Bridgeland Stability Manifold

• Bridgeland stability manifold is the space of stability conditions on a triangulated category defined by a complex central charge and a slicing satisfying Harder–Narasimhan and support property constraints.
• Local models show that the manifold is a complex space locally isomorphic to Hom(Λ, ℂ), with deformation of central charges governing wall-crossing behavior.
• The manifold interconnects derived categories, moduli spaces, and autoequivalence symmetries, offering practical tools for analyzing birational geometry and stability variations.

Stability conditions on triangulated categories were introduced by Bridgeland as a “continuous” generalisation of $t$-structures, and the set of locally finite stability conditions on a triangulated category forms the Bridgeland stability manifold (Jorgensen et al., 2011). In its standard form, the manifold is the space of pairs consisting of a complex-valued central charge and a slicing, subject to Harder–Narasimhan and support-property constraints; when nonempty, it is locally modeled on a complex vector space of central charges, so that it carries the structure of a complex manifold (Bayer, 2016). The subject sits at the intersection of derived categories, wall-crossing, birational geometry, quiver mutation, moduli theory, and, in several examples, hyperplane arrangements, quadratic differentials, and Fourier–Mukai symmetries.

1. Definition and local model

A Bridgeland stability condition on a triangulated category $\mathcal D$ can be presented as a pair

$\sigma=(Z,\mathcal P),$

where $Z:K(\mathcal D)\to \mathbb C$ is the central charge and $\mathcal P(\phi)$ is a slicing by semistable phase. The slicing satisfies

$\mathcal P(\phi+1)=\mathcal P(\phi)[1],$

$$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$

and every nonzero object has a finite Harder–Narasimhan filtration by semistable factors of strictly decreasing phases. Compatibility with the central charge is encoded by

$$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$

Equivalently, one may specify a heart $\mathcal A=\mathcal P((0,1])$ of a bounded $t$-structure together with a stability function on $\mathcal D$0 satisfying Harder–Narasimhan (Barbieri, 2024).

For a fixed finite-rank lattice $\mathcal D$1, one considers stability conditions whose central charge factors through $\mathcal D$2. Bridgeland’s deformation theorem yields the fundamental local model: $\mathcal D$3 and this map is a local homeomorphism. Hence, whenever nonempty, $\mathcal D$4 is a complex manifold of dimension $\mathcal D$5 (Bayer, 2016). In the form recalled in representation-theoretic expositions, this means that the stability manifold is locally isomorphic to $\mathcal D$6, so local variation is controlled entirely by deformation of the central charge (Barbieri, 2024).

The manifold is not merely a receptacle for central charges. The paper on the co-stability manifold emphasizes that the Bridgeland stability manifold decomposes into pieces corresponding to bounded $\mathcal D$7-structures in the category (Jorgensen et al., 2011). This suggests that the manifold simultaneously records analytic data, via $\mathcal D$8, and homological data, via the bounded hearts underlying the slicing.

2. Support property and deformation theory

The support property is the condition that makes the local manifold structure effective. In the quadratic-form formulation, a pre-stability condition $\mathcal D$9 relative to $\sigma=(Z,\mathcal P),$0 satisfies the support property with respect to a quadratic form $\sigma=(Z,\mathcal P),$1 on $\sigma=(Z,\mathcal P),$2 if $\sigma=(Z,\mathcal P),$3 is negative definite on $\sigma=(Z,\mathcal P),$4 and

$\sigma=(Z,\mathcal P),$5

for every semistable object $\sigma=(Z,\mathcal P),$6. The same paper formulates the support property as the key “Bogomolov-Gieseker type” inequality preserved under wall-crossing (Bayer, 2016).

The associated deformation theorem is strong. If $\sigma=(Z,\mathcal P),$7 satisfies the support property with respect to $\sigma=(Z,\mathcal P),$8, then there exists an open neighborhood $\sigma=(Z,\mathcal P),$9 such that the central-charge projection

$Z:K(\mathcal D)\to \mathbb C$0

restricts to a covering map onto the region

$Z:K(\mathcal D)\to \mathbb C$1

In particular, any path $Z:K(\mathcal D)\to \mathbb C$2 of central charges with $Z:K(\mathcal D)\to \mathbb C$3 lifts uniquely to a path of stability conditions once the initial point is fixed, and every nearby stability condition still satisfies the support property with respect to the same $Z:K(\mathcal D)\to \mathbb C$4 (Bayer, 2016).

The topology on slicings is measured by Bridgeland’s generalized metric

$Z:K(\mathcal D)\to \mathbb C$5

A useful equivalent form recalled in the deformation proof reduces estimates to semistable objects. Quantitatively, if

$Z:K(\mathcal D)\to \mathbb C$6

with $Z:K(\mathcal D)\to \mathbb C$7 the $Z:K(\mathcal D)\to \mathbb C$8-orthogonal projection and $Z:K(\mathcal D)\to \mathbb C$9, then $\mathcal P(\phi)$0 yields a unique nearby stability condition, and the resulting slicing satisfies

$\mathcal P(\phi)$1

This is the analytic mechanism behind the local manifold structure (Bayer, 2016).

A common misconception is that the manifold property is a purely formal consequence of the definition. The deformation theorem shows that the support property is the decisive hypothesis: without it, local lifting of the central charge need not behave as required. The modern literature repeatedly uses this theorem as the foundation for wall-crossing, moduli-space variation, and autoequivalence actions.

3. Hearts, walls, and symmetries

For a fixed heart $\mathcal P(\phi)$2, the corresponding chamber in the stability manifold can be very explicit. If $\mathcal P(\phi)$3 is finite with simple objects $\mathcal P(\phi)$4, then any choice

$\mathcal P(\phi)$5

gives a stability condition, and

$\mathcal P(\phi)$6

Walls between such chambers are described by tilting at torsion pairs; in the simple-tilt situation, the codimension-one locus where a simple object has phase $\mathcal P(\phi)$7 is precisely the common boundary of adjacent hearts related by a backward tilt (Barbieri, 2024). This chamber structure ties the geometry of $\mathcal P(\phi)$8 to the exchange graph of hearts.

The stability manifold carries two standard symmetries. Autoequivalences act by

$\mathcal P(\phi)$9

and there is a right action of $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$0 which linearly transforms the central charge and shifts phases. In many constructions this action does not change the semistable objects themselves, only their phase bookkeeping and the complex linear coordinates (Barbieri, 2024). Several geometric papers use this action to normalize stability conditions or to compare an autoequivalence image with a standard geometric chamber.

Wall-crossing is described by equality of slopes. In threefold problems, a numerical wall for a class $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$1 is the locus where two classes have the same stability slope, and an actual wall is one where there exists an exact sequence in the relevant heart

$\mathcal P(\phi+1)=\mathcal P(\phi)[1],$2

with $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$3 of class $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$4 and $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$5 semistable, so semistability changes across the wall (Schmidt, 2015). On surfaces, wall geometry can be made very explicit inside slices $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$6; for $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$7, the walls in the plane $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$8 are nested semicircles centered on the $\mathcal P(\phi+1)=\mathcal P(\phi)[1],$9-axis, and the wall equation on the $$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$0 plane is

$$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$1

The paper on line bundles emphasizes that actual destabilizing walls are the left hyperbolas in this conic classification (Arcara et al., 2014).

For threefolds, the geometry is more intricate. Tilt-stability walls are semicircles with center on the $$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$2-axis or vertical rays, and distinct tilt walls cannot intersect unless they coincide. By contrast, Bridgeland walls involve $$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$3, so they become real degree-$$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$4 curves and can intersect. One of the main tools in threefold examples is therefore comparison between Bridgeland walls and tilt walls near the hyperbola $$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$5 (Schmidt, 2015).

4. Computed examples and global topology

Several categories admit explicit descriptions of all or large parts of their stability manifolds. For a smooth projective curve $$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$6 of genus $$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$7, the known description used in the holomorphic-triples paper is

$$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$8

This serves as a building block for more complicated categories (Martínez-Romero et al., 2019). In the derived category of holomorphic triples over a genus $$\phi_1>\phi_2 \implies \operatorname{Hom}(A_1,A_2)=0 \quad (A_j\in \mathcal P(\phi_j)),$$9 curve,

$$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$0

the numerical Grothendieck group has rank $$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$1, and the full stability manifold is described as

$$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$2

a connected $$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$3-dimensional complex manifold (Martínez-Romero et al., 2019).

The topology can also be completely controlled in representation-theoretic settings. For a contraction algebra $$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$4 associated to a $$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$5-fold flop, the main theorem states that

$$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$6

is a regular covering map, where $$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$7 is the complexification of a simplicial hyperplane arrangement, and since the total space is contractible, it is the universal cover (August et al., 2019). In special cases this arrangement is an ADE root system, so the stability manifold is literally the universal cover of an arrangement complement governed by braid-group combinatorics.

In type $$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$8 quiver-with-potential categories, after quotienting the preferred component by spherical twists,

$$0\neq E\in \mathcal P(\phi)\quad\Longrightarrow\quad Z([E])=m(E)e^{i\pi\phi},\qquad m(E)>0.$$9

there is a natural local biholomorphism

$\mathcal A=\mathcal P((0,1])$0

to the type $\mathcal A=\mathcal P((0,1])$1 cluster Poisson variety. The construction passes through framed quadratic differentials and the ODE

$\mathcal A=\mathcal P((0,1])$2

whose asymptotic monodromy data provide the cluster coordinates (Allegretti, 2017). This identifies the stability space, locally, with a cluster variety endowed with the same mutation and chamber combinatorics.

For smooth projective varieties with finite Albanese morphism, the global structure can become rigid. Every numerical stability condition is geometric, meaning that all skyscraper sheaves are stable of a common phase; for irregular surfaces and for abelian threefolds with Picard rank one, the stability manifolds are connected and contractible (Fu et al., 2021). The same paper proves for polarized abelian threefolds that every stability condition lies in the Bayer–Macrì–Stellari component, so in Picard rank one the entire manifold is explicitly controlled (Fu et al., 2021).

A further geometric comparison appears for Kummer surfaces. The maximal connected component $\mathcal A=\mathcal P((0,1])$3 of an abelian surface embeds topologically into the distinguished connected component $\mathcal A=\mathcal P((0,1])$4 of the associated Kummer surface, with the map induced on the base of central charges by a lattice embedding compatible with the orbifold construction (Engenhorst, 2012).

5. Moduli spaces, wall-crossing, and geometric applications

The stability manifold is used to organize actual birational transformations of moduli spaces. On a smooth projective surface $\mathcal A=\mathcal P((0,1])$5 containing a smooth rational curve

$\mathcal A=\mathcal P((0,1])$6

a family of stability conditions is constructed on a wall of the geometric chamber of $\mathcal A=\mathcal P((0,1])$7. Along that wall, for $\mathcal A=\mathcal P((0,1])$8, the skyscraper sheaf destabilizes through

$\mathcal A=\mathcal P((0,1])$9

and after wall-crossing the moduli space of objects of class $t$0 becomes

$t$1

with $t$2 embedded in $t$3 as a rational normal curve (Tramel et al., 2017). The paper explicitly verifies the support property and shows that the new stability conditions lie on a wall of the geometric chamber.

On $t$4, Bridgeland stability on threefolds is accessed through tilt stability. For the twisted cubic class

$t$5

the full wall-crossing along a suitable path produces the sequence

$t$6

where $t$7 is a blow-up of $t$8 in a smooth locus and $t$9 is a $\mathcal D$00-bundle over $\mathcal D$01 (Schmidt, 2015). This re-proves the classical description of the Hilbert scheme of twisted cubics via derived wall-crossing.

The geometry of the manifold can also detect classical Brill–Noether theory. For the bounded derived category of coherent systems on a smooth projective curve, the paper describes an open locus

$\mathcal D$02

and on the tilting side the quotient by $\mathcal D$03 is exactly

$\mathcal D$04

The wall-and-chamber structure in the $\mathcal D$05-plane is controlled by the Brill–Noether function $\mathcal D$06, and classical $\mathcal D$07-stability of coherent systems reappears as a large-volume limit (Feyzbakhsh et al., 3 Nov 2025).

On abelian threefolds, the main achievement of the Donaldson–Thomas paper is that the double-tilt stability conditions satisfy the full support property with respect to the full numerical lattice, so the stability manifold is non-empty: $\mathcal D$08 This is then used to establish a Gieseker chamber and to prove invariance of reduced Donaldson–Thomas invariants under derived autoequivalences, up to explicit wall-crossing terms (Oberdieck et al., 2018).

The manifold also interacts with combinatorics and autoequivalence dynamics. For a Weierstraß elliptic surface, solving

$\mathcal D$09

reduces to

$\mathcal D$10

whose formal power-series solution is governed by Catalan numbers: $\mathcal D$11 with radius of convergence

$\mathcal D$12

This yields an explicit quantitative region in which the Fourier–Mukai transform preserves geometric form up to $\mathcal D$13 (Lo et al., 2020).

A family-theoretic extension is provided by the theory of stability conditions in families. For a flat projective family $\mathcal D$14, the paper develops $\mathcal D$15 as a complex manifold locally modeled on $\mathcal D$16, with openness of stability, semistable reduction over Dedekind bases, a uniform support property, and boundedness of semistable objects. This framework underlies relative moduli problems, deformation invariance of Donaldson–Thomas invariants, and applications to cubic fourfolds and Kuznetsov components (Bayer et al., 2019).

6. Empty loci, compactifications, and reduced forms

The stability manifold is not automatically nonempty. A standard example is

$\mathcal D$17

whose Bridgeland stability manifold is empty (Jorgensen et al., 2011). This is one reason for introducing co-stability conditions, a mirror theory based on co-slicings and bounded co-$\mathcal D$18-structures. For a triangulated category $\mathcal D$19 satisfying the stated finiteness hypotheses, the space $\mathcal D$20 of co-stability conditions satisfying condition $\mathcal D$21 is a topological manifold of dimension $\mathcal D$22, where $\mathcal D$23; for $\mathcal D$24, the co-stability manifold is $\mathcal D$25 (Jorgensen et al., 2011). A plausible implication is that the Bridgeland picture is only one half of a broader stability geometry on triangulated categories.

Boundary behavior motivates compactification. The local-compactification paper observes that the standard Bridgeland metric does not see sequences with massless objects as finite-distance boundary points: such sequences are infinitely far away, and the stability manifold is already complete in that metric. To remedy this, it equips $\mathcal D$26 with the pullback Riemannian metric $\mathcal D$27 from the central-charge space, forms the metric completion

$\mathcal D$28

and associates to a Cauchy sequence $\mathcal D$29 the thick subcategory

$\mathcal D$30

of asymptotically massless objects. The completion injects into

$\mathcal D$31

via

$\mathcal D$32

so a boundary point is encoded by a massless thick subcategory and a genuine stability condition on the quotient category (Bolognese, 2020).

A related partial compactification allows semistable objects of zero mass but still assigns them a phase. These are lax stability conditions. The massless subcategory $\mathcal D$33 is thick, the induced massive part descends to a genuine stability condition on the Verdier quotient $\mathcal D$34, and after quotienting lax stability conditions by deformation-equivalence inside a fixed charge fiber one obtains a stratified partial compactification by strata

$\mathcal D$35

with

$\mathcal D$36

under the stated support-propagation hypotheses (Broomhead et al., 2022). The paper on $\mathcal D$37 shows that this compactification is selective: not every degeneration of charges appears, only those compatible with convergent slicings and the support condition (Broomhead et al., 2022).

A different simplification is the real reduction

$\mathcal D$38

where

$\mathcal D$39

The quotient $\mathcal D$40 is a real, possibly non-Hausdorff, manifold of real dimension $\mathcal D$41, and the local coordinate map

$\mathcal D$42

is a local homeomorphism (Li, 27 Jun 2025). This reduced space preserves wall-and-chamber structure, and the full stability manifold can be reconstructed from $\mathcal D$43, the order relation $\mathcal D$44, and the real part of the central charge. This suggests a systematic separation between the “real core” of the stability geometry and the extra data needed to recover the full complex manifold (Li, 27 Jun 2025).