Bridgeland's stability conditions and applications to the geometry of surfaces.

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Universit`

a degli Studi di Pisa

FACOLT `A DI MATEMATICA Corso di Laurea magistrale in Matematica

Tesi di laurea magistrale

Bridgeland’s stability and its application to the gemoetry of

surfaces

Candidato:

Luigi Pagano

Matricola 508489

Relatori:

Chiar.mo Prof. Enrico Arbarello

Chiar.ma Prof.ssa Rita Pardini

Controrelatore:

Chiar.mo Prof. Marco Franciosi

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Introduction

During this survey we will study the notion of Bridgeland stability on triangulated categories, as introduced in [7]. We shall give the definition of Bridgeland stability condition.

Definition 0.0.1 (Stability condition). A stability condition σ = (Z, P), is given by a gropup homomorphism Z : K(D) → C between the Grotendieck group K(D) of a triangulated category D and the additive group C, which is also called central charge of σ, and by a slicing Pof D which consists on the data of a full additive subcategory P(ϕ) ⊆ D for each ϕ ∈ R, satisfying the following axioms:

1. if E ∈ P(ϕ), then Z(E) = m(E) · eiπϕ_{, for some m(E) ∈ R;}

2. ∀ϕ ∈ R we have P(ϕ + 1) = P(ϕ)[1];

3. If ϕ1 > ϕ2 and Aj ∈ Pj, then Hom(A1, A2) = 0;

4. ∀E ∈ D, there are real numbers ϕ1 > ϕ2 > · · · > ϕn and objects Aj ∈ P(ϕj) such

that one can write a diagram

0 = E0 E1 E2 · · · En−1 En= E

A1 A2 An

where Ej−1 → Ej → Aj → Ej−1[1] are triangles in D and the Aj will be called

Harder-Narasimhan factors of E.

Equivalently one can replace the datum of P by the datum of a bounded t-structure over D, we shall see how this is done. After a brief general discussion about the set of stability conditions on a given category, we will focus on the subset of the locally-finite stability conditions, which will be denoted as Stab(D), and we will set up a distance on this subset (it would be better saying a generalized distance as there could be couples of stability condition whose distance is +∞), this will yield a topology with many nice properties which will be useful during our study. Moreover we will see how this space,

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Chapter 0. Introduction

under some technical hypoteses on our category, has a natural complex structure: indeed, each connected component of the space will, be a complex manifold. The space of stability conditions carries a natural group action of fGL+(2, R), the universal cover of GL+(2, R); if two stability conditions lie in the same orbit (with respect to this gruop action) they give the same information about the object and the morphisms of D.

We will focus our study on the following model. Let us fix an algebraic variety X over an algebrically closed field k and consider the category Coh(X) of the coherent sheaves of modules over the structure sheaf OX; its derived category D(X) is a triangulated

category and our object of study will be the space Stab(X) of the locally finite stability conditions over D(X). The structure of Stab(X) for a general variety X is not well understood yet; while for specific cases one can construct explicitely some subspace of Stab(X). As an example, if X is a smooth projective curve of positive genus over C it is well known that Stab(X) = σ fGL+(2, R) ∼= fGL+(2, R) is generated by a stability condition σ through the (free and transitive) action of fGL+(2, R); while Stab(P1_{) ∼}₌

C2. If X is a surface, the space Stab(X) becomes more complicated; in [8] Bridgeland provides a description of a connected component of Stab(X), when X is a K3 surface, while Macr`ı and Schmidt in [20], through a similar idea, provide a description of a subspace of Stab(X) for any surface X (when X is a K3, they do not get the same space as Bridgeland, hence their result cannot be considered a generalization of Bridgeland’s one).

Even though Bridgeland’s idea arose from string theory, the study of stability is a very useful tool in algebraic geometry. As an example some classical vanishing theorems can be stated and proved in terms of this theory. In [11] Feyzbakhsh uses the construcions from [8] and especially the wall-crossing technique to establish an important theorem which generalizes Mukai’s program.

Wall crossing is one of the most useful techniques which arise from the study of the space of stability conditions. To explain how it works, it is important to understand the structure, or at least part of it, of the stability condition space on a variety. We will follow Bridgeland construction for the K3 case (as in [8]).

One of the first question one should answer is if the classical slope stability can be turned into a Bridgeland stability; in the case of a curve C the answer is yes and it is given by the Bridgeland stability condition whose central charge is, for any E ∈ D(C), Z(E) = − deg(E) + i rk(E) and whose associated t−structure’s heart is Coh(C) (see [17] for more details). When trying to put a similar stability condition on a surface X (naturally, in order to define the slope of E one needs to use ω · c1(E) for a fixed

line bundle ω instead of deg(E)); in this way Coh(X) can never be the heart of a t−structure, as one would get Z(Ox) = 0 for any skyscreaper sheaf Ox. Bridgeland

considers an ample line bundle ω and a line bundle β and he constructs the hearth of a t−structure via the technique of tilting with respect to (T , F) (which is a torsion pair on Coh(X)), where T consists of the sheaves whose torsion-free parts have µω−semistable

Harder-Narasimhan (HN) factors of (ω−)slope greater than β · ω, while F consists on the torsion-free sheaves whose HN factors have slope less or equal to β · ω; by setting Z(E) := exp(β + iω) · v(E) (where v(E) is the Mukai vector of E) it turns out that one

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gets a stability condition on X for almost any couple (β, ω). Altough this construction does not define a Bridgeland condition equivalent to a classical condition, changing the couple (β, ω) by (β, λω), for λ ∈ (0+∞) and letting λ → +∞, the central charge behaves similarly to the one costructed on a curve, so one can see the slope condition as a ”limit” of Bridgeland conditions; we will see some result which will set up a formal apparatus based on the idea of limit; those result will give an important connection between the old and the new world. Bridgeland’s construction inspired Macr`ı and Schmidt in [20] to construct some stability conditions on P3 _{via the second tilt; this suggests that it is}

possible to construct stability conditions for any variety by tilting an adequate number of times. Next, we are going to study a connected component Stab∗ of Stab(X) which contains a stability condition whose central charge is of the form constructed above. The most astonishing result is that, once we fix an object E ∈ D(X), there exists a (locally finite) set of real hypersurfaces Wγ of Stab∗, which we will call walls, such that

for any connected component C of Stab∗\S

γWγ, i.e. C is a region bounded by some

of these hypersurfaces, then if E is unstable, resp. semistable, for some σ0 ∈ C then

E is unstable, resp. semistable, for any σ ∈ C. The wall crossing principle consists on studying all the phenomena arising from examining a path of stability conditions which sarts in a chamber C and ends into an adjacent one, hitting, thus, a wall W.

In the last chapter of this survey we will show an application of the wall-crossing in a classical problem: Mukai’s program for curves in K3 surfaces.

Let us consider the moduli space KCg of triplets (X, H, C), where (X, H) is a

polar-ized K3 surface and C ∈ |H| is a smooth curve of genus g and the moduli space Mg of

smooth curves of genus g. There is a forgetful map KCc → Mg which sends the triple

(X, H, C) to the curve C; such map is birational onto its image if g ≥ 11 and g 6= 12. The program introduced by Mukai consists in finding the rational inverse of such map.

Mukai firstly showed that if g = 11 and C ,→ X is an embedding of a curve C in the K3 surface X, then all vector bundles in the Brill-Noether locus T = B7

C(2, KC), i.e.

the set of slope-semistable rank 2 vector bundles on C with the canonical sheaf KC as

determinant and having at least 7 linearly independent global sections, are restriction of vector bundles on X having Mukai’s vector equal to v = (2, H, 5). This proves that T is a K3 surface. The original K3 surface can be reconstructed as a Fourier-Mukai partner of T .

Arbarello, Bruno and Sernesi generalized Mukai’s work by extending his construction to curves having odd genus g = 2s + 1 ≥ 11, where the Fourier-Mukai partner of X could be found inside the Brill-Noether locus B_Cs+2(2, KC).

The first application of wall-crossing appears in Feyzbakhsh’s paper [11], where the author show that any K3 surface with Pic(X) = H · Z, i.e. the general K3 surface of genus g, can be reconstructed by a Brill-Noether locus on C.

While explaining Feyzbakhsh’s article [11] we will consider a K3 surface X of Picard number equal to 1, so that Pic(X) = ZH for a certain line bundle H. Suppose we have the curve C ∈ |H| belonging to its linear system such that g(C) ≥ 11 and g(C) 6= 12. We will consider separately two cases: in the first one we write g = rs + 1, for some integers 2 ≤ r ≤ s with also s ≥ 5, while in the second one we have g = p + 1, for some

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Chapter 0. Introduction

odd number p. Notice that those two cases overlap when r, s are odd, but they cover all the genera we aim to consider. Let, then, T = B_Cr+s(r, 2rs) or T = B_Cp+4(4, 4p) (the choice depends in a naural way on the case we are considering) be the Brill-Noether locus of slope-semistable sheaves on C whose rank is r, respectively 4, whose degree is 2rs, respectively 4p, and which has at least r +s, resp. p+4 independent global sections; let then N be the moduli space of H−Gieseker semistable sheaves with Mukai vector (r, H, s), respectively (4, 2H, p). It turns out that N is a K3 surface in both cases. The aim is to show that T and N are isomorphic via the restriction morphism. Repeating again the construction, starting from the K3 surface T = N with a generic polarization, we get another K3 surface which, in turns, will be isomorphic to the starting surface X. Feyzbakhsh, then, analyzes the connection between the Bridgeland stability and the classical notions of H−Gieseker and H−slope stability; we will see, in particular, that for sheaves whose support coincide with a curve the old fashoned notions are equivalent to the Bridgeland stability for condition in a specific chamber (as the ”limit” behavior outlined before would suggest).

Feyzbakhsh gave an upper bound for the number of global section of a sheaf on X in terms of its Harder-Narasimhan filtration relative to a certain stability condition; here the wall and chamber structure of the space of stability conditions will be crucial, as we will be able to find a small region of stability conditions whose point induce the same HN filtration of the considered sheaf.

Thus she showed that the restriction map N → T is well defined and bijective via the study of the same HN filtration we got before. The most astonishing fact is that the injectivity of the restrinction map is an immediate consequence of the ”continuity” of the HN filtration; while in previous attempt, showing the injectivity of the map was considerably more involved.

Finally she concluded by showing that the map induces an isomorphism of tangent spaces and hence it is an isomorphism of algebraic varieties.

In this survey we will follow Feyzbakhsh work by dropping the hypothesis on the Picard number of X. We will show that for a primitive class H ∈ NS(X) such that the generic curve C ∈ |H| is a smooth curve of genus g ≥ 11 such that g 6≡ 0 (mod 4), then the restriction map from a suitable moduli space MX,H(v) of vector bundles on

X is injective in the Brill-Noether locus B_Cs+2(2, 4s) of rank 2 vector bundles of degree 4s, or in the Brill-Noether locus Bp+4_C (4, 4p), depending on the fact that g = 2s + 1 or g = p + 1 for some odd integer p. As a consequence such Brill-Noether loci relative to curves lying on a K3 surface have dimension at least 2, against any prediction obtained by a computation of their virtual dimension. Finally, following [9], we will explain under what condition on the curve C the forgetful map

KC_g → M_g (X, H, C) 7→ C

has positive-dimensional fibre. We will give, then, an explicit construction of a curve of genus 23 whose Brill-Noether locus B13

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Chapter 1

Stability on triangulated

categories

In this chapter, following Bridgeland approach in [7], we will introduce the main topic of this discussion. We will give two definitions of the stability conditions, showing, their equivalence; the rest of the thesis will usually refer to the first definition, while we can consider the second one as a way to construct stability conditions.

After giving the definitions, we will see how the set of the stability condition can be endowed with a natural distance; we will focus our study on this topologycal space.

Through this chapter let D be a triangulated category and let K(D) be its Groethendieck group, i.e. the abelian group obtained as a quotient of the free abelian group spanned by the objects of D with respect to the subgroup generated by exact sequences: M = A + B if 0 → A → M → B → 0 is exact.

1.1 Definition and basic properties

The idea of the definition of a stability condition has been obtained by abstracting the notion of Harder-Narasimhan filtration which arises from the classical slope stability over an algebraic variety.

Definition 1.1.1. A stability condition σ = (Z, P) on D is the datum of a group homomorphism Z : K(D) → C, called the central charge, and a slicing P of D, i.e. a collection of full additive subcategories P(ϕ) ⊆ D for each ϕ ∈ R satysfying the fillowing axioms (to define a slicing of D one only needs axioms 2-4, while the first axiom is necessary to define the ”slicing of a stability condition”):

1. if E ∈ P(ϕ), then Z(E) = m(E) · eiπϕ_{, for some m(E) ∈ R;}

2. ∀ϕ ∈ R we have P(ϕ + 1) = P(ϕ)[1];

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Chapter 1. Stability on triangulated categories

4. ∀E ∈ D, there are real numbers ϕ1 > ϕ2 > · · · > ϕn and nonzero objects Aj ∈

P(ϕj) such that one can write a diagram

0 = E0 E1 E2 · · · En−1 En= E

A1 A2 An

where Ej−1 → Ej → Aj → Ej−1[1] are triangles in D and the Aj will be called

Harder-Narasimhan factors of E. We will say that an HN filtration has length n if it has n HN factors.

Now, given a stability condition σ = (Z, P), we call any nonzero object E ∈ P (ϕ) a semistable object of phase ϕ with respect to σ; moreover we will say that E is stable if it is simple in P(ϕ), i.e. if it has no subobject in that category.

Now it is easy to see that if E ∈ P(ϕ) is semistable, then 0 = E0 ,→ E1 = E is an

Harder-Narasimhan filtration of E; moreover, if there is a different filtration, it must have at least two factors:

0 = F0 F1 F2 · · · Fn−1 Fn= E

A1 A2 An

then we have the following inequalities chain: ϕ ≥ ϕ(A1) > ϕ(An) ≥ ϕ where by

abuse of notation we will denote by ϕ(A) the phase of a semistable object. The first inequality holds true because there is a non-zero morphism 0 6= A1 = F1 → E and the

last one because of the non-zero morphism E → An.

Then, if E ∈ D is semistable, there is a unique HN filtration of E; moreover, if an object has an HN filtration of length one it has to be semistable.

We can say more:

Proposition 1.1.2. Any E ∈ D has a unique HN filtration.

Proof. We will proceed by induction on the length n of the shortest HN filtration of E ∈ D, as we already know that the statement holds true if n = 1.

Now let

0 = E0 E1 E2 · · · En−1 En= E

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1.1. Definition and basic properties

be the minimal HN filtration of E and let

0 = F0 F1 F2 · · · Fm−1 Fm = E

B1 B2 Bm

be another filtration.

As 0 6= F1 ⊆ E, there is 1 ≤ k ≤ n such that F1 ⊆ Ek and F1 6⊆ Ek−1, hence there

is a nonzero morphism F1 → Ak, which means ϕ(F1) ≤ ϕ(Ak) ≤ ϕ(A1). Viceversa we

have that ϕ(E1) ≤ ϕ(B1) and so A1 and B1 have the same phase. Thus we cannot have

nonzero morphism F1 → Ak or E1 → Bk for k > 1, so F1 ⊆ E1 ⊆ F1 and E1 = F1.

Now consider the cone of E1 → E which has an HN filtration of length n − 1 with

subobjects Cone(E1→ Ej) and factors A2, . . . , An as well as a filtration of length m − 1

whose subobjects are Cone(E1 → Fj) and factors are B2, . . . , Bm. By the inductive

hypothesis these two filtration are the same, moreover we have m = n, Aj = Bj, E1= F1

and the triangles Aj[−1] → Ej → Ej+1, Bj[−1] → Fj → Fj+1. Therefore the two initial

filtration must be equal.

We use the previous statement to conclude that the two quantities ϕ+_P(E) := ϕ(A1)

and ϕ−_P(E) := ϕ(An) are well defined for any 0 6= E ∈ D; those numbers are called

maximum and minimum phase of E with respect to the slicing P. Another well defined notion is that of mass of E with respect to the stability condition σ:

mσ(E) = n

X

i=1

|Z(Ai)| .

Now, given a slicing P and an interval I ⊆ R, we can define P(I) ⊆ D as the full subcategory containing 0 and all objects E ∈ D with ϕ−_P(E), ϕ+_P(E) ∈ I, which is equivalent to saying that all the phases of HN factors of E belong to I, i.e. P(I) is the extension closed subcategory of D generated by P(ϕ) : ϕ ∈ I.

Remark. E ∈ Dis a stable object of phase ϕ with respect to a certain stability condition σ if and only if ϕ+_{(E) = ϕ}−_{(E) = ϕ.}

Notice, also, that Z(E) = Z(A1) + · · · + Z(An), so we have that m(E) = Z(E) if and

only if E is semistable, or if its factors have the same phases up to translation in 2Z. We use these quantities to define the generalized distance function between two stability conditions σ, τ in the following way:

d(σ, τ ) = sup

06=E∈D

|ϕ+_σ(E) − ϕ+_τ(E)|, |ϕ−_σ(E) − ϕ−_τ(E)|, logmσ(E) mτ(E) .

This defines a generalized distance in the sense that there could be stability conditions σ, τ such that d(σ, τ ) = +∞, but this function actually satisfies all the remaining axioms defining a distance.

We shall not study the metric space associated to the set of all the stability conditions, we shall, instead, consider a suitable subset.

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1.2 Stability funcions on abelian categories and t-structures

Throughout this section let A be an abelian category and K(A) be its Grotendieck gorup. We shall study a notion of stability function over A and see how this leads to a stability condition over its (bounded) derived category. Moreover we will see that for any stability condition σ on D can be obtained in this way, if we consider a suitable abelian subcaegory of D.

In order to perform the construction we need to identify an abelian category embed-ded in our triangulated category; we will give the notion of t-structure over D and see how this will detect an abelian category in a natural way.

1.2.1 Stability functions

Definition 1.2.1. A stability function on A is a group homomorphism Z : K(A) → C such that for 0 6= E ∈ A, we have

Z(E) ∈ {r · exp(iπϕ) : r > 0, 0 < ϕ ≤ 1} .

Similarly to the previous section, we can define the phase of a nonzero object E ∈ A. Since Z takes value only on the upper half-plane, we can define the phase ϕ(E) of any object E ∈ A as the real number 0 < ϕ(E) ≤ 1 such that Z(E) = r exp(iπϕ(E).

An object 0 6= E ∈ A will be (semi)stable if for any subobject 0 6= F ⊆ E we have ϕ(F ) < (≤)ϕ(E). Note that if f : E → F is a nonzero map between semistable objects, then we have ϕ(ker f ) ≤ ϕ(E), and since Z(E) = Z(ker(f ))+ Z(im(f )), ϕ(E) ≤ ϕ(f (E)) ≤ ϕ(F ); therefore the ”slicing” induced by this notion of stability has similar properties to those of the slicing on triangulated categories.

Definition 1.2.2. Let 0 6= E ∈ A. An Harder-Narasimhan filtration with respect to a stability function Z is a chain 0 = E0 ⊂ E1 ⊂ · · · ⊂ En = E whose nonzero factors

Fj := Ej/Ej−1 are semistable objects with decreasing phases:

ϕ(F1) > ϕ(F2) > · · · > ϕ(Fn) .

We will say that Z has the Harder-Narasimhan property if any 0 6= E ∈ A admits an HN filtration.

Proposition 1.2.3. Let Z be a stability function on A satisfying the followings proper-ties:

1. there are no infinite sequences of subobjects

· · · ⊂ E_j ⊂ E_j−1⊂ · · · ⊂ E₁ with ϕ(Ej+1) > ϕ(Ej) ∀j.

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1.2. Stability funcions on abelian categories and t-structures

2. there are no infinite sequences of quotients

E1 · · · Ej Ej+1 . . .

with ϕ(Ej+1) < ϕ(Ej) ∀j.

Then Z has the HN property.

For the proof see Proposition 2.4 in [7].

1.2.2 t-structures

Definition 1.2.4. A t-structure on D is a full subcategory F ⊆ D such that 1. F[1] ⊆ F;

2. for every object E ∈ D there is a triangle F → E → G → F [1] with F ∈ F and

G ∈ F⊥ := {A ∈ D : HomD(F, A) = 0 , ∀F ∈ F} .

Moreover we will say that the t-structure is bounded if we can write

D = [

i,j∈Z

F [i] ∩ F⊥[j] .

Example 1.2.5. If we consider any abelian category A and its derived category D = D(A), then we can find a natural t-structure by

F = {E ∈ D(A) : Hi_{(E) = 0 , ∀i > 0} .}

Moreover, if we take the bounded derived category instead of D(A), then we get a bounded t-structure in the same way.

Definition 1.2.6. The heart of a t-structure F is the full subcategory A:= F ∩ F⊥[1] ⊂ D .

In [6] it has been proved that the heart of a t-structure is an abelian category, see also [12] Theorem IV.4.4 ; we will see soon that this is the category we were looking for. Remark. A bounded t-structure is determined by its heart. Suppose we are given the heart A ⊂ D of the t-structure F, then we can reconstruct F as the extension closed subcategory of D generated by A[j], j ≥ 0. We can reconstruct F⊥ _{as the sbcategory}

generated by A[j], for j < 0.

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Lemma 1.2.7. Let A ⊆ D a full additive subcategory. Then A is the heart of a t-structure F if and only if the following hold:

1. if k1 > k2 are integers, then HomD(A[k1], B[k2]) = 0, ∀A, B ∈ A;

2. for every object 0 6= E ∈ D, there are integers k1 > k2 > · · · > kn and objects

Aj ∈ A[kj], j = 1, . . . , n such that there is a filtration in triangles:

0 = E0 E1 E2 · · · En−1 En= E

A1 A2 An _.

Proof. If A is the heart of the t-structure F and k1 > k2 are integers, then A[k1] =

F [k1] ∩ F⊥[k1+ 1] ⊆ F[k1] and A[k2] = F[k2] ∩ F⊥[k2 + 1] ⊆ F⊥[k2+ 1] ⊆ F⊥[k1].

Hence the first condition is satisfied. Now, for any object 0 6= E ∈ D, we have two objects F ∈ F, G ∈ F⊥ and a triangle F → E → G; the previous Remark tells us that there are filtrations

0 = E0 E1 E2 · · · Er−1 Er = F

A1 A2 Ar _,

with Aj ∈ A[kj] for some kj ≥ 0

0 = E_r0 E_r+10 E_r+20 · · · E_n−10 E_n0 = G

Ar+1 Ar+2 An _,

with Aj ∈ A[kj] for some kj <0.

Notice that F → E → G leads us to G[−1] → F → E, so we have the maps E_j0[−1] ⊆ G[−1] → F (for j > r) whose cone E_j0[−1] → F → Ej allow us to lift the

E_j0 to Ej ⊆ E (notice that we could say j ≥ r instead of j > r as the cone of 0 → Er

is Er itself); the cones of Ej−1 → Ej must be isomorphic to Aj = C(Ej−10 → Ej0).

Since E_n0 = G and G[−1] → F → E is a triangle, we have En = E and therefore this

construction yields the desired filtration.

Let us suppose A satisfies the condition above and let F be the extension closed subcategory of D generated by A[j], for j ≥ 0. Let, then, G be the extension closed subcategory of E generated by the A[j], for j < 0; of course we have G ⊆ F⊥_.

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1.2. Stability funcions on abelian categories and t-structures

If we show that for any E ∈ D there are F ∈ F and G ∈ G such that F → E → G is a triangle we can infer F is a t-structure; moreover it will be a bounded t-structure, because for any E ∈ D we have integers k1, . . . , knand objects A1, . . . , Anas above, then

E ∈ F[kn] ∩ F⊥[k1+ 1].

But now, for E ∈ D just take a filtration as above and suppose kr≥ 0 > kr+1, then

let F = Er and G be the cone of Er→ E, then of course F ∈ F, and G has the factors

Ar+1, . . . , An, thus we also infer that G ∈ G and we are done.

1.2.3 Construction of a slicing from a t-structure and viceversa

In this paragraph we will, at last, relate the notion of stability introduced at the begin-ning of the chapter with the notion of t-structure and stability function we are introduc-ing in this section.

Remark. If A ⊆ D is the heart of a boundet t-structure, then every object in D is generated (via extension) by objects of A and K(A) = K(D), thus there is a natural link between the central charge of a stability condition and a stability funcion over A. Our problem consists on constructing P from A, Z and A from P.

Now, suppose we have a slicing P satisfying the requirements 2-4 in Definition1.1.1, then we will set a t-structure on D given by F = P((0, +∞)), F⊥ _{= P((−∞, 0]) and}

A = P((0, 1]). Obviously A is a full additive subcategory of D and its non negative translated generates (via extension) F, so, in order to check that F is actually a bounded t-structure, we just need to check P((0, 1]) satisfies the two conditions of the criterion

1.2.7; the first condition is an immediate consequence of the axiom 3 in Definition 1.1.1, while the second is a consequence of the forth axiom: just take the HN resolution 0 = E0 ⊆ E1 ⊆ · · · ⊆ En= E and factors of phases ϕ1 > · · · > ϕnand get the resolution

0 = F0 ⊆ F1 ⊆ · · · ⊆ Fr = E with Fl = Ejl, where ∀ l, jl is an integer such that there

exist m ∈ Z satisfying ϕjl > m ≥ ϕjl+1 and ∃m

0

∈ Z such that m0 ≥ ϕjl−1+1 and

ϕjl > m

0_{− 1. Moreover, for any object E ∈ A, its HN factors A}

1, . . . , An are still in A

and for any stability condition σ(Z, P) whose slicing is P, we have that Z(Aj) = rjeiπϕj,

so Z(E) = Z(A1) + · · · + Z(An) belongs to the upper half plane (or to the negative half

line), therefore the central charge of σ descends to a stability function on A with the HN property.

Viceversa, if Z is a stability function with the HN property over the heart of a t-structure A ⊆ D, then we can define the slicing P as follows: ∀ϕ ∈ (0, 1] ⊆ R, let P(ϕ) be the set of semistable object of A of phase ϕ with respect to Z, while, for n ∈ Z let P(n + ϕ) = P(ϕ)[n]. Now we shall check that P is actually a slicing.

By construction it is obvious P satisfies the translation axiom; moreover E ∈ P(ϕ) ⇒ Z(E) = reiπϕ_{certainly holds if E ∈ A, but it also holds in general because E[1] = −E ∈}

K(D).

If n ∈ Z and ϕ > n ≥ ψ are real numbers, then P(ϕ) ⊆ F[n] and P(ψ) ⊆ F⊥_[n];

so for A ∈ P(ϕ) and B ∈ P(ψ) we have HomD(A, B) = 0. If n ∈ Z and n + 1 ≥

ϕ > ψ > n are real numbers, then P(ϕ), P(ψ) ⊆ A[n]; so for A ∈ P(ϕ) and B ∈ P(ψ) we have HomD(A, B) = HomD(A[−n], B[−n]) = 0, where last equality holds because

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A[−n], B[−n] ∈ A are semistable and the phase of the former is greater than the phase of the latter.

At last, given E ∈ D, by the criterion 1.2.7, we get a filtration

0 = E0 E1 E2 · · · En−1 En= E

A0₁ A2 A0n _,

with A0_j ∈ A[kj] for some integer kj. A0j is not necessairely semistable, but since Z

has the HN property, then A0_j[−kj] ∈ A admit a HN filtration with quotients Aj,i ∈ A

(i = 1, . . . , nj) of decreasing phase and the Aj,i[kj] will give a HN filtration of E as

desired, therefore the last axiom is satisfied too.

1.3 The topological space Stab(D)

The aim of this section is to study the properties of the generalized distance we intro-duced in the first section. We are not interested in the topology it induces on the space of all stability condition, we will restrict to the space of locally finite stability conditions. Before introducing the notion of locally finite stability condition we shall open a small digression about quasi-abelian categories.

1.3.1 Quasi-abelian categories.

Previously we said that a slicing P of D induces an abelian category P((ϕ, ϕ + 1]): the heart of the t-structure P((ϕ, +∞)) (we showed it for ϕ = 0, but the same arguments yields this result too); P([ϕ, ϕ + 1)) is abelian as well. We cannot say the same about P(I), if I ⊆ R is an interval of length less than 1. It turns out that it lacks one property, but it still has the structure of a quasi-abelian category, as we are going to define in this section.

We will follow [24] for basic definitions.

Definition 1.3.1. Let C be and additive category with kernels and cokernels. Given a morphism f : A → B we say f is strict if the induced map

coim f → im f is an isomorphism.

Remark. If f is a strict monomorphism (respectively epimorphism), then f is a kernel (respectively a cokernel). Viceversa, for any morphism ψ : A → B, the canonical maps

ker ψ → A and B →coker f are strict.

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1.3. The topological space Stab(D)

A morphism f is strict if and only if it is the composition of a strict monomorphism iand a strict epimorphism q:

f = i ◦ q .

Definition 1.3.2. An additive category C is quasi-abelian if and only if every pullback of a strict epimorphism is a strict epimorphism and every pushout of a strict monomorphism is a strict monomorphism.

The following lemma provides an useful characterization of quasi-abelian categories: Lemma 1.3.3. An additive category C is quasi-abelian if and only if there exist abelian categories C], C[ together with fully faithful embeddings C ⊆ C], C ⊆ C[ satisfying:

• If A → E is a monomorpism in C] _{and E ∈ C, then also A ∈ C;}

• If E → B is an epimorpism in C[ _{and E ∈ C, then also B ∈ C;}

For a proof see Proposition 1.2.34 of [24].

Example. A certainely well known example in algebraic geometry of a quasi-abelian category is the category of torsion-free sheaves over a projective variety X. In this case one may take C] _{= C}[ _{= Coh(X). Monomorphism in C are the same as in Coh(X), while}

epimorphism are maps in Coh(X) whose cokernel is a torsion sheaf.

The following lemma is the main reason for introducing this notion in our discusison. Lemma 1.3.4. Let P be a slicing of a triangulated category D. Let I ⊂ R an interval such that, for a= inf I and b = sup I, we have b−a < 1. Then the subcategory P(I) ⊆ D is quasi-abelian.

The strict short exact sequences in P(I), i.e. composition of two maps where the former is kernel of the latter and the latter is cokernel of the former, are in a one-to-one correspondence with triangles in D whose vertices are objects of P(I).

Proof. It will be enough to apply the previous Lemma to the faithful inclusions P(I) ⊂ P([a, a+1)) and P(I) ⊆ P([b−1, b)). Now take a monomorphism A → E with E ∈ P(I) and A ∈ P([a, a + 1)), we want to prove that ϕ+_{(A) ∈ I; the first HN factor of A is a}

subobject of A and by composition it is also a subobject of E in P([a, a + 1)), hence ϕ(A1) ≤ ϕ+(E) ∈ I and so we are done. It follows from a similar argument that

P([b − 1, b)))P(I)[_.

The last statement is quite obvious.

We conclude the discussion with the following definition:

Definition 1.3.5. A quasi-abelian category C is said to be noetherian if every chain of strict epimorphisms

A0 A1 · · ·

sabilizes. It is said to be artinian if every chain of strict monomorphisms · · · ,→ A1 ,→ A0

stabilizes.

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1.3.2 Locally finite stability conditions

Definition 1.3.6. A slicing P of D is said to be locally finite if there exists η > 0 such that for all t ∈ R the quasi-abelian category P(t − η, t + η) is of finite length.

A stability condition σ = (Z, P) is locally finite if its slicing P is.

Remark. If σ = (Z, P) is a locally finite stability condition, then P(ϕ) is of finite length for any ϕ ∈ R, therefore any semistable object admits a finite Jordan-H¨older filtration as in the following definition.

Definition 1.3.7. Given a semistable object E ∈ P(ϕ) a Jordan-H¨older filtration (JH) of E is a finite chain of subobjects 0 = ˜E0 ⊆ ˜E1 ⊆ · · · ⊆ ˜En= E such that the quotients

Ei:= ˜Ei/ ˜Ei−1are stable of phase ϕ.

By a similar argument as the one for HN factors, one can see that the quotients Ei

are unique up to changin order; they are called JH factors of E.

We will study the set Stab(D) of all locally finite stability conditions. We may endow this set with the distance

d(σ, τ ) = sup

06=E∈D

|ϕ+_σ(E) − ϕ+_τ(E)|, |ϕ−_σ(E) − ϕ−_τ(E)|, logmσ(E) mτ(E)

we introduced before, but we will, instead, set a topology (which in turns will be equivalet to the one induced by the distance) on Stab(D) in a different way and prove some result using this definition.

Firstly let us consider the following generalized distance of slicings:

δ(P, Q) = sup 06=E∈D |ϕ+ P(E) − ϕ + Q(E)|, |ϕ − P(E) − ϕ − Q(E)|

Note that if δ(P, Q) = 0, then for any E ∈ P(ϕ) we have ϕ+_Q(E) = ϕ = ϕ−_Q(E), hence P ⊆ Q and we get as well Q ⊆ P, so P = Q. The triangular inequality and the symmetric properties are trivial, therefore δ is actually a distance.

Lemma 1.3.8. If P, Q are slicing of D, then δ(P, Q) = inf

ε∈R≥0{Q(ϕ) ⊆ P([ϕ − ε, ϕ + ε]) , ∀ϕ ∈ R} .

For the proof one can see Lemma 6.1 of [7].

Now fix σ = (Z, P) ∈ Stab(D) and set the following (generalized) norm on the (possibly infinite-dimensional) vector space Hom_Z(K(D), C):

||U ||σ = sup

|U (E)|

|Z(E)|: E is semistable in σ

.

Now for ∈ (0,1_/₈_{) consider the set}

B(σ) = {τ = (W, Q) ∈ Stab(D) : ||W − Z||σ <sin(π) and δ(P, Q) < } .

Our goal is to show that these sets (as σ and vary) form the basis for a topology on Stab(D). It is enough to show that τ ∈ B(σ) ⇒ ∃η > 0 : Bη(τ ) ⊆ B(σ).

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Lemma 1.3.9. If τ = (W, Q) ∈ B(σ), then there are constants k1, k2>0 such that

k1||U ||σ < ||U ||τ < k2||U ||σ, ∀ U ∈ HomZ(K(D), C) .

Remark. This statement ensure that while a stability condition moves inside B(σ), then

the norm || · ||τ remains equivalent to || · ||σ. In particular, after showing that the sets

B(σ) are open sets of a topology on Stab(D) we will conclude that the topology induced

by σ on Hom_Z(K(D), C) depends only on the connected component σ ∈ Σ ⊆ Stab(D). Proof. First notice that ∀σ(Z, P) ∈ Stab(D) and ∀η : 0 < η < 1₂, one has

|U (E)| < ||U ||σ

cos(πη)|Z(E)| , (1.3.1) for U ∈ Hom_Z(K(D), C) and E ∈ D with ϕ+

σ(E) − ϕ−σ(E) < η.

That’s true because Z(E) = Z(A1) + · · · + Z(An), U (E) = U (A1) + · · · + U (An)

where the Ai are HN factors of E with respect to σ; now the Ai are semistable, so by

definition we have |U (Ai)| ≤ ||U ||σ|Z(E)| and the vectors |Z(Ai)| lie inside a sector of

C bounded by an angle of πη, since η > ϕ+(E) − ϕ−(E).

Now, our hypotheses say that δ(P, Q) < and that ||W − Z||σ <sin(π), so (1.3.1)

with U = W − Z and η = 2 gives

|W (E) − Z(E)| < sin(π)

cos(2π)|Z(E)|

for any E semistable with respect to τ ; we can apply the previous inequality with η = 2 since the Lemma 1.3.8ensures that if E is τ −semistable of phase ϕ, then ϕ+

σ(E) < ϕ+,

ϕ−_σ(E) > ϕ − .

It follows, then, that there exists κ such that ∀E τ −semistable we have |Z(E)| < κ|W(E)|, otherwise for any n ∈ N we could find Ensemistable with respect to τ such that

|Z(En)| > n|W (En)|. Letting n → +∞ leads to a contradiction, since sin(π) < cos(2π)

for <1_/₈_{. Now we can use the last inequality and apply (}_1.3.1_{) to any U ∈ Hom}_Z_{(D, C)}

to get ||U ||τ < k2||U ||σ. The other inequality is similar.

Now, if we choose η < − δ(P, Q), then δ(Q, R) < η ⇒ δ(P, R) < . Moreover, for ρ= (U, R) stability condition, we have

||U − Z||σ ≤ ||U − W ||σ+ ||W − Z||σ <

1 k1

||U − W ||τ+ ||W − Z||σ,

so if η small enough (i.e. s.t. sin(πη) < k1(sin(π − ||W − Z||σ)), we can conclude that

(U, R) ∈ Bη(τ ) ⇒ (U, R) ∈ B(σ).

Hence we have constructed a topology on Stab(D). If we fix a connected component Σ ∈ Stab(D), the norms || · ||σ are all equivalent and they define a linear subspace

V = V(Σ) ⊆ Hom_Z(K(D), C) where ||U||σ <+∞ , ∀ U ∈ V ; therefore || · ||σ is actually

a norm on V . Now, if σ = (Z, P) ∈ Σ, then ||Z||σ = 1, therefore Z ∈ V and we get a

map Z : Σ → V which associate to any stablity condition in the connected component its central charge.

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1.3.3 The complex manifold Σ.

We are going to show that Z is a local homeomorphism, this will endow Σ with the structure of a complex manifold.

Lemma 1.3.10. Suppose σ= (Z, P), τ = (Z, Q) ∈ Stab(D) share the central charge. If δ(P, Q) < 1, then σ = τ . In particular Z is locally injective.

Proof. If, by contradiction σ 6= τ , then there is 0 6= E ∈ P(ϕ) such that E /∈ Q. Now, if E ∈ Q([ϕ, +∞)), then E ∈ Q([ϕ, ϕ + 1)) because δ(P, Q) < 1; but then Z(E) = Z(A1) + · · · + Z(An), where Aj are the HN factors with respect to Q, moreover, since

E /∈ Q(ϕ), there must be at least one Ai with phase different from ϕ. On the other hand

Z(E) = reiπϕ _{because it is σ−semistable of phase ϕ. This yields a contradiction. One}

cannot have E ∈ Q((∞, ϕ]) as well. Therefore, there is a triangle A → E → B → A[1] with 0 6= A ∈ Q((ϕ, ϕ + 1)) and 0 6= B ∈ Q((ϕ − 1, ϕ]). A similar argument tell us A /∈ P((−∞, ϕ]), so there is ϕ < ψ ∈ R, an object C ∈ P(ψ) and a nonzero map f: C → A. Since E ∈ P(ϕ), we have that any map C → E must be 0, in particular the composition C → A → E is 0 and f factors through B[−1]. Now, A ∈ Q((ϕ, ϕ + 1)) and B[−1] ∈ Q((∞, ϕ − 1]). This means B[−1] → A is 0 and so f = 0, a contradiction.

We state the following theorem, its proof is the Section 7 of [7].

Theorem 1.3.11. Let σ= (Z, P) ∈ Stab(D). Then ∃ 0 >0 such that given 0 < < 0,

and W ∈Hom_Z(K(D), C) with

|W (E) − Z(E)| < sin(π)|Z(E)|

for any σ−semsitable object E, there exists a slicing Q with τ = (W, Q) ∈ Stab(D) and δ(P, Q) < .

In particular, if < 1_/₂ _{the stability condition τ with central charge W and close}

enough to σ is unique.

Remark. It is enough to choose 0 <1/8, such that P(t − 40, t+ 40) is of finite length.

Note that Q(t − , t + ) ⊆ P(t − 2, t + 2), therefore if we choose 0 as we just said,

then Q is automatically locally finite; moreover if τ is constructed as in the theorem, then the real0/2 is suitable to repeat the construction starting from τ .

The previous results yield the following theorem:

Theorem 1.3.12. Fix a connected component Σ ⊆ Stab(D). Then the map Z : Σ → V(Σ) defined above is a local homeomorphism.

Proof. Given a point σ ∈ Σ, let Z = Z(σ) ∈ V = V(Σ) and consider 0 ∈ R as in

the previous remark. Now consider the open sets B0(σ) ⊆ Σ and {W ∈ V : |W −

Z| <sin(π0)|Z|} ⊆ V . By Theorem 1.3.11, we may infer that the former open set is

mapped, via Z, bijectively onto the latter one; this map is continuous, while the inverse map is continuous at least in Z. Also, up to changing 0 7→ ₂0 we make Z a local

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It follow immediately that each connected component Σ ⊆ Stab(D) is a ”Banach manifold”, i.e. a topological space locally modelled on the complete normed vector space V(Σ).

For the last part of this paragraph we assume D has a few additional hypotheses. These hypotheses will always hold true in the geometrical situation we are going to study in the following chapters.

Firstly we assume D is linear over a field k. We, thus, require, that HomD(A, B) is

a vector space over k for any two objects of D and that the composition map HomD(A, B) × HomD(B, C) → HomD(A, C)

is k−bilinear.

We assume also that D is of finite type over k, i.e. ⊕_i∈ZHomD(E, F [i]) is finite

dimensional. Then we can define a bilinear pairing on K(D), known in literature as the Euler form, via the formula

χ(E, F ) :=X

i∈Z

(−1)idimkHomD(E, F [i]) .

Now consider the numerical Groethendieck group N (D) := K(D)/K(D)⊥; if it has finite rank (as a Z−module), we will say D is numerically finite.

If D is a triangulated category of finite type, we can consider the set of numerical stability conditions StabN(D) ⊆ Stab(D), i.e. the set of stability condition whose central

charge factrors via N (D). Now, if D is numerically finite, each connected component Σ ⊆ StabN(D) yields a finite dimensional complex vector space V(Σ).

This discussion allows us to state the following

Corollary 1.3.13. If D is a numerically finite k−linear triangulated category over some field k. Each connected component Σ ⊆ StabN(D) is a finite dimensional complex

manifold whose complex structure is naturally inherited by the local homeomorphism Z : Σ → V(Σ).

1.3.4 The equivalence of the two topologies

Firstly we shall show that the function d(σ, τ ) defined above is actually a generalized distance.

The triangular inequality and the symmetric property are trivial and follow by a separate check on every single object in D.

It remains to show that d(σ, τ ) = 0 ⇒ σ = τ . Let σ = (Z, P), τ = (W, Q) be at distance 0; of course their slicing have to be the same (the argument is the same we used for δ). Moreover, if mσ(E) = mτ(E) for any E ∈ D, then it follows, in particular, that

|Z(A)| = |W (A)| for any A semistable (we do not state with respect to which stability condition A is semistable because the slicing are the same); so Z, W have the same phase and the same norm on semistable objects, hence they coincide on semistable objects and so they do on every object.

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Theorem 1.3.14. The topology of the metric space(Stab(D), d) is the same we described above.

Proof. To show that Bε(σ) are open in the topology induced by d it is enough to show

that there is a small ball centered in σ and contained in Bε(σ) for any σ, ε.

Now, if τ = (W, Q) is close enough to σ (let us say d(σ, τ ) < η) and E is σ−semistable, we have that the τ −HN factors of E lie inside and angle small enough, therefore mτ(E) is close enough to |W (E)|, i.e. |mτ(E) − |W (E)|| < (1 − cos(πη))|mτ(E)|, but

exp(−η)mσ(E) < mτ(E) < exp(η)mσ(E) and mσ(E) = |Z(E)|; therefore η sufficiently

small yields |W (E) − Z(E)| < sin(πε)|Z(E)|.

Now, for the reverse implication, we can again just check that any d−ball centered in σ contains a B(σ) for small enough.

Let r be the radius of a ball centered in sigma; of course if < r, the slicing of any τ ∈ B(σ) are close enough to P.

Now, if E is semistable with respect to σ and τ = (W, Q) ∈ B(σ), we have |W (E) −

Z(E)| < sin(π)|Z(E)|, moreover mσ(E) = |Z(E)|. Also, for c ∈ (1, +∞), we can

choose small enough such that we have mτ(E) < c|W (E)|, so mτ(E) < c|W (E)| <

c(1 + sin(π))|Z(E)| = c(1 + sin(π))mσ(E), for E σ−semistable.

If E is not σ−semistable and A1, . . . , An are its HN factors, we have that mτ(E) ≤

mτ(A1) + · · · + mτ(An), so we get

mτ(E) ≤ mτ(A1)+· · ·+mτ(An) < c(1+sin(π))(mσ(A1)+· · ·+mσ(An)) = c(1+sin(π)mσ(E) .

We reach conclusion because we can choose c, arbitrarily close to 1 and 0 respec-tively. (An inequality of the form mσ(E) < c0mτ(E) holds via a similar argument).

This theorem has a very important consequence: for any object E the functions ϕ±•(E), m•(E) : Stab(D) → R are continuous.

It follows

Corollary 1.3.15. Let E be an object of D, then the set of stability conditions σ such that E is σ−semistable is closed in Stab(D).

Proof. That set is just the locus ϕ+

σ(E) − ϕ−σ(E) = 0.

1.3.5 The action of fGL+(r, R) on Stab(D).

In this paragraph we will study the actions of two groups on Stab(D).

There is, obviously, a left action of Aut(D), the group of exact autoequivalences of D: if Φ is an exact autoequivalence of D, it induces an automorphism φ of K(D), therefore, for σ = (Z, P) ∈ Stab(D) we can define Φ(σ) = (Z ◦ φ−1,Φ∗(P)), where

Φ∗(P)(ϕ) = Φ(P(ϕ)). This group acts via isometries on Stab(D).

The second action we should study concerns fGL+(2, R): the universal cover of GL+(2, R). As one can see in Lemma 2.14 of [15], fGL+(2, R) can be thought as the set of the couples (T, f ) where T ∈ GL+_{(2, R) and f : R → R is an increasing function}

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such that f (t+1) = f (t)+1 and is linked to T via the following relation (just use R2 ∼_{= C}

and think T as an orientation preserving R−linear map on C):

T(r · exp(iπt)) = r0· exp(iπf (t)) .

Now, the right action of this group on Stab(D) can be described as (Z, P) · (T, f ) = (T−1◦ Z, f∗(P)), where f∗(P)(t) = P(f (t)).

Now it is possible to check that also this one is an action via isometries: Proposition 1.3.16. The two actions above commute.

Proof. Keep the same notation we introduced at the beginning of the paragraph, (Φ(Z, P))(T, f ) = (Z ◦ φ−1,Φ∗(P))(T, f ) = (T−1◦ Z ◦ φ−1, f∗(Φ∗(P))) ,

Φ((Z, P)(T, f )) = Φ(T−1◦ Z, f∗(P)) = (T−1◦ Z ◦ φ−1,Φ∗(f∗(P))) .

But

∀ t ∈ R f∗(Φ∗(P))(t) = Φ∗(P)(f (t)) = Φ(P(f (t))) = Φ(f∗(P)) = Φ∗(f∗(P))(t) ,

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Chapter 2

Stability conditions on algebraic

varieties

During the first part of this chapter we will denote by X a smooth projective variety over C, however the base field is not important throughout our discussion. The general discussion will be very short and soon we will restrict to a specific case and X will be a K3 surface. We are going to study the theory we developed in the previous chapter in the case when D(X) = Db_{(Coh(X)) is the bounded derived category of coherent sheaves}

of OX−modules.

The category D(X) is always of finite type (as defined in Paragraph 1.3.3) and therefore we can define the bilinear form over the Groethendieck group K(D(X))

χ(E, F ) =X

i

(−1)idim_CHomX(E, F [i]) ,

which will be called the Euler characteristic. Note that χ(OX, E) coincide with the

usual notion of Euler characteristic χ(E) =X

i

(−1)iHi(X, E) ,

moreover χ(E, F ) = χ(E∨_{⊗ F ).}

Again we will consider the numerical Groethendieck group N (X) = K(D(X))/K(D(X))⊥ which, as a consequence of the Riemann-Roch theorem, will be a finite rank free abelian group, thus our category will always be numerically finite. We will only consider numer-ical stability conditions, i.e. we will assume that the central charge Z factros through N (X); we will denote by Stab(X) the space of locally finite numerical stability condition, which corresponds to StabN(D(X)) with the notation of the previous chapter.

2.1 Basic results

Now we are going to restate one of the results we discussed in the previous chapter. But before we should note that, since the vector space N (X) ⊗_Z C is finite

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dimen-Chapter 2. Stability conditions on algebraic varieties

sional and endowed with a non-degenerate pairing χ(·, ·), there is a natural isomor-phism Hom_C(N ⊗ C, C) ∼= N ⊗ C. Hence, the central charge of a stability condition σ= (Z, P) factors through N (X) if anf only if there is a vector π(σ) ∈ N ⊗ C such that Z(E) = −χ(π(σ), E).

Theorem 2.1.1. For each connected componentΣ ⊆ Stab(X) there is a linear subspace V ⊆ N_{(X) ⊗ C such that the map we just introduced}

π: Σ → V , is a local homeomorphism onto an open subset of V .

In particular Σ is a finite dimensional manifold.

The previous theorem is just the translation of Corollary 1.3.13 from the general setting into the geometric one.

Remark. No example is known where the subspace from the previous theorem V is different from N ⊗ C. However we will usually assume V coincide with the whole space, thus we will need the following definition.

Definition 2.1.2. A connected component Σ ⊆ Stab(X) is full if the subspace V of the Theorem 2.1.1is equal to N (X) ⊗ C.

A stability condition σ will called full if it belongs to a full connected component. Now we are going to prove that the constant 0 in Theorem 1.3.11 can be chosen

uniformly in a full connected component. But before we will consider a particular case. Definition 2.1.3. A stability condition σ = (Z, P) on a derived triangulated category D is called discrete if Z(K(D)) ⊆ C is a discrete subgroup.

Lemma 2.1.4. If σ= (Z, P) is a discrete stability condition over a triangulated category D and 0 < < 1

2 is a fixed real number, then ∀ϕ ∈ R the quasi abelian category

P(ϕ − , ϕ + ) is of finite length.

In particular a discrete stabilty condition is locally finite.

Proof. Fix ϕ ∈ R and let A = P(ϕ − , ϕ + ). Then for any object A ∈ A its central charge lies inside the sector

S= {z = r · exp(iπψ) ∈ C : r > 0, ϕ − < ψ < ϕ + } ,

which is strictly contained in an half-plane in C. As a consequence we have that the function f : A → R defined as

f(A) = <(exp(−iπϕ) · Z(A))

is positive for any nonzero object A ∈ A. Moreover it extend to a group homomorphism K(A) → R, so for any exact sequence

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2.2. The case of curves

we have that f (B) = f (A) + f (C).

It follows that if E ∈ A, then any subobject or any quotient F of E is such that f(F ) < f (E), then its central charge lies in the compact region

{z ∈ S : <(exp(−iπϕ) · z) < f (E)} .

Therefore E has a finite number of subobject and quotient (a discrete contained in a compact set is finite), therefore any chain of subobjects or quotients starting from E must stabilize after finitely many steps.

Now we will show that an analogue result holds for any good stability condition σ without requiring that it is discrete.

Lemma 2.1.5. Let σ ∈Stab(X) be a full stability condition and let 0 < < 1

2 be a real

number.

Then for any ϕ ∈ R, the quasi abelian category P(ϕ − , ϕ + ) is of finite length. Proof. Let Σ ⊆ Stab(X) be the full connected component containing σ.

Firstly notice that if π(σ) ∈ N (X) ⊗ Q[i], then σ is discrete and the thesis holds. However N (X) ⊗ Q[i] is dense in N (X) ⊗ C, therefore we can choose a point in v ∈ N(X) ⊗ Q[i] arbitrarily close to π(σ). By the fullness of Σ and by the Theorem

1.3.11, for any η ∈ R>0 small enough there exist a (unique) stability condition τ =

(W, Q) ∈ Stab(X) such that π(τ ) = v and d(σ, τ ) < η. Just choose η + < 1

2 and P(ϕ − , ϕ + ) ⊆ Q(ϕ − − η, ϕ + + η) will be of finite

length.

It follows that, if σ is a full stability condition and < 1

8, then P(ϕ − 4, ϕ + 4) is

of finite length, therefore

Corollary 2.1.6. If σ ∈Σ is a full stability condition, one can always choose 0= 1₈ in

the setting of Theorem1.3.11.

2.2 The case of curves

During this section X will denote any smooth projective curve. We will follow the discussion in §2 of [17]

We will recall the classical notion of slope stability and we will see how it is related to our formalism.

Definition 2.2.1. If E is a vector bundle on a curve X, its slope is the real number µ(E) := deg E

rk E .

If E is a coherent sheaf, we can define its slope in the same way as we did for vector bundles, setting µ(E) := +∞ if rk E = 0.

A coherent E is said to be slope-(semi)stable if for any non trivial subbundle F ⊆ E we have the inequality

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Chapter 2. Stability conditions on algebraic varieties

In analogy with the Bridgeland’s stability conditions, any coherent sheaf has a unique Harder-Narasimhan filtration, i.e. a chain of subobjects

0 = E0⊆ E1⊆ · · · ⊆ En= E ,

such that any quotient Qi := Ei/Ei−1 is slope-semistable and

µ+(E) := µ(Q1) > · · · > µ(Qn) =: µ−(E) .

Now we can see that, at least in the case of curves, the notion of stability we intro-duced in the previous chapter is a natural gerealization of the classical notion we just defined.

Remark. The abelian category Coh(X) is, obviously, the heart of a bounded t-structure of D(X) = Db_(Coh(X)).

Let Z : Coh(X) → C be the stability function described by Z(E) = −deg(E) + irk(E). Note that, since X is a curve, any nonzero coherent sheaf E must have at least one of rk E or deg E different from 0.

Now we will explain how the stability condition we just constructed is equivalent to the slope stability.

If E is a coherent sheaf on X, then Z(E) = r · exp(iπϕ) with ϕ ∈ (0, 1] and its phase is related with the slope by the equation

µ(E) = − cot(πϕ) .

It follows that there is an increasing bijection between the slope of E and its phase ϕ ∈ (0, 1], hence any slope-(semi)stable sheaf is also (semi)stable with respect to the stability function and vice versa.

We will now discuss separately what happen for curves of genus g ≥ 1 and for P1_;

in the case of positive genus we will find out that all the numerically finite stability conditions are essentially the same as the slope stability (we will show that all of them are in the same fGL+(2, R)−orbit generated by the condition constructed above), while in the other case it turns out Stab(P1_{) ∼}_{= C}2_{, as shown in [}₂₂_].

2.2.1 Curves of positive genus.

The technical result that allow us to get our results for curves of positive genus is Lemma 7.2 of [13]:

Lemma 2.2.2. Let X be a projective curve of genus g(X) > 1. Suppose E ∈ Coh(X) is included in a triangle

A → E → B → A[1] , withHom≤0(A, B) = 0. Then A, B ∈ Coh(X)

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2.3. First results forK3 surfaces.

Theorem 2.2.3. If X has genus g(X) > 1, then the action of fGL+(2, R) on Stab(X) is free and transitive, therefore

Stab(X) ∼= fGL+(2, R) .

Proof. Fix any σ = (Z, P) ∈ Stab(X). If x ∈ X is a point and Ox its structure

sheaf, then Ox is semistable, otherwise there would be a triangle A → Ox → B with

A semistable of phase ϕ(A) = ϕ+_(O

x) and ϕ+(B) < ϕ(A), therefore we would have

Hom≤0(A, B) = 0, so by the previous lemma A, B ∈ Coh(X) and the triangle would correspond to an exact sequence in Coh(X)

0 → A → Ox → B → 0 ,

with A, B 6= 0, this is impossible and Ox is semistable.

Moreover a simiar argument shows that if Ox is not stable, then its stable factors

are all isomorphic to a single object K ∈ Coh(X); this mean K ∼= Ox and therefore Ox

is stable.

A similar argument lead to the conclusion that every line bundle is stable too. Let σ = (Z, P) ∈ Stab(X). Note, that saying σ is a numerical stability condition means it factors via Chern character. It may be thought, therefore as a map which associate to a couple (rk, deg) a complex number z ∈ C, hence we can see it as a map from the singular cohomology H∗_{(X, R) ∼}_{= R}2 _{to C. Now suppose it is not an isomorphism,}

therefore im(Z) ⊆ l, where l ⊆ C is a real line. If H is an hyperplane section on X, we have a nonzero map L(−H) → L for any line bundle L, but L, L(−H) are stable objects, so ϕ(L) > ϕ(L(−H)), but Z(K(D)) is inside a line; it means ϕ(H) ∈ Z+and so

ϕ(L + nH) → +∞ when n → +∞. But ∀n ∈ N there are nonzero morphisms L + nH → Ox and this is impossible because the former is an object with phase aribitrarily high,

and the latter a stable object with fixed phase.

Take a line bundle L and the structure sheaf Ox of some point on X, they will be

stable of certain phases ϕ(L), ϕ(Ox). There is of course a nonzero map L → Ox, i.e

Hom(L, Ox) 6= 0; we have also, by Serre’s duality, Hom(Ox, L[1]) ∼= Hom(L ⊗ KX, Ox) 6=

0. It follows the double inequality ϕ(Ox) − 1 ≤ ϕ(L) ≤ ϕ(Ox). This means that Z is an

orientation preserving isomorphism.

Therefore, up to acting by an element of fGL+(2, R) one can assume that there is a point such that Ox has phase 1 (hence for any line bundle L, 0 ≤ ϕ(L) ≤ 1 holds) and

the central charge is Z(E) = − deg(E) + i rk(E). Moreover, for any y ∈ X, we have Z(Oy) = −1, so ϕ(Oy) = 1 (it must be an odd integer, but the previous inequality forces

it to be 1).

This means that Coh(X) = (0, 1], therefore this stability condition coincides with that one we illustrated before.

2.3 First results for K3 surfaces.

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2.3.1 Classical stability notions

Among the notions of stability commonly used in literature, we will recall slope and Gieseker stability conditions.

Let ω ∈ NS(X) be an ample divisor class, or more in general, let ω ∈ Amp(X) be in the ample cone.

For a torsion-free sheaf E we will define the ω−slope as

µω(E) =

c1(E) · ω

rk(E) .

A torsion-free sheaf is said to be µω−(semi)stable if for every torsion-free subsheaf A ,→

E one has µω(A) < (≤)µω(E).

This definition can be extended to torsion sheaves as well by setting µω(E) = +∞.

Similarly to the previous cases, one can define HN and JH filtrations; likewise for maximum and minimum phases ϕ+_{, ϕ}−_{, maximum and minum slope µ}+

ω, µ−ω are defined

as the slope of the first and last HN factors of a sheaf.

Unlike the case of curves, slope stability does not arise from a stability condition on D(X). Indeed, the function Z(E) = −c1(E) · ω + i rk(E) is zero on skyscreaper sheaves,

which belong to Coh(X).

Now consider the map which sends an object E ∈ D(X) into its Mukai’s vectror: Definition 2.3.1. Given E ∈ Coh(X) (and extending by linearity to the Groethendieck group K(D(X))), the Mukai’s vector of E is

v(E) := ch(E) ·pTd(X) = (rk(E), ch1(E), ch2(E) + rk(E)) ∈ H∗(X, Z) .

The Mukai’s vector, actually gives the identification

v(E) ∈ H∗(X, Z) ∩ Ω⊥∼= Z ⊕ NS(X) ⊕ Z = N (X) , where Ω is the symplectic form on the K3 surface X.

If v = (r, ∆, s), w = (r0_,_∆0_{, s}0_{) ∈ N (X) ⊗ C, we will consider the Mukai’s pairing}

hv, wi:= ∆∆0_{− rs}0_{− r}0_{s. For any E, F ∈ Coh(X), it satisfies}

hv(E), v(F )i =X

i

(−1)i+1dim_CHomi_X(E, F ) = −χ(E, F ) .

Note that this pairing is nondegenerate, by definition of N (X).

For semplicity we will denote homi_X(E, F ) = dim_CHomX(E, F ) and hi(· · · ) =

dim_CHi(· · · ).

Using the Mukai’s vector we can define the Gieseker stability notion. Again fix ω ∈Amp(X).

The Hilbert polynomial of an object E ∈ D(X) is

Pω(E, t) :=

rk(E)ω2

2 t

2_{+ ch}

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2.4. Construction of stability conditions

as one can notice, this depends only on the Mukai’s vector of E.

The reduced Hilbert polynomial pω(E, t) is obtained by the previous one by dividing

it for its leading coefficent, which will be rk(E)ω2

/2if E has positive rank.

A sheaf E is ω−Gieseker (semi)stable if for any subsheaf A ⊆ E one has p(A, t) < (≤)p(E, t) for t >> 1 .

As for slope stability, we cannot see Gieseker stability as a Bridgeland’s condition. But later, when we will study the ”large volume limit”, we will see that there are similarities between the two notions.

2.3.2 Sheaves on K3 surfaces.

In this paragraph we will study some generalities about sheaves on X and more in general about objects in the derived category. We first introduce an important class of objects in D(X), that constitute the only obstructions to constructing our stability conditions. Definition 2.3.2. E ∈ D(X) is a spherical object if HomX(E, E) ∼= Hom2X(E, E) ∼= C

and Homi

X(E, E) = 0 for i 6= 0, 2.

When studying the sheaves on a X, it is essential to study the functor Hom together with its derived functors.

Serre’s duality, in the K3 surface case, i.e. the isomorphism Homi(E, F ) ∼= Hom2−i(F, E)∨, together with the Euler characteristic and Riemann-Roch theorem, often carry enough information to determine completely Homi

X(E, F ). As we can see in the following results:

Lemma 2.3.3. If E ∈ D(X) is stable in some stabilty condition σ ∈ Stab(X), or is a µω−stable sheaf for some ω ∈ Amp(X), then

2 + v(E)2 _{= hom}1

X(E, E) ≥ 0 ,

with the equality reached if and only if E is spherical.

Proof. Since E is stable, it is automatically satisfied HomX(E, E) ∼= C; indeed a nonzero

endomorphism ψ of a stable object (no matter which notion of stability holds) with non-trivial kernel and image, must satisfy ϕ+_{(ker ψ) < ϕ(E) < ϕ}−_{(im ψ) ≤ ϕ}+_{(im ψ) <}

ϕ(E) (and an analogue with µω instead of ϕ) which are impossible. Thus End(E) =

Aut(E) ∪ {0}, hence the finitely generated C−algebra End(E) is a field, so we must have End(E) ∼= C. Moreover Hom2X(E, E) = C by Serre’s duality.

Now, v(E)2 _{= hom}1_{(E, E) − hom}0_{(E, E) + hom}2_{(E, E) = hom}1_{(E, E) − 2 and we}

are done by recalling that hom1(E, E) = 0 if and only if E is spherical.

2.4 Construction of stability conditions

We will follow the construction of Bridgeland in §6 of [8] to give some examples of stability conditions over an algebraic K3 surface X.

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Consider the following pair of subcategories of D(X) (T , F) = (T (β, ω), F(β, ω)), where

T = {E ∈ Coh(X) : ∀E F with F torsion-free, µω(F ) > β · ω}

F = {E ∈ Coh(X) : E is torsion free and ∀F ,→ E, µω(F ) ≤ β · ω}

This pair of subcategories is a torsion pair, as defined in the following

Definition 2.4.1. A pair (T , F) of full subcategory of an abelian category A is called torsion pair if the following are satisfied:

1. T =⊥F and F = T⊥;

2. For any A ∈ A there are F ∈ F, T ∈ T such that 0 → T → A → F → 0 is exact. We can now construct an abelian category A(β, ω) ⊆ D(X) as

A(β, ω) = {E ∈ D(X) : Hi(E) = 0 for i /∈ {0, 1}, H−1(E) ∈ F and H0(E) ∈ T } .

The abelian category A(β, ω) is the heart of a bounded t-structure over D. Note that the hearth of the t-structure does not change if we multiply ω for a positive real number.

Now we will define a central charge as

Zβ,ω(E) := hexp(β + iω), (r, ∆, s)i = ∆ · β − s −

r 2(β

2_{− ω}2_{) + i(∆ − rβ)ω .}

The following lemma says when Zβ,ω defines actually a stability function over A.

Lemma 2.4.2. Take a pair β, ω as above. Then the group homomorphism defined above is a stability function over A providing that for any spherical object E ∈ D(X) one has Z(E) /∈ R≤0. In particular, if ω2 >2, then Zβ,ω is always a stability function.

Proof. Take any E ∈ A(β, ω). If E is a sheaf supported on a curve, then Z(E) always lies in the upper half plane, while if E is a skyscreaper sheaf, Z(E) ∈ R<0.

Let E be a torsion-free µω−stable sheaf; if µω(E) > β · ω, then Z(E) belongs to the

upper half plane, while if µω(E) < β · ω, then Z(E[1]) belongs to the upper half-plane.

If µω(E) = β · ω, then Z(E[1]) ∈ R; we only have to check that in this case Z(E) ∈ R>0.

Let v(E) = (r, ∆, s); by Lemma 2.3.3, we have that v(E)2 _{≥ −2 with equality if and}

only if E is spherical, but if E is not spherical v(E)2 _{≥ 0. Now (∆ − rβ)ω = 0, hence}

by Hodge index theorem it follows (∆ − rβ)2_{≤ 0, so}

Z(E) = 1 2r (∆

2_{− 2rs) + r}2_ω2_{− (∆ − rβ)}2_≥ 1

2r(−2 + r

2_ω2_{) ,}

moreover, if E is not spherical, Z(E) ≥ 1 2rω

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2.4. Construction of stability conditions

As we can see from the previous proof, the central charge of any skyscreaper sheaf Ox belongs to the negative half real line, moreover it is a torsion sheaf, hence it belongs

to the heart of the t-structure. It follows that Ox is always semistable with respect to

the stability function Z on A (and it will be semistable on D(X) once we prove that Z has the Harder Narasimhan property as in Definition 1.2.2). The following lemma shows that Ox is actually stable:

Lemma 2.4.3. For any β, ω ∈ NS(X), with ω ∈ Amp(X) and any point x ∈ X, the skyscreaper sheaf Ox is a simple object in the abelian category A(β, ω).

Proof. If, by contradiction, there is a short exact sequence 0 → A → Ox → B → 0,

then we have also the following exact sequence (which arises by taking the long exact sequence in cohomology):

0 → H−1(B) → H0(A) → Ox → H0(B) → 0 .

Since Ox is a torsion sheaf, H−1(B) and H0(A) must have the same slope stable factors,

which is impossible since H−1(B) ∈ F and H0_{(A) ∈ T , unless both of them are torsion}

sheaves. Moreover H−1(B) ∈ F must be torsion-free, hence H−1(B) = 0. Therefore the objects in the previous short exact sequence are coherent sheaves; this means Ox is

simple in A.

To prove that the stability function defined above satisfies the HN property, we need to show it for a particular case (it would be enough to show it for a couple (β, ω)); after having studied the geometry of Stab(X) we will be able to extend the result to the remaining couples (β, ω).

Lemma 2.4.4. If β, ω ∈NS(X) ⊗ Q, then the stability function Z as above has the HN property.

Moreover those stability conditions are locally finite.

Remark. Since β, ω are rational, the image of the central charge Z is a lactice of C. The proof follows proposition 7.1 of [8].

Proof. The first part will follow after having showed that the hypotheses of Criterion

1.2.3are satisfied.

Fix E ∈ A suppose we have a chain of subobjects in A: · · · ⊆ Ei+1⊆ Ei⊆ · · · ⊆ E0= E

with increasing phases ϕi+1 > ϕi. Consider the short exact sequences 0 → Ei+1 →

Ei → Fi → 0 in A; since =Z(Fi) ≥ 0, we have that =Z(Ei+1) ≤ =Z(Ei) and since the

set of values for =Z(·) is disctete, we have that =Z(Ei+1) = =Z(Ei) for all i >> 1.

Hence up to dropping the first terms of the sequence we may assume =Z(Fi) = 0, ∀i.

But Fi ∈ A ⇒ <Z(Fi) < 0, hence =Z(Ei) = =Z(Ei+1) and <Z(Ei) < =Z(Ei+1), this

Bridgeland's stability conditions and applications to the geometry of surfaces.

Universit`

a degli Studi di Pisa

Bridgeland’s stability and its application to the gemoetry of

surfaces

Contents

Introduction

Chapter 1

Stability on triangulated

categories

1.1

Definition and basic properties

1.2

Stability funcions on abelian categories and t-structures

1.3

The topological space Stab(D)

Chapter 2

Stability conditions on algebraic

varieties

2.1

Basic results

2.2

The case of curves

2.3

First results for K3 surfaces.

2.4

Construction of stability conditions