VECTOR BUNDLES ON ALGEBRAIC SURFACES
KŌTA YOSHIOKA

Contents
0. Introduction
0.1. Stability
0.2. Classical results
0.3. Recent development
1. Construction of 3 fold flops
1.1. Idea of the construcion
1.2. Perverse coherent sheaves
2. Bridgeland’s stability condition
2.1. Definition of stability condition
2.2. Group actions on Stab(X)2.3. Examples of stability conditions
2.4. Birational geometry of MH (v)2.5. Abelian surface with NS(X) = ZH and (H 2 ) = 2
2.6. Related results and some problems
References

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0. Introduction
In this survey, we shall explain some properties of moduli spaces of stablesheaves on smooth projective surfaces. We first explain classical results on themoduli spaces briefly. Then we explain recent development in detail. Thus weshall explain geometry associated to the derived category of coherent sheaves,which was started by Mukai and developed by Orlov, Bridgeland and other people.
For other aspects of derived category of coherent sheaves, we recommend excellent
articles [20], [38].
Due to lack of the author’s ability and also enough time to write, this article is
not written well. Moreover there will be many misunderstanding on the results inparticular references. So if you are interested in these topic, it is better to checkthe detail as students do before seminar.
0.1. Stability. Let X be a smooth projective surface over C. For a coherent
sheaf E on X, v(E) denotes a topological invariant of E. Typical topologicalinvariants are(i) the Chern character v(E) = ch(E) ∈ H 2∗ (X, Q),
2010 Mathematics Subject Classification. 14D20.
The author is supported by the Grant-in-aid for Scientific Research (No. 26287007), JSPS.
1√(ii) Mukai vector v(E) = ch(E) tdX ∈ H 2∗ (X, Q), if KX is numericallytrivial.
(iii) the class in the numerical Grothendieck group
K(X)num = K(X)/ ker(ch),where ch : K(X) → H 2∗ (X, Q) is the Chern character map or(iv) v(E) = (rk E, c1 (E), χ(E)) ∈ Z ⊕ NS(X) ⊕ Z.
Definition 0.1. Let H be an ample divisor on X and β ∈ NS(X)Q . A coherent
sheaf E is β-twisted semi-stable if E is torsion free andχ(F (−β + nH))χ(E(−β + nH))
(0.1)≤
(n ≫ 0)
rk F
rk E
for all non-trivial subsheaf F of E. If β = 0, then β-semi-stability is nothing but
the semi-stability of Gieseker.
Remark 0.2. Roughly speaking, torsion freeness means locally free except finitely
many points of X, since X is smooth of dimension 2. Although we are mainlyinterested in locally free sheaves, in order to get a compact moduli space, we needto add torsion free sheaves in the boundary.
Theorem 0.3 (Gieseker [16], Matsuki-Wentworth [27]). There is a coarse moduli
scheme MHβ (v) of β-twisted semi-stable sheaves with topological invariant v. It isa projective scheme.
If β = 0, then we denote the moduli space by MH (v). If H is general in
Amp(X), then MHβ (v) is independent of the choice of β, and hence MHβ (v) =MH (v). β-twisted semi-stability is a generalization of a more restrictive notion
slope stability, which is defined by looking the coeﬃcient of n in (0.1). The
famous Kobayashi-Hitchin correspondence connects the algebro-geometric notion
with the diﬀerential geometric notion:(0.2) slope stable vector bundles ⇔ irreducible Hermite-Einstein connections.
For smooth 4-manifolds, Donaldson constructed a very powerful invariant called
Donaldson invariant by using the moduli of Hermite-Einstein connections. Thenthe Kobayashi-Hitchin correspondence says that the computations of Donaldson
invariants of Kähler surfaces can be reduced to study MH (v). By this reason, thestructure of MH (v) was extensively studied until a more useful invariant called
Seiberg-Witten invariant appeared in 1995.
0.2. Classical results. We first pick up some general results on the structure ofMH (v) which were obtained until 1995. There are many examples of smooth andirreducible moduli spaces if X is not of general type. However MH (v) is singularin general, and may be non-reduced and reducible. On the other hand, if c2 ≫ 0,then the singular locus of MH (v) is higher codimension, MH (v) is irreducible andlocally complete intersection [34], [17]. In particular, MH (v) is a normal variety.
If the geometric genus pg is positive, then Jun Li [Li] and O’Grady proved that
MH (v) is of general type under mild conditions. In order to prove the claim, asection of H 0 (X, KX ) is used to construct many canonical sections on a resolutionof MH (v). Hence the proof does not work if pg = 0. Indeed there is no general
results of the moduli spaces for the case pg = 0, as far as I know.
2

Problem 0.4. Study MH (v) for a surface of general type with pg = 0.
For special surfaces such as P2 , K3 surfaces, abelian surfaces, or elliptic surfaces, there are many works on the structure of MH (v). They were started in early
80’s and continued after 1995. For these cases, there are many examples of smoothor normal moduli spaces, and we can expect beautiful properties of the moduli
spaces. As examples, we pick (1) Drezet and Le Potier’s result for P2 [15] and(2) Mukai’s fundamental results for abelian and K3 surfaces [31],[32]. (1) In [15],they described the condition for the existence of stable sheaves. Although they
only treated the case of a projective plane, their method1 are powerful enough tobe applied to other surfaces. We mention in section 2.6 recent developement on
their work. For Mukai’s works and related results, we shall explain them in next
paragraph. For other results, I would like to recommend to see Huybrechts andLehn’s book and its references [21].
Assume that X is an abelian surface or a K3 surface. Then we usually use
Mukai vector v(E) ∈ H 2∗ (X, Z)alg = Z ⊕ NS(X) ⊕ ZϱX , where ϱX is the fundamental class of X. We have an integral bilinear form ⟨ , ⟩ on H 2∗ (X, Z)alg :(0.3)⟨(x0 , x1 , x2 ), (x′0 , x′1 , x′2 )⟩ = (x1 , x′1 ) − x0 x2 − x2 x′0 ∈ Z.

If v = (r, ξ, a) is primitive, i.e., gcd(r, ξ, a) = 1 and H is a general ample divisor,
then Mukai showed MH (v) is a holomorphic symplectic manifold of dimension⟨v 2 ⟩ + 2. Moreover the deformation class of MH (v) is determined by ⟨v 2 ⟩ [42].
The main idea to determine the deformation class is to use the symmetry of X.
For abelian surfaces and K3 surfaces, we have big symmetries to determine thedeformation classes. More concretely, we use the following symmetry:(i) Deformation of the pair (X, H): For a polarized deformation of X, wehave a monodromy representation on H 2 (X, Z). By this action, we canchange ξ.
(ii) Equivalences of the derived category (Fourier-Mukai transforms): By
these actions, we can change r.
Let us explain (ii) more. A non-trivial example of equivalences Φ : D(X) →D(X ′ ) of derived categories was first constructed by Mukai for abelian varieties[30]. Later similar equivalences were constructed for other varieties, e.g., a K3
surface [36], [10]. Here non-trivial implies the category of coherent sheaves are
not preserved under Φ and the rank of objects change. Although these kind ofequivalences are quite useful, technically it is not so easy to deal with: Indeed
we need to find deformations (X ′ , H ′ ) of (X, H) and Fourier-Mukai transforms
Φ : D(X ′ ) → D(X ′′ ) which induce isomorphisms MH ′ (v ′ ) ∼= MH ′′ (v ′′ ). Thiscan be achieved by solving some indeterminate equations on Mukai vectors. It
was elementary but not interesting, since there was no systematic treatment, andwhich may make the topic of stable sheaves on surfaces unattractive.
0.3. Recent development. A method to improve the situation is to appreciate
the symmetries of the derived category and apply them to classical problems.
For this purpose, we may need to consider complexes and their moduli spaces.
1The author imagines that they were influenced by the work of Atiyah and Bott [3], andHarder and Narasimhan to compute the Betti numbers of the moduli spaces on curves.
3However derived categories of coherent sheaves themselves are too big to control, it is desirable to introduce good subcategories which is easier to control andcapture rich geometric and algebraic structures. Bridgeland’s theory of stability
conditions is very suitable for this purpose. Before explaining Bridgeland’s stability conditions and their applications, let us start with related results before
Bridgeland’s theory of stability conditions appeared.
First of all, we note that Beilinson constructed a derived equivalence(0.4)
D(Pn ) ∼= D(A),
where A is a non-commutative algebra and D(A) is the derived category of Amodules. In the theory of vector bundles, it is famous as Beilinson’s spectral
sequence, which was a very useful tool to study vector bundles on projective
spaces. As we mention in section 2.6, instead of Beilinson’s spectral sequence,the equivalence (0.4) itself plays important roles recently.
Similar equivalences are also found for some rational varieties by Russian mathematicians. Moreover many interesting results on the derived categories wereproved. For the relation with this survey, Bondal and Orlov’s contributions([8],[9]) are noteworthy. They proved that the underlying manifold X can berecovered from D(X) if KX or −KX is ample. They also studied the relation of
3-fold flops with the derived categories, and conjectured that 3 fold flops preserves
the derived categories. Bridgeland [11] solved this conjecture by using the geometry of derived categories. Thus he constructed 3 fold flops as moduli spaces ofsome objects in D(X) and used the technique of Fourier-Mukai transforms which
was developped mainly by Bridgeland. Since this work is the first well-known
application of moduli problem of objects in D(X), we shall explain the construction in detail. Derived category of coherent sheaves also appears in Homologicalmirror symmetry, which also influenced recent development.
1. Construction of 3 fold flops
1.1. Idea of the construcion. Let us explain the idea of Bridgeland construction of 3 fold flops. Let φ : X → Z be a flopping contraction of a smooth 3-fold.
Let X ′ be a moduli space of stable 0-dimensional sheaves E with v(E) = v(Cx )(x ∈ X). Then E ∼= Cx for some x ∈ X, and we get X ′ ∼= X. By perturbing thestability condition, we would like to get a diﬀerent moduli space giving the flop of
X. For simplicity, we first assume that φ is a contraction of (−1, −1)-curve, that
is, a contraction of a smooth rational curve C with the normal bundle OC (−1)⊕2 .
Since Cx is an irreducible object of Coh(X), there is no choice to modify thestability. For x ∈ C, we have an exact sequence in Coh(X).
(1.1)

0 → OC (−1) → OC → Cx → 0

In the derived category, it can be understand as an exact triangle(1.2)

0 → OC → Cx → OC (−1)[1] → OC [1]

If there is an abelian category C such that OC , OC (−1)[1] ∈ C, then we may
have a stability condition such that Cx is not stable. Bridgeland introduceda modification of Coh(X) called a category of perverse coherent sheaves, andconstructed the flop as a moduli space of stable perverse coherent sheaves. As
the name indicates, this notion is influenced by Beilinson, Bernstein and Deligne’s
4

article [7]. In their paper, they introduced the notion of t-structure, which has
been useful in representation theory and the theory of non-commutative algebra.
By this work of Bridgeland, the importance of t-structure may be noticed bygeneral algebraic geometer.
In the remaining of this section, let us explain a more detail of the constructionof the flop. So it is possible to skip to the next section.
1.2. Perverse coherent sheaves. Let φ : X → Z be a flopping contraction ofa smooth 3-fold X. Then we haveRφ∗ (OX ) = OZ , dim φ−1 (z) ≤ 1 for any z ∈ Z.
For simplicity, we assume that there is one singular point p of Z. Let C1 , ..., Cn
be the irreducible components of the exceptional locus φ−1 (p). Ci are smooth
rational curves. We set
T :={E ∈ Coh(X) | Hom(E, OCi (−1)) = 0},(1.3)
F :={E ∈ Coh(X) | E is generated by OCi (ai ), ai ≤ −1}.
Then (T, F ) is a torsion pair of Coh(X), that is, there is a unique exact sequence
0 → E1 → E → E2 → 0(1.4)

such that E1 ∈ T and E2 ∈ F for any E ∈ Coh(X). Let −1 Per(X/Z) be thetilting: For E ∈ −1 Per(X/Z),
{
}
i
H
(E)
=
0,
i
=
̸
−1,
0,
(1.5) −1 Per(X/Z) = ⟨T, F [1]⟩ = E ∈ D(X).
H −1 (E) ∈ F, H 0 (E) ∈ T
Thus we have an exact sequence in −1 Per(X/Z):(1.6)

0 → H −1 (E)[1] → E → H 0 (E) → 0.

There is a locally free sheaf G such that Rφ∗ (G∨ ⊗ E) ∈ Coh(Z) for E ∈−1
Per(X/Z) and Rφ∗ (G∨ ⊗ E) = 0 if and only if E = 0. G is called a local
projective generator of −1 Per(X/Z) and Rφ∗ (G∨ ⊗ •) induces a Morita equivalence [39]
−1

(1.7)

Per(X/Z) → CohA (Z),where A = φ∗ (G∨ ⊗ G) is a sheaf of OZ -algebras. Then we define the dimensionof E ∈ −1 Per(X/Z) by the dimension of Rφ∗ (G∨ ⊗ E). X ′ is constructed as amoduli space of A-modules on Z.
We choose a divisor D on X. For 0-dimensional objects of −1 Per(X/Z), we
have a notion of stability.
Definition 1.1. A 0-dimensional object E is D-semi-stable, if(1.8)

χ(G, E1 (−D))χ(G, E(−D))≤χ(G, E1 )χ(G, E)for all subobject E1 of E.
Then there is a moduli space M D (v) of D-twisted semi-stable objects E of−1
Per(X/Z), where v is the Chern character of E [43, Prop. 1.5.4] (the assumption of dim X in [43, Prop. 1.5.4] is not needed).
5

We note that χ(G, E(−D)) = χ(G, E) − (ch2 (E), D). Hence (1.8) is equivalentto−(ch2 , D)−(ch2 (E), D)≤.
χ(G, E1 )χ(G, E)(1.9)

Let ϱX ∈ H 3 (X) be the fundamental class of X. Then ϱX = v(Cx ). Forv(E) = ϱX , (1.9) implies E is D-semi-stable iﬀ (ch2 (F ), D) ≥ 0 for all subobject
F of E.
From now on, we assume that −D is φ-ample. Then
Lemma 1.2. Cx is not D-twisted semi-stable for x ∈ φ−1 (p).
Proof. We have an exact sequence in Coh(X)(1.10)

0 → c → φ∗ (φ∗ (Cx )) → Cx → 0,

where Rφ∗ (c) = 0. We note that φ∗ (φ∗ (Cx )) = Oφ−1 (p) and c is generated byOCi (−1). Since ch2 (Oφ−1 (p) ) = −φ−1 (p) and ch2 (OCi ) = −Ci ,(1.11)(ch2 (Oφ−1 (p) ), D) > 0, (ch2 (OCi [1]), D) < 0□

for all i. Hence the claim holds.

In the paper [11], Bridgeland constructed the moduli space of perverse coherent
subsheaves I of OX with v(OX /I) = v(Cx ). As we shall see below, these twomoduli spaces are the same.
Lemma 1.3. Let E be a perverse coherent sheaf with v(E) = v(Cx ). E is Dsemi-stable if and only if there is a surjective morphism f : OX → E.
Proof. Assume that E is a quotient of OX . Let I be the kernel. Since
Hom(OX , OCi (−1)[1]) = Hom(I, OCi (−1)) = 0,
E satisfies Hom(E, OCi (−1)[1]) = 0. Hence if v(E) = v(Cx ), then it is D-twisted
stable.
Conversely for a D-stable object E with v(E) = v(Cx ), χ(E) = 1 implies there
is a morphism f : OX → E. Since OCi [1] and Oφ−1 (p) are the irreducible objects
supported on φ−1 (p), we have∑
im f =
ni OCi [1] + nOφ−1 (p) , (ni , n ≥ 0)i

(1.12)
coker f =∑mi OCi [1] + mOφ−1 (p) , (mi , m ≥ 0).

i

Since 1 = χ(E) = χ(im f ) + χ(coker f ), we get (n, m) = (1, 0) or (0, 1). Since
Hom(im f, OCi (−1)[1]) = 0 for all i, (n, m) = (1, 0). By the D-stability of E,
coker f must be zero. Thus f is surjective.
□
We set X ′ := M D (ϱX ).
Remark 1.4. There is a universal family E on X ′ × X. Since L := pX ′ ∗ (E) is aline bundle on X ′ , replacing E by E ⊗ p∗X ′ (L∨ ), we have a surjective morphism
OX ′ ×X → E in −1 Per(X ′ × X/X ′ × Z). Thus E is a quotient of OX ′ ×X .
6

Since π∗ (E) (E ∈ X ′ ) is a structure sheaf of a point of Z, we have the following
diagram:X′

X↘

(1.13)↙
Z

Then Bridgeland proved that the universal family E induces an equivalence D(X) ∼=′′D(X ) and also proved that X → Z is the flop of X → Z.
2. Bridgeland’s stability condition
2.1. Definition of stability condition. Bridgeland introduced a notion of stability for objects of D(X). Moreover he showed that the space Stab(X) of stability
conditions has a structure of a complex manifold. In this section, we shall briefly
explain stability condition ([12], [13]). We start with the definition.
Definition 2.1. A stability condition σ = (Zσ , Pσ ) on D(X) consists of a grouphomomorphism Zσ : K(X) → C and full additive subcategories Pσ (ϕ) ⊂ D(X)
for all ϕ ∈ R satisfying the following conditions:(i) For E ∈ Pσ (ϕ) \ {0}, we have√Zσ (E) = m(E)eπ −1ϕ(2.1)

with some m(E) ∈ R>0 .
(ii) Pσ (ϕ + 1) = Pσ (ϕ)[1] for all ϕ ∈ R.
(iii) If ϕ1 > ϕ2 and Ei ∈ Pσ (ϕi ) (i = 1, 2), then HomD(X) (E1 , E2 ) = 0.
(iv) For any E ∈ D(X) \ {0}. we have the following collection of triangles
0 = EcG0

GG
GG
GG[1] GG

A1

/ E1
`AA
}
AA
}}
AA
}
}
}
[1] AA
~}}

A2

/ E2
}
}}
}}
}
~}}

/ · · · En−1
dHH
HH
HH
H
[1] HHAn/ En = E
vv
vv
v
vv
vz v

such that Ai ∈ Pσ (ϕi ) with ϕ1 > ϕ2 > · · · > ϕn .
A non-zero object E ∈ Pσ (ϕ) is called σ-semi-stable of phase ϕ.
Definition 2.2. M̃σ (v) denotes the moduli space of σ-semi-stable objects E with
v(E) = v in a suitable sense (e.g. a scheme, an algebraic space or a stack). Since
the phase is not determined by v, shift functor [2] acts on M̃σ (v). We denote thequotient by Mσ (v).
We explain another formulation of the stability condition, which resemble
Gieseker semi-stability: For a stability condition σ, let Aσ := Pσ (0, 1] denotes
the extension closed full subcategory of D(X) generated by E ∈ Pσ (ϕ) withϕ ∈ (0, 1]. Then Aσ is an abelian category and the phase ϕ(E) ∈ (0, 1] is defined for a non-zero object E ∈ Aσ by (2.1). Moreover E ∈ Aσ is σ-semi-stableiﬀ ϕ(F ) ≤ ϕ(E) for any subobject F of E in Aσ . Conversely for an abeliansubcategory A (which is the heart of a t-structure) and a group homomorphism
Z : K(X) → C such that√

Z(A \ {0}) ⊂ {R>0 eπ −1ϕ | ϕ ∈ (0, 1]} = H ∪ R<07and satisfying some conditions like the Harder-Narasimhan property, we have astability condition σ = (Zσ , Pσ ) with Zσ = Z and A = Pσ (0, 1]. Therefore wehave one to one correspondence:(2.2)

σ = (Zσ , Pσ ) ←→ (Zσ , Aσ ).

As in the β-twisted stability in Definition 0.1, the phase ϕ plays the role of aslope function on the abelian category Aσ . Thus we have the correspondence(Aσ , ϕ(•)) ←→ (Coh(X), χ(•(−β+nH))).
rk(•)
Remark 2.3. Gieseker semi-stability is not a stability condition in the sense of
Definition 0.1 if dim X ≥ 2.
The space Stab(X) of stability conditions has a structure of complex manifold.
For a stability condition σ, we have Π(σ) ∈ H ∗ (X, C)alg such that Zσ (•) =⟨Π(σ), •⟩. Then Π : Stab(X) → H ∗ (X, C)alg is a covering map over the image.
We require that im Π is an open subset from now on.2
Proposition 2.4. We fix a Mukai vector v. There is wall/chamber structure on
Stab(X) and Mσ (v) is constant on each chamber.
2.2. Group actions on Stab(X). On the space of stability conditions, there+f+is a natural right action of√ the universal+ cover GL (2, R) of GL (2, R). By an
identification C = R + R −1, g ∈ GL (2, R) acts on Zσ : K(X) → C by thecomposition g −1 ◦ Zσ . In order to get an action on the phase function ϕ, we needto take the universal cover of GL+ (2, R). Here we only explain the action of af + (2, R):
subgroup C ⊂ GL
f + (2, R)
C ⊂ GL↓↓C∗ ⊂ GL+ (2, R).

(2.3)
For λ ∈ C,
(2.4)

√

λ(σ) = (e−π −1λ Zσ , P ′ ), P ′ (ϕ) = Pσ (ϕ + Re(λ)).

Remark 2.5. The second formulation of the stability condition may be familiar tof + (2, R) is notwhom working on vector bundles. However the description of GL
easier for this formulation. Indeed it seems there is no simple relation of abeliancategories P ′ (0, 1] and Pσ (0, 1] unless Re(λ) ∈ Z, where λ ∈ C.
There is also an action of Aut(D(X)). For an equivalence Φ : D(X) → D(X ′ )and a stability condition σ = (Zσ , Pσ ) on X, we define a stability condition Φ(σ)
on X ′ by(2.5)

Φ(σ) := (Zσ ◦ Φ−1 , PΦ(σ) ), PΦ(σ) (ϕ) = Φ(Pσ (ϕ)).

By this property (and the first formulation of stability condition), Φ induces anisomorphism(2.6)

Φ : Mσ (v) → MΦ(σ) (Φ(v)).

2Thus we require the support property.

8Assume that X is a K3 surface or an abelian surface, so that there are enough
Fourier-Mukai transforms. The stability conditions of Matsuki-Wentworth (classical stability) are parameterized by(2.7)(β, H) ∈ NS(X)Q × Amp(X)Q .

We would like to extend the notion of stability which behaves well in the derived
category. However we don’t want to generalize the notion of stability blindly,
since it makes hard to analyse. In order to minimize our consideration, we would
like to impose the following conditions for our parameter space S of stability
conditions:(a) For each Mukai vector v, there is a wall/chamber structure on S and thestability is constant on each chamber.
(b) For a general (β, H), there is σ ∈ Stab(X) such that MHβ (v) = Mσ (v).
(c) Stability is preserved under any Fourier-Mukai transformΦ : D(X) → D(X ′ ).
Thus for σ ∈ S, there is Φ(σ) ∈ S such that E ∈ D(X) is σ-semi-stable
if and only if Φ(E) is Φ(σ)-semi-stable.
(d) For each chamber C, there is a Fourier-Mukai transform Φ which induces′′
an isomorphism Mσ (v) → MHβ ′ (w), where σ ∈ C and MHβ ′ (w) is a moduli
space of β ′ -twisted semi-stable sheaves.
Theorem 2.6. Bridgeland stability conditions satisfy (a)–(d).
For (a), (b), see [13]. (c) is (2.6). (d) is in [5] and [26].
Remark 2.7. For other surfaces, we cannot expect property (d) in general.
2.3. Examples of stability conditions. Let us explain examples of stability
conditions in [13]. Let X be a K3 surface or an abelian surface. Assume that(β, ω) ∈ NS(X)R × Amp(X)R . For E ∈ K(X), we set√Z(β,ω) (E) :=⟨eβ+ −1ω , v(E)⟩.
Let A(β,ω) be the tilt of Coh(X) with respect to the torsion pair (T(β,ω) , F(β,ω) )defined by(i) T(β,ω) is generated by β-twisted stable sheaves with Z(β,ω) (E) ∈ H ∪ R<0 .
(ii) F(β,ω) is generated by β-twisted stable sheaves with −Z(β,ω) (E) ∈ H∪R<0 .
A(β,ω) is the abelian category in [13] and it depends only on β and the ray Q>0 ω.
Then the pair σ(β,ω) = (Z(β,ω) , A(β,ω) ) satisfies the requirement of stabilityf + (2, R) and Aut(D(X)), a generalconditions on D(X)3 [13]. By the action of GL
stability condition is represented by σ(β,ω) . Moreover if X is an abelian surface,thenf + (2, R) ∼(2.8)
Stab(X)/GL= NS(X)R × Amp(X)R .
Every object E ∈ A(β,ω) fits in an exact sequence in A(β,ω) :(2.9)

0 → H −1 (E)[1] → E → H 0 (E) → 0

3We need one more condition if X is a K3 surface.

9where H −1 (E) ∈ F(β,ω) and H 0 (E) ∈ T(β,ω) . Thus E is a two-term complex. If
X is a K3 surface, more complicated complexes appear by applying autoequivalences. On the other hand for an abelian surface, (2.8) implies only 2-terms
complexes appear.
Definition 2.8. For a Mukai vector v, M(β,ω) (v) denotes the moduli stack ofσ(β,ω) -semi-stable objects E of A(β,ω) with v(E) = v. M(β,ω) (v) denotes the moduli
scheme of the S-equivalence classes of σ(β,ω) -semi-stable objects E of A(β,ω) with
v(E) = v, if it exists.
As we claimed, there is a region called large volume limit, where Bridgelandstability coincides with the classical stability.
Proposition 2.9 (Large volume limit). For a Mukai vector v = (r, ξ, a) with r ≥0, assume that (ω 2 ) ≫ 0 (depending on v). If (ξ − rβ, ω) > 0, then M(β,ω) (v) =Mωβ (v).
2.4. Birational geometry of MH (v). In this subsection, we shall explain the
birational geometry of MH (v) in [6], [26], [44]. These results are the significant application of Bridgeland stability conditions to the structure of the moduli spaces.
We start with some definitions.
Definition 2.10. Let f : M → S be a projective morphism.
(1) D ∈ Pic(M ) is f -movable, if
codim coker(f ∗ f∗ (OM (D)) → OM (D)) ≥ 2.
(2) A relative movable cone Mov(M/S) is a cone generated by f -movable
divisors.
We shall consider relative cones over the albanese map.
Definition 2.11.
(1) Two albanese maps a′ : M ′ → Alb(M ′ ) and a : M →Alb(M ) are equivalent if there is a birational map f : M ′ · · · → M and
an isomorphism g : Alb(M ′ ) → Alb(M ) with a commutative diagram(2.10)

M′


′
ay

f

−−−→

M

a
y

g

Alb(M ′ ) −−−→ Alb(M )(2) For a normal variety M , Amprel (M ) denotes the relative ample cone of thealbanese map a : M → Alb(M ). We set Movrel (M ) := Mov(M/ Alb(M )).
Movrel (M )0 denotes the interior of Movrel (M ).
For the moduli space MH (v), all the (relative) minimal models are the moduli
spaces of Bridgeland’s stable objects. Thus by extending our consideration fromsheaves to complexes, we get a desirable description of birational properties.
We take σ ∈ C. Then we have a map ξσ : Stab(X) → P + (Mσ (v)). Up tothe action of R>0 , ξσ is surjective. Then ξσ (C) = Amprel (Mσ (v)). So we set
CAmp := C. There is another chamber decomposition of Stab(X) such that each10

chamber CMov containing σ corresponds to the movable cone via ξσ .
ξσ(2.11)

Stab(X) −→P + (Mσ (v))∪∪
CMov−→ Movrel (Mσ (v))0∪∪
CAmp−→ Amprel (Mσ (v))

We also have a decomposition.
(2.12)∪

Movrel (Mσ (v)) =Amprel (M ′ ).

M ′ ···→Mσ (v)By the characterization of walls for nef cones of each birational models, we get
the following result.
Corollary 2.12 ([24]). Kawamata and Morrison’s Nef cone conjecture holds:
There is a rational finite polyhedral fundamental domain for the action of Aut(Kσ (v))
on Nef(Kσ (v)), where Kσ (v) is a fiber of the albanese map.
2.5. Abelian surface with NS(X) = ZH and (H 2 ) = 2.
2.5.1. Stability condition. In order to give an easy example of Stab(X), we assume
that X is a principally polarized abelian surface with NS(X) = ZH, i.e., (H 2 ) = 2.
In this case, we have an identification
NS(X)R × Amp(X)R ∼H√=(s, t)↔ s + t −1
Then the action of the autoequivalence group Aut(D(X)) on H factors throughthe natural action of SL(2, Z):/ Aut(H)
r9
r
rr
r
r
r
rrr

Aut(D(X))

NNN
NNN
NNN
NN'

SL(2, Z)

So we havef + (2, Z) ∼SL(2, Z)\ Stab(X)/GL= SL(2, Z)\H ∼= C.
By adding a cusp4∞, we get a compactification.
SL(2, Z)\(H ∪ P1 ) ∼= P1 .
Q

f + (2, R) ∼Remark 2.13. If n = (H 2 )/2 > 1, we still have Stab(X)∗ /GL= H, andAut(D(X)) acts as the action of Γ0 (n) ⊂ SL(2, Z).
The large volume limit (Proposition 2.9) corresponds to a neighborhood of∞ ∈ P1C . Mz (r, dH, a) = MH (r, dH, a) for |z| ≫ 0 and z < d/r (Mz (r, dH, a) ={E ∨ | E ∈ MH (r, −dH, a)} for |z| ≫ 0 and z > d/r). Each wall is a semi-circle C
in H ∪ {∞}. If ∞ ∈ C, then C is a half line (rz − d = 0). Walls do not intersect
each other.
4Cusps are related to Foourier-Mukai transforms [23]11We illustrate walls for v = (1, 0, −3) (see Figure 1). C0 is a wall passing ∞.
By the action of linear fractional transformation52z − 3f (z) =,−z + 2C0 is transformed to C1 and C1 is transformed a semi-circle C2 which is inside of C1 . Applying f n (z), we have infinitely many walls Cn = f n (C0 ) (n ∈Z). We also have infinitely many walls Wn = f n (W0 ). Cn (n > 0) and Wn(n ≥ 0) are semi-circles in the left hand side of C0 . Let C0− be the chamberbounded by C0 and C1 , and C0+ the chamber bounded by C0 and C−1 . We set
Cn± := f n (C0± ). These are the walls and chambers for v = (1, 0, −3). The walls
and chambers are symmetric with respect to t-axis. Each bounded chamber Cn±(n ̸= 0) is annulus. There are two unbounded chambers C0± . The chamber C0−
parameterizes MH (1, 0, −3) and the chamber C0+ parameterizes the dual. Forq∈ Cn± , by a Fourier-Mukai transform Φ inducing an isomorphism f (z) of P1pwith f (q/p) = ∞, Cn± is transformed to an unbounded chamber, which gives
the claim (d). In this example, the movable chamber is a single chamber. In
particular, Movrel (Mσ (v))0 = Amprel (Mσ (v)) = ξσ (Cn± ), where σ ∈ Cn± .
t
C0−

1

W0
C1+
−3

1
4

C1
−2

− 23

−1

C0+
C0

o

s

Figure 1. Walls for v = (1, 0, −3) between C0 and C1 .
2.5.2. Relation with binary quadratic form. Let Q be the space of binary quadratic forms:(2.13)

Q := {rx2 + 2dxy + ay 2 | r, d, a ∈ Z}.

If X is a principally polarized abelian surface with NS(X) = ZH, then the
Mukai lattice H ∗ (X, Z)alg is isomorphic to Q via(2.14)H ∗ (X, Z)alg →Q(r, dH, a) 7→ rx2 + 2dxy + ay 2 .

Then for v = (r, dH, a), ⟨v 2 ⟩/2 = d2 − ra is the discriminant of rx2 + 2dxy +ay 2 . Under this identification, the cohomological action of the Fourier-Mukai
transforms factors through the natural action of SL(2, Z) on Q.
5f (z) is the generator of the action of D(X) on H preserving ±(1, 0, −3). It is related to the√fundamental unit of Z[ 3].

12Theorem 2.14 ([40],[25],[44]). Let (X, H) be a principally polarized abelian surface with NS(X) = ZH.
(1) The birational equivalence class of MH (v) depends on the equivalenceclass of the associated quadratic form. In particular,(2.15)#{MH (v) | gcd(r, d, a) = 1, ⟨v 2 ⟩/2 = D}/(birational equiv.) ≤ hD ,where hD is the class number of quadratic forms of discriminant D.
(2) There is a birational map⟨v 2 ⟩/2

MH (v) · · · → X × HilbX
if and only if

rx2 + 2dxy + ay 2 = ±1
has a solution.
Remark 2.15. Finally we would like to remark that almost all results on MH (v)in this subsection are found or conjectured by Mukai around 1980 ([28],[29]).
It is really surprizing to the author that these discoveries were done without
Bridgeland stability condition.
2.6. Related results and some problems. The example of stability condition
σ(β,ω) in section 2.3 is generalized to arbitrary surface by Arcara and Bertram[1]. If X = P2 , then Arcara, Bertram, Coskun and Huizenga [2] studied moduli
spaces M(β,ω) (v) of stable objects. By the equivalence (0.4), we can use thetheory of quiver in order to study Bridgeland stability condition. In particular,
we can use King’s result to construct the moduli space M(β,ω) (v) as a projective
scheme. Moreover they showed that M(β,ω) (v) is a log Fano variety, which implies
M(β,ω) (v) is a Mori dream space. In particular, the movable cone is rational finite
polyhedral cone. Moreover for a non-commutative P2 , Li and Zhao [22] studied
moduli spaces of stable objects, and showed that they are Fano varieties. See
also [14] for recent development.
For the stability condition on Enriques surface, Nuer [33] studied moduli of
stable objects.
Finally we list the following problems.
Problem 2.16.
(1) Construction of moduli space.
(2) Smoothness of the moduli spaces. In birational geometry, smooth varieties
do not form a good category. So it may not be so important.
As we explained, Bridgeland stability condition is related to birational geometry of moduli spaces. So it is desirable to have projective moduli spaces. Notethat Inaba [18] constructed moduli of simple complexes M(v)spl . Let Mσ (v) be
the subset consisting of σ-stable objects. On the other hand, Bayer and Macri[5] constructed nef line bundle Lσ on Mσ (v). For all known cases, Lσ is ample.
For example, if X is a K3 surface or an abelian surface, Lσ is nothing but ξσ in(2.11). It is interesting to show the ampleness of Lσ for general cases.
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Department of Mathematics, Faculty of Science, Kobe University, Kobe, 657,
Japan
E-mail address: yoshioka@math.kobe-u.ac.jp15