use Lemmas 4.7 and 4.9 to obtain Therefore, Corollary 3.6 implies the existence of HN filtrations for Z 1 on A.
Continuity. So far, we have constructed a pre-stability condition σ t = (A, Z t ) for each t ∈ [0, 1].
Lemma 4.11. The map t → σ t = (A, Z t ) defines a continuous path in the space of pre-stability conditions.
Proof. Let P t denote the associated slicing. It is enough to prove that for t sufficiently small, the distance d(P 0 , P t ) becomes arbitrarily small. We will apply Lemma 2.8. Thus consider a P 0semistable object E ∈ D; up to shift, we may assume E ∈ A. Let A ֒→ E be the leading HN filtration factor of E with respect to Z t . Write Z 0 (A) = a + x where a ∈ C has the same phase as Z 0 (E) and x ≥ 0, see fig. 5. By convexity, m Z 0 (A) ≤ |a| + x. Therefore Note that π · (ψ + (E) − φ(E)) is the argument of 1 a Z t (A); hence Combined with an analogous argument for ψ − (E) we obtain d ′ (P 0 , P t ) ≤ 1 π t as claimed.

PRESERVATION OF THE QUADRATIC INEQUALITY
It remains to show that the pre-stability condition (Z 1 , A) satisfies the support property with respect to Q, i.e. that that Q(v(E)) ≥ 0 for all E ∈ A that are Z 1 -stable. The basic reason is that the quadratic inequality is preserved by wall-crossing: Lemma 5.1. Let σ = (Z, P) be pre-stability condition. Assume that Q is a non-degenerate quadratic form on Λ R of signature (2, rk Λ−2) such that Q is negative definite on Ker Z. If E is strictly σ-semistable and admits a Jordan-Hölder filtration with factors E 1 , . . . , E m , and if Q(v(E i )) ≥ 0 for i = 1, . . . , m, then Q(v(E)) = 0.
Proof. We apply Lemma 4.2; then Q(v) ≥ 0 is equivalent to |Z(v)| ≥ p(v) . We obtain where the first equality holds since the central charges of all E i are aligned, the first inequality holds by assumption, and the second inequality is the triangle inequality.
The proof strategy is thus clear: if E ∈ A is Z 1 -stable with Q(v(E)) < 0, then it must be Z 0unstable; wall-crossing gives a t ∈ [0, 1) such that E is strictly Z t -semistable; by the Lemma, one of its Jordan-Hölder factors will also violate the inequality, and we proceed by induction. To make this argument work, we have to show that we can find such a wall, and that this process terminates.   Consider the polygon whose vertices are the extremal points of HN Z 0 (E) on the left; we will call this the truncated HN polygon of E, see fig. 6. Note that if A ⊂ E is a subobject with φ 0 (A) ≥ φ 0 (E), then Z 0 (A) is contained in the truncated HN polygon of E; by Lemmas 4.9 and 4.7 there are only finitely many classes v(A) of such subobjects.

Lemma 5.2. Given two objects A, E ∈ A, denote their phases with respect to
Proof. Otherwise, E must be Z 0 -unstable. By Lemma 5.2 and the following observation, there are only finitely many classes v(A) of subobjects A ֒→ E that destabilise E with respect Z t for any t ∈ [0, 1]. Hence there is a wall t 1 ∈ (0, 1] such that E is strictly semistable with respect to Z t 1 , and moreover E admits a Jordan-Hölder filtration with respect to Z t 1 . By Lemma 5.1, there are subobjects G 1 ֒→ F 1 ֒→ E of the same phase, such that Q(v(F 1 /G 1 )) < 0.
Lemma 5.2 gives φ t 2 (F 1 ) ≥ φ t 2 (E) and φ t 2 (G 1 ) ≥ φ t 2 (E). Since the central charge of Z t 2 (F 2 ) lies on the line segment connecting Z t 2 (F 1 ) and Z t 2 (G 1 ), we also have φ t 2 (F 2 ) ≥ φ t 2 (E) (and therefore φ t (F 2 ) ≥ φ t (E) for all t ∈ [0, t 2 ]; similarly for G 2 . Continuing this argument by induction, we see that Z 0 (F i ) and Z 0 (G i ) are all contained in the truncated HN polygon of E. Thus this process terminates.
By Lemma 4.5, this concludes the proof of Theorem 1.2 whenever Assumption 4.1 holds.

REDUCTIONS
Finally, we will show that we can always reduce the general situation to the case where Assumption 4.1 holds. By abuse of language, we call a quadratic form degenerate or non-degenerate if the associated symmetric bilinear form is degenerate or non-degenerate, respectively. Lemma 6.1. Assume that the quadratic form Q on Λ R is degenerate. Then there exists an injective map Λ R ֒→ Λ of real vector spaces and a non-degenerate quadratic form Q on Λ, extending Q, such that any central charge Z : Λ R → C whose kernel is negative definite with respect to Q extends to a central charge Z : Λ → C whose kernel is negative definite with respect to Q.
Proof. Let N ֒→ Λ R be the null space of Q; we will only treat the case dim R N = 1 (otherwise, we can iterate the construction that follows). Choose a splitting Λ R ∼ = N ⊕ C; then for n ∈ N, c ∈ C, we have Q(n ⊕ c) = Q(c). Let Λ R := N ⊕ N ∨ ⊕ C, let q be the canonical quadratic form on the hyperbolic plane N ⊕ N ∨ , and set Q := q ⊕ Q| C .
Given Z as above, the restriction Z| N is injective, and we may assume that Z maps N to the real line. Let n ∈ N be such that Z(n) = 1, and let n ∨ ∈ N ∨ be the dual vector with (n, n ∨ ) = 1. We claim that for α ≫ 0, the extension of Z defined by Z ′ (n ∨ ) = α has the desired property.
Let K := Ker Z; then the kernel of Z ′ is contained in N ⊕ N ∨ ⊕ K, and given by vectors of the form a · n − a α · n ∨ + k for k ∈ K, a ∈ R. For such vectors, we have This is a quadratic function in a with negative constant term; its discriminant is negative if is a positive definite form on K).
Replacing Λ by Λ ⊕ Z and v by we can therefore restrict to the case where Q is non-degenerate: given a path Z t of central charges in Hom(Λ R , C) that are negative definite with respect to Q, we can choose extensions Z t as in the Lemma that form a continuous path in Hom(Λ, C). If we can lift the latter path to a path of stability conditions σ t = (Z t , P t ) that satisfy the support property with respect to Q, then σ t := (Z t , P t ) is a path of stability conditions satisfying the support property with respect to Q. The reduction to the case where Q has signature (2, rk Λ − 2) works similarly: Lemma 6.2. Assume that Q is non-degenerate and of signature (p, rk Λ − p) for p ∈ {0, 1}. Let Λ := Λ R ⊕ R, and let Q be the extension given by Q(v, α) = Q(v) + α 2 for v ∈ Λ R and α ∈ R.
Then any central charge Z on Λ R whose kernel is negative definite with respect to Q extends to a central charge Z on Λ whose kernel is negative definite with respect to Q.
Proof. We claim that there exists z ∈ C such that for all v ∈ Λ R with Z(v) = z, we have Q(v) < −1. Indeed, let K ⊂ Λ R be the kernel of Z, and let K ⊥ be its orthogonal complement. Then clearly we may assume v ∈ K ⊥ . Since the restriction of Z to K ⊥ is injective, and since K ⊥ either has rank one, or has signature (1, −1) with respect to Q, the claim is evident.
This concludes the proof of Theorem 1.2.
Let σ = (Z, P) be a stability condition with Z ∈ P 0 (D). By the same argument as in Lemma 4.2 we may assume where p : Λ R → Ker Z is the orthogonal projection onto the kernel of Z, and where · denotes the norm on Ker Z induced by the negative of the Mukai pairing. We claim that (1) C := inf {|Z(δ)| : δ ∈ Λ, (δ, δ) = −2} > 0.
Therefore, σ satisfies the support property with respect to Q. Theorem 1.2 gives an open neighbourhood P Z (Q) ⊂ Hom(Λ, C) of Z and an open neighbourhood U σ of σ such that U σ Z − → P Z (Q) is a covering.
By compactness, any path Z t in P 0 (D) is contained in a finite number of such sets P Zt i (Q i ), where Q i the quadratic form associated to Z t i . Thus Z t lifts uniquely to a path in Z −1 (P 0 (D)).