Stability conditions on higher dimensional projective varieties
by Yucheng Liu
B.S. in Mathematics, Zhejiang University M.S. in Mathematics, Zhejiang University
A dissertation submitted to
The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy
April 15th, 2020
Dissertation directed by
Emanuele Macr`ı Professor of Mathematics
1 Acknowledgements
First and foremost, I would like to acknowledge my supervisor Emanuele Macr`ıfor his patient guidance and continued support. I am very grateful to him for taking me as a student when I barely know anything in algebraic geometry. Thanks go to Jonathan Mboyo Esole, Alina Marian, and Alexander Martsinkovsky for joining my defense committee and for their comments and suggestions for my dissertation. I am indebted to Alex Perry for his careful reading of Chapter 1 and first two sections of Chapter 3. His comments on grammar and mathematics are very beneficial. I would also like to thank Arend Bayer, Aaron Bertram, Chunyi Li, Paolo Stellari, Yukinobu Toda and Xiaolei Zhao for helpful discussions and suggestions. Special thanks to Hongyu Shi, without her love and support, this journey would be much less interesting and funny. Finally, I would like to thank my parents and brother for the consistent inspiring support and care.
2 Abstract of Dissertation
In this thesis, we discuss the theory of stability conditions on higher dimensional projec- tive varieties. By the observation of polynomial structure on the global heart constructed by Abramovich and Polishchuk in [AP06] and [Pol07], we are able to establish some inequalities on the coefficients of the polynomial. These inequalities turn out to be useful in constructing stability conditions.
≤1 We provide a construction of stability conditions on DS (see 2.2.1 and 3.3.2 and for the notion). In particular, this gives us a family of stability conditions on X × C from a stability condition on X, where C is an integral curve. This establish the existence of stability conditions on product of curves over any algebraically closed field.
3 Table of Contents
Acknowledgements2
Abstract of Dissertation3
Table of Contents4
Disclaimer 5
Introduction6
1 Stability conditions and sheaves of t-structures 10 1.1 Bridgeland Stability conditions...... 10 1.2 Sheaf of t-structures and polynomial functions...... 16
2 Some mathematical structures on AS 30 2.1 Positivity of the coefficitens...... 30
2.2 Abelian subcategories in AS ...... 34 2.3 Weak stability conditions and torsion pairs...... 42
3 Constructing stability conditions 50 3.1 Product over a curve...... 50 3.2 Large volume limit and support property...... 58
≤1 3.3 Stability conditions on DS ...... 68
4 Disclaimer
Chapter 1 and the first two sections of Chapter 3 are adapted from the paper [Liu19]: “Stability conditions on product varieties”. To appear in Journal f¨urdie reine und ange- wandte Mathematik.
5 Introduction
The notion of stability conditions was introduced by Bridgeland in [Bri07], motivated by Douglas’s work on D-branes and Π-stability (see [Dou02]). The theory was further studied by Kontsevich and Soibelman in [KS08]. In general, stability conditions are very difficult to construct: while we have a very good knowledge in the case of curves and surfaces (see [Mac07], [Bri08] and [AB13]), starting from 3-folds no example was known on varieties of general type or Calabi-Yau varieties in dimension 4 or higher (for Calabi-Yau threefolds, see [MP15], [BMS16] and [Li19]). In this thesis, we try to construct stability conditions for triangulated categories on higher dimensional projective varieties. In this thesis, all algebraic varieties are defined over a field k. The field k can be of any characteristics. We always assume that k is algebraically closed, but I believe this condition is not necessary, we will remark on how to prove the same results in the case when k is not algebraically closed, especially when k is a finite field. Let X be a projective variety, S be a smooth projective variety. And let σ = (A,Z) be a stability condition on the bounded derived category of coherent sheaves D(X). The following theorem is the main theorem of this thesis.
Main Theorem. Assume that the image of the central charge Z is discrete. Then there ex-
≤1 ists a continuous family of stability conditions on DS , parametrized by R>0 ×R>0, associated with σ.
≤1 ≤1 For the notion of DS , please see Definition 3.3.2. In the case when S is a curve, DS = D(X ×S), the bounded derived category of coherent sheaves on X ×S, and we get the main
6 theorem in [Liu19]. There is one thing worth noticing, the main theorem includes the case when X is singular. In that case, though the triangulated category of perfect complexes P erf(X × S) is a very interesting subject to study, We will focus on D(X × S), the bounded derived categories of coherent sheaves on X × S. This is because of the obstruction of existence of bounded t-structure on P erf(X × S) (see [AGH19]). Theorem holds in more general situations. For instance, the triangulated category D(X) is replaced by an admissible subcategory D ⊂ D(X) and D(X × C) is replaced by the base
+ change category DC (see [BLMS17] and [BLM 19]). Special cases in dimension three were studied in [Kos18]. One of the interesting Corollaries is the following one.
Main Corollary. Let C1, ··· ,Cn be smooth projective curves. Then stability conditions exist on D(C1 × · · · × Cn).
In the case when n = 3, some related results appeared in [Sun19a] and [Sun19b]. The techniques are completely different. An important special case of Corollary is when all curves are elliptic curves. This gives examples of stability conditions on Calabi-Yau varieties of any dimension. In this case, the mirror symmetry version of this statement, for Fukaya categories of products of elliptic curves has been announced by Kontsevich in [Kon15]. The starting point of our construction is the construction by Abramovich and Polishchuk in [AP06] and [Pol07] of a heart of bounded t-structure on D(X × S), where S is any quasi- projective variety of finite type. Then we observed that there exists polynomial structure on the global heart. According to the degree of the polynomial, we find some abelian subcategories in the global heart. The next step is using the global slicing constructed by Bayer and Macr`ıin [BM14a]. This will give us some quadratic inequalities, which can be viewed as a generalization of Bogomolov-Gieseker inequalities on product varieties. In fact, we can use that method to
7 establish some Bogomolov-Gieseker inequalities in new cases. But unfortunately, I do not have enough time to include this application in the thesis, because I need to graduate by May this year. Besides the generalized Bogomolov-Gieseker inequalities, there are some other interesting mathematical structures on the global heart, for example, some abelian subcategories and torsion pairs in the global heart. In particular, we provide some interesting cuts of the global heart into torsion pairs. We can get some interesting inequalities after tilting with respect to that kind of torsion pairs. This is the content of chapter 2. The last part of this thesis is devoted to constructing stability conditions. We firstly give a construction of stability conditions in the case when S is a curve. In this case, the polynomial is a linear polynomial. Using the quadratic inequality in Chapter2, we can construct stability conditions on X × S. Then we apply the construction on some triangulated subcategories on the product variety when S is of higher dimension.
Outline of the thesis
The first chapter is almost identical to the first sections of paper [Liu19], it is devoted to introduce some basic notions and results on stability conditions, sheaves of t-structures and the polynomial structure. In the second chapter, we discuss some mathematical structures on the global heart and the generalized Hilbert polynomials. This includes some positivity of the coefficients of the polynomial, abelian subcategories of the global heart and torsion pairs. The construction of stability conditions is included in the last chapter. The main con- struction is included in Section 3.1. We prove a family of quadratic inequalities in Section 3.2, these quadratic inequalities are stronger than what we used in the construction.
8 Notations and Conventions
In this paper, all varieties are integral algebraic varieties over a field k, a curve is such a variety of dimension 1. We will use D(X) rather than the usual notation Db(CohX) to denote
the bounded derived categories of coherent sheaves on X. We set H = {z ∈ C | Im(z) > 0}. All functors are derived unless otherwise specified. We use k(x) to denote the skyscraper sheaf supported on the point x. We set Arg(z), Im(z), and Re(z) to be the argument, the imaginary part, and the real part of a complex number z respectively.
9 Chapter 1
Stability conditions and sheaves of t-structures
1.1 Bridgeland Stability conditions
In this section, we review the definition and some basic results on weak stability conditions (See [BBD82], [Pol07], [Bri08], [KS08] and [BMT14]).
Definition 1.1.1. Let D be an triangulated category. A t-structure on D is a pair of full subcategories (D≤0, D≥0) satisfying the condition (i) and (ii) below. We denote D≤n =
D≤0[−n], D≥n = D≥0[−n] for every n ∈ Z. Then the conditions are: (i) Hom(X,Y ) = 0 for every X ∈ D≤0 and Y ∈ D≥1; (ii) D≤−1 ⊂ D≤0 and D≥1 ⊂ D≥0. (iii) every object X ∈ D fits into an exact triangle
τ ≤0X → X → τ ≥1X → · · ·
with τ ≤0X ∈ D≤0, τ ≥1X ∈ D≥1. The heart of the t-structure is A = D≤0 ∩ D≥0. It is an abelian category (see [HT07, Theorem 8.1.9]). The associated cohomology functors are defined by H0 = τ ≤0τ ≥0, Hi(X) =
10 H0(X[i]).
A t-structure on D gives a Z-grading on D. The following definition of slicing is a refinement of t-structure, and it gives us a R-grading on D.
Definition 1.1.2. A slicing on a triangulated category D consists of full subcategories
P(φ) ⊂ D for each φ ∈ R, satisfying the following axioms: (a) for all φ ∈ R, P(φ + 1) = P(φ)[1],
(b) if φ1 > φ2 and Aj ∈ P(φj) then HomD(A1,A2) = 0, (c) for every 0 6= E ∈ D there is a sequence of real numbers
φ1 > φ2 > ··· > φm
and a sequence of morphisms
f1 f2 fm 0 = E0 −→ E1 −→· · · −→ Em = E
such that the cone of fj is in P(φj) for all j.
Definition 1.1.3. A weak stability condition on D consists of a pair (P,Z), where P is a
slicing and Z : K(D) → C is a group homomorphism such that the following conditions are satisfied:
(a) If 0 6= E ∈ P(φ) then Z(E) = m(E)exp(iπφ) for some m(E) ∈ R≥0. v g (b) (Support property) The central charge Z factors as K(D) −→ Λ −→ C, where Λ is a finite rank lattice, v is a surjective group homomorphism and g is a group homomorphism,
and there exists a quadratic form Q on Λ ⊗ R such that Q|ker(g) is negative definite, and Q(v(E)) ≥ 0, for any object E ∈ P(φ).
Remark 1.1.4. If we require m(E) to be strictly positive in (a), then the pair (P,Z) is called a stability condition. By [Bri08, Lemma 2.2], there is a S1 action on the space of stability conditions. Specifically, for any element eiθ ∈ S1, eiθ · (Z, P) = (Z0, P0) by setting Z0 = eiθZ and P0(φ) = P(φ − θ).
11 There is an equivalent way of defining a stability condition, which will be more frequently used in this paper. Firstly, we need to define what is a (weak) stability function Z on an abelian category A.
Definition 1.1.5. Let A be an abelian category. We call a group homomorphism Z :
K(A) → C a weak stability function on A if, for E ∈ A, we have Im(Z(E)) ≥ 0,with Im(Z(E)) = 0 =⇒ Re(Z(E)) ≤ 0. If moreover, for E 6= 0, Im(Z(E)) = 0 =⇒ Re(Z(E)) < 0, we say that Z is a stability function.
Definition 1.1.6. A weak stability condition on D is a pair σ = (A,Z) consisting of the
heart of a bounded t-structure A ⊂ D and a weak stability function Z : K(A) → C such that (a) and (b) below are satisfied: (a) (HN-filtration) The function Z allow us to define a slope for any object E in the heart A by
Re(Z(E)) − Im(Z(E)) if Im(Z(E)) > 0, µσ(E) := +∞ otherwise.
The slope function gives a notion of stability: An nonzero object E ∈ A is σ semi-stable if for every proper subobject F , we have µσ(F ) ≤ µσ(E). We require any object E of A to have a Harder-Narasimhan filtration in σ semi-stable ones, i.e., there exists a unique filtration
0 = E0 ⊂ E1 ⊂ E2 ⊂ · · · ⊂ Em−1 ⊂ Em = E
such that Ei/Ei−1 is σ semi-stable and µσ(Ei/Ei−1) > µσ(Ei+1/Ei) for any 1 ≤ i ≤ m. (b) (Support property) Equivalently as in Definition 1.1.3, the central charge Z factors
v g as K(D) −→ Λ −→ C. And there exists a quadratic form Q on ΛR such that Q|ker(g) is negative definite and Q(v(E)) ≥ 0 for any σ semi-stable object E ∈ A.
12 Remark 1.1.7. Similarly, we call (A,Z) a stability condition if Z is a stability function on A. Sometimes, we call the pair (A,Z) a weak pre-stability condition when they just satisfy condition (a). The equivalence of these two definitions is given by setting P(φ) to the full subcategory consisting of σ semi-stable objects in A.
If Z has discrete image in C, and A is Noetherian, then condition (a) is satisfied auto- matically.
There is an important operation called tilting with respect to a torsion pair, which is very useful for constructing stability conditions.
Definition 1.1.8. A torsion pair in an abelian category A is a pair of full subcategories
(T , F) of A which satisfy HomA(T,F ) = 0 for T ∈ T and F ∈ F, and such that every object E ∈ A fits into a short exact sequence
0 → T → E → F → 0
for some pair of objects T ∈ T and F ∈ F.
Remark 1.1.9. In this paper, most torsion pairs are coming from weak stability conditions σ = (A,Z). In fact, let
T = {E ∈ A | µσ,min(E) > 0} and F = {E ∈ A | µσ,max(E) ≤ 0}
be a pair of full subcategories, where µσ,min(E) is the slope of the last HN-factor of E and
µσ,max(E) is the slope of the first HN-factor of E. It is easy to see this is a torsion pair.
Lemma 1.1.10 ([HRS96, Proposition 2.1]). Suppose A is the heart of a bounded t-structure on a triangulated category D, (T , F) is a torsion pair in A. Then A# = hT , F[1]i is a heart of a bounded t-structure on D.
In this paper, we are interested in the case when D is the bounded derived category of coherent sheaves on an algebraic variety X or an admissible component of it. From now on,
13 X will be a smooth projective variety over an algebraically closed field k, and D(X) will be the bounded derived category of coherent sheaves on X. In the case when D is the bounded derived category of coherent sheaves on X, there is one special family of stability conditions worth noticing.
Definition 1.1.11. A stability condition σ on X is called geometric if all skyscraper sheaves k(x) are σ-stable with the same phase.
We conclude this section by providing some examples of stability conditions.
Example 1.1.12. Let C be a smooth projective curve, then the pair (Coh(C), −deg+i·rank) is a stability condition on D(C). This is a geometric stability condition.
Example 1.1.13. Let Q be a quiver, and A be the category of representations of Q. Let
Z : K(A) → C, such that Z(kv) ∈ H ∪ R<0 for all vertexes v where kv is the simple module supported on the vertex v. Then the pair (A,Z) is a stability condition on Db(A).
Example 1.1.14. A slightly nontrivial example of stability conditions are stability condi- tions on surfaces, it was firstly constructed by Bridgeland in [Bri08] for K3 surfaces, the construction was then generalized by Arcara and Bertram to any smooth projective surfaces in [AB13]. Let X be a smooth projective surface, we consider central charge of the following form
Z −(B+iw) ZB,w(E) = − e ch(E) S where w is an ample R-divisor and B is an arbitrary R-divisor. Let σw = (Coh(S), −c1(E) · w + i · rk(E)) be a weak stability condition on D(S), then we get a torsion pair (T , F) given by
T = {E ∈ A | µσw,min(E) > B · w} and F = {E ∈ A | µσw,max(E) ≤ B · w}.
This gives us a tilted heart AB,w = hT , F[1]i, then we have the following theorem in [AB13].
14 Theorem 1.1.15. [AB13, Corollary 2.1] Each pair (AB,w,ZB,w) is a Bridgeland stability condition.
Proof. We only prove that ZB,w is a stability function on AB,w. The proof is in [AB13], we include the proof for reader’s convenience. For each E ∈ AB,w, we have a short exact sequence 0 → H−1(E)[1] → E → H0(E) → 0 with H−1(E) ∈ F and H0(E) ∈ T . It suffices to prove
(1) ZB,w(E) ∈ H ∪ R<0 for all nonzero torsion sheaves on S,
(2) ZB,w(E) ∈ H ∪ R<0 for all σw-stable sheaves with µσw > B · w,
(3) ZB,w(E[1]) ∈ H ∪ R<0 for all σw-stable sheaves with µσw ≤ B · w. We can compute that
B2 w2 Z (E) = (−ch (E) + B · c (E) − rk(E)( − ) + iw · (c (E) − rk(E)B). B,w 2 1 2 2 1
In (1), either E is supported in dimension 1 and Im(ZB,w(E)) = c1(E) · w > 0 since c1(E) is effective, or else E is supported in dimension 0, in which case Im(ZB,w(E)) = 0, but Re(ZB,w(E)) = −ch2(E) < 0. So (1) is proved.
In (2), Im(ZB,w(E)) = w · (c1(E) − rk(E)B) = rk(E)(µσw (E) − w · B) > 0. Similarly, in
(3), if µσw (E) < B ·w, then Im(ZB,w(E)) < 0, so Im(ZB,w(E[1]) > 0. Finally, if µσw = B ·w and E is σw semi-stable. then we have
c2(E) B2 w2 Re(Z (E)) ≥ − 1 + B · c (E) − rk(E)( − ) B,w 2 · rk(E) 1 2 2 1 w2 = − (rk(E)B − c (E))2 + rk(E) . 2 · rk(E) 1 2
But µσw = B · w implies (rk(E)B − c1(E)) · w = 0, hence Hodge index theorem implies that Re(ZB,w(E)) > 0. Hence (3) is proved.
15 Remark 1.1.16. As we can see from the proof, in the expression of ZB,w
B2 w2 Z (E) = (−ch (E) + B · c (E) − rk(E)( − ) + iw · (c (E) − rk(E)B), B,w 2 1 2 2 1
w2 the term 2 can be replaced by any positive real numbers.
We only proved that ZB,w is a stability function on AB,w. The proof of HN property can be found in [BM11, Section 4].
1.2 Sheaf of t-structures and polynomial functions
The notion of t-structures was introduced in [BBD82] to study the derived category of constructible sheaves. This notion turns out to very interesting on D(X), since Bridgeland introduced a refined notion of slicing (see Definition 1.1.2) in his paper [Bri07]. Abramovich and Polishchuk studied the family of t-structures in [AP06] and [Pol07]. Suppose there exists a stability condition (A,Z) on D(X), where we assume A is Noethe- rian, and the image of Z is discrete. Let S be a quasi-projective variety of finite type, and O(1) be an ample line bundle on
S. Abramovich and Polishchuk defined a sheaf of t-structures and a global heart AS for D(X × S) in their papers [AP06] and [Pol07]. Here we summarize some of their beautiful properties we need in the next.
(1) The global heart AS is independent of the choice of ample line bundle. (2) If S is projective, then
∗ AS = {E ∈ D(X × S) | p∗(E ⊗ q (O(n))) ∈ A for all n 0}, where p, q are projections from X × S to X and S respectively. (3) The functor p∗ : D(X) → D(X × S) is t-exact, where p is the projection from X × S
16 to X.
(4) For every closed immersion iT : T,−→ S, the functor iT ∗ : D(X × T ) → D(X × S) is
∗ t-exact and iT is t-right exact.
(5) The heart AS is Noetherian. We also need some definitions from [AP06, Section 3].
Definition 1.2.1. We call an object E ∈ AS to be S-torsion if it is the push forward of an object E0 ∈ D(X × T ) for some closed subscheme T ⊂ S.
An object E ∈ AS is torsion free with respect to a closed subscheme T if it contains no nonzero torsion subobject supported on T . In this case we say that E is T -torsion free. We say that E is torsion free if it contains no torsion subobject, i.e., it is torsion free with respect to any closed subscheme in S.
Definition 1.2.2. The object E ∈ AS is called t-flat if Es ∈ A for arbitrary closed point s ∈ S.
r In the construction of the global heart AS, the most important case is when S is P . In [AP06, Section 2.3], Abramovich and Polishchuk use Koszul complex to decompose D(X ×
Pr) and construct a global t-structure on it. The Koszul complex can be expressed as follows
r 0 → OPr → Λ V ⊗ OPr (1) → · · · → V ⊗ OPr (r) → OPr (r + 1) → 0,
0 r where V = H (P , OPr (1)). It is not only useful in decomposing derived categories, it is also numerically interesting. Indeed, since the dimensions of ΛiV are binomial coefficients,
∗ Koszul complex implies a polynomial structure of Z(p∗(E ⊗ q (O(n))) for any E ∈ APr . Motivated by this observation, we are able to construct a global weak stability condition on D(X × S) for any projective variety S of finite type.
Theorem 1.2.3. For any smooth projective variety S of finite dimension r, we define
(AS,ZS) as below:
17 ∗ AS = {E ∈ D(X × S) | p∗(E ⊗ q (O(n))) ∈ A for all n 0}
∗ Z(p∗(E ⊗ q (O(n)))r! ZS(E) = lim , n→+∞ nrvol(O(1)) where vol(O(1)) is the volume of O(1). Then this pair is a weak pre-stability condition on D(X × S).
Proof. It is easy to see that (AS,ZS) does not change if we change O(1) to O(N) for N ∈ N>0, so we can assume that O(1) is very ample.
The definition of AS is just taken from [Pol07]. We need to prove that ZS is a weak stability function on AS.
∗ Suppose E ∈ AS and set LE(n) := Z(p∗(E ⊗ q (O(n))), we claim that LE(n) is a polynomial of degree no more than r, and its leading coefficient lies in H ∪ R≤0 for n 0. We will prove it by induction on r. When r = 0, the claim is obvious. We assume the claim is true for r ≤ i − 1, then prove it for r = i. As k is algebraically closed, we can take a general smooth divisor H ∈ |O(1)|. Since AS is Noetherian, as in [AP06, Corollary 3.1.3] we have the following exact sequence
0 → F → E → E¯ → 0, where F is the maximal torsion subobject of E supported over H, and E¯ is H-torsion free.
By induction, LF (n) is a polynomial of degree strictly less than i. Therefore, we can assume E is torsion free with respect to H. By the sequence
0 → O(n − 1) → O(n) → O(n)|H → 0, we have the following exact sequence
∗ ∗ ∗ 0 → q O(n − 1) → q O(n) → q O(n)|H → 0
18 by flatness of q, which gives us a triangle
∗ ∗ ∗ ∗ [1] ∗ p∗(E ⊗ q O(n − 1)) → p∗(E ⊗ q O(n)) → p∗(E ⊗ q c∗c O(n)) −→ p∗(E ⊗ q O(n − 1))[1] where c : H → S is the natural inclusion. Note that we have the following commutative diagram
H × X c×id S × X
q|H q H c S .
By derived flat base change and projection formula (see [Huy06, Section 3.3]), we know
∗ ∗ ∗ ∗ ∗ ∗ ∗ that p∗(E ⊗q c∗c O(n)) = p∗(E ⊗(c×id)∗q|H c O(n)) = p∗(c×id)∗((c×id) E ⊗q|H c O(n)).
∗ Since E is H-torsion free, by [AP06, Corollary 3.1.3], we have (c × id) E ∈ AH . Therefore, this triangle is a short exact sequence in A for n sufficiently large. We get
∗ LE(n)−LE(n−1) = LE|H (n) where E|H := (c×id) E. By induction, LE|H (n) is a polynomial of degree not bigger than i − 1, so the degree of LE(n) is not bigger than i. The leading
coefficient of LE|H (n) is an integral multiple of the leading coefficient of LE(n), therefore, they both lie in H ∪ R≤0 by induction. So we proved that ZS is a weak stability function.
Now for HN-filtration, we know that AS is Noetherian by our assumption and [Pol07,
Theorem 3.3.6]. Then it suffices to prove that the image of ZS is discrete. This can be done similarly by induction on the dimension of S. When dim(S) = 0, it is our assumption
that image of Z is discrete. From the equation LE(n) − LE(n − 1) = LE|H (n), we get
ZS(E) = ZH (E|H ), hence the inductive step holds. Therefore, (AS,ZS) is a weak pre- stability condition.
Remark 1.2.4. In the proof, we used Bertini’s theorem. In the case when k is a finite field, the Bertini’s theorem is proved by Poonen and Charles in [Poo04] and [CP16]. The proof and definition of the weak stability function is similar to the way we define
19 the Hilbert polynomial and take its leading exponential coefficient. For instance, if we take
X = Spec(C), A is the category of C-vector spaces and Z(V ) = i · dim(V ) for any finite dimensional C vector space. Then, the heart is the category of coherent sheaves on S, and
LE(n) = i · HilbE(n) for n 0.
Now, we can use this weak stability condition to induce weak stability condition on X×A, where A is quasi-projective scheme of finite type(in particular, affine scheme of finite type)
over Spec(C). If we take a projective completion S of A, and denote the inclusion map by i : A → S. By abuse of notation, we use the same i to denote the morphism A×X → S ×X too, we have the following theorem.
Proposition 1.2.5. We can define a pair (AA,ZA) on X ×A as follows, we take a projective
completion S of A, and a weak stability condition (AS,ZS) on S by theorem 2.3, then A :=
∗ ∗ i AS, ZA(i E) = ZS(E) is a weak stability condition on X ×A. Moreover, this weak stability condition is independent of the projective completion.
Proof. By [Pol07, Lemma 2.3.1], we know i∗ is essentially surjective. and by [AP06], we know that the definition of AA is what we need. We only need to prove that ZA is well
∗ ∗ defined, i.e., if i E1 = i E2, we need to show that ZS(E1) = ZS(E2), we will prove this by
∗ showing that ZS(Ej) = ZS(i∗i (Ej)), for j = 1, 2.
f ∗ We have the unit map Ej −→ i∗i Ej, we can complete it to a triangle
f ∗ Ej −→ i∗i Ej → Cone(f)
Then Cone(f) is numerically equivalent to an object supported on complement of A on
S, which has strictly lower dimension than S, so ZS(Cone(f)) = 0 by definition. Therefore,
∗ we have ZS(Ej) = ZS(i∗i (Ej)), which implies ZS(E1) = ZS(E2).
The heart AA is independent of the projective completion follows from [AP06, Proposition 3.4.1]. The stability function is independent of the projective completion follows from the proof of the same proposition and the argument above.
20 Remark 1.2.6. This proposition is the only place in this thesis where we require our base field to be of characteristic 0 (e.g., the field of complex numbers). Because the independence
of AA relies on the resolution of singularities.
There is one more thing worth noticing; the lower terms of LE(n) is not well defined in the quasi-projective case. We will need all terms to construct stability conditions later. Therefore, we always assume that S is projective from now on.
Corollary 1.2.7. Let S be a smooth projective variety of finite type. (a) If E = p∗F ⊗ q∗L, where F ∈ A and L is an arbitrary line bundle over S, then
ZS(E) = Z(F ). Moreover, ZS is independent of the choice of ample line bundle O(1).
∗ ∗ (b) If E ∈ AS and {s ∈ S|isE ∈ A and isE 6= 0} contains an open dense subset in S, where is : X × {s} → X × S is the natural inclusion, then ZS(E) 6= 0.
Proof. We deal with the untwisted case E = p∗F first. The following equation hold:
∗ ∗ ∗ Z(p∗(p F ⊗ q (O(n))))r! ZS(p F ) = lim n→+∞ nrvol(O(1)) Z(F ⊗ H0(S, O(n)))r! = lim n→+∞ nrvol(O(1)) Z(F ) V ol(O(1)) nrr! = lim r! n→+∞ nrvol(O(1)) = Z(F ).
The second equation comes from projection formula, and the third equation follows from Asymptotic Riemann-Roch and Serre vanishing (see [Laz04, Corollary 1.1.25]). It is easy to see that twisting p∗F by q∗L will not effect this equation. This proves the first half of (a).
For the independence of ZS on the choice of O(1), we know the line bundles generate K(D(S)) for S smooth. Hence, the objects of the form p∗F ⊗ q∗L span the group K(D(X ×
∗ ∗ ∗ ∗ S)). Therefore, ZS is determined by its value on p F ⊗ q L. Since ZS(p F ⊗ q L) = Z(F ) is
independent of the choice of O(1), we proved the independence of ZS on the choice of O(1).
21 For (b), similarly as in previous theorem, we have the following sequence
0 → F → E → E¯ → 0, where F is the maximal torsion subobject of E, and E¯ is torsion free. It is easy to see that
∗ ¯ ∗ ¯ {s ∈ S|isE ∈ A and isE 6= 0} contains an open dense subset in S. Therefore, we can assume
E is torsion free. Since ZS is independent of the choice of the ample line bundle, we can choose O(1) to be globally generated. Because of the smoothness of S we are able to find a
∗ ∗ smooth divisor D in the linear system |O(1)| such that D ∩ {s ∈ S|isE ∈ A and isE 6= 0} is open and dense in D. Since E is torsion free, we have E|D ∈ AD, and we know that the
leading coefficient of LE(n) is the leading coefficient of LE|D (n) times a nonzero constant
(the constant is the reciprocal of dimension of S, which implies ZS(E) = ZD(E|D)). This finishes the proof by induction.
Remark 1.2.8. Though ZS is independent of the choice of ample line bundle O(1), LE(n) is definitely dependent of the choice of ample line bundle O(1). We suppress this dependence in our notation for simplicity.
Example 1.2.9. Let C be a smooth projective cure over C, σ = (Coh(C), −deg+i·rank) be a stability condition on D(C), and S be an elliptic curve. Then the global heart is Coh(C ×S), the category of coherent sheaves on C × S.
To calculate the polynomial LE(n), let us denote the Chern characters of E by ch(E) =
2 0 0 2 (r, m1l1 + m2l2 + δ, v), where l1 ∈ H (C, Z) ⊗ H (S, Z), l2 ∈ H (C, Z) ⊗ H (S, Z), δ ∈
1 1 H (C, Z) ⊗ H (S, Z) and m1, m2 ∈ Q. Then by Grothendieck-Riemann-Roch, we have
22 ∗ ∗ ch(p∗(E ⊗ q (O(n)))) = p∗(ch(E ⊗ q (O(n))))
= p∗((r, m1l1 + m2l2 + δ, v) · (1, nl1))
= p∗(r, rnl1 + m1l1 + m2l2 + δ, v + nm2)
= (rn + m1, v + nm2) since the Todd class of elliptic curve is trivial. Therefore, the polynomial can be written as
LE(n) = −(v + n · ch1(E)l1) + i(r · n + ch1(E)l2).
Recall a global slicing constructed in [BM14a]. Given a stability condition σ = (Z, P) on D(X) and a phase φ ∈ R, then we have its associated t-structure P(> φ) = D≤−1,
≥0 P(≤ φ) = D . By Abramovich and Polishchuk’s construction, we get PS(> φ), PS(≤ φ) as t-structure on D(X × S). Then we have the following lemma in [BM14a].
Lemma 1.2.10 ([BM14a, Lemma 4.6]). Assume σ = (Z, P) is a stability condition as in
b our setup, and PS(> φ), PS(≤ φ) defined as above. There is a slicing PS on D (X × S) defined by
PS(φ) = PS(≤ φ) ∩ ∩ PS(> φ − ). >0
Proof. We include the proof for reader’s convenience. It is sufficient to construct a HN
filtration for any object E ∈ AS = PS(0, 1]. For any φ ∈ (0, 1], we have PS(φ, φ + 1] ⊂ hAS, AS[1]i. By [Pol07, Lemma 1.1.2], this induces a torsion pair Tφ, Fφ on AS with
Tφ = AS ∩ PS(φ, φ + 1] and Fφ = AS ∩ PS(φ − 1, φ].
0 Let Tφ ,→ E Fφ be the induced short exact sequence in AS. Assume φ < φ ; since
23 Fφ ⊂ Fφ0 , the surjection E Fφ factors via E Fφ0 Fφ. Since AS is Noetherian, the set ∼ of induced quotients {Fφ : φ ∈ (0, 1]} of E must be finite. In addition, if Fφ = Fφ0 , we must
∼ 00 0 also have Fφ00 = Fφ0 for any φ ∈ (φ, φ ). Thus, there exist real numbers
φ0 = 1 > φ1 > φ2 > ··· > φl > φl+1 = 0
such that Fφ is constant for φ ∈ (φi+1, φi), and Fφi− 6= Fφi+ for any small positive real ∼ number . Let us assume for simplicity that Fφ1+ = E. For i = 1, ··· , l we set
i 1. F := Fφi−,
i i 2. E := Ker(E F ), and
3. Ai = E i/E i−1.
i i−1 >φi+ i i We have E ∈ PS(> φi − ) and E = τS E for all > 0. Hence the quotient A satisfies the following
i 1. A ∈ PS (> φi − ),
i 2. A ∈ PS (≤ φi + ).
i l The latter implies A ∈ PS(≤ φi) by [BM14a, Lemma 4.5]. We also have F ∈ PS(0, 1]∩
l i PS(≤ ) for all > 0, which implies F = 0. Thus the E induces a HN filtration of E.
For any E ∈ AS, we can write
r k LE(n) := Σk=0(ak(E) + ibk(E))n .
From the polynomial LE(n), we have two ways to define a slope of an object E ∈ AS.
24 (1) The first one only cares about ZS(E). We define µ1(E) as
Re(ZS (E)) − if Im(ZS(E)) > 0, Im(ZS (E)) µ1(E) := +∞ otherwise.
(2) The second one is the slope of the first nonzero coefficient of LE(n):
Re(L (n)) − lim E if it is well defined, n→+∞ Im(LE (n)) µ2(E) := +∞ otherwise.
We use −cot−1(µ (E)) ψ(E) = 2 π to denote the phase of E. Then 0 < ψ(E) ≤ 1 for E ∈ AS.
Remark 1.2.11. In the second case, unlike the usual slope function, any subobjects have smaller or equal slope is not equivalent to that any quotient objects have bigger or equal slope. Therefore, we define E to be semi-stable with respect to µ2, if for any subobject
F ⊂ E, we have µ2(F ) ≤ µ2(E) and for any quotient objects G of E, we have µ2(G) ≥ µ2(E).
The semi-stability with respect to µ2 is closely related to the slicing constructed in [BM14a, Section 4].
More specifically, given a stability condition σ = (Z, P) on D(X) and a phase φ ∈ R, then we have its associated t-structure P(> φ) = D≤−1, P(≤ φ) = D≥0. By Abramovich
and Polishchuk’s construction, we get PS(> φ), PS(≤ φ) as a t-structure on D(X × S). Then we have the following lemma in [BM14a, Section 4].
Lemma 1.2.12 ([BM14a, Lemma 4.6]). Assume σ = (Z, P) is a stability condition as in
b our setup, and PS(> φ), PS(≤ φ) defined as above. There is a slicing PS on D (X × S) defined by
25 PS(φ) = PS(≤ φ) ∩ ∩ PS(> φ − ). >0
Lemma 1.2.13. If E ∈ PS(φ), then ψ(E) = φ.
Proof. By definition of PS(φ) in Lemma 1.2.12, we know that for n 0, the phases of the
∗ HN-factors of p∗(E ⊗ q (O(n))) lie in (φ − , φ]. Hence we have Arg(LE(n)) ∈ ((φ − )π, φπ] for n 0. This implies φ ≥ ψ(E) ≥ φ − for arbitrary > 0. Therefore, ψ(E) = φ.
Then, we have the following proposition.
Proposition 1.2.14. Suppose E ∈ AS, then E ∈ PS(φ) if and only if E is semi-stable of
phase φ with respect to µ2.
Proof. If E is semi-stable of phase φ with respect to µ2, then take the HN-filtration of E with respect to the slicing. We get
0 = E0 ⊂ E1 ⊂ E2 ⊂ · · · ⊂ En−1 ⊂ En = E,
where Ei/Ei−1 ∈ PS(φi) and
φ1 > φ2 > ··· > φn.
Since E is semi-stable with respect to µ2, we get φ1 ≤ φ and φn ≥ φ. Therefore,
E ∈ PS(φ).
Assume E ∈ PS(φ) and is not semistable with respect to µ2. Let Q be a quotient object
of E in AS with ψ(Q) < ψ(E) = φ by Lemma 1.2.13. Take the HN-filtration
0 = Q0 ⊂ Q1 ⊂ Q2 ⊂ · · · ⊂ Qn−1 ⊂ Qn = Q,
of Q with respect to the slicing, where Qi/Qi−1 ∈ PS(φi) . Then we have φn ≤ ψ(Q) < φ
by see-saw principle. Therefore the nontrivial morphism E → Q → Q/Qn−1 contradicts the definition of slicing. Similarly, we can draw a contradiction for subobject case.
26 Corollary 1.2.15. If E ∈ AS, then E admits HN filtration with respect to µi for i = 1, 2.
Proof. For µ1, this follows from the fact that AS is Noetherian and the image of ZS is discrete.
For µ2, it follows from Proposition 1.2.14.
Lemma 1.2.16. If S is a smooth projective variety of finite type and E ∈ AS is t-flat over
S, then ZS(E) = Z(Es) for any point s ∈ S.
Proof. We can prove it by induction on the dimension of S. If the dimension of S is 0, then the statement is trivial. Now for the inductive step, we use the same argument in the proof of part (b) in Corollary 1.2.7. Indeed, for any point s ∈ S, there exists a smooth divisor D such that s ∈ D, and ZS(E) = ZD(E|D). Hence, Z(Es) = ZD(E|D) = ZS(E) by induction.
Proposition 1.2.17. Let S be a smooth projective variety of finite type and E ∈ AS be t-flat. Consider the following propositions.
(1) E is semi-stable of phase φ with respect to µ2.
(2) Es ∈ P(φ), for arbitrary s ∈ S. Then (1) implies (2).
Proof. If E = 0, the statement is obvious.
Now assume E ∈ AS is a nonzero object and t-flat, then we can deduce that ZS(E) 6= 0.
Otherwise we have Z(Es) = 0, which implies Es is the zero object in A for all s ∈ S. Hence
E = 0, which contradicts our assumption. Now we have Z(Es) 6= 0 and the phase of Es is also φ by Lemma 1.2.16.
On the other hand, the assumption that E is semi-stable of phase φ with respect to µ2 is equivalent to
E ∈ PS(φ) = PS(≤ φ) ∩ ∩ PS(> φ − ) >0
27 ∗ by Lemma 1.2.12 and Proposition 1.2.14. Since is is t-right exact, it sends objects in PS(>
φ − ) to objects in P(> φ − ), hence Es ∈ P(> φ − ). The assumption that E is t-flat
implies Es ∈ A. Hence we have Es ∈ P(> φ − ) ∩ P(≤ 1) for all > 0.
Combing these two facts, we get Es ∈ P(φ).
Proposition 1.2.18. If S is a smooth projective variety of finite type, and E ∈ AS is semi-stable with respect to µ1 of phase φ and ZS(E) 6= 0, then there exists a short exact sequence
0 → K → E → Q → 0
such that K ∈ PS(φ), Q ∈ PS(< φ) and ZS(Q) = 0, where Q could be zero.
Proof. The sequence comes from the HN filtration of E with respect to µ2, or equivalently, the global slicing PS. Indeed, suppose
0 = E0 ⊂ E1 ⊂ E2 ⊂ · · · ⊂ En−1 ⊂ En = E
is the filtration.
We claim that ZS(Ei/Ei−1) = 0 for all i > 1. Otherwise there exists i0 > 1 such that
ZS(Ei0 /Ei0−1) 6= 0, then µ1(Ei0 /Ei0−1) = µ2(Ei0 /Ei0−1) and E1 would destabilize E with
respect to µ1 by see-saw principle of ZS. Therefore, ZS(E1) = ZS(E) 6= 0.
Then 0 → E1 → E → E/E1 → 0 is the sequence we need.
Example 1.2.19. Take X = S = P1 and σ = (CohX, Z), where Z(E) = − deg(E) + i ·
1 1 rk(E). Then the ideal sheaf I of a closed point (x0, s0) in P ×P is an example of an object
that is semi-stable with respect to µ1 but not semi-stable with respect to µ2. This is because
28 Is is semi-stable for every s ∈ S except s0 ∈ S. Indeed, we have
OP1×s if s 6= s0, Is = 1 OP ×s0 (−1) ⊕ k(x0, s0) if s = s0,
where k(x0, s0) is the skyscraper sheaf of the point (x0, s0). In this case, the sequence in Proposition 1.2.18 is
1 0 → O(0, −1) → I → OP ×s0 (−1) → 0.
1 1 One can check that O(0, −1) is of phase 2 with LO(0,−1)(n) = i · n, OP ×s0 (−1) is torsion 1 and of phase with LO 1 (−1)(n) = 1 + i. 4 P ×s0
29 Chapter 2
Some mathematical structures on AS
2.1 Positivity of the coefficitens
Suppose we have a stability condition σ = (A,Z) on D(X) with A Noetherian, then by results in Chapter 1, we have a global heart AS for any projective variety S. Moreover, for any object E ∈ AS, we have a polynomial LE(n) whose degree is no more than the dimension of S.
As in Section 1.2, for any E ∈ AS, we can write
r k LE(n) := Σk=0(ak(E) + ibk(E))n .
Then we have the following positivity of coefficients ai and bi.
Proposition 2.1.1. For any E ∈ AS, we have the following inequalities.
(1) br(E) ≥ 0.
(2) If br(E) = 0, then ar(E) ≤ 0 and br−1(E) ≥ 0. (3) In general, if
br(E) = ar(E) = br−1(E) = ··· = ai(E) = bi−1(E) = 0,
30 then ai−1(E) ≤ 0 and bi−2(E) ≥ 0 for any 2 ≤ i ≤ r. (4) Moreover, if E is a nonzero object and
br(E) = ar(E) = br−1(E) = ··· = a1(E) = b0(E) = 0,
then a0(E) < 0.
Proof. Since E ∈ AS = PS(0, 1], we have the HN-filtration of E with respect to the slicing
PS. We get
0 = E0 ⊂ E1 ⊂ E2 ⊂ · · · ⊂ En−1 ⊂ En = E,
where Ei/Ei−1 ∈ PS(φi) and φi ∈ (0, 1]. Therefore, by Lemma 1.2.13, we know that the argument of the first nonzero coefficient
of LEi (n) is in (0, π]. Hence (1), (2) and (3) are proved.
∗ For (4), it is because LE(n) = 0 implies p∗(E ⊗ q (O(n))) = 0 for n 0, which implies E = 0.
Lemma 2.1.2. Suppose E is an object in PS(φ), then we have the following inequalities;
(1) br(E)ar−1(E) − br−1(E)ar(E) ≥ 0.
(2) If br(E)ar−1(E) − br−1(E)ar(E) = 0, then br(E)ar−2(E) − br−2(E)ar(E) ≥ 0. (3) In general, for any 1 ≤ i ≤ r, if
br(E)ar−1(E) − br−1(E)ar(E) = ··· = br(E)ai(E) − bi(E)ar(E) = 0,
Then, br(E)ai−1(E) − bi−1(E)ar(E) ≥ 0.
Proof. Recall the definition
PS(φ) = PS(≤ φ) ∩ ∩ PS(> φ − ). >0
We know that Arg(ar(E) + ibr(E)) is πφ by lemma 1.2.13. By the definition of PS(φ),
31 we know that for any small enough > 0, there exists a positive integer N0, such that for
any positive integer n > N0, Arg(LE(n)) is in (π(φ − ), πφ].
Hence the complex number ar−1(E) + ibr−1(E) is not on the left side of the line passing
from 0 to ar(E)+ibr(E) in the complex plane. Otherwise, we could find a positive integer N1,
such that for any integer n > N1, the Arg(LE(n)) is strictly bigger that πφ, this contradicts
to the fact E ∈ PS(φ). This implies
br(E)ar−1(E) − ar(E)br−1(E) ≥ 0.
If br(E)ar−1(E)−ar(E)br−1(E) = 0, then by the same reasoning, the following inequality holds.
br(E)ar−2(E) − ar(E)br−2(E) ≥ 0.
Inductively using the same argument, we proved (3).
Remark 2.1.3. Under the assumption of Lemma 2.1.2, if ar(E) = br(E) = 0, then the in- equalities in Lemma 2.1.2 becomes vacuous. In this case, we can derive the similar quadratic inequalities on ak(E), bk(E), ak−1(E), ··· , b0(E), where ak(E) + ibk(E) is the first nonzero
coefficient of LE(n). In the case when X is a point, and σ = (V ect, −dim), these inequalities are also vacuous. Indeed, all the coefficients are in the same ray in complex plane.
Proposition 2.1.4. If E ∈ AS is semi-stable with respect to µ1, then the following inequal- ities are satisfied.
(1) br(E)ar−1(E) − br−1(E)ar(E) ≥ 0.
(2) If br(E)ar−1(E) − br−1(E)ar(E) = 0, then br(E)ar−2(E) − br−2(E)ar(E) ≥ 0. (3) In general, if
br(E)ar−1(E) − br−1(E)ar(E) = ··· = br(E)ai(E) − bi(E)ar(E) = 0,
32 Then, br(E)ai−1(E) − bi−1(E)ar(E) ≥ 0 for any 1 ≤ i ≤ r.
Proof. If ar(E) = br(E) = 0, then the statement is trivial.
Hence we can assume ar(E) + ibr(E) 6= 0. Then by Proposition 1.2.18, there exists a short exact sequence in AS 0 → K → E → Q → 0
such that K ∈ PS(φ), Q ∈ PS(< φ) and ZS(Q) = 0. If Q = 0, the statement follows from Lemma 2.1.2.
Then we can assume Q is a nonzero object in PS(< φ). Hence we have
−a (E) −Re(L (n)) r > Q br(E) Im(LQ(n)) for n 0.
This implies that if ar−1(Q) + ibr−1(Q) 6= 0, we have
−a (E) −a (Q) r > r−1 . br(E) br−1(Q)
Lemma 2.1.2 and Proposition 1.2.18 imply the following inequality.
br(E)ar−1(K) − br−1(E)ar(K) = br(K)ar−1(K) − br−1(K)ar(K) ≥ 0.
Combing this two inequalities, we get (1). Now suppose
br(E)ar−1(E) − br−1(E)ar(E) = 0,
This implies
br(K)ar−1(K) − br−1(K)ar(K) = 0
and ar−1(Q) = br−1(Q) = 0.
33 Then by Lemma 2.1.2, we get
br(K)ar−2(K) − ar(K)br−2(K) ≥ 0.
The fact Q ∈ PS(< φ) implies that if ar−2(Q) + ibr−2(Q) 6= 0, then
−a (E) −a (Q) r > r−2 . br(E) br−2(Q)
Hence, we have
br(E)ar−1(E) − br−1(E)ar(E) ≥ 0
which proves (2). Inductively using the same argument, we get (3).
2.2 Abelian subcategories in AS
This section is devoted to investigate some abelian subcategories in AS.
Firstly, let us look at some full subcategories of AS.
≤j Definition 2.2.1. For any integer 1 ≤ j ≤ r. Let AS be the subcategory in AS satisfying the following properties.
≤j (1) E ∈ AS ⇐⇒ deg(LE(n)) ≤ j. ≤j (2) For any two objects E,F ∈ AS , we have
≤j Hom (E,F ) = HomAS (E,F ). AS
≤j Lemma 2.2.2. Suppose E is an object in AS , and Q is a subquotient object of E in AS. ≤j Then Q ∈ AS .
34 ≤j Proof. Assume E is an object in AS , let
0 → K → E → Q → 0
be a short exact sequence in AS.
Suppose k, q are the maximal integers such that ak(K)+ibk(K) 6= 0 and aq(Q)+ibq(Q) 6= 0 respectively. If k > j, then we get
ak(K) + ibk(K) = −ak(Q) − ibk(Q).
By Lemma 2.1.1, we get bk(K) ≥ 0, and if bk(K) = 0, we have ak(K) < 0 by the
definition of k. Then bk(Q) ≤ 0, and if bk(Q) = 0, we have ak(Q) > 0. Therefore, we have q > k by Lemma 2.1.1. Similarly, we can prove that k > q, a contradiction.
≤j Therefore, k, q can not exceed j, which implies K,Q ∈ AS .
≤j Corollary 2.2.3. The subcategory AS is an abelian category for any 0 ≤ j ≤ r.
≤j Proof. Suppose we have two objects E,F ∈ AS , and a morphism
f : E → F
in AS. Let K,I,Q be the kernal, image and cokernal of f in AS respectively. It suffices to
≤j show that K,I,Q are objects in AS , which follows directly from Lemma 2.2.2.
>j Proposition 2.2.4. There exists subcategory F in AS for any 1 ≤ j ≤ r, such that the
≤j >j pair (AS , F ) is a torsion pair of AS.
Proof. We claim that for any object E ∈ AS, there exists a unique maximal subobject
≤j Ej ∈ AS of E.
To prove the claim, we firstly show that for any two subobjects E1,E2 ⊂ E and E1,E2 ∈
≤j ≤j AS . There exsits a third subobjects E3 ∈ AS such that E1,E2 ⊂ E3 ⊂ E.
35 Suppose we have two injections
f g E1 ,−→ E, and E2 ,−→ E.
Then we have a morphism (f,g) E1 ⊕ E2 −−→ E.
≤j Let E3 be the image of this morphism, by Lemma 2.2.2, we know that E3 ∈ AS . And
E1,E2 ⊂ E3 ⊂ E by construction. Hence it suffices to prove that the increasing sequence
E1 ⊂ E2 ⊂ · · · ⊂ E
≤j of subobjects Ek ∈ AS stabilizes after finite step. This follows directly from the fact AS is Noetherian. Let F >j be the full subcategory consisting of the objects whose maximal subobjects in
≤j ≤j >j AS is 0. Then it is easy to show (AS , F ) is a torsion pair.
Corollary 2.2.5. There exists a filtration
0 = E0 ⊂ E1 ⊂ · · · ⊂ Er = E
≤j for any object E ∈ AS, such that Ej ∈ AS .
Proof. By the proof of Proposition 2.2.4, we can take Ej to be the maximal subobject in
≤j ≤j ≤k AS . Then since AS is a full subcategory of AS for any integers 1 ≤ j < k ≤ r. We get
Ej ⊂ EK , hence the filtration as well.
To achieve the full generality, we take a look at abelian subcategories from a general weak stability condition σ = (A,Z) on a triangulated category D.
36 Lemma 2.2.6. Let A0 be the full subcategory consisting of objects E whose central charge is 0, then A0 is an abelian subcatgory.
Proof. It is very easy to see that A0 is closed under subobjects and quotient objects. In particular, A0 is an abelian subcategory.
Remark 2.2.7. Corollary 2.2.3 can also be viewed as a direct consequence of Lemma 2.2.6.
≤j Indeed, (AS , aj +ibj) is a weak stability condition for 1 ≤ j ≤ r, hence we can apply Lemma 2.2.6 inductively.
Let Pσ(φ) be the subcategory of semi-stable objects whose phase is φ. By definition, we have A0 ⊂ Pσ(1).
Lemma 2.2.8. If we have a short exact sequence in A
0 → K → E −→f Q → 0,
where K ∈ Pσ(φ) and Q ∈ A0. Then we have the following short exact sequence in A
0 → Q0 → E → K0 → 0,
0 0 where Q ∈ A0 and K ∈ Pσ(φ).
0 Proof. If K ∈ Pσ(1), then we know that E is in Pσ(1) as well. Hence we can take K = E and Q0 = 0. Now we can assume that 0 < φ < 1. By the standard fact in the theory of stability
0 conditions, there exist a unique maximal subobject Q ∈ Pσ(1) with a natural inclusion i : Q0 ,→ E. Hence we have the following short exact sequence
0 → Q0 −→i E −→g K0 → 0.
0 0 We claim that Z(Q ) = 0 and K ∈ Pσ(φ).
37 0 0 0 If Z(Q ) 6= 0, then we have Z(Q ) ∈ R<0. Let G := ker(f ◦ i): Q → Q, then we have
0 Z(G) = Z(Q ) ∈ R<0 and G,→ K. This contradicts the facts K ∈ Pσ(φ) and φ < 1. Hence, we get Z(Q0) = 0. Now we know that Z(E) = Z(K0). Suppose that K0 is not semi-stable, let F 0 be a destabilizing subobject of K0 and F = g−1(F 0). Then we have a morphism f 0 : F → Q which is the composition of f and the inclusion from F to E. Let H := ker(f 0), we know that Z(H) = Z(F ) and H is a subobject of K. Since Z(K) = Z(K0) = Z(E), H destabilize K, which contradicts the assumption that K is semi-stable. Therefore, we get the short exact sequence.
Remark 2.2.9. In the proof of this lemma, we also proved Q0 is a subobject of Q.
Definition 2.2.10. We call A0 the abelianizer of the weak stability condition σ = (A,Z). We say that E is quasi-semi-stable if there exists a short exact sequence
0 → Q0 → E → K0 → 0,
0 0 where Q ∈ A0 and K ∈ Pσ(φ).
The following Proposition is the justification of naming A0 the abelianizer of σ.
Proposition 2.2.11. Let
Aσ(φ) := {E ∈ A|E is quasi − semistable and Z(E) ∈ exp(iπφ) · R≥0}.
Then Aσ(φ) is an abelian subcategory.
Proof. If φ = 1, then Aσ(1) = Pσ(1). Then Aσ(1) is an abelian category since it is closed under subobjects and quotient objects.
Now we can assume 0 < φ < 1. If we have a morphism f : E1 → E2 in A between two
objects E1,E2 ∈ Aσ(φ). There exists the following diagram in A.
38 0 Q1 E1 K1 0
f
0 Q2 E2 K2 0 where Qi ∈ A0 and Ki ∈ Pσ(φ) for i = 1, 2.
Since 0 < φ < 1 and Qi ∈ A0, there is no nontrivial morphism from Q1 to K2. Hence the diagram can be completed by the following commutative diagram.
0 Q1 E1 K1 0
h f g
0 Q2 E2 K2 0
It suffices to show that ker(f), im(f) and coker(f) are quasi-semi-stable. By snake lemma, we have the following long exact sequence
0 → ker(h) → ker(f) → ker(g) −→δ coker(h) → coker(f) → coker(g) → 0
This can be decomposed into two short exact sequences
0 → ker(h) → ker(f) → ker(δ) → 0 and 0 → coker(δ) → coker(f) → coker(g) → 0.
We know that im(g) is a subobject of K2 and a quotient object of K1, where K1,K2 ∈
Pσ(φ). This implies that im(g) ∈ Pσ(φ), hence ker(g) ∈ Pσ(φ) as well. Since coker(h) is a quotient object of Q2, it is an object in A0 by Lemma 2.2.2. Therefore, Z(ker(δ)) =
Z(ker(g)) and ker(δ) ⊂ ker(g) imply that ker(δ) ∈ Pσ(φ). Thus we get that ker(f) is quasi-semi-stable.
Now we want to show that coker(f) ∈ Aσ(φ). It is easy to see that coker(g) ∈ Aσ(φ). Indeed, if F is a subobject of coker(g) that destabilize coker(g). Let H := coker(g)/F ,
39 then if Z(F ) 6= 0, we have Z(H) 6= 0, this implies µσ(H) < µσ(coker(g)) = µσ(K2). This
contradicts to the assumption K2 ∈ Pσ(φ). Hence we proved that any destabilizing subobject
of coker(g) is in A0. This is equivalent to coker(g) ∈ Pσ(φ). Therefore, we have the following commutative diagram.
0 coker(δ) Q Q0 0
id 0 coker(δ) coker(f) coker(g) 0
0 0 K id K 0
0 where Q ∈ A0 and K ∈ Pσ(φ), and moreover every column is a short exact sequence. Hence
Q ∈ A0, and coker(f) ∈ Pσ(φ).
It is left to prove that im(f) ∈ Aσ(φ). By snake lemma, we have the following commu- tative diagram.
0 ker(h) ker(f) ker(δ) 0
0 Q1 E1 K1 0
0 im(h) im(f) J 0
where J := K1/ker(δ), im(h) ∈ A0 and every column is a short exact sequence. Hence it
40 suffices to prove that J ∈ Aσ(φ). This follows from the following commutative diagram.
0
0 0 im(δ)
0 ker(δ) K1 J 0
id
0 ker(g) K1 im(g) 0
im(δ) 0 0
0
where im(δ) ∈ A0 and im(g) ∈ Pσ(φ), and every column is a short exact sequence. Hence
J ∈ Aσ(φ), the proof is complete.
Remark 2.2.12. When σ = (A,Z) is a stability condition, we know that A0 is trivial. In this case, Proposition 2.2.11 is [Bri07, Lemma 5.2].
≤j We end this section by including some quadratic inequalities for objects in AS .
≤j Lemma 2.2.13. For any E ∈ AS , we have the following inequalities.
(1) bj(E) ≥ 0.
(2) If bj(E) = 0, then aj(E) ≤ 0 and bj−1(E) ≥ 0. (3) In general, if
bj(E) = aj(E) = bj−1(E) = ··· = ai(E) = bi−1(E) = 0,
then ai−1(E) ≤ 0 and bi−2(E) ≥ 0 for any 2 ≤ i ≤ j. (4) Moreover, if E is a nonzero object and
bj(E) = aj(E) = bj−1(E) = ··· = a1(E) = b0(E) = 0,
41 then a0(E) < 0.
Proof. This is a direct corollary from Lemma 2.1.1.
≤j Let σj := (AS , aj + ibj), then we have the following quadratic inequalities.
≤j Proposition 2.2.14. If E ∈ AS is semi-stable with respect to σj, then the following in- equalities are satisfied.
(1) bj(E)aj−1(E) − bj−1(E)aj(E) ≥ 0.
(2) If bj(E)aj−1(E) − bj−1(E)aj(E) = 0, then bj(E)aj−2(E) − bj−2(E)aj(E) ≥ 0. (3) In general, if
bj(E)aj−1(E) − bj−1(E)aj(E) = ··· = bj(E)ai(E) − bi(E)aj(E) = 0,
Then, bj(E)ai−1(E) − bi−1(E)aj(E) ≥ 0 for any 1 ≤ i ≤ j.
Proof. The proof is similar to the proof in Proposition 2.1.4.
2.3 Weak stability conditions and torsion pairs
Proposition 2.1.1 and Proposition 2.2.11 give us a lots weak stability conditions and hence
abelian subcategories in AS. For the simplicity of our statements and arguments, we introduce the following definition.
Definition 2.3.1. If (A,Z) is a stability condition, and the image of Z lies in Q ⊕ Qi, we call (A,Z) a rational stability condition. We use RStab(X) to denote the set of rational stability conditions on D(X).
Remark 2.3.2. By [AP06, Proposition 5.0.1], we know the heart A of a rational stability
condition is Noetherian. And in this case, the images of aj, bj are rational. We focus on the rational stability conditions just for the simplicity of statements and arguments. All results
42 and proofs in the rest of this thesis can be easily adapted to the stability conditions whose central charge have discrete image.
From now on, we always assume our original stability condition σ on X is a rational stability condition.
t t ≤j t Lemma 2.3.3. Let Zj(E) = aj(E)t − bj−1(E) + ibj(E), then σj := (AS ,Zj) is a weak
stability condition for any t ∈ Q≥0 and any integer 0 ≤ j ≤ r.
t ≤j Proof. By Lemma 2.1.1, Zj is a weak stability function on AS . Then by the rationality of
≤j t bj, aj and the Noetherianity of AS , we see that σj admits HN property.
t Apply Proposition 2.2.11 on σj, we have the following abelian subcategories in AS.
t t Definition 2.3.4. For simplicity of the notation, let A (φ) to denote A t (φ). Then A (φ) j σj j is an abelian subcategory by Proposition 2.2.11.
Lemma 2.3.5. The pair
0 1 σt,t = (At ( ), a t0 − b + ib ) j j 2 j−1 j−2 j
0 is a weak stability condition for any t ∈ Q≥0.
t 1 t 1 t Proof. If E ∈ Aj( 2 ) and bj(E) = 0. Then by the definition of Aj( 2 ), we have Zj(E) = 0, hence bj(E) = aj(E) = bj−1(E) = 0. By Lemma 2.1.1, this implies that aj−1(E) ≤ 0 and
0 bj−2(E) ≥ 0. Therefore, we know that aj−1t − bj−2 + ibj is a weak stability function on
t 1 Aj( 2 ). The HN property following from rationality of central charge and the Noetherianity of
t 1 Aj( 2 ).
We can construct weak stability conditions inductively by Lemma 2.1.1.
43 t1,t2 Lemma 2.3.6. (1). Let Aj (φ) := A t1,t2 (φ), then the pair σj
1 σt1,t2,t3 = (At1,t2 ( ), a t − b + ib ) j j 2 j−2 3 j−3 j
is a weak stability condition for any t3 ∈ Q≥0.
t1,t2,··· ,tk (2). In general, let Aj (φ) := A t1,t2,··· ,tk (φ), then the pair σj
1 σt1,t2,··· ,tk+1 = (At1,t2,··· ,tk ( ), a t − b + ib ) j j 2 j−k k+1 j−k−1 j
is a weak stability condition for any tk+1 ∈ Q≥0.
t1,t2,··· ,tk 1 Proof. By induction on k, if E ∈ Aj ( 2 ) and bj(E) = 0, then bj(E) = aj(E) =
bj−1(E) = ··· = ai−k+1(E) = bj−k(E) = 0. Then by Lemma 2.1.1, we have bj−k−1(E) ≥ 0
t1,t2,··· ,tk 1 and aj−k(E) ≤ 0. Hence aj−ktk+1 −bj−k−1 +ibj is a weak stability condition on Aj ( 2 ). The HN property following from rationality of central charge and the Noetherianity of
t1,t2,··· ,tk 1 Aj ( 2 ).
In the rest of this section, we will see a finer cut of AS into torsion pairs.
Definition 2.3.7. Let
1 t1,t2,··· ,tk t1,t2,··· ,tk−1 T := {E ∈ Ar ( )|µ t1,t2,··· ,tk > 0} 2 σr ,min
1 t1,t2,··· ,tk t1,t2,··· ,tk−1 F := {E ∈ Ar ( )|µ t1,t2,··· ,tk ≤ 0} 2 σr ,max
and 1 t1,t2,··· ,tk t1,t2,··· ,tk−1 G := {E ∈ Ar ( )|µ t1,t2,··· ,tk < 0}. 2 σr ,max
t1,t2,··· ,tk where r is the dimension of S, and µ t1,t2,··· ,tk is the slope function coming from σj . σj
Definition 2.3.8. If we have two full subcategories T and F in an abelian category, we let hT , Fi to denote the full subcategory of extension of T and F, i.e., if H is an object in
44 hT , Fi, then we have a short exact sequence
0 → T → H → F → 0 where T ∈ T and F ∈ F.
Remark 2.3.9. This definition is order sensitive. In general, hT , Fi is not necessarily equiv- alent to hF, T i.
Then we have the following lemma.
Proposition 2.3.10. Assume T t1,··· ,tk , F t1,··· ,tk and Gt1,··· ,tk are defined as in Definition 2.3.7. Let
t1,··· ,tk Tt1,··· ,tk := hTt1,··· ,tk−1 , T i
t1,··· ,tk Ft1,··· ,tk := hF , Gt1,··· ,tk−1 i and
t1,··· ,tk Gt1,··· ,tk := hG , Gt1,··· ,tk−1 i
Then the pair (Tt1,··· ,tk , Ft1,··· ,tk ) is a torsion pair of AS.
Proof. Firstly, I claim that if T ∈ Tt1,t2,··· ,tk , F ∈ Ft1,t2,··· ,tk , then HomAS (T,F ) = 0. This claim is trivial when k = 1. Then we assume the claim is true for k = n, then since
HomAS (Tt1,··· ,tn , Ft1,··· ,tn ) = 0
and Ft1,··· ,tn+1 is a subcategory of Ft1,··· ,tn , we have
HomAS (Tt1,··· ,tn , Ft1,··· ,tn+1 ) = 0.
45 And by definition, we have
t1,··· ,tn+1 t1,··· ,tn+1 HomAS (T , F ) = 0.
The last piece is by considering the semi-stablity, we get
t1,··· ,tn+1 HomAS (T , Gt1,··· ,tn ) = 0.
Hence by definition, we have
HomAS (Tt1,··· ,tn+1 , Ft1,··· ,tn+1 ) = 0 which completes the proof of the claim.
Now it is left to show that for any object E in AS, we can decompose E into the following short exact sequence.
0 → T → E → F → 0
where T ∈ Tt1,t2,··· ,tk and F ∈ Ft1,t2,··· ,tk . We prove this statement by induction on k as well. If k = 1, this statement is trivial. We assume the statement is true for k = n, hence we have the following short exact sequence 0 → T → E → F → 0
where T ∈ Tt1,t2,··· ,tn and F ∈ Ft1,t2,··· ,tn . Then by definition of Ft1,t2,··· ,tn , we have the following short exact sequence
t1,··· ,tn 0 → F → F → Gt1,··· ,tn−1 → 0
t1,··· ,tn t1,··· ,tn t1,··· ,tn where F ∈ F and Gt1,··· ,tn−1 ∈ Gt1,··· ,tn−1 . Then by the HN-filtration of σr ,
46 we have the following short exact sequence
0 → P → F t1,··· ,tn → Gt1,··· ,tn → 0
1 t1,··· ,tn t1,··· ,tn where P ∈ P t1,··· ,tn ( ) and G ∈ G . Hence we have the following commutative σr 2 diagram
0
0 0 Gt1··· ,tn
0 P F Gt1,··· ,tn 0
id
t1,··· ,tn 0 F F Gt1,··· ,tn−1 0
Gt1,··· ,tn 0 0
0
where Gt1,··· ,tn ∈ Gt1,··· ,tn by definition.
t1,··· ,tn+1 By HN-filtration of σr , we have the following short exact sequence
0 → T t1,··· ,tn+1 → P → F t1,··· ,tn+1 → 0
where T t1,··· ,tn+1 ∈ T t1,··· ,tn+1 and F t1,··· ,tn+1 ∈ F t1,··· ,tn+1 . Therefore, we have the following
47 commutative diagram
0
0 0 F t1··· ,tn+1
t1··· ,tn+1 0 T F Ft1,··· ,tn+1 0
id
0 P F Gt1,··· ,tn 0
F t1,··· ,tn+1 0 0
0
where Ft1,··· ,tn+1 ∈ Ft1,··· ,tn+1 by definition. Hence we have the following commutative diagram
0
0 0 T t1··· ,tn+1
0 T E F 0
id
0 Tt1,··· ,tn+1 E Ft1,··· ,tn+1 0
T t1,··· ,tn+1 0 0
0
where T ∈ Tt1,t2,··· ,tn , hence Tt1,··· ,tn+1 ∈ Tt1,··· ,tn+1 . Therefore, we get the short exact sequence
0 → Tt1,··· ,tn+1 → E → Ft1,··· ,tn+1 → 0
where Tt1,··· ,tn+1 ∈ Tt1,cdots,tn+1 and Ft1,··· ,tn+1 ∈ Tt1,··· ,tn+1 . This completes the proof.
48 t1,··· ,tk Suppose (Tt1,··· ,tk , Ft1,··· ,tk ) is a torsion pair of AS, then we denote AS to the tilted
heart. hFt1,··· ,tk [1], Tt1,··· ,tk i.
t1,t2,··· ,tk Theorem 2.3.11. For any object E in AS , we have the following inequalities.
(1) −ar(E)t1 + br−1(E) ≥ 0. (2) For any positive integer 0 < l < k, if
−ar(E)t1 + br−1(E) = ··· = −ar+1−l(E)tl + br−l(E) = 0,
then we have −ar−l(E)tl+1 + br−l−1(E) ≥ 0.
t1,t2,··· ,tk Proof. By the definition of AS , we have the following short exact sequence
0 → F [1] → E → T → 0.
where F ∈ Ft1,t2,··· ,tk and T ∈ Tt1,t2,··· ,tk . By induction, we can easily show that if
−ar(E)t1 + br−1(E) = ··· = −ar+1−l(E)tl + br−l(E) = 0,
t1,t2,··· ,tl 1 t1,t2,··· ,tl 1 then we have T ∈ Ar ( 2 ) and F ∈ Ar ( 2 ). Hence by definition, we have F ∈
F t1,t2,··· ,tl+1 and T ∈ T t1,t2,··· ,tl+1 . Therefore, we have
−ar−l(T )tl+1 + br−l−1(T ) ≥ 0
and
−ar−l(F )tl+1 + br−l−1(F ) ≤ 0.
This completes the proof.
49 Chapter 3
Constructing stability conditions
3.1 Product over a curve
In this section, let us consider the case when S is a smooth projective curve. Then the polynomial LE(n) become a linear polynomial, one can write the linear polynomial in the form
LE(n) := a(E)n + b(E) + i(c(E)n + d(E)), where a, b, c, d are group homomorphisms from K(AS ) to R. Please notice that this notation is a little bit different from the notation we used in Section 2.1.
By Theorem 1.2.3, we know that a + ic is a weak stability function on AS, and LE(n) will lie in H ∪ R<0 for nonzero object E ∈ AS and n 0. By this observation, we have the following lemma.
Lemma 3.1.1. For a nonzero object E ∈ AS, we have the following inequalities. (i) c(E) ≥ 0. (ii) If c(E) = 0, then d(E) ≥ 0 and a(E) ≤ 0. (iii) If c(E) = a(E) = d(E) = 0, then b(E) < 0.
Proof. This is the special case of r = 1 in Lemma 2.1.1.
50 Now we can restate the Positivity Lemma from [BM14b, Lemma 3.3] in terms of a, b, c, d.
Lemma 3.1.2 (Restatement of Positivity Lemma). If E ∈ AS is t-flat and Es is semi-stable for any point s ∈ S, then b(E)c(E) − a(E)d(E) ≥ 0.
−a(E) Proof. It is easy to see that Es is of slope c(E) for any point s ∈ S from Lemma 1.2.16.
Now we rotate σ = (A,Z) by angle θ to make Es of phase 1. Since Es is in the rotated
iθ iθ global heart e ·A, then E is in the corresponding global heart e ·AS by [AP06, Corollary
iθ 3.3.3]. Therefore, Im(e LE(n)) ≥ 0 for n >> 0. This means that if
» eiθ(a(E) + ic(E)) = − a(E)2 + c(E)2,
then Im(eiθ(b(E) + id(E))) ≥ 0.
This implies Im(ic(E) − a(E))(b(E) + id(E)) ≥ 0, which is equivalent to b(E)c(E) − a(E)d(E) ≥ 0.
Lemma 3.1.3. If E ∈ AS is semi-stable with respect to µ1, then b(E)c(E) − a(E)d(E) ≥ 0.
Proof. It is a special case of Proposition 2.1.4.
Now we assume that σ = (A,Z) is a rational stability condition (see Definition 2.3.1). Then for any positive rational number t, we can define the following slope function, coming from the weak stability function Zt(E) = a(E)t − d(E) + ic(E)t.
−a(E)t+d(E) c(E)t if c(E) 6= 0, νt(E) = +∞ otherwise.
51 By part (i) and (ii) of Lemma 3.1.1, we know that Zt is a weak stability function on
AS. Since t is a fixed positive rational number, the pair σt = (AS,Zt) admits HN property because of the facts that AS is Noetherian and Zt is discrete. Then AS can be decomposed into two parts, torsion part T = {E ∈ AS | νt,min(E) > 0} and torsion free part F = {E ∈
AS | νt,max(E) ≤ 0}. Now, we can apply tilting method on this heart to get a new heart
t AS = hT , F[1]i.
s,t Proposition 3.1.4. For arbitrary s, t ∈ R>0, ZS (E) = c(E)s + b(E) + i(−a(E)t + d(E))
t is a stability function on AS.
t t Proof. It is easy to see that −a(E)t + d(E) ≥ 0 for E ∈ AS from the definition of AS. Now
t we need to prove that if −a(E)t + d(E) = 0, then c(E)s + b(E) < 0 for nonzero E ∈ AS. We have the following short exact sequence
0 → F [1] → E → T → 0
where F ∈ F,T ∈ T . Therefore, we have to deal with the following two cases.
Firstly, if −a(T )t + d(T ) = 0. By definition of T , we have c(T ) = 0 and νt(E) = +∞. Therefore, in this case −a(E)t + d(E) = 0 is equivalent to a(E) = d(E) = 0 by Lemma 3.1.1, which also implies b(E) < 0. Now we deal with F . By definition of F, we know that c(F ) > 0 if F is nonzero. Then
F ∈ F and −a(F )t + d(F ) = 0 implies that F is semi-stable with respect to σt. Therefore, it suffices to prove c(F )s + b(F ) > 0 in this case. Take
0 = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fl−1 ⊂ Fl = F
as the HN filtration of F with respect to µ1. We let Qk = Fk/Fk−1 be the k-th HN factor of F , for 1 ≤ k ≤ l. We have
−a(Q ) −a(Q ) k > k+1 (3.1) c(Qk) c(Qk+1)
52 by the property of HN filtration. If c(Q1) = 0, then F1 will destabilize F with respect to νt.
Hence c(Q1) > 0, which implies c(Qk) > 0 for 1 ≤ k ≤ l. Moreover, c(Qk) > 0 and Qk is
semi-stable with respect to µ1 implies Qk is torsion free, which is equivalent to being t-flat since S is a curve. Applying Lemma 3.1.3, we get
b(Qk)c(Qk) ≥ a(Qk)d(Qk) (3.2)
for 1 ≤ k ≤ l. The last piece of data is that F is semi-stable of slope 0 with respect to νt. We have j l Σk=1(−a(Qk)t + d(Qk)) Σk=j(−a(Qk)t + d(Qk)) j ≤ 0 ≤ l (3.3) Σk=1c(Qk)t Σk=jc(Qk)t for any 1 ≤ j ≤ l. Using (1), (2) and (3), we are able to prove the following inequality.
l l a(Qk)d(Qk) b(F ) = Σk=1b(Qk) ≥ Σk=1 c(Qk)
a(Ql) l−1 j a(Qj+1) a(Qj) = d(F ) − Σj=1Σk=1d(Qk)( − ) c(Ql) c(Qj+1) c(Qj)
a(Ql) l−1 j a(Qj+1) a(Qj) ≥ a(F )t − Σj=1Σk=1a(Qk)t( − ) c(Ql) c(Qj+1) c(Qj) 2 l ta(Qk) = Σk=1 c(Qk) ≥ 0
The first inequality is from (2) and the fact c(Qk) > 0, the second equality is Abel’s sum- mation formula. The second inequality comes from (1) and the left side of (3). The last equality is Abel’s summation formula.
Therefore c(F )s + b(F ) > 0 for s ∈ R>0. The proof is complete.
Remark 3.1.5. The proof is similar to the proof in [Lan04]. We can also put positive coeffi- + cients in front of b and d, but this can be induced by the action of GL˜ (2, R), so this makes no essential difference.
53 Theorem 3.1.6. If (A,Z) is a rational stability condition on D(X), then the pair σs,t =
t s,t (AS,ZS ) is a rational stability condition on D(X × S) for s, t ∈ Q>0.
t Proof. Firstly, we need to prove that AS is Noetherian. The idea of the proof is essentially the same as in [PT19]. Readers should consult [PT19, Section 2.3] for details.
t Suppose there exists an object E ∈ AS and an infinite sequence of surjections
E E1 E2 ··· .
s,t t s,t Since a, d are discrete and Im(ZS (F )) ≥ 0 for any F ∈ AS, we may assume Im(ZS (Ei)) =
s,t t Im(ZS (E)) for all i. Then consider the following short exact sequences in AS
0 → Fi → E → Ei → 0.
s,t We have Im(ZS (Fi)) = 0 by assumption. And by the Noetherianity of AS, we can as- sume that H0 (E) = H0 (E ) and H−1 (F ) is independent of i. By setting V = H−1 (E)/H−1 (F ), AS AS i AS i AS AS i we have the following short exact sequence in AS
0 → V → H−1 (E ) → H0 (F ) → 0. AS i AS i
Then we look at the short exact sequences
0 → Fi → Fj → Fij → 0, for i < j.
Since H0 (F ), H0 (F ) ∈ T and AS i AS ij
Im(Zs,t(H0 (F ))) = Im(Zs,t(H0 (F ))) = 0, S AS i S AS ij
54 by the proof of Theorem 3.1.4, we get
a(H0 (F )) = c(H0 (F )) = d(H0 (F )) = 0 AS i AS i AS i
and a(H0 (F )) = c(H0 (F )) = d(H0 (F )) = 0. AS ij AS ij AS ij
Hence, we have c(H−1 (F )) = c(H−1 (F )) − c(H−1 (F )) = 0, AS ij AS j AS i
and H−1 (F ) ∈ F. Then from the definition of F we know that H−1 (F ) = 0. Therefore, AS ij AS ij we have H0 (F ) ⊂ H0 (F ) ··· , AS 1 AS 2 which gives us an infinite filtration in F
H−1 (E ) ⊂ H−1 (E ) ··· AS 1 AS 2
where H−1 (E )/H−1 (E ) = H0 (F ). AS j AS i AS ij Therefore, by Lemma 3.1.1, we know that b(H0 (F )) < 0 if H0 (F ) 6= 0. Hence AS ij AS ij b(H−1 (E )) < b(H−1 (E )) for any i < j if H0 (F ) 6= 0, which is equivalent to F 6= 0. AS j AS i AS ij ij Let us use Q to denote H−1 (E ). As in the proof of [PT19, Lemma 2.15], we can assume i AS i
Q1 is semi-stable with respect to σt by induction on the number of HN factors of Q1. Hence
by [PT19, Sublemma 2.16], Qi is semi-stable with respect to σt for all i. Then by Lemma 3.2.4, we know that
b(Qi)c(Qi) − a(Qi)d(Qi) ≥ 0,
where a, d, c are constant on Qi and b decreases as i grows. Since b is discrete, the inequalities hold for all i only if
b(Qi) = b(Qi+1)
55 for i 0, or
c(Qi) = 0.
The first case implies Fij = 0 for i sufficiently large, the second case combining the fact
Qi ∈ F force Qi = 0. In either case, the filtration terminates after finite steps. By Lemma 3.2.7, we also have the support property.
Remark 3.1.7. In fact, our construction also works with analogue proofs for stability condi- tions on Kuznetsov components; please see [BLMS17] and [BLM+19].
We conclude this section by providing a lemma, which might be useful in characterizing geometric stability conditions.
Lemma 3.1.8. Suppose E is an object in AS. If a(E) = c(E) = d(E) = 0 and b(E) is
t minimal in the image of the real part of Z. Then E is a simple object in AS.
t Proof. Since c(E) = 0, we have E ∈ T , hence E ∈ AS. Suppose we have a short exact sequence 0 → K → E → Q → 0
t in AS. Then taking cohomology with respect to AS gives us an exact sequence
0 → H−1 (Q) → H0 (K) → E → H0 (Q) → 0 AS AS AS
in A . By assumption we know that a(E) = c(E) = d(E) = a(H0 (Q)) = c(H0 (Q)) = S AS AS d(H0 (Q)) = 0, hence H−1 (Q) and H0 (K) are of the same slope with respect to ν . This AS AS AS t contradicts the definition of T and F unless H−1 (Q) = 0. Therefore, we have the following AS short exact sequence 0 → H0 (K) → E → H0 (Q) → 0 AS AS in T . Since b(E) is minimal, we know that either K or Q must be zero.
56 Hence, we have the following corollary.
Corollary 3.1.9. If σ = (A,Z) is a geometric stability condition, and Z(k(x)) ∈ R<0 is minimal in the image of the real part of Z, then σs,t is a geometric stability condition for any two positive rational numbers s, t.
Proof. Suppose that E is the skyscraper sheaf on X × S, then we have a(E) = c(E) = 0 and b(E) + id(E) = Z(k(x)). Hence, it follows directly from Lemma 3.1.8.
The following is a trivial case of our construction.
Example 3.1.10. Let X be a point, σ = (V ect, −dim) a stability condition on D(X), and S be an elliptic curve. Then we have the global heart equivalent to Coh(S), and the
polynomial LE(n) = −rk(E)·n−deg(E). Hence we have a(E) = −rk(E), b(E) = −deg(E), c(E) = d(E) = 0. Our construction provides (Coh(S), −deg(E) + rk(E)t) as stability conditions on D(S).
To provide a slightly nontrivial example, let us consider the product of two curves. The following is the continuation of Example 1.2.9.
Example 3.1.11. Let X be a smooth projective cure over C, σ = (Coh(X), −deg + i · rank) be a stability condition on D(X), and S be an elliptic curve. Then the global heart is Coh(X × S), the category of coherent sheaves on X × S.
As in Example 1.2.9, we denote denote the Chern characters of E by ch(E) = (r, m1l1 +
2 0 0 2 1 m2l2 + δ, v), where l1 ∈ H (X, Z) ⊗ H (S, Z), l2 ∈ H (X, Z) ⊗ H (S, Z), δ ∈ H (X, Z) ⊗
1 H (S, Z) and m1, m2 ∈ Q. Then the polynomial can be written as
LE(n) = −(v + n · ch1(E)l1) + i(r · n + ch1(E)l2).
Hence, we have a(E) = −ch1(E)l1, b(E) = −v, c(E) = r, d(E) = ch1(E)l2. Our positivity
57 lemma in this case is
b(E)c(E) − a(E)d(E) = −v · r + ch1(E)l1 · ch1(E)l2
= m1 · m2 − r · v 1 = (ch (E)2 − δ2 − 2ch (E)ch (E)) ≥ 0. 2 1 0 2
This inequality can also be deduced form Bogomolov-Gieseker inequality
2 ch1(E) − 2ch0(E)ch2(E) ≥ 0
2 and Hodge index theorem. Indeed, Hodge index theorem implies that δ ≤ 0 since δ·(l1 +l2) =
0 and l1 + l2 is ample. In this case, the weak stability condition σt is the pair
(Coh(X × S), −ch1(E)(t · l1 + l2) + i · rk(E)).
This weak stability condition coincides with the weak stability condition σw in Example 1.1.14,
t where w = t · l1 + l2. Thus, the tilted heart AS is A0,w, and the central charge
s,t ZS (E) = s · r − v + i(t · ch1(E)l1 + ch1(E)l2).
If s = t, this is the central charge Z0,t·l1+l2 . In general, by Remark 1.1.16, we know that
σs,t is a stability condition for any s, t > 0.
3.2 Large volume limit and support property
Suppose we have a rational stability condition σ = (A,Z) on D(X). By Definition 1.1.3, Z
v g factors as K(A) = K(D(X)) −→ Λ −→ C. We assume σ satisfies the support property with respect to the quadratic form Q on Λ ⊗ R. There is an equivalent definition of support property.
58 Definition 3.2.1 ([KS08, Section 1.2]). Pick a norm k k on Λ⊗R. The stability condition σ satisfies the support property if there exists a constant C > 0 such that for all σ-semistable objects 0 6= E ∈ D(X), we have
kv(E)k ≤ C|Z(E)|.
Then the quadratic form Q can be written as Q(w) := C2|Z(w)|2 − kwk2.
s,t We have the following factorization of ZS .
s,t Lemma 3.2.2. The central charge ZS factors as
T (v1,v2) (sIm(g)−itRe(g),g) K(AS) −−−−→ Λ ⊕ Λ −−−−−−−−−−−→ C,
where
∗ ∗ v1(E) = v(p∗(E ⊗ q O(n))) − v(p∗(E ⊗ q O(n − 1))),
∗ v2(E) = v(p∗(E ⊗ q O(n))) − n · v1(E)
T for n 0, and the superscript T in (v1, v2) stands for transpose. We also have Im(g)◦v1 =
c, Re(g) ◦ v1 = a, Im(g) ◦ v2 = d, Re(g) ◦ v2 = b.
Proof. Assume we have a short exact sequence
0 → K → E → Q → 0
in AS. This gives a triangle
∗ ∗ ∗ [1] ∗ p∗(K ⊗ q O(n)) → p∗(E ⊗ q O(n)) → p∗(Q ⊗ q O(n)) −→ p∗(K ⊗ q O(n))[1]
59 in D(X). Since v : K(A) = K(D(X)) → Λ is a group homomorphism, we get
∗ ∗ ∗ v(p∗(E ⊗ q O(n))) = v(p∗(K ⊗ q O(n))) + v(p∗(Q ⊗ q O(n)))
for any n 0. This proves that v1(E) = v1(K) + v1(Q). Therefore, v1 is a group homomor-
phism. Similarly, we can prove v2 is a group homomorphism too.
We also need to prove that v1 and v2 are independent of n when n 0. For v1, the independence follows from the following triangle
∗ ∗ [1] ∗ p∗(E ⊗ q O(n − 1)) → p∗(E ⊗ q O(n)) → p∗(E|D) −→ p∗(E ⊗ q O(n − 1))[1],
where D ∈ |O(1)| is a 0 dimensional subscheme of finite length and E|D is the derived
pull-back of E along X × D. Hence we have v1(E) = v(p∗(E|D)), which is independent of n.
The independence of v2 follows from the independence of v1 by simple calculation.
∗ ∗ Apply g on v1, we get g ◦ v1(E) = g ◦ v(p∗(E ⊗ q O(n))) − g ◦ v(p∗(E ⊗ q O(n − 1))) =
LE(n) − LE(n − 1) = a(E) + ic(E). Hence we get Re(g) ◦ v1 = a and Im(g) ◦ v1 = c.
Similarly, we can prove that Im(g) ◦ v2 = d, Re(g) ◦ v2 = b. Therefore, we have
T s,t (sIm(g) − itRe(g), g) ◦ (v1, v2) (E) = sc(E) − ita(E) + b(E) + id(E) = ZS (E).
Definition 3.2.3. We call w ∈ Λ a semi-stable vector if w = v(E) for some semi-stable object E ∈ A.
Lemma 3.2.4. If E ∈ AS is semi-stable with respect to µ1, then v1(E) is a semi-stable vector.
60 Proof. We can take the short exact sequence
0 → F → E → E¯ → 0
where F is the maximal torsion subobject of E, E¯ is torsion free hence t-flat. It is easy to ¯ see that v1(E) = v1(E). Therefore, we can assume E is t-flat. Now we consider the short exact sequence in Proposition 1.2.18
0 → K → E → Q → 0.
Here Q is torsion and K is torsion free since E is torsion free, hence K is t-flat and semi-
stable with respect to µ2. By Proposition 1.2.17, K is fiberwisely semi-stable. Therefore, it
is easy to see v1(E) = v1(K) is a semi-stable vector.
Lemma 3.2.5. If E ∈ AS is semi-stable with respect to the weak stability condition σt =
(AS,Zt) for a fixed t ∈ Q>0, then
b(E)c(E) − a(E)d(E) + ηQ(v1(E)) ≥ 0
t for 0 ≤ η ≤ C2 .
Proof. If c(E) = 0, then b(E)c(E) − a(E)d(E) ≥ 0 by Lemma 3.1.1 and Q(v1(E)) ≥ 0 by last lemma. Therefore, the statement is true in this case. We assume c(E) > 0, take the HN filtration
0 = E0 ⊂ E1 ⊂ E2 ⊂ · · · ⊂ El−1 ⊂ El = E,
and use Qk to denote Ek/Ek−1 for 1 ≤ k ≤ l. Then we have the same inequalities (1) and (2) as in the proof of Theorem 3.1.4, and we have the following inequalities
61 j l Σk=1(−a(Qk)t + d(Qk)) −a(E)t + d(E) Σk=j(−a(Qk)t + d(Qk)) j ≤ ≤ l . (3.4) Σk=1c(Qk)t c(E)t Σk=jc(Qk)t
We have c(Qk) > 0 by same reason in the proof of Theorem 3.1.4. Similarly we get
l a(Qk)d(Qk) b(E) ≥ Σk=1 c(Qk)
a(Ql) l−1 j a(Qj+1) a(Qj) = d(F ) − Σj=1Σk=1d(Qk)( − ) c(Ql) c(Qj+1) c(Qj)
a(Ql) l−1 j −a(E)t + d(E) j a(Qj+1) a(Qj) ≥ d(F ) − Σj=1(Σk=1 c(Qk) + Σk=1a(Qk)t)( − ) c(Ql) c(E) c(Qj+1) c(Qj)
l a(Qk) −a(E)t + d(E) = Σk=1 ( c(Qk) + a(Qk)t) c(Qk) c(E) 2 −a(E)t + d(E) l a(Qk) t = a(E) + Σk=1 . c(E) c(Qk)
Therefore, we have
2 l l a(Qk) t 2 b(E)c(E) − a(E)d(E) ≥ Σk=1c(Qk)Σk=1 − a(E) t c(Qk)
a(Qi) » a(Qj) » 2 = tΣ1≤i On the other hand, let wi := v1(Qi). Then l l Q(v1(E)) = Q(Σi=1wi) = Σi=1Q(wi) + 2Σ1≤i By Lemma 3.2.4, we know that Q(wi) ≥ 0. Therefore, it suffices to prove that a(Qi) » a(Qj) » 2 t(» c(Qj) − » c(Qi)) + 2ηQ(wi, wj) ≥ 0 c(Qi) c(Qj) t for 0 ≤ η ≤ C2 and arbitrary 1 ≤ i < j ≤ l. 62 We have 2Q(wi, wj) = Q(wi + wj) − Q(wi) − Q(wj) 2 2 2 2 2 2 2 2 2 = C |Z(Qi) + Z(Qj)| − kwi + wjk − C |Z(Qi)| + kwik − C |Z(Qj)| + kwjk 2 2 2 2 2 2 2 = C ((a(Qi) + a(Qj)) + (c(Qi) + c(Qj)) ) − a(Qi) − c(Qi) − a(Qj) − c(Qj) ) 2 2 2 − kwi + wjk + kwik + kwjk 2 ≥ 2C (a(Qi)a(Qj) + c(Qi)c(Qj)) − 2kwikkwjk 2 » 2 2» 2 2 ≥ 2C (a(Qi)a(Qj) + c(Qi)c(Qj) − a(Qi) + c(Qi) a(Qj) + c(Qj) ). The first inequality is from Cauchy-Schwarz inequality. The second inequality comes from the definition of support property. Hence, it suffices to prove that a(Qi) » a(Qj) » 2 t(» c(Qj) − » c(Qi)) c(Qi) c(Qj) 2 » 2 2» 2 2 ≥ 2ηC ( a(Qi) + c(Qi) a(Qj) + c(Qj) − a(Qi)a(Qj) − c(Qi)c(Qj)). By Cauchy’s inequality, the right hand side is nonnegative. Hence, it is enough to prove the t inequality for η = C2 . In this case, the inequality is equivalent to 2 2 a(Qi) a(Qj) c(Qj) + c(Qi) + 2c(Qi)c(Qj) c(Qi) c(Qj) » 2 2» 2 2 ≥ 2 a(Qi) + c(Qi) a(Qj) + c(Qj) which becomes trivial if we divide both sides by c(Qi)c(Qj). Therefore, the lemma is proved. t s,t Now, we consider the stability conditions σs,t = (AS,ZS ). t Lemma 3.2.6. If t is a fixed positive rational number and E ∈ AS is semi-stable with respect t s,t to σs,t = (AS,ZS ) for all s sufficiently large. Then b(E)c(E) − a(E)d(E) + ηQ(v1(E)) ≥ 0 t for 0 ≤ η ≤ C2 . 63 Proof. We have that E sits in the following short exact sequence 0 → F [1] → E → T → 0 where F ∈ F,T ∈ T . We claim that one of the following cases is true: (i) F = 0, and T is semi-stable with respect to σt. (ii) c(T ) = a(T ) = d(T ) = 0, and F is semi-stable with respect to σt. The proof of the claim is essentially the same as in [BMS16, Lemma 8.9]. We sketch the proof for reader’s convenience. If F = 0, it is easy to see T is semi-stable with respect to σt. Therefore, the inequality holds by Lemma 3.2.5. t s,t Now we can assume F 6= 0. Since E is semi-stable with respect to σs,t = (AS,Zs ) for s sufficiently large. We have c(F )s + b(F ) (c(F ) − c(T ))s + b(F ) − b(T ) ≤ a(F )t − d(F ) (−a(T ) + a(F ))t + d(T ) − d(F ) for s sufficiently large. But we have (−a(T ) + a(F ))t + d(T ) − d(F ) ≥ a(F )t − d(F ) ≥ 0, and c(F ) > 0, c(T ) ≥ 0. As a consequence, the inequality can hold if only if c(T ) = a(T ) = d(T ) = 0, which implies that T is a torsion object, hence v1(T ) = 0. In this case, it is easy to see F is semi-stable with respect to σt. 64 Then b(E)c(E) − a(E)d(E) = (b(T ) − b(F ))(−c(F )) − (−a(F ))(−d(F )) = −b(T )c(F ) + b(F )c(F ) − a(F )d(F ) ≥ b(F )c(F ) − a(F )d(F ) and Q(v1(E)) = Q(v1(F )) because v1(T ) = 0. Hence, the inequality follows from Lemma 3.2.5. t t s,t Lemma 3.2.7. If E ∈ AS is semi-stable with respect to σs,t = (AS,ZS ) for fixed s, t ∈ Q>0. Then b(E)c(E) − a(E)d(E) + ηQ(v1(E)) ≥ 0 min{s,t} for any 0 ≤ η ≤ C2 . Proof. The idea of this proof is essentially from [BMS16, Lemma 8.8]. We first prove that s,t the kernel of ZS is negative semi-definite with respect to bc − ad + ηQ min{s,t} for 0 ≤ η ≤ C2 . s,t If ZS (E) = 0, we have 2 2 b(E)c(E) − a(E)d(E) + ηQ(v1(E)) = −sc(E) − ta(E) + ηQ(v1(E)) ≤ −sc(E)2 − ta(E)2 + ηC2(a(E)2 + c(E)2) ≤ 0 min{s,t} for 0 ≤ η ≤ C2 . Since the image of −at+d is discrete, we can prove the lemma by induction on −a(E)t+ d(E). If −a(E)t + d(E) = 0 or −a(E)t + d(E) is minimal in the image of imaginary part, then it is easy to see E is semi-stable for m sufficiently large. Therefore, the inequality holds by Lemma 3.2.6. Now we assume it is true for objects whose imaginary part is less than 65 N0 > 0. Let t be a positive rational number. Suppose E is semi-stable with respect to σs0,t for a positive rational number s0 and −a(E)t + d(E) = N0. By [BMS16, Lemma A.6], we can assume E is stable. If E remains semi-stable with respect to σs,,t, for all s > s0, then this follows from Lemma 3.2.6. Otherwise, suppose E is unstable with respect to σs1,t for a positive rational number s1 bigger than s0. Let s1,t W := {ZS (F )|0 6= F ⊂ E and µs1,t(F ) > µs1,t(E)} where µs1,t is the associated slope function of σs1,t. Then by [MS17, Lemma 4.9] and dis- s1,t creteness of ZS , we know that W is a finite subset in C. For any element w ∈ W , we set s1,t Mw := {F | F ⊂ E and ZS (F ) = w}. s0,t Then since ZS is discrete, we can find Fw ∈ Mw such that µs0,t(Fw) = max µs0,t(F ). F ∈Mw Since a, b, c, d are linear and rational, we can find a positive rational number sw, such that s0 < sw < s1 and sw,t sw,t ZS (Fw)/ZS (E) ∈ R>0. Moreover, by the definition of Fw, we know that µsw,t(F ) ≤ µsw,t(E) for all F ∈ Mw. 0 Therefore, if we take s = minw∈W sw, which is also a positive rational number and s0 < 0 s < s1. One can easily check that E is strictly semi-stable with respect to σs0,t. Then by induction, all its Jordan-H¨olderfactor (with respect to σs0,t) satisfy the inequality. Therefore, the inequality holds for E by [BMS16, Lemma A.6]. 66 Remark 3.2.8. Sometimes we only use the case η = 0, like in the following theorem. The difference is that when η = 0, we only get the support property on a rank 4 quotient lattice min{s,t} Λ/ker(g) ⊕ Λ/ker(g). While for 0 < η < C2 , we get the stronger support property on s,t the lattice Λ ⊕ Λ/ker(g). Indeed, for E ∈ ker(ZS ), we have b(E)c(E) − a(E)d(E) = −sc(E)2 − ta(E2) ≤ 0. The equality holds if and only if a(E) = b(E) = c(E) = d(E) = 0. min{s,t} While for 0 < η < C2 , we have 2 2 b(E)c(E) − a(E)d(E) + ηQ(v1(E)) = −sc(E) − ta(E) + ηQ(v1(E)) ≤ −sc(E)2 − ta(E)2 + ηC2(a(E)2 + c(E)2) ≤ 0. The equality holds if and only if a(E) = b(E) = c(E) = d(E) = 0 and v1(E) = 0 since Q|ker(g) is negative definite and v1(E) ∈ ker(g). Theorem 3.2.9. We have a map η : RStab(X) × R>0 × R>0 → Stab(X × S), where S is an integral smooth projective curve. Moreover, the stability conditions in the image satisfy the support property. Proof. The image satisfy the support property because of Lemma 3.2.7. 0 By Theorem 3.1.6, we have the map η : RStab(X)×Q>0 ×Q>0 → Stab(X ×S). We only 0 need to prove that η is continuous at the factor Q>0 × Q>0 for any given rational stability condition σ ∈ RStab(X). We look at rational stability conditions σs0,t0 , where s0, t0 ∈ Q>0. We assume that s0 ≥ t0 1 1 (the case s0 < t0 is similar). Then if |s − s0| < sin( 10 π)t0 and |t − t0| < sin( 10 π)t0, we have 67 1 |Zs,t(E) − Zs0,t0 (E)| ≤ sin( π)|Zs0,t0 (E)| S S 10 S for E semi-stable with respect to σs0,t0 . More specifically, by Lemma 3.2.7, we have s ,t 1 0 0 2 2 2 2 2 2 2 |ZS (E)| = (c(E) s0 + b(E) + 2s0b(E)c(E) + a(E) t0 + d(E) − 2t0a(E)d(E)) 1 2 2 2 2 2 2 2 ≥ (c(E) (s0 − (s0 − t0) ) + a(E) t0 + d(E) + 2t0(b(E)c(E) − a(E)d(E))) 1 2 2 2 2 2 ≥ (c(E) t0 + a(E) t0) and s,t s ,t 1 0 0 2 2 2 2 2 |ZS (E) − ZS (E)| = ((t − t0) a(E) + (s − s0) c(E) ) 1 2 2 2 2 1 ≤ sin( π)(c(E) t + a(E) t ) 2 . 10 0 0 Therefore, by [Bri07, Theorem 7.1], we get the map. ≤1 3.3 Stability conditions on DS In this section, we assume that S is a smooth projective algebraic variety of arbitrary di- mension r. We will generalize the construction in Section 3.1 and Section 3.2 to triangulated subcategory in D(X × S). Given a t-structure on a triangulated category D, we get the heart A ⊂ D. On the converse direction, one can show that the heart determines the t-structure as well. The following result is included in [Bri08, Section 3] (one can also consult [BBD82], [GM13] and [HT07]). Lemma 3.3.1 ([Bri08, Lemma 3.1]). A bounded t-structure is determined by its heart. More- over, if A ⊂ D is a full additive subcategory of a triangulated category D, then A is the heart 68 of a bounded t-structure on D if and only if the following conditions hold: (a) if A and B are objects of A then HomD(A, B[k]) = 0 for k < 0, (b) for every nonzero object E ∈ D there are integers m < n and a collection of triangles 0 = Em Em+1 Em+2 ··· En−1 En = E Am+1 Am+2 An with Ai[i] ∈ A for all i. Proof. See [Bri08, Lemma 3.1]. ≤j Definition 3.3.2. We let the triangulated subcategory in D(X × S) generated by AS to ≤j be denoted by DS . ≤j ≤j Lemma 3.3.3. There exists a t-structure on DS , and its heart is AS . ≤j Proof. We only need to prove that AS satisfy the conditions (a) and (b) in Lemma 3.3.1. ≤j For (a), if A and B are two objects in AS , they are objects in AS as well. Hence HomD(A, B[k]) = 0 for k < 0. ≤j ≤j Since DS is the sub-triangulated category generated by AS , (b) follows automatically. By Lemma 2.2.13, we know that t ≤j σ≤j = (AS , ajt − bj−1 + ibj) ≤j is a weak stability condition on DS . Hence, we have a torsion pair (T , F), given by t ≤j T = {E ∈ A | µ t (E) > 0} ≤j S σ≤j ,min and t ≤j F = {E ∈ A | µ t (E) ≤ 0}. ≤j S σ≤j ,max ≤j,t t t We let AS to denote the tilted heart hF≤j[1], T≤ji. 69 ≤j,t Proposition 3.3.4. For any two positive numbers s, t > 0, the pair (AS , sbj + aj−1 + ≤j,t i(−ajt + bj−1)) is a weak stability condition. Moreover, if E ∈ AS with trivial central j−2 charge, then E ∈ AS . Proof. This follows from 2.2.13, Proposition 2.2.14, Section 3.1 and Section 3.2. Hence , we have the following corollary. ≤1,t Corollary 3.3.5. For any two positive numbers s, t > 0, the pair (AS , sb1 + a0 + i(−a1t + b0)) is a stability condition. ≤j Remark 3.3.6. For j > 1, this procedure just make it less weak than (AS , aj + ibj). One ≤j ≤j−2 may expect to get a stability condition on the quotient triangulated category DS /DS . ≤j ≤j−2 But the problem is that we do not have a natural t-structure on DS /DS whose heart is j j−2 equivalent to AS/AS . This problem can be solved if the quotient functor ≤j ≤j ≤j−2 Q : DS → DS /DS admits a right joint, i.e., a section functor ≤j ≤j−2 ≤j P : DS /DS → DS . We conclude this section by providing a quadratic equality, which suffices to construct a stability condition from data (A; a, b, c, d) satisfying Lemma 3.1.1. Definition 3.3.7. Let D be a triangulated category, suppose we have the data (A; a, b, c, d). Here A is the heart of a bounded t-structure on D, and a, b, c, d : K(A) → Q are group homomorphisms from K(A) to Q. We call the data (A; a, b, c, d) pre-tilt data if A is Noetherian and they satisfy Lemma 3.1.1. If (A; a, b, c, d) is pre-tilt data, then the pair σt = (A, a − dt + ic) is a weak stability condition for any positive number t ∈ R≥0. 70 Proposition 3.3.8. If for any σ0-semi-stable object E ∈ A with c(E) 6= 0, the quantity b(E)c(E) − a(E)d(E) c(E)2 t is bounded below, and suppose the lower bound is s0. Then (A , sc + b + i(−a + dt)) is a t stability condition for any s > s0, t ∈ R>0, where A is the heart tilted with respect to σt. Proof. Using the proof in Proposition 3.1.4, we can show that sc+b+i(−a+dt) is a stability function on At. Then we argument as in Section 3.2 to show the HN-property and support property. 71 Bibliography [AB13] Daniele Arcara and Aaron Bertram. Bridgeland-stable moduli spaces for K- trivial surfaces. J. Eur. Math. Soc. (JEMS), 15(1):1–38, 2013. With an appendix by Max Lieblich. [AGH19] Benjamin Antieau, David Gepner, and Jeremiah Heller. 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