A short proof of the deformation property of Bridgeland stability conditions

Abstract

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 an effective version of this deformation property.

1 Introduction

Stability conditions on triangulated categories, introduced in [9], have been hugely influential, due to their connections to physics [13, 14], to mirror symmetry [11] and to representation theory [2], and due to their applications within algebraic geometry, for example to Donaldson–Thomas invariants [23], to the derived category itself [4, 15], or to the birational geometry of moduli spaces [1, 3, 6, 18, 20].

Their distinguishing property, crucial in all applications, is a strong deformation property: by [9, Theorem 1.2], there is a complex manifold of stability conditions, with a map to a vector space that is a local isomorphism. We give a short proof of an effective version of this result.

Result.  We refer to Sect. 2 for complete definitions; here we review notation and the support property. Let ${\mathcal {D}}$ be a triangulated category, and let $v :K({\mathcal {D}}) \rightarrow \Lambda$ be a homomorphism from its K-group to a finitely generated free abelian group $\Lambda$. A pre-stability condition on ${\mathcal {D}}$ with respect to v is a pair $\sigma = (Z, {\mathcal {P}})$, where ${\mathcal {P}}$ is a slicing (see Definition 2.1) and $Z :\Lambda \rightarrow {\mathbb {C}}$ is a compatible (see Definition 2.2) group homomorphism.

Definition 1.1

[9, 17] Let $Q :\Lambda _{\mathbb {R}}\rightarrow {\mathbb {R}}$ be a quadratic form. We say that a pre-stability condition $(Z, {\mathcal {P}})$ satisfies the support property with respect to Q if

• 1. the kernel ${\mathrm {Ker}}Z \subset \Lambda _{\mathbb {R}}$ of the central charge is negative definite with respect to Q, and

• 2. for any semistable object E, i.e. $E \in {\mathcal {P}}(\phi )$ for some $\phi \in {\mathbb {R}}$, we have $Q(v(E)) \ge 0$.

In this case, we call $\sigma$ a stability condition. Let ${\mathrm {Stab}}_\Lambda ({\mathcal {D}})$ denote the topological space (see Sect. 2) of stability conditions, and ${\mathcal {Z}}:{\mathrm {Stab}}_\Lambda ({\mathcal {D}}) \rightarrow {\mathrm {Hom}}(\Lambda , {\mathbb {C}})$ the map ${\mathcal {Z}}(Z, {\mathcal {P}}) = Z$.

Theorem 1.2

Let Q be a quadratic form on $\Lambda \otimes {\mathbb {R}}$, and assume that the stability condition $\sigma = (Z, {\mathcal {P}})$ satisfies the support property with respect to Q. Then:

• 1. There is an open neighbourhood $\sigma \in U_\sigma \subset {\mathrm {Stab}}_\Lambda ({\mathcal {D}})$ such that the restriction ${\mathcal {Z}}:U_\sigma \rightarrow {\mathrm {Hom}}(\Lambda , {\mathbb {C}})$ is a covering of the set of $Z'$ such that Q is negative definite on ${\mathrm {Ker}}Z'$.

• 2. All stability conditions in $U_\sigma$ satisfy the support property with respect to Q.

In other words, ${\mathrm {Stab}}_\Lambda ({\mathcal {D}})$ is a manifold, and any path $Z_t \in {\mathrm {Hom}}(\Lambda , {\mathbb {C}})$ for $t \in [0,1]$ with $Z_0 = Z$ and ${\mathrm {Ker}}Z_t$ negative definite for all $t \in [0,1]$ lifts uniquely to a continuous path $\sigma _t = (Z_t, {\mathcal {P}}_t)$ in the space of stability conditions starting at $\sigma _0 = \sigma$.

Part (1) is an effective variant of [9, 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 [7, Proposition A.5] with an indirect proof based on reduction to [9, Theorem 1.2].

Remarks.  The support property can be a deep and interesting property in itself: a quadratic Bogomolov-Gieseker type inequality for Chern classes of semistable objects which, by Theorem 1.2, is preserved under wall-crossing.

Theorem 1.2 was crucial in [7] 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 [8]). 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. [5, Section 4]).

Theorem 1.2 of [9] also allows for components of the space of stability conditions modelled on a linear subspace $L \subset {\mathrm {Hom}}(\Lambda , {\mathbb {C}})$. When L is defined over ${\mathbb {Q}}$, we can recover that statement by replacing $\Lambda$ with $\Lambda /{\mathrm {Ker}}L$ (see [19] for examples where this is not satisfied; however, to achieve well-behaved wall-crossing one has to assume that L is defined over ${\mathbb {Q}}$).

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 Sect. 3.

This avoids the need for quasi-abelian categories, and some of the more technical arguments of [9, Section 7]. We still need a few arguments similar to ones in [9], which we have reproduced for the convenience of the reader.

Application.  Assume that ${\mathcal {D}}$ is a 2-Calabi–Yau category, i.e. for all $E, F \in {\mathcal {D}}$ we have a bi-functorial isomorphism ${\mathrm {Hom}}(E, F) = {\mathrm {Hom}}(F, E[2])^\vee$. Let $\Lambda$ be the numericalK-group of ${\mathcal {D}}$, and assume that $\Lambda$ is finitely generated. Then there is a surjection $v :K({\mathcal {D}}) \rightarrow \Lambda$, and $\Lambda$ admits a non-degenerate bilinear form $(\_, \_)$, called Mukai-pairing, with

$$\begin{aligned} \chi (E, F) = -\bigl (v(E), v(F) \bigr ). \end{aligned}$$

Let ${\mathcal {P}}_0({\mathcal {D}}) \subset {\mathrm {Hom}}(\Lambda , {\mathbb {C}})$ be the set of central charges Z such that ${\mathrm {Ker}}Z$ is negative definite with respect to the Mukai pairing, and such that ${\mathrm {Ker}}Z$ contains no root $\delta \in \Lambda , (\delta , \delta ) = -2$.

Corollary 1.3

The restriction ${\mathcal {Z}}^{-1}\left( {\mathcal {P}}_0({\mathcal {D}})\right) \xrightarrow {{\mathcal {Z}}} {\mathcal {P}}_0({\mathcal {D}})$ is a covering map.

The proof, given in Sect. 8, is fairly similar to the case of K3 surfaces [10, 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 [12, 16, 22]. In each of these cases, there is a connected component of ${\mathrm {Stab}}({\mathcal {D}})$ covering a connected component of ${\mathcal {P}}_0(X)$; such statements rely crucially on non-emptiness of moduli spaces of stable objects.

2 Review: definitions and basic properties

Throughout, ${\mathcal {D}}$ will be a triangulated category, equipped with a group homomorphism

$$\begin{aligned} v :K({\mathcal {D}}) \rightarrow \Lambda \end{aligned}$$

from its K-group to an abelian group $\Lambda \cong {\mathbb {Z}}^m$.

Definitions.  We first recall the main definitions from [9].

Definition 2.1

A slicing${\mathcal {P}}$ on ${\mathcal {D}}$ is a collection of full subcategories ${\mathcal {P}}(\phi )$ for all $\phi \in {\mathbb {R}}$ with

• 1. ${\mathcal {P}}(\phi +1) = {\mathcal {P}}(\phi )[1]$ for all $\phi \in {\mathbb {R}}$;

• 2. for $\phi _1 > \phi _2$ and $E_i \in {\mathcal {P}}(\phi _i), i = 1,2$, we have ${\mathrm {Hom}}(E_1, E_2) = 0$; and

• 3. for any $E \in {\mathcal {D}}$ there is a sequence of maps $0 = E_0 \xrightarrow {i_1} E_1 \rightarrow \dots \xrightarrow {i_m} E_m = E$ and of real numbers $\phi _1> \phi _2> \dots > \phi _m$ such that the cone of $i_j$ is in ${\mathcal {P}}(\phi _j)$ for $j = 1, \dots , m$.

The non-zero objects of ${\mathcal {P}}(\phi )$ are called semistable of phase$\phi$; its simple objects are called stable. The sequence of maps in (3) is called the Harder–Narasimhan (HN) filtration of E.

Definition 2.2

A pre-stability condition on ${\mathcal {D}}$ is a pair $\sigma = (Z, {\mathcal {P}})$ where $Z :\Lambda \rightarrow {\mathbb {C}}$ is a group homomorphism and ${\mathcal {P}}$ a slicing, such that for all $0 \ne E \in {\mathcal {P}}(\phi )$, we have $Z(v(E)) \in {\mathbb {R}}_{>0}\cdot e^{i \pi \phi }$.

We will abuse notation and write Z(E) instead of Z(v(E)).

Basic properties.  Let ${\mathrm {GL}}_2^+({\mathbb {R}})$ denote the group of real $2 \times 2$-matrices with positive determinant. Since ${\mathrm {GL}}_2^+({\mathbb {R}})$ acts on $S^1$, its universal cover ${\widetilde{{\mathrm {GL}}}}_2^+({\mathbb {R}})$ acts on the universal cover ${\mathbb {R}}\rightarrow S^1$ given explicitly by $\phi \mapsto e^{i\pi \phi }$. For ${\tilde{g}} \in {\widetilde{{\mathrm {GL}}}}_2^+({\mathbb {R}})$ we will write g for the corresponding element of ${\mathrm {GL}}_2^+({\mathbb {R}})$, and ${\tilde{g}}.\phi$ for the given action on ${\mathbb {R}}$.

Proposition 2.3

There is a natural action of ${\widetilde{{\mathrm {GL}}}}_2^+({\mathbb {R}})$ on the set of pre-stability conditions given by ${\tilde{g}}.(Z, {\mathcal {P}}) = (g \circ Z, {\mathcal {P}}')$ where ${\mathcal {P}}'({\tilde{g}}.\phi ) = {\mathcal {P}}(\phi )$.

The heart of a bounded t-structure is a full subcategory ${\mathcal {A}}\subset {\mathcal {D}}$ such that

$$\begin{aligned} {\mathcal {S}}(\phi ) := {\left\{ \begin{array}{ll} {\mathcal {A}}[\phi ] &{}\quad \text {if} \, \phi \in {\mathbb {Z}}\\ 0 &{} \quad \text {if} \, \phi \notin {\mathbb {Z}}\end{array}\right. } \end{aligned}$$

is a slicing (see [9, Lemma 3.2]). It is automatically an abelian subcategory; and stability conditions on ${\mathcal {D}}$ can be constructed from slope-stability in ${\mathcal {A}}$.

Definition 2.4

A stability function Z on an abelian category ${\mathcal {A}}$ is a morphism $Z :K({\mathcal {A}}) \rightarrow {\mathbb {C}}$ of abelian groups such that for all $0 \ne E \in {\mathcal {A}}$, the complex number Z(E) is in the semiclosed upper half plane

$$\begin{aligned} {\mathbb {H}}:= \left\{ z \in {\mathbb {C}}\,:\,\mathfrak {I}z > 0\ \text {or} \ z \in {\mathbb {R}}_{<0}\right\} . \end{aligned}$$

For $0 \ne E \in {\mathcal {A}}$ we define its phase by $\phi (E) := \frac{1}{\pi }\arg Z(E) \in (0,1]$. An object $E \in {\mathcal {A}}$ is called Z-semistable if for all subobjects $A \hookrightarrow E$, we have $\phi (A) \le \phi (E)$.

Definition 2.5

We say that a stability function Z on an abelian category ${\mathcal {A}}$ satisfies the HN property if every object $E \in {\mathcal {A}}$ admits a Harder–Narasimhan (HN) filtration: a sequence $0 = E_0 \hookrightarrow E_1 \hookrightarrow E_2 \hookrightarrow \cdots \hookrightarrow E_m = E$ such that $E_i/E_{i-1}$ is Z-semistable for $i = 1, \dots , m$, with

$$\begin{aligned} \phi \left( E_1/E_0\right)> \phi \left( E_2/E_1\right)> \cdots > \phi \left( E_m/E_{m-1}\right) . \end{aligned}$$

Proposition 2.6

[9, Proposition 5.3] To give a pre-stability condition on ${\mathcal {D}}$ is equivalent to giving a heart ${\mathcal {A}}$ of a bounded t-structure, and a stability function Z on ${\mathcal {A}}$ with the HN property.

Here we tacitly assume that Z factors via $K({\mathcal {A}}) = K({\mathcal {D}}) \xrightarrow {v} \Lambda$. Given $(Z, {\mathcal {A}})$, the slicing is determined by setting ${\mathcal {P}}(\phi )$ to be the Z-semistable objects $E \in {\mathcal {A}}$ of phase $\phi$ for $\phi \in (0, 1]$. Conversely, given $(Z, {\mathcal {P}})$, the heart ${\mathcal {A}}= {\mathcal {P}}(0, 1]$ is the smallest extension-closed subcategory of ${\mathcal {D}}$ containing ${\mathcal {P}}(\phi )$ for $\phi \in (0, 1]$. More generally, ${\mathcal {P}}(\phi , \phi +1]$ is a heart for every $\phi \in {\mathbb {R}}$.

Definition 2.7

A stability condition $\sigma$ 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 $\Lambda \otimes {\mathbb {R}}$.

Topology and local injectivity.  There is a generalised metric, and thus a topology, on the set of slicings ${\mathrm {Slice}}({\mathcal {D}})$ given as follows. Given two slicings ${\mathcal {P}}, {\mathcal {Q}}$, we write $\phi ^{\pm }(E)$ and $\psi ^{\pm }(E)$ for the largest and smallest phase in the associated HN filtration of an object E for ${\mathcal {P}}$ and ${\mathcal {Q}}$, respectively. Then we define the distance of ${\mathcal {P}}$ and ${\mathcal {Q}}$ by

$$\begin{aligned} d({\mathcal {P}}, {\mathcal {Q}}) := \sup \left\{ \left|\phi ^+(E) - \psi ^+(E)\right|, \left|\phi ^-(E) - \psi ^-(E)\right|\,:\,E \in {\mathcal {D}}\right\} \in [0, +\infty ]. \end{aligned}$$

We recall that this distance can be computed by considering ${\mathcal {P}}$-semistable objects alone:

Lemma 2.8

[9, Lemma 6.1] We have $d({\mathcal {P}}, {\mathcal {Q}}) = d'({\mathcal {P}}, {\mathcal {Q}})$, where the latter is defined by

$$\begin{aligned} d'({\mathcal {P}}, {\mathcal {Q}}) := \sup \left\{ \psi ^+(E) - \phi , \phi - \psi ^-(E)\,:\,\phi \in {\mathbb {R}}, 0 \ne E \in {\mathcal {P}}(\phi )\right\} . \end{aligned}$$

Proof

The inequality $d({\mathcal {P}}, {\mathcal {Q}}) \ge d'({\mathcal {P}}, {\mathcal {Q}})$ is immediate. For the converse, consider $E \in {\mathcal {D}}$, and let $A_i$ be one of its HN factors with respect to ${\mathcal {P}}$. Then $\psi ^+(A_i) \le \phi (A_i) + d'({\mathcal {P}}, {\mathcal {Q}}) \le \phi ^+(E) + d'({\mathcal {P}}, {\mathcal {Q}})$. Hence E admits no non-zero maps from ${\mathcal {Q}}$-stable objects of phase bigger than $\phi ^+(E) + d'({\mathcal {P}}, {\mathcal {Q}})$, and so $\psi ^+(E) \le \phi ^+(E) + d'({\mathcal {P}}, {\mathcal {Q}})$. The analogous inequality $\psi ^-(E) \ge \phi ^-(E) - d'({\mathcal {P}}, {\mathcal {Q}})$ follows similarly.

Finally, for $E \in {\mathcal {D}}$ we have a non-zero map $A_1 \rightarrow E$ with $A_1 \in {\mathcal {P}}(\phi ^+(E))$. Therefore, $\psi ^+(E) \ge \psi ^-(A_1) \ge \phi ^-(A_1) - d'({\mathcal {P}}, {\mathcal {Q}}) = \phi ^+(E) - d'({\mathcal {P}}, {\mathcal {Q}})$, and so $\left|\psi ^+(E) - \phi ^+(E)\right| \le d'({\mathcal {P}}, {\mathcal {Q}})$. Combined with the same inequality for $\psi ^-$, the claim follows. $\square$

The topology on ${\mathrm {Stab}}_{\Lambda }({\mathcal {D}})$ is the coarsest topology such that both forgetful maps

$$\begin{aligned} {\mathrm {Stab}}_{\Lambda }({\mathcal {D}})&\rightarrow {\mathrm {Slice}}({\mathcal {D}}), \quad (Z, {\mathcal {P}}) \mapsto {\mathcal {P}}\\ {\mathcal {Z}}:{\mathrm {Stab}}_{\Lambda }({\mathcal {D}})&\rightarrow {\mathrm {Hom}}(\Lambda , {\mathbb {C}}), \quad (Z, {\mathcal {P}}) \mapsto Z \end{aligned}$$

are continuous.

The following Lemma is a variant of [9, Lemma 6.4]:

Lemma 2.9

Assume that $\sigma = (Z, {\mathcal {P}})$ and $\tau = (W, {\mathcal {Q}})$ are two pre-stability conditions such that $\sigma$ satisfies the support property with respect to Q, such that

$$\begin{aligned} \frac{\left|W(v)-Z(v)\right|}{\left|Z(v)\right|} < \sin \pi \epsilon \quad \text {for all} \, v \in \Lambda \, \text {with} \, Q(v) \ge 0, \end{aligned}$$

(1)

and such that either $d({\mathcal {P}}, {\mathcal {Q}}) < \frac{1}{4}$, or that $\sigma , \tau$ have the same heart ${\mathcal {P}}(0,1] = {\mathcal {Q}}(0,1]$. Then $d({\mathcal {P}}, {\mathcal {Q}}) < \epsilon$.

Of course, (1) means in particular that the phases of W(v) and Z(v) differ by at most $\epsilon$.

Proof

We want to apply Lemma 2.8, so let us consider an object $E \in {\mathcal {P}}(\phi )$. In the first case, $d({\mathcal {P}}, {\mathcal {Q}}) < \frac{1}{4}$, we apply this assumption twice to see that every HN filtration factor of E with respect to ${\mathcal {Q}}$ is contained in ${\mathcal {Q}}(\phi - \frac{1}{4}, \phi + \frac{1}{4}) \subset {\mathcal {P}}(\phi -\frac{1}{2}, \phi +\frac{1}{2}] =: {\mathcal {A}}$. In particular, the first HN filtration factor $E_1 \rightarrow E$ of E with respect to ${\mathcal {Q}}$ is a subobject of E in the abelian category A. The analogous claim is obvious in the second case for ${\mathcal {A}}= {\mathcal {P}}(0,1]$. Therefore, every HN factor F of $E_1$ with respect to ${\mathcal {P}}$ has phase at most $\phi$. By (1), it follows that W(F) has phase less than $\phi + \epsilon$, and thus the same holds for the phase $\psi ^+(E)$ of $W(E_1)$.

A similar argument shows $\psi ^-(E) > \phi (E) - \epsilon$, thus proving the claim. $\square$

Corollary 2.10

The map ${\mathcal {Z}}:{\mathrm {Stab}}_{\Lambda }({\mathcal {D}}) \rightarrow {\mathrm {Hom}}(\Lambda , {\mathbb {C}})$, see Definition 2.1, is locally injective.

Moreover, consider a section $U \rightarrow {\mathrm {Stab}}_\Lambda ({\mathcal {D}}), Z \mapsto \sigma _Z = (Z, {\mathcal {P}}_Z)$ of ${\mathcal {Z}}$ defined on a subset $U \subset {\mathrm {Hom}}(\Lambda , {\mathbb {C}})$, such that every $\sigma _Z$ satisfies the support property with respect to Q. Assume that U can be covered by open subsets $V_i$ such that for all $Z, Z' \in V_i$ we have $d({\mathcal {P}}_Z, {\mathcal {P}}_{Z'}) < \frac{1}{4}$. Then this section is continuous.

Proof

For the first statement, we just set $Z = W$ in the Lemma, to obtain $d({\mathcal {P}}, {\mathcal {Q}}) = 0$.

For the second statement, we only need to show that (1) holds in a neighbourhood of $Z \in {\mathrm {Hom}}(\Lambda , {\mathbb {C}})$ when Q is negative definite on ${\mathrm {Ker}}Z$. Choose any metric on $\Lambda _{\mathbb {R}}$; then clearly (1) only needs to be checked for vectors with unit length. Since $Q(\_) \ge 0$ defines a compact subset of the unit sphere, on which $\left|Z(v)\right|$ is a positive continuous function, the claim follows. $\square$

Morever, Proposition 2.3 gives a continuous action of ${\widetilde{{\mathrm {GL}}}}_2^+({\mathbb {R}})$ on ${\mathrm {Stab}}_\Lambda ({\mathcal {D}})$.

3 Harder–Narasimhan filtrations via the Harder–Narasimhan polygon

Throughout this section, let ${\mathcal {A}}$ be an abelian category with a stability function Z.

Definition 3.1

The Harder–Narasimhan polygon${\mathrm {HN}}^Z(E)$ of an object $E \in {\mathcal {A}}$ is the convex hull of the central charges Z(A) of all subobjects $A \subset E$ of E.

(The trivial subobjects $A = 0$ or $A = E$ are included in the definition.) The idea to consider this convex set in the context of slope-stability goes back at least 40 years [21].

Definition 3.2

We say that the Harder–Narasimhan polygon ${\mathrm {HN}}^Z(E)$ of an object $E \in {\mathcal {A}}$ is polyhedral on the left if the set has finitely many extremal points $0 = z_0, z_1, \dots , z_m = Z(E)$ such that ${\mathrm {HN}}^Z(E)$ lies to the right¹ of the path $z_0z_1z_2\dots z_m$; see Fig. 1.

Fig. 1

Polyhedral on the left

In other words, the intersection of ${\mathrm {HN}}^Z(E)$ with the closed half-plane to the left of the line through 0 and Z(E) is the polygon with vertices $z_0, z_1, \dots , z_m$. Our proof of Theorem 1.2 is based on the following well-known statement; we provide a proof for completeness:

Proposition 3.3

The object E has a Harder–Narasimhan filtration with respect to Z if and only if its Harder–Narasimhan polygon ${\mathrm {HN}}^Z(E)$ is polyhedral on the left.

Assume that ${\mathrm {HN}}^Z(E)$ is polyhedral on the left. For each $i = 1, \dots , m$, choose a subobject $E_i \subset E$ such that $Z(E_i) = z_i$. (This exists as $z_i$ is extremal.)

Lemma 3.4

This is a filtration, i.e. $E_{i-1} \subset E_{i}$ for $i = 1, \dots , m$.

Fig. 2

Lemma 3.4

Proof

Let $A := E_{i-1} \cap E_i \subset E$ be the intersection of two subsequent objects, and $B := E_{i-1} + E_i \subset E$ be their span inside E. Then there is a short exact sequence

$$\begin{aligned} A \hookrightarrow E_{i-1} \oplus E_i \twoheadrightarrow B. \end{aligned}$$

Hence the midpoint of Z(A) and Z(B) is also the midpoint of $z_{i-1}$ and $z_i$, see also Fig. 2.

On the other hand, Z(A), Z(B) lie in ${\mathrm {HN}}^Z(E)$; by convexity and the choice of $z_{i-1}, z_i$, they both have to lie either in the open half-plane to the right of the line $\left( z_{i-1}z_i\right)$, or on the line segment $\overline{z_{i-1}z_i}$. The former would be a contradiction to the previous paragraph, and so $Z(A) \in \overline{z_{i-1}z_i}$.

Since $A \subset E_{i-1}$, this implies $Z(A) = z_{i-1}$ and $A \cong E_{i-1}$; therefore, $E_{i-1} \subset E_i$. $\square$

Fig. 3

Lemma 3.5

Lemma 3.5

The filtration quotient $E_i/E_{i-1}$ is semistable.

Proof

Otherwise, there is an object A with $E_{i-1} \subset A \subset E_i$ such that $A/E_{i-1}$ has bigger phase than $E_i/E_{i-1}$, see Fig. 3. It follows that Z(A) lies to the left of the line segment $\overline{z_{i-1}z_i}$. Since $A \subset E$ and hence $Z(A) \in {\mathrm {HN}}^Z(E)$, this is a contradiction. $\square$

Proof of Proposition 3.3

The phase of $E_i/E_{i-1}$ is determined by the argument of $z_i - z_{i-1}$; by convexity this shows $\phi (E_1/E_0)> \dots > \phi (E_m/E_{m-1})$, and so the $E_i$ form a HN filtration.

Conversely, assume that we are given a HN filtration $0 = E_0 \hookrightarrow E_1 \hookrightarrow \cdots \hookrightarrow E_m$ and a subobject $A \hookrightarrow E$. We have to show that Z(A) lies to the right of the path $z_0z_1 \dots z_m$ with vertices $z_i := Z(E_i)$. By induction on m, we may assume that $Z(A \cap E_{m-1})$ lies to the right of the path $z_0z_1 \dots z_{m-1}$. On the other hand, $A/\left( A \cap E_{m-1}\right)$ is a subobject of $E_m/E_{m-1}$, which is semistable; thus its central charge $Z\bigl (A/\left( A \cap E_{m-1}\right) \bigr )$ lies to the right of the line segment from 0 to $z_m - z_{m-1}$. Therefore, $Z(A) = Z(A \cap E_{m-1}) + Z(A/\left( A \cap E_{m-1}\right)$ lies to the right of the path $z_0z_1\dots z_m$ as claimed.

Corollary 3.6

Given $E \in {\mathcal {A}}$, assume that there are only finitely many classes v(A) of subobjects $A \subset E$ with $\mathfrak {R}Z(A) < \max \left\{ 0, \mathfrak {R}Z(E) \right\}$. Then E admits a HN filtration.

4 Linear algebra lemmas

Throughout Sects. 4, 5 and 6, we fix a quadratic form Q on $\Lambda _{\mathbb {R}}$ with:

Assumption 4.1

The quadratic form Q has signature $(2, {\mathrm {rk}}\Lambda - 2)$.

Lemma 4.2

Let $Z :\Lambda \rightarrow {\mathbb {C}}$ be a group homomorphism such that Q is negative definite on $K := {\mathrm {Ker}}Z \subset \Lambda _{\mathbb {R}}$. Let $\left\| \cdot \right\|$ be the norm on K associated to $-Q|_K$, and let $p :\Lambda _{\mathbb {R}}\rightarrow K$ be the orthogonal projection with respect to Q. After replacing Z by a ${\mathrm {GL}}_2^+({\mathbb {R}})$-translate, we have

$$\begin{aligned} Q(v) = \left|Z(v)\right|^2 - \left\| p(v)\right\| ^2. \end{aligned}$$

(2)

Proof

Let $K^\perp$ be the orthogonal complement of K. As $Z|_{K^\perp }$ is injective, Assumption 4.1 can only hold if Q is positive definite on $K^\perp$, and if $Z|_{K^\perp } :K^\perp \rightarrow {\mathbb {C}}$ is an isomorphism. Up to the ${\mathrm {GL}}_2^+({\mathbb {R}})$-action, we may assume this to be an isometry. Then the the claim follows. $\square$

Remark 4.3

In [10], this normalisation is used with the Mukai quadratic form replacing Q.

Consider the subset in ${\mathrm {Hom}}(\Lambda , {\mathbb {C}})$ of central charges whose kernel is negative definite with respect to Q; let ${\mathcal {P}}_Z(Q)$ be its connected component containing Z.

Lemma 4.4

Let Q, Z and K be as in Lemma 4.2.

• 1. For each $Z' \in {\mathcal {P}}_Z(Q)$ there exists a unique $g \in {\mathrm {GL}}_2^+({\mathbb {R}})$ and a linear map $u :K \rightarrow {\mathbb {C}}$ with $\left\| u\right\| < 1$ such that

  $$\begin{aligned} g Z' = Z + u \circ p. \end{aligned}$$

• 2. Up to the action of ${\mathrm {GL}}_2^+({\mathbb {R}})$, we can break up this deformation of Z into a pure real and a purely imaginary part with analogous properties: there exist $u_{\mathbb {R}}$, $u_{i{\mathbb {R}}}$, $Z_1$ and $g_1, g_2 \in {\mathrm {GL}}_2^+({\mathbb {R}})$, depending continuously on $Z'$, such that

  • (a) $u_{\mathbb {R}}:K \rightarrow {\mathbb {R}}$ and $u_{i{\mathbb {R}}} :{\mathrm {Ker}}Z_1 \rightarrow {\mathbb {R}}$ satisfy $\left\| u_{\mathbb {R}}\right\| < 1$, $\left\| u_{i{\mathbb {R}}}\right\| < 1$,

  • (b) $Z_1 := Z + u_{\mathbb {R}}\circ p$,

  • (c) Eq. (2) holds with Z and p replaced by $g_1 Z_1$ and $p_1 :\Lambda _{\mathbb {R}}\rightarrow {\mathrm {Ker}}Z_1$, and

  • (d) $g_2 Z' = g_1 Z_1 + i u_{i{\mathbb {R}}} \circ p_1$, where $p_1 :\Lambda _{\mathbb {R}}\rightarrow {\mathrm {Ker}}Z_1$ is the orthogonal projection.

Proof

The restriction of $Z'$ to the orthogonal complement $K^\perp$ is an isomorphism for all $Z' \in {\mathcal {P}}_Z(Q)$. Hence there exists a unique $g \in {\mathrm {GL}}_2({\mathbb {R}})$ such that $gZ'|_{K^\perp } = Z|_{K^\perp }$; since ${\mathcal {P}}_Z(Q)$ is connected, in fact $g \in {\mathrm {GL}}_2^+({\mathbb {R}})$. Let $u:= gZ'|_{{\mathrm {Ker}}Z}$; then $g Z' - Z = u \circ p$ holds both on K and $K^\perp$, and thus on all of $\Lambda _{\mathbb {R}}$.

Next, we prove $\left\| u\right\| < 1$. Otherwise, let $v \in K$ with $\left\| v\right\| = 1$ and $\left|u(v)\right| \ge 1$, and let ${\overline{v}} \in K^\perp$ be such that $Z({\overline{v}}) = u(v)$. Then $Z'(v - {\overline{v}}) = 0$, but $Q(v - {\overline{v}}) = \left|u(v)\right|^2 - \left\| v\right\| ^2 \ge 0$ in contradiction to Q being negative definite on ${\mathrm {Ker}}Z'$. This completes the proof of (1).

Now let $u_{\mathbb {R}}:= \mathfrak {R}u$, and let $Z_1$ be as above. If we write $\Lambda _{\mathbb {R}}= K \oplus K^\perp$ and identify $K^\perp$ with ${\mathbb {C}}$ via Z, then ${\mathrm {Ker}}Z_1$ is the graph of $-u_{\mathbb {R}}$, and its orthogonal complement is the graph of ${\mathbb {C}}\rightarrow K, z \mapsto \mathfrak {R}z \cdot u^\vee$, where $u^\vee$ corresponds to u under the identification $K \cong K^\vee$ induced by the symmetric form associated to $\left\| \cdot \right\|$. A straightforward computation shows that $g_1 \circ Z_1$ induces an isometry $({\mathrm {Ker}}Z_1)^\perp \rightarrow {\mathbb {C}}$ for $g_1 := \begin{pmatrix} \left( 1 + \left\| u_{\mathbb {R}}\right\| ^2\right) ^{-\frac{1}{2}} &{} 0 \\ 0 &{} 1 \end{pmatrix}\in {\mathrm {GL}}_2^+({\mathbb {R}})$; by the proof of Lemma 4.2 this implies (2c). Moreover, $g_1 g Z' - g_1 Z_1 = g_1 \circ \mathfrak {I}u \circ p$ is completely imaginary. Applying part (1) to $g_1 g Z'$ and $g_1 Z_1$ then shows part (2d). $\square$

5 Real variations of the central charge

The key lemma, proved in this section and the next, treats the case where only the real part of the central charge is varying:

Lemma 5.1

Consider a stability condition $\sigma = (Z, {\mathcal {P}})$, satisfying the support property with respect to Q, and assume that (2) holds with p and $\left\| \cdot \right\|$ as defined in Lemma 4.2. Given $u :{\mathrm {Ker}}Z \rightarrow {\mathbb {R}}$ with $\left\| u\right\| < 1$, there is a stability condition $\tau = (W, {\mathcal {Q}})$ with $W = Z + u \circ p$, satisfying the support property with respect to Q, and with $d({\mathcal {P}}, {\mathcal {Q}}) < \frac{\left\| u\right\| }{2}$.

Proof of Theorem 1.2, assuming Lemma 5.1 and Assumption 4.1

Lemma 5.1 automatically also applies to purely imaginary variations of the central charge as in Lemma 4.4.(2d), due to the ${\widetilde{{\mathrm {GL}}}}_2^+({\mathbb {R}})$-action. We may assume that Q and the central charge satisfy (2); then Lemmas 4.4 and 5.1 combined with the ${\widetilde{{\mathrm {GL}}}}_2^+({\mathbb {R}})$-action give a set-theoretic section ${\mathcal {P}}_Z(Q) \rightarrow {\mathrm {Stab}}_\Lambda ({\mathcal {D}})$.

Since ${\mathrm {GL}}_2^+({\mathbb {R}})$ acts continuously on the compact set $S^1$, there exists an open neighbourhood $B_\epsilon (1)$ of $1 \in {\widetilde{{\mathrm {GL}}}}_2^+({\mathbb {R}})$ such that $\left|{\tilde{g}}.\phi - \phi \right| < \epsilon$ for all ${\tilde{g}} \in B_\epsilon (1)$. It follows that if $({\mathcal {Q}}, W) = {\tilde{g}}.({\mathcal {P}}, Z)$, then $d({\mathcal {Q}}, {\mathcal {P}}) < \epsilon$. Then there is an open neighborhood $U_\epsilon (Z)$ of Z in ${\mathcal {P}}_Z(Q)$ where, in the notation of Lemma 4.4, $\left\| u_{{\mathbb {R}}}\right\|< \epsilon , \left\| u_{i{\mathbb {R}}}\right\| < \epsilon$, and $g_1, g_2$ lift to $\tilde{g}_1, {\tilde{g}}_2 \in B_\epsilon (1)$. Then our lift $\tau = (Z, {\mathcal {Q}})$ of any $Z'$ in this open neighbourhood satisfies $d({\mathcal {Q}}, {\mathcal {P}}) < \frac{\epsilon }{2} + \epsilon + \frac{\epsilon }{2} + \epsilon$.

By Corollary 2.10, our section is continuous on $U_\epsilon (Z)$ for $\epsilon \le \frac{1}{12}$. Since the neighbourhoods $U_{\frac{1}{12}}(Z')$ for $Z' \in {\mathcal {P}}_Z({\mathcal {Q}})$ cover ${\mathcal {P}}_Z({\mathcal {Q}})$, and since the lifts constructed on each such neighbourhood agree on the overlaps by the local injectivity of ${\mathcal {Z}}$, this proves the Theorem. $\square$

Lemma 5.1 is simpler to prove since it allows (and forces) us to leave the heart ${\mathcal {A}}:= {\mathcal {P}}(0, 1]$ unchanged: we will apply Proposition 2.6 and prove that $({\mathcal {A}}, W)$ produces a stability condition.

Lemma 5.2

In the situation of Lemma 5.1, $W = Z + u \circ p$ is a stability function on ${\mathcal {A}}$.

Proof

Consider $E \in {\mathcal {A}}$; if $\mathfrak {I}Z(E) = \mathfrak {I}W(E) > 0$, there is nothing to prove. Otherwise, E is semistable with $Z(E) \in {\mathbb {R}}_{<0}$ and thus $\left\| p(E)\right\| \le - Z(E)$. From $\left\| u\right\| < 1$ we conclude

$$\begin{aligned} W(E) = Z(E) + u\circ p(E) \le Z(E) + \left\| u\right\| \left\| p(E)\right\| < Z(E) - Z(E) = 0. \end{aligned}$$

$\square$

We will use Proposition 3.3 and Corollary 3.6 to prove that $({\mathcal {A}}, W)$ satisfies the HN property.

Let us define the mass$m^Z(E)$ of $E \in {\mathcal {A}}$ with respect to Z as the length of the boundary of ${\mathrm {HN}}^Z(E)$ on the left between 0 and Z(E).

Lemma 5.3

For all $E \in {\mathcal {A}}$ we have $\left\| p(E)\right\| \le m^Z(E)$.

Proof

If E is semistable, then $0 \le Q(E) = \left|Z(E)\right|^2 - \left\| p(E)\right\| ^2 = \left( m^Z(E)\right) ^2 - \left\| p(E)\right\| ^2$, which is exactly the claim. Otherwise, consider the HN filtration $E_0 \hookrightarrow E_1 \hookrightarrow \dots \hookrightarrow E_m = E$. Combined with the triangle inequality, this gives

$$\begin{aligned} \left\| p(E)\right\|\le & {} \sum _i \left\| p(E_i/E_{i-1})\right\| \le \sum _i \left|Z(E_i/E_{i-1})\right|\\= & {} \sum _i \left|Z(E_i) - Z(E_{i-1})\right| = m^Z(E). \end{aligned}$$

$\square$

The following Lemma needs no proof:

Lemma 5.4

If $A \subset E$, then ${\mathrm {HN}}^Z(A) \subset {\mathrm {HN}}^Z(E)$.

Lemma 5.5

Given any subobject $A \subset E$, we have

$$\begin{aligned} m^Z(A) - \mathfrak {R}Z(A) \le m^Z(E) - \mathfrak {R}Z(E). \end{aligned}$$

Proof

This follows from the previous Lemma, convexity and a picture, see Fig. 4. Indeed, choose $x > \mathfrak {R}Z(A), \mathfrak {R}Z(E)$; let $a = x + i \mathfrak {I}Z(A)$ and $e = x + i \mathfrak {I}Z(E)$. Let $\gamma _A$ be the path that follows the boundary of ${\mathrm {HN}}^Z(A)$ from 0 to Z(A), and then continues horizontally to a; similarly $\gamma _E$ follows the boundary of ${\mathrm {HN}}^Z(E)$ and then continues to e. Their lengths are given as

$$\begin{aligned} \left|\gamma _A\right| = m^Z(A) + x - \mathfrak {R}Z(A), \quad \left|\gamma _E\right| = m^Z(E) + x - \mathfrak {R}Z(E). \end{aligned}$$

On the other hand, convexity and Lemma 5.4 imply $\left|\gamma _A\right| \le \left|\gamma _E\right|$; for example, if $\gamma _I$ denotes the intermediate path that follows the boundary of ${\mathrm {HN}}^Z(E)$ up to height $\mathfrak {I}Z(A)$ and then goes horizontally to a, we clearly have $\left|\gamma _A\right| \le \left|\gamma _I\right| \le \left|\gamma _E\right|$. $\square$

Fig. 4

Proof of Lemma 5.5

Lemma 5.6

Given $C \in {\mathbb {R}}$, the set of $v(A) \in \Lambda$ for subobjects $A \subset E$ with $\mathfrak {R}W(A) < C$ is finite.

Proof

Given any such A, we use Lemmas 5.3 and 5.5 to obtain

$$\begin{aligned} C&> \mathfrak {R}W(A) \ge \mathfrak {R}Z(A) - \left\| u\right\| \left\| p(A)\right\| \\&> (1 - \left\| u\right\| )\mathfrak {R}Z(A) - \left\| u\right\| \left( m^Z(A) - \mathfrak {R}Z(A)\right) \\&\ge (1 - \left\| u\right\| )\mathfrak {R}Z(A) - \left\| u\right\| \left( m^Z(E) - \mathfrak {R}Z(E)\right) . \end{aligned}$$

Since $\left\| u\right\| < 1$, this bounds $\mathfrak {R}Z(A)$ from above. On the other hand, $Z(A) \in {\mathrm {HN}}^Z(E)$, and thus Z(A) is constrained to lie in a compact region of ${\mathbb {C}}$. Using Lemmas 5.5 and 5.3 again, this gives an upper bound first for $m^Z(A)$ and consequently for $\left\| p(A)\right\|$. Hence v(A) is contained in a compact region of $\Lambda \otimes {\mathbb {R}}$ depending only on E and C, and the claim follows. $\square$

Lemma 5.6 and Corollary 3.6 imply the existence of HN filtrations for W on ${\mathcal {A}}$, and thus yield a pre-stability condition $\tau = ({\mathcal {A}}, W)$ by Proposition 2.6; write ${\mathcal {Q}}$ for the associated slicing.

Lemma 5.7

The pre-stability condition $\tau = (W, {\mathcal {Q}})$ satisfies $d({\mathcal {P}}, {\mathcal {Q}}) < \frac{\left\| u\right\| }{2}$.

Proof

We apply Lemma 2.9: by construction, by (2), and by the assumption $\left\| u\right\| < 1$ we have

$$\begin{aligned}&\frac{\left|W(v)-Z(v)\right|}{\left|Z(v)\right|} \\&\quad = \frac{\left|u \circ p(v)\right|}{\left|Z(v)\right|} \le \left\| u\right\| \frac{\left\| p(v)\right\| }{\left|Z(v)\right|} \le \left\| u\right\| < \sin \pi \frac{\left\| u\right\| }{2} \ \text {for} \, v \in \Lambda \, \text {with} \, Q(v) \ge 0. \end{aligned}$$

$\square$

6 The support property is preserved

It remains to show that $({\mathcal {A}}, W)$ satisfies the support property with respect to Q, i.e. $Q(v(E)) \ge 0$ for all W-stable $E \in {\mathcal {A}}$. The basic reason is that this inequality preserved by wall-crossing:

Lemma 6.1

Let $\sigma = (Z, {\mathcal {P}})$ be pre-stability condition. Assume that Q is a non-degenerate quadratic form on $\Lambda _{\mathbb {R}}$ of signature $(2, {\mathrm {rk}}\Lambda -2)$ such that Q is negative definite on ${\mathrm {Ker}}Z$. If E is strictly $\sigma$-semistable and admits a Jordan-Hölder filtration with factors $E_1, \dots , E_m$, and if $Q(v(E_i)) \ge 0$ for $i = 1, \dots , m$, then $Q(v(E)) \ge 0$.

Proof

We apply Lemma 4.2; then $Q(v) \ge 0$ is equivalent to $\left|Z(v)\right| \ge \left\| p(v)\right\|$. We obtain

$$\begin{aligned} \left|Z(E)\right| = \sum _i \left|Z(E_i)\right| \ge \sum _i \left\| p(v(E_i))\right\| \ge \left\| \sum _i p(v(E_i))\right\| = \left\| p(v(E))\right\| \end{aligned}$$

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. $\square$

The proof strategy is thus clear: we use wall-crossing for the path of stability functions $Z_t = Z + t\cdot u \circ p$ on ${\mathcal {A}}$, for $0 \le t \le 1$. If $E \in {\mathcal {A}}$ is $Z_1$-stable with $Q(v(E)) < 0$, then it must be $Z_0$-unstable; wall-crossing gives a $t \in [0, 1)$ 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 conclude, we have to show that we can find such a wall, and that this process terminates.

Lemma 6.2

Given two objects $A, E \in {\mathcal {A}}$, denote their phases with respect to $Z_t$ by $\phi ^t(A), \phi ^t(E)$, respectively. If the set of $t \in [0,1]$ with $\phi ^t(A) \ge \phi ^t(E)$ is non-empty, then it is a closed subinterval of [0, 1] containing one of its endpoints.

Proof

The condition is equivalent to $\frac{-\mathfrak {R}Z_t(A)}{\mathfrak {I}Z_t(A)} \ge \frac{-\mathfrak {R}Z_t(E)}{\mathfrak {I}Z_t(E)}$, which is a linear inequality in t. $\square$

Fig. 5

The truncated HN polygon

Consider the polygon whose vertices are the extremal points of ${\mathrm {HN}}^{Z_0}(E)$ on the left; we will call this the truncated HN polygon ofE, see Fig. 5. Note that if $A \subset E$ is a subobject with $\phi ^0(A) \ge \phi ^0(E)$, then $Z_0(A)$ is contained in the truncated HN polygon of E; by Lemmas 5.5 and 5.3 there are only finitely many classes v(A) of such subobjects.

Lemma 6.3

Every $Z_1$-semistable object $E \in {\mathcal {A}}$ satisfies $Q(E) \ge 0$.

Proof

Otherwise, E must be $Z_0$-unstable. By Lemma 6.2 and the following observation, there are only finitely many classes v(A) of subobjects $A \hookrightarrow E$ that destabilise E with respect $Z_t$ for any $t\in [0,1]$. Hence there is a wall $t_1 \in (0, 1]$ 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 6.1, there are subobjects $G_1 \hookrightarrow F_1 \hookrightarrow E$ of the same phase, such that $F_1/G_1$ is $Z_{t_1}$-stable with $Q(v(F_1/G_1)) < 0$.

Applying the same logic to $F_1/G_1$, we obtain $t_2 \in (0, t_1)$ and subobjects $G_1 \subset G_2 \subset F_2 \subset F_1 \subset E$ such that $F_2/G_1, G_2/G_1$ and $F_1/G_1$ all have the same phase with respect to $t_2$, and such that $Q(v(F_2/G_2)) < 0$. Continuing by induction, we obtain a sequence $t_1> t_2> t_3 > \dots$ in (0, 1) and chains of subobjects $G_1 \subset G_2 \subset G_3 \subset \dots \subset E$ and $E \supset F_1 \supset F_2 \supset F_3 \supset \dots$

Lemma 6.2 gives $\phi ^{t_2}(F_1) \ge \phi ^{t_2}(E)$ and $\phi ^{t_2}(G_1) \ge \phi ^{t_2}(E)$. Since $Z_{t_2}(F_2)$ lies on the line segment connecting $Z_{t_2}(F_1)$ and $Z_{t_2}(G_1)$, we also have $\phi ^{t_2}(F_2) \ge \phi ^{t_2}(E)$ (and therefore $\phi ^{t}(F_2) \ge \phi ^t(E)$ for all $t \in [0, t_2]$); similarly for $G_2$. Continuing by induction, we conclude $\phi ^0(F_i) \ge \phi ^0(E)$. This is a contradiction to the observation above that there are only finitely many classes v(A) of subobjects $A \subset E$ with $Z_0(A)$ lying in the truncated HN polygon. $\square$

This concludes the proof of Lemma 5.1, and thus of Theorem 1.2 under the Assumption 4.1.

7 Reductions

Finally, we will show that we can always reduce the 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 7.1

Assume that the quadratic form Q on $\Lambda _{\mathbb {R}}$ is degenerate. Then there exists an injective map $\Lambda _{\mathbb {R}}\hookrightarrow {\overline{\Lambda }}$ of real vector spaces and a non-degenerate quadratic form ${\overline{Q}}$ on ${\overline{\Lambda }}$, extending Q, such that any central charge $Z :\Lambda _{\mathbb {R}}\rightarrow {\mathbb {C}}$ with kernel negative definite with respect to Q extends to a central charge ${\overline{Z}} :{\overline{\Lambda }} \rightarrow {\mathbb {C}}$ with kernel negative definite with respect to ${\overline{Q}}$.

Proof

Let $N \hookrightarrow \Lambda _{\mathbb {R}}$ be the null space of Q; we will only treat the case ${\mathrm {dim}}_{\mathbb {R}}N = 1$ (otherwise, we can iterate the construction that follows). Choose a splitting $\Lambda _{\mathbb {R}}\cong N \oplus C$; then for $n \in N, c \in C$, we have $Q(n \oplus c) = Q(c)$. Let $\overline{\Lambda _{\mathbb {R}}} := N \oplus N^\vee \oplus C$, let q be the canonical quadratic form on the hyperbolic plane $N \oplus N^\vee$, and set ${\overline{Q}} := q \oplus Q|_C$.

Given Z as above, the restriction $Z|_N$ is injective, and we may assume that Z maps N to the real line. Let $n \in N$ be such that $Z(n) = 1$, and let $n^\vee \in N^\vee$ be the dual vector with $(n, n^\vee ) = 1$. We claim that for $\alpha \gg 0$, the extension of Z defined by $Z'(n^\vee ) = \alpha$ has the desired property.

Let $K := {\mathrm {Ker}}Z$; then the kernel of $Z'$ is contained in $N \oplus N^\vee \oplus K$, and given by vectors of the form $a \cdot n - \frac{a}{\alpha }\cdot n^\vee + k$ for $k \in K, a \in {\mathbb {R}}$. For such vectors, we have

$$\begin{aligned} Q\left( a\cdot n - \frac{a}{\alpha }\cdot n^\vee + k \right) = - \frac{2a^2}{\alpha } - \frac{2a}{\alpha }(n^\vee , k) + Q(k). \end{aligned}$$

This is a quadratic function in a with negative constant term; its discriminant is negative if

$$\begin{aligned} \alpha > \max \left\{ \frac{(n^\vee , k)^2}{-Q(k)}\,:\,k \in K, k \ne 0\right\} \end{aligned}$$

(which is finite since $-Q(\cdot )$ is a positive definite form on K). $\square$

Replacing $\Lambda$ by $\Lambda \oplus {\mathbb {Z}}$ and v by

$$\begin{aligned} K({\mathcal {D}}) \xrightarrow {v} \Lambda \hookrightarrow \Lambda \oplus {\mathbb {Z}}\end{aligned}$$

we can therefore restrict to the case where Q is non-degenerate: given a path $Z_t$ of central charges in ${\mathrm {Hom}}(\Lambda _{\mathbb {R}}, {\mathbb {C}})$ that are negative definite with respect to Q, we can choose extensions ${\overline{Z}}_t$ as in the Lemma that form a continuous path in ${\mathrm {Hom}}({\overline{\Lambda }}, {\mathbb {C}})$. If we can lift the latter path to a path of stability conditions ${\overline{\sigma }}_t = ({\overline{Z}}_t, {\mathcal {P}}_t)$ that satisfy the support property with respect to ${\overline{Q}}$, then $\sigma _t := (Z_t, {\mathcal {P}}_t)$ is a path of stability conditions satisfying the support property with respect to Q. The reduction to the case where Q has signature $(2, {\mathrm {rk}}\Lambda -2)$ works similarly:

Lemma 7.2

Assume that Q is non-degenerate and of signature $(p, {\mathrm {rk}}\Lambda -p)$ for $p \in \{ 0, 1\}$. Let ${\overline{\Lambda }} := \Lambda _{\mathbb {R}}\oplus {\mathbb {R}}$, and let ${\overline{Q}}$ be the extension given by ${\overline{Q}}(v, \alpha ) = Q(v) + \alpha ^2$ for $v \in \Lambda _{\mathbb {R}}$ and $\alpha \in {\mathbb {R}}$. Then any central charge Z on $\Lambda _{\mathbb {R}}$ whose kernel is negative definite with respect to Q extends to a central charge ${\overline{Z}}$ on ${\overline{\Lambda }}$ whose kernel is negative definite with respect to ${\overline{Q}}$.

Proof

We claim that there exists $z \in {\mathbb {C}}$ such that for all $v \in \Lambda _{\mathbb {R}}$ with $Z(v) = z$, we have $Q(v) < -1$. Indeed, let $K \subset \Lambda _{\mathbb {R}}$ be the kernel of Z, and let $K^\perp$ be its orthogonal complement. Then clearly we may assume $v \in K^\perp$. Since the restriction of Z to $K^\perp$ is injective, and since $K^\perp$ either has rank one, or has signature $(1, -1)$ with respect to Q, the claim is evident.

Using the claim, we can set ${{\overline{Z}}}(v, \alpha ) := Z(v) + \alpha z$. $\square$

This concludes the proof of Theorem 1.2.

8 Application

Proof of Corollary 1.3

Using the same arguments as in the previous section, we may assume that the Mukai pairing on $\Lambda$ has signature $(2, {\mathrm {rk}}\Lambda -2)$.

By Serre duality, any $\sigma$-stable object $E \in {\mathcal {D}}$ satisfies ${\mathrm {Hom}}(E, E[i]) = 0$ for $i < 0$ or $i > 3$ and ${\mathrm {Hom}}(E, E) = {\mathbb {C}}= {\mathrm {Hom}}(E, E[2])$; therefore, $(v(E), v(E)) \ge -2$. Moreover, Serre duality induces a non-degenerate symplectic form on ${\mathrm {Ext}}^1(E, E)$, and it has even dimension; thus v(E) is a root, or $(v(E), v(E)) \ge 0$.

Let $\sigma = (Z, {\mathcal {P}})$ be a stability condition with $Z \in {\mathcal {P}}_0({\mathcal {D}})$. As in Lemma 4.2 we may assume

$$\begin{aligned} (v, v) = \left|Z(v)\right|^2 - \left\| p(v)\right\| ^2, \end{aligned}$$

where $p :\Lambda _{\mathbb {R}}\rightarrow {\mathrm {Ker}}Z$ is the orthogonal projection onto the kernel of Z, and where $\left\| \cdot \right\|$ denotes the norm on ${\mathrm {Ker}}Z$ induced by the negative of the Mukai pairing. We claim that

$$\begin{aligned} C:= {\mathrm {inf}}\left\{ \left|Z(\delta )\right|\,:\,\delta \in \Lambda , (\delta , \delta ) = -2\right\} > 0. \end{aligned}$$

(3)

Indeed, if $\left|Z(\delta )\right| \le 1$, then $\left\| p(\delta )\right\| \le \sqrt{3}$; as $\left|Z(\cdot )\right| + \left\| p(\cdot )\right\|$ is a norm on $\Lambda _{\mathbb {R}}$, there are only finitely many integral classes satisfying both inequalities. Since $Z(\delta ) \ne 0$ by assumption, the claim follows.

Now set

$$\begin{aligned} Q(v) := (v, v) + \frac{2}{C^2} \left|Z(v)\right|^2. \end{aligned}$$

Clearly Q is negative definite on ${\mathrm {Ker}}Z$ and depends only on Z. Moreover, $Q(\delta ) \ge 0$ for all roots $\delta$, and $Q(v) \ge 0$ for all classes with $(v, v) \ge 0$; therefore, any stability condition $\sigma ' = (Z', {\mathcal {P}}')$ with $Z' \in {\mathcal {P}}_Z(Q)$ satisfies the support property with respect to Q.

Theorem 1.2 shows that the restriction of ${\mathcal {Z}}$ to the preimage of ${\mathcal {P}}_Z(Q)$ is a covering of ${\mathcal {P}}_Z(Q)$. Since the neighbourhoods ${\mathcal {P}}_Z(Q)$ cover ${\mathcal {P}}_0({\mathcal {D}})$, this completes the proof. $\square$