Moduli of Bridgeland semistable holomorphic triples
aa r X i v : . [ m a t h . AG ] F e b MODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES
DOMINIC BUNNETT AND ALEJANDRA RINCÓN-HIDALGOA
BSTRACT . We prove that the moduli stack of Bridgeland semistable holomorphic triples over acurve of g ( C ) ≥ C and admits a proper good moduli space. We prove that this also holds for a class of Bridgelandstability conditions on the category of holomorphic chains T C , n .In the process, we construct an explicit geometric realisation of T C , n and prove the open heartproperty for noetherian hearts in admissible categories of D b ( X ) , where X is a smooth projectivevariety over C , whose orthogonal complements are geometric triangulated categories.
1. I
NTRODUCTION
The purpose of this paper is to study the moduli of Bridgeland semistable holomorphic chains.Holomorphic chains were first introduced by Álvarez-Consúl and García-Prada in [ ] . An n -holomorphic chain is a chain of morphisms E ϕ −→ E ϕ −→ · · · ϕ n − −−→ E n ,where E i ∈ Coh ( X ) with X a smooth projective variety over C . Moduli spaces of holomorphicchains of vector bundles were constructed by Schmitt [ ] using geometric invariant theory (GIT).These moduli spaces have played an important role in the study of Higgs bundles [
9, 16 ] .We denote the abelian category of holomorphic chains by Q X , n and its derived category T X , n .We also refer to objects of T X , n as holomorphic chains. When n = T X : = T X ,2 and referto it as the category of holomorphic triples.Stability conditions on triangulated categories were introduced by Bridgeland [ ] and play avery important role in algebraic geometry via the study of moduli spaces and wall-crossings. In [ ] the second author, Martínez-Romero and Rüffer completely described the stability manifoldof T C for a curve C .The construction of moduli spaces of Bridgeland semistable objects is highly non-trivial. Build-ing on the work of Lieblich [ ] , Toda carried out this construction for the derived category of aK3 surface X [ ] . In particular, Toda proved that the moduli space of σ -semistable objects in D b ( X ) is an algebraic stack of finite type over C for some σ ∈ Stab ( X ) . Toda and Piyaratne conjectured in [
30, Conjecture 1.1 ] that the same holds for the derived cate-gory of any smooth projective variety. This conjecture has been confirmed for K3 surfaces, three-folds satisfying the Bogomolov-Gieseker inequality, and where replaces the derived category withthe Kuznetsov component of a cubic fourfold [
33, 30, 5 ] . The main result of this paper confirmsthis conjecture when D b ( X ) is replaced by T C .Moduli spaces of objects in D b ( X ) were studied by Lieblich in [ ] and by Abramovich andPolishchuk in [ ] . In [ ] , motivated by the study of the Kuznetsov component, Bayer, Lahoz, Macrì,Nuer, Perry and Stellari studied moduli problems associated to full admissible subcategories of D b ( X ) and of relative moduli spaces building on work of Lieblich, Piyaratne, and Toda [
23, 33, 30 ] .In order to construct the moduli space of Bridgeland semistable holomorphic chains, we firstneed to embed T X , n into D b ( Y X , n ) for a smooth projective variety Y X , n . This is precisely the con-tents of Theorem 3.5 which constructs an explicit embedding. We refer to this embedding by T X , n , → D b ( Y X , n ) as a geometric realisation of T X , n .The construction of Y X , n is a generalisation of the work of Orlov [ ] , the key difference beingthat semiorthogonal decompositions are needed in place of strong exceptional collections. Inconstructing the embedding we characterise T X , n via its semiorthogonal components and the as-sociated gluing functor. Gluing semiorthogonal components (as in [ ] ) requires that one workson the level of dg-categories.To understand families of objects in T X , n we look to the base change (cid:0) T X , n (cid:1) S ⊂ D b ( Y X , n × S ) ,where S is a base scheme, defined by Kuznetsov [ ] . We then consider local t-structures [ ] on ad-missible subcategories in D b ( X ) (so-called sheaves of t-structures by Abramovich and Polishchuk [ ] ).As laid out in [
1, Section 6.1 ] , to construct moduli spaces of Bridgeland semistable objects withrespect to an algebraic stability conditions three problems remain:(1) the generic flatness property ;(2) the open heart property , and;(3) the boundedness of semistable objects of fixed type β and phase φ .Let us first address boundedness. If boundedness of a moduli space associated to a stabilitycondition can be shown to hold, then the same is true for any stability condition in its connectedcomponent. Thus, via the well-studied GIT moduli space of holomorphic chains of vector bun-dles [ ] , we can conclude boundedness for the connected component containing GIT-stabilityconditions. Moreover, by [
26, Theorem 1.1 ] , boundedness follows for the entire stability manifoldwhen n = ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 3
For a noetherian heart in D b ( X ) the ‘open heart property’ (Definition 4.13) is already known [
1, Proposition 3.3.2 ] . In the case of T X , n , we appeal to the structure inherited from the geometricrealisation. For a noetherian heart A ⊆ T X , n , our approach is to pass down properties alreadyproved for hearts in D b ( X ) . To this end, we construct a bigger noetherian heart via recollement(as in [ ] ) f A ⊆ D b ( Y X , n ) containing A . In Proposition 4.17, we prove in a general setting, thatbeing a recollement heart is stable under base change.More generally, we use the same strategy to prove the open heart property for a heart A in anadmissible subcategory T ⊆ D b ( X ) under the condition that T ⊥ is geometric (see Remark 4.22).We first prove the generic flatness property (Definition 4.24) for algebraic stability conditionsconstructed via gluing (Remark 2.12). By adapting the results of [ ] , we can then extend this toany algebraic stability condition in the Ý GL + ( R ) -orbit of a gluing stability condition. Further-more, given that the support property is satisfied for triples, we obtain the following theorem. Theorem 1.1 (Theorem 4.35 and 4.39) . Let σ ∈ Stab ( T C ) be a stability condition. The stack M β , φ ( σ ) is an algebraic stack of finite type over C admitting a proper good moduli space. Further, we obtain a partial result in the case of holomorphic chains in Proposition 4.37 andTheorem 4.39.For holomorphic triples, one has a complete picture of the wall and chamber structure in theclassical situation [ ] . By studying the wall and chamber decomposition of the stability mani-fold, we get a full picture of the moduli of holomorphic triples. We expect that the moduli spacesare projective and their that their birational geometry is dictated by the wall and chamber struc-ture - this will be pursued elsewhere. Layout.
The layout of the paper is as follows. In Section 2, we provide preliminaries, fixing no-tation and providing results central to this paper. In Section 3, we study geometric realisationsof T X , n . This involves first studying the structure of T X , n in the setting of dg-categories and sub-sequently explicitly constructing a geometric realisation as a tower of projective bundles withfoundation X . Finally, in Section 4 we study the moduli stacks themselves. Notation and conventions.
We always work over C . A curve is a smooth irreducible projectivevariety of dimension 1. We denote by C -dgm the dg-category of complexes of C -vector spaces.2. P RELIMINARIES
Bridgeland stability conditions.
DOMINIC BUNNETT AND ALEJANDRA RINCÓN-HIDALGO
Definition 2.1. A t-structure on a triangulated category T consists of a pair of full additive sub-categories ( T ≤ , T ≥ ) satisfying the following properties. We write T ≤ i : = T ≤ [ − i ] and T ≥ i : = T ≥ [ − i ] for i ∈ Z .(1) Hom T ( T ≤ , T ≥ ) = E ∈ T , there is a distinguished triangle G → E → F → G [ ] with G ∈ T ≤ and F ∈ T ≥ .(3) T ≤ ⊂ T ≤ and T ≥ ⊃ T ≥ .A t-structure is bounded if every E ∈ T is contained in T ≤ n ∩ T ≥− n for some n >
0. The heart ofa bounded t-structure ( T ≤ , T ≥ ) is defined as A : = T ≤ ∩ T ≥ . Remark 2.2.
The heart A of a bounded t-structure on T is an abelian category and K ( A ) = K ( T ) . Definition 2.3. A slicing P on T is a collection of full subcategories P ( φ ) for all φ ∈ R satisfying:(1) P ( φ )[ ] = P ( φ + ) , for all φ ∈ R .(2) If φ > φ and E i ∈ P ( φ i ) , i =
1, 2, then Hom T ( E , E ) = E ∈ T there exists a finite sequence of maps0 = E f −→ E f −→ · · · → E m − f m − −−→ E m = E and of real numbers φ > · · · > φ m − such that the cone of f j is in P ( φ j ) for j = · · · , m − I ⊆ R we define P ( I ) to be the extension-closed subcategory generated bythe subcategories P ( φ ) with φ ∈ R . Definition 2.4.
Let A be a heart. We say that a group homomorphism Z : K ( A ) → C is a stabilityfunction on A if the image of Z is contained in the semi-closed upper half plane H = { α ∈ C |ℑ ( α ) ≥ ℑ ( α ) =
0, then ℜ ( α ) < } .We now fix a finite rank Z -lattice Λ and a surjective homomorphism v : K ( T ) ։ Λ . When T isnumerically finite, we have that the numerical Grothendieck group N ( T ) is a finite rank Z -lattice.We often choose Λ = N ( T ) and v as the natural projection.We consider a group homomorphism Z : Λ → C , such that Z ◦ v : K ( A ) → C is a stability func-tion on A . We define the slope by µ σ ( E ) = − ℜ ( Z ( E )) ℑ ( Z ( E )) if ℑ ( Z ( E )) = + ∞ otherwise , ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 5 where Z ( E ) : = Z ( v ([ E ])) . We say that a non-zero E ∈ A is σ -semistable (stable) if for all propersubobjects F ⊆ E , we have that µ σ ( F ) ≤ µ σ ( E )( < ) . We also define the phase of E as φ ( E ) = arg ( Z ( E )) π ∈ (
0, 1 ] . Definition 2.5. A pre-stability condition on T is a pair σ = ( Z , A ) , where A ⊆ T is the heartof a bounded t-structure and Z : Λ → C is a group homomorphism such that Z ◦ v : K ( A ) → C is a stability function on A and every E ∈ A has a Harder-Narasimhan (HN) filtration with σ -semistable factors. If additionally σ satisfies the support property i.e. there is a symmetric bilinearform Q on Λ R : = Λ ⊗ R such that Q ( v ( E ) , v ( E )) ≥ σ -semistable objects E ∈ A and it isnegative definite on the kernel of Z , then σ is called a Bridgeland stability condition with respectto Λ . Remark 2.6 ( [
10, Proposition 5.3 ] ) . To give a pre-stability condition σ on T is equivalent to givinga slicing P and a group homomorphism Z : Λ → C such that for every non-zero E ∈ P ( φ ) , wehave that Z ( E ) ∈ R > · e i πφ . The objects of P ( φ ) are precisely the σ -semistable objects of phase φ .The set of Bridgeland stability conditions with respect to ( Λ , v ) is denoted by Stab Λ ( T ) andmoreover, Stab Λ ( T ) admits the structure of a complex manifold [ ] and is referred to as the sta-bility manifold. If Λ = N ( T ) and v the natural projection, then the set of stability conditions isdenoted by Stab ( T ) . If T = D b ( X ) , we write by Stab ( X ) . Definition 2.7.
We call a stability condition σ = ( Z , A ) ∈ Stab ( T ) algebraic if the image of Z : N ( T ) → C is contained in Q ⊕ Q i .As in [
10, Lemma 8.2 ] , we consider the right action of Ý GL + ( R ) on the stability manifold. If σ = ( Z , A ) is a stability condition and g = ( T , f ) ∈ Ý GL + ( R ) , then we define σ · g = ( Z ′ , P ′ ) to be Z = T − ◦ Z and P ′ ( φ ) = P ( f ( φ )) , where P and P ′ are the slicings of Z and Z ′ respectively. Notethat the Ý GL + ( R ) -action preserves the semistable objects, but relabels their phases.Note that by [
24, Theorem 2.7 ] , if g ( C ) ≥ ( C ) ∼ = Ý GL + ( R ) . Let us consider the groupAut Λ ( T ) of autoequivalences Φ on T whose induced automorphism φ ∗ of K ( T ) is compatiblewith the map v : K ( T ) → Λ . We define a left action of the group Aut Λ ( T ) on the set of stabilityconditions. For Φ ∈ Aut Λ ( T ) of T . We define Φ ( σ ) = ( Z ′ , P ′ ) as Z ′ = Z ◦ φ − ∗ and P ′ ( φ ) = Φ ( P ( φ )) .Note that if E is a σ -semistable object, then Φ ( E ) is Φ ( σ ) -semistable.2.2. Bridgeland stability conditions on the category of holomorphic chains.
Let X be a smoothprojective variety over C . An n -holomorphic chain is a chain of morphisms E ϕ −→ E ϕ −→ · · · ϕ n − −−→ E n . DOMINIC BUNNETT AND ALEJANDRA RINCÓN-HIDALGO where E i ∈ Coh ( X ) . We denote the abelian category of such chains by Q X , n and its derived cate-gory by T X , n : = D b ( Q X , n ) .The special case Q X ,2 is the abelian category of holomorphic triples over X as in [ ] . In this case,we write T X : = T X ,2 .Recall the description of T X given in [
26, Section 3.1 ] : There is a semiorthogonal decomposi-tion T X = 〈 D , D 〉 where D j ∼ = T X is the image of the fully faithful embeddings i : D b ( X ) , → T X E ( E → ) , i : D b ( X ) , → T X E ( → E ) respectively.Note that we have a semiorthogonal decomposition of the form T X , n = 〈 D , T X , n − 〉 by see-ing T X , n − as the subcategory of objects E ∈ T X , n with E =
0. Inductively, we get the standardsemiorthogonal decomposition of T X , n . Definition 2.8.
We denote the semiorthogonal decomposition of T X , n by T X , n = 〈 D , . . ., D n 〉 ,where D j ∼ = D b ( X ) and is given by the image of the functor i j : D b ( X ) , → T X , n defined by sending E to the chain satisfying E j = E and E l = l = k . Remark 2.9.
From these semiorthogonal decompositions, we conclude that K ( T X , n ) = L ni = K ( D b ( X )) .Moreover, if C is a curve, we have that N ( T C , n ) = Z n with the isomorphism given by sending [ E ] to ( d , r , . . . , d n , r n ) with d i : = deg ( E i ) and r i = rank ( E i ) .2.2.1. CP-gluing and recollement.
Let T be a triangulated category equipped with a semiortho-gonal decomposition T = 〈D , D 〉 and let i j : D j → T be full embeddings for j =
1, 2. Throughoutthe whole paper, we assume that semiorthogonal components are admissible i.e. the functors i j have a left adjoint i ∗ j : T → D j and a right adjoint i ! j : T → D j . Proposition 2.10 ( [
14, Lemma 2.1 ] ) . With the above notations, assume that we have t-structures ( D ≤ i , D ≥ i ) with hearts A i in D i , for i =
1, 2 , such that (1) Hom ≤ D ( i A , i A ) = Then there is a t-structure on T with the heart (2) gl ( A , A ) = { E ∈ D | i !2 E ∈ A , i ∗ E ∈ A } . Moreover, i k A k ⊂ A : = gl ( A , A ) for k =
1, 2 . ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 7
Let us consider T = 〈 D , . . ., D n 〉 and the full embeddings i j : D j → T . Let A i ⊆ D i be the heartsof a bounded t-structures for i =
1, . . ., n . Assume the hearts satisfy the gluing condition Hom ≤ T ( i l A l , i j A j ) = < l < j ≤ n . We define B : = gl ( A n − , A n ) and B j : = gl ( A n − j , B j − ) . By Proposi-tion 2.10, we have that B j is a heart of a bounded t-structure on the triangulated subcategory 〈 D n − j , . . ., D n 〉 ⊆ T for j =
1, . . ., n − Definition 2.11.
We define gl ( A , . . ., A n ) : = B n − ⊆ T . We refer to gl ( A , . . ., A n ) as a gluingheart of T with respect to the semiorthogonal decomposition T = 〈 D , . . . , D n 〉 . Remark 2.12. (1) Since the gluing of two noetherian hearts is again a noetherian heart, itfollows that a gluing heart in the sense of Definition 2.11 is noetherian.(2) Note that if E ∈ A = gl ( A , . . ., A n ) if and only if there is a sequence of triangles0 E E · · · E n − EA n A n − · · · A .with A j ∈ i j A j , for j =
1, . . . , n .(3) gl ( A , A , A ) = gl ( gl ( A , A ) , A ) in T = 〈 D , D , D 〉 .Let σ i = ( Z i , A i ) ∈ Stab ( X ) . If the hearts A i satisfy the gluing condition, then by recursivelyapplying Proposition 2.10, there is a stability function on A = gl ( A , . . ., A n ) given by Z ( E ) = P ni = Z i ( A i ) , with A i as in Remark 2.12. Moreover, if σ i are algebraic stability conditions, we ob-tain that gl ( σ , . . ., σ n ) : = ( Z , A ) is a locally finite pre-stability condition, see [
11, Lemma 4.4 ] . Ifa stability condition σ ∈ Stab ( T X , n ) is equal to gl ( σ , . . . , σ n ) , we refer to σ as a gluing stabilitycondition .The next example defines α -stability as studied in [ ] . Example 2.13. ( α -stability) Let C be a curve and ( α j ) j = n ∈ Q n . Consider σ α j = ( Coh ( C ) , Z α j ) ∈ Stab ( C ) where Z α j ( d , r ) = − d − α j r + i r . We define σ α = gl ( σ α , . . ., σ α n ) is a locally finite pre-stability condition. Theorem 2.14 ( [
7, Theorem 1.4.10 ] ) . Let ( D ≤ , D ≥ ) and ( D ≤ , D ≥ ) be the bounded t-structureswith hearts A and A respectively and T = 〈 D , D 〉 . Then there is a t-structure ( T ≤ , T ≥ ) in T defined by: T ≤ : = { T ∈ T | i ∗ T ∈ D ≤ , i ∗ T ∈ D ≤ }T ≥ : = { T ∈ T | i ∗ T ∈ D ≥ , i T ∈ D ≥ } . DOMINIC BUNNETT AND ALEJANDRA RINCÓN-HIDALGO
We denote by rec ( A , A ) : = T ≤ ∩T ≥ . We say that rec ( A , A ) is a recollement heart with respectto the semiorthogonal decomposition T = 〈 D , D 〉 . Remark 2.15. By [
25, Proposition 2.8.12 ] , if A j ⊆ D j satisfy the gluing condition, then rec ( A , A ) = gl ( A , A ) .Let C be a curve with g ( C ) ≥
1. Under the assumption that all the pre-stability conditionsconstructed in [ ] satisfy the support property, we have the following result. Theorem 2.16. [
26, Theorem 1.1 and Theorem 4.47 ] The stability manifold
Stab ( T C ) is a connected -dimensional complex manifold. Moreover, if σ = ( Z , A ) ∈ Stab ( T C ) then, up to autoequivalence,it satisfies one of the following properties:(1) σ is a gluing stability condition with respect to the semiorthogonal decomposition T C = 〈 D , D 〉 . (2) There is g ∈ Ý GL + ( R ) such that, σ ′ = σ g is a gluing stability condition in (1).(3) There is a g ∈ Ý GL + ( R ) and σ ′′ = ( Z ′′ , A ′′ ) satisfying (2), such that for σ g = ( Z ′ , A ′ ) wehave that A ′′ = A ′ .2.3. Base change for semiorthogonal decompositions and hearts.
The base change of a semi-orthogonal decomposition was introduced by Kuznetsov in [ ] and generalised in [ ] . The basechange a heart of D b ( Y ) for a smooth projective variety was first given in [ ] and greatly gener-alised in the relative setting in [ ] . Definition 2.17.
Let
T ⊆ D ( QCoh ( X )) be a triangulated subcategory. If Y is a scheme and Φ : T → D ( QCoh ( Y )) is a triangulated functor, we say that Φ has cohomological amplitude [ a , b ] if Φ € T ∩ D [ p , q ] qc ( X ) Š ⊆ D [ p + a , q + b ] qc ( Y ) for all p , q ∈ Z , where ( D qc ( X ) ≤ , D ≥ ( X )) is the standard t-structure of D ( QCoh ( X )) . We say Φ has left finite cohomological amplitude if a can be chosen finite, right finite cohomological amplitude if b can be chosen finite, and finite cohomological amplitude if a and b can be chosen finite. Wesay that a semiorthogonal decomposition T = 〈 D , . . ., D n 〉 is of (right or left) finite cohomologicalamplitude if its projection functors have (right or left) finite cohomological amplitude. Remark 2.18.
In order to apply the results of [ ] and [ ] , we require a full admissible triangu-lated subcategory T ⊆ D b ( Y ) which is a strong semiorthogonal component of a semiorthogonaldecomposition of finite cohomological amplitude. Indeed this is the case under the conditionthat Y is a smooth projective variety by [
21, Lemma 2.9 ] . Moreover, the admissibility of T thenimplies the admissibility of T ⊥ in this case and thus we only require that T is admissible. ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 9
We start by giving the base change of an admissible semiorthogonal component. Let Y be asmooth projective variety and S be a quasi-projective variety. Let p : X × S → X and q : X × S → S be the projections. Proposition 2.19. [
21, Corollaries 5.7 and 5.9 ] Let
T ⊆ D b ( Y ) be an admissible subcategory, thenthe category T S = { F ∈ D b ( Y × S ) | R p ∗ ( F ⊗ q ∗ G ) ∈ Ò T for all G ∈ D perf ( S ) } , is an admissible subcategory in D b ( Y × S ) such that the corresponding projection functor has fi-nite cohomological amplitude. Here Ò T is the minimal triangulated subcategory of D ( QCoh ( Y )) containing T closed under arbitrary direct sums. Additionally, we have that p ∗ ( T ) ⊆ T S and R p ∗ ( T S ) ⊆ T , if S is projective. Moreover, D b ( Y × S ) = 〈T ⊥ S , T S 〉 .Let us consider the inclusion i s : Y × { s } , → Y × S and we denote E s : = L i ∗ s ( E ) . Lemma 2.20. [
5, Lemma 9.3 ] Let S be a quasi-compact C -scheme with affine diagonal and letE ∈ D b ( Y × S ) . Then(1) E ∈ T S if and only if E s ∈ T for every s ∈ S , where i s : X × { s } , → X × S . (2) The set { s ∈ S | E s ∈ T } is open. Theorem 2.21. [
5, Theorem 5.7 ] Let Y be a smooth projective variety and
T ⊆ D b ( Y ) be a fulladmissible subcategory. Let ( T ≤ , T ≥ ) be a bounded t-structure on T with a noetherian heart A .If S is a smooth quasi-projective variety, then(1) (S -local ) For every open U ⊆ S , there exists a t-structure ( T ≤ U , T ≥ U ) on T U with heart A U such that the restriction functor i ∗ : T S → T U , induced by i : U , → S , is t-exact.(2) A S is noetherian.(3) If S is projective and L is an ample line bundle, then ( T [ a , b ] ) S = { E ∈ T S | R p ∗ ( E ⊗ q ∗ ( L ) n ) ∈ T [ a , b ] for all n ≫ } . (4) Let S ′ be a smooth quasiprojective variety. For a morphism f : S ′ → S and its induced mor-phism f ′ : S ′ × Y → S × Y , we have that f ′∗ : T S → T S ′ is right t-exact. Moreover, if f is flat,then f ′∗ is t-exact and if f is finite, then f ′∗ : T S ′ → T S is t-exact.
3. G
EOMETRIC REALISATIONS A geometric realisation of a triangulated category T is a fully faithful embedding T , → D b ( X ) for some C -scheme X . In general, we will be interested in the case where X is a smooth projectivevariety. In order to use [ ] , we are going to prove the existence of a geometric realisation T X , n , → D b ( Y X , n ) of T X , n for a smooth projective variety Y X , n . dg-enhancements and gluing. A general reference for dg-categories is [ ] and we refer thereader there for definitions of dg-categories and dg-bimodules. We use the conventions and no-tation of [ ] .In this section we prove that a triangulated category T with a semiorthogonal decomposition,such that T and the semiorthogonal components admit unique dg-enhancements, is completelydetermined by the semiorthogonal components and the gluing functor. This result is well-knownand is a consequence of [
22, Proposition 4.10 ] . To the best of the authors’ knowledge, a proof doesnot occur in the literature and so one is included for the sake of completeness.Suppose that D is a dg-category. We denote its homotopy category [ D ] . Recall that a dg-category is pretriangulated if [ D ] is a triangulated category. Definition 3.1. An enhancement of a triangulated category T is a pretriangulated dg-category D with an equivalence T ∼ = [ D ] of triangulated categories.A triangulated category T has a unique dg-enhancement, if any two dg-enhancements D , D ′ of T are quasiequivalent. Lemma 3.2.
The category T X , n has a unique dg-enhancement. We denote this enhancement D X , n . The existence follows from standard arguments, see [ ] .If D ( X ) is the dg-enhancement of D b ( X ) .It could be also explicitly given by Mor ( D ( X )) , the dg-category of morphisms as defined in [ ] . Proof.
We proceed by induction on n . For n =
1, we have that T X ,1 ∼ = D b ( X ) and this case iswell known [
13, Corollary 7.2 ] . Now take n > T X , n has unique dg-enhancement.Recall the semiorthogonal decomposition of T X , n given in Definition 2.8. Let D n + be a dg-enhancement of T X , n + = 〈 D , T X , n 〉 , where D ∼ = D b ( X ) . Take D and D n to be the full dg-subca-tegories of D n + having the same objects as D and T X , n respectively. By induction, we have that D and D n are the unique dg-enhancements of D b ( X ) and T X , n respectively. Moreover, by [ ] we have the existence of a dg-module Φ given by Φ : D op n ⊗ D −→ C − dgm ( E , E ) Hom D n + ( i ( E ) , j ( E )[ ]) ,where i : D → D and j : D n → D n + are the corresponding inclusions, such that D × Φ D n ∼ = D n + is the gluing of D and D n along Φ , see [
22, Definition 4.1 ] . ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 11
Analogously, if we have another dg-enhancement D ′ n + of T X , n + , there is a D op n ⊗D -bimodule Φ ′ , such that D ′ n + ∼ = D × Φ ′ D n . By [
22, Lemma 4.7 ] , it is enough to show that Φ and Φ ′ are quasi-isomorphic. This is indeed the case, since H i ( Hom D n + ( E , F )) = Hom i T X , n + ( E , F ) and H i ( Hom D ′ n + ( E , F )) = Hom i T X , n + ( E , F ) ,for all E , F ∈ T X , n + . As a consequence, we have D ′ n + ∼ = D n + . (cid:3) Consider triangulated categories T and T ′ and suppose that we have full triangulated subcate-gories and embeddings i : T → T , j : T → T and i ′ : T → T ′ , j ′ : T → T ′ . Suppose additionallythat we have semiorthogonal decompositions T = 〈 i ( T ) , j ( T ) 〉 and T ′ = 〈 i ′ ( T ) , j ′ ( T ) 〉 .If T or T ′ admit a dg-enhancement, we deduce that T and T admit dg-enhancements [ ] . Lemma 3.3.
Suppose that T and T ′ are triangulated categories admitting dg-enhancements D and D ′ respectively and that T i has a unique dg-enhancement D i for i =
1, 2.
Assume that thegluing functors coincide, that is, i ! j = i ′ ! j ′ : T → T . Then
T ∼ = T ′ .Proof. By [
22, Proposition 4.10 ] , we have that D is quasiequivalent to D × Φ D , where D × Φ D isthe glued dg-category along the bimodule Φ : D op2 ⊗ D −→ C − dgm ( E , E ) Hom D ( i ( E ) , j ( E )[ ]) .Analogously, we have that D ′ is quasiequivalent to D × Φ ′ D where Φ ′ : D ′ op2 ⊗ D ′ −→ C − dgm ( E , E ) Hom D ( i ′ ( E ) , j ′ ( E )[ ]) .Moreover, we also have that the homotopy category [ D × Φ D ] is equivalent to T and [ D × Φ ′ D ] is equivalent to T ′ . Claim 3.4.
The bimodules Φ and Φ ′ are quasiisomorphic. Proof.
It suffices to show that H i ( Hom D ( i ( E ) , j ( E )[ ])) ∼ = H i ( Hom D ′ ( i ′ ( E ) , j ′ ( E )[ ])) for all E ∈D and E ∈ D . Note that H i ( Hom D ( i ( E ) , j ( E )[ ])) = Hom i T ( i ( E ) , j ( E )[ ])) . By adjunction, we have that Hom i T ( i ( E ) , j ( E )[ ])) ∼ = Hom i T ( E , i ! j ( E )[ ])) . In the same way weget that H i ( Hom D ′ ( i ′ ( E ) , j ′ ( E )[ ])) = Hom i T ( E , i ′ ! j ′ ( E )[ ])) .As i ! j = i ′ ! j ′ , we obtain that Φ and Φ ′ are quasiisomorphic. (cid:3) By [
22, Lemma 4.7 ] , we finally conclude that D × Φ D is quasiequivalent to D × Φ ′ D and there-fore T and T ′ are equivalent. (cid:3) Explicit geometric realisations.
Our main goal in this section is to construct smooth projectivevarieties Y X , n , depending on X , and a fully faithful functors T X , n , → D b ( Y X , n ) . We start by find-ing geometric realisations for the bounded derived categories of representation of A n -quivers byfollowing the steps of [ ] .In [ ] , Orlov gives geometric realisations for so-called geometric noncommutative schemes,this also encapsulates the categories in which we are interested. However, our purposes requirean explicit construction. Theorem 3.5. [
28, Theorem 2.6 ] Let Q be a quiver with n ordered vertices. Then there exists asmooth projective variety Y Q and an exact functor u : Rep C ( Q ) , → Coh ( Y Q ) such that the followingconditions hold.(1) The induced derived functor u : D b ( Rep C ( Q )) −→ D b ( Y Q ) is fully faithful.(2) The variety Y Q is a tower of projective bundles and has a full exceptional collection.(3) Simple modules S i go to line bundles L i , n on Y Q under u.(4) any representation M goes to a vector bundle on Y Q . The following example is important for our purposes.
Remark 3.6.
Let us consider the A n quiver. After applying Theorem 3.5 we obtain a tower ofprojective bundles Y n π n −−→ Y n − −→ . . . Y π −−→ P and embeddings u n : D b ( A n ) , → D b ( Y n ) , such that the simple modules S i are sent to line bundles L n , i ∈ Coh ( Y n ) for i =
1, . . . , n . Following [
28, Theorem 2.6 ] , we have an inductive description L n , i = π ∗ n ( L n − i − ) for i =
2, .., n with L n ,1 = O Y n ( − ) and L n , n = O Y n .We write P i for the projective indecomposable corresponding to S i for i =
1, . . . , n . Then we havethe semiorthogonal decompositions: D b ( A n ) = 〈 S , D b ( A n − ) 〉 = 〈 D b ( A n − ) , P 〉 , where i : D b ( A n − ) → D b ( A n ) is the fully faithful functor induced by adjoining a vertex on the left. We now consider the ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 13 triangle M n −→ P −→ S induced by the semiorthogonal decomposition D b ( A n ) = 〈 S , D b ( A n − ) 〉 for P . Applying u n to the triangle above, we have π ∗ n f M n −→ E n −→ O Y n ( − ) ,where M n ∈ D b ( A n − ) such that π ∗ n ( f M n ) = u n ( M n ) and E n = u n ( P ) .Following Orlov’s steps we now define a smooth projective variety Y X , n , depending on X , anda fully faithful functor v n , i : D b ( X ) , → D b ( Y X , n ) such that the triangulated category generated bythe images of the v n , i has a semiorthogonal decomposition with components v n , i ( D b ( X )) andthe gluing functors coincide with the one of T X , n . This allows us to apply Lemma 3.3 and obtainthe result. Definition 3.7.
We define Y X , n : = Y n × X and e L n , i : = q ∗ n L n , i where Y n and L n , i are as in Remark3.6 and q n : Y X , n → Y n is the projection.There is the following commutative diagram:(3) η n : Y X , n η n / / q n (cid:15) (cid:15) Y X , n − η n − / / q n − (cid:15) (cid:15) · · · / / (cid:15) (cid:15) Y X ,1 η / / q (cid:15) (cid:15) X π n : Y n π n / / Y n − π n − / / · · · / / Y = P .By [
28, Proposition 2.5 ] Y n = P Y n − ( K n ) , therefore we get that Y X , n = P Y X , n − ( q n − ∗ K n ) , where K n fitsinto the following short exact sequences: For n =
2, we get K = O P ( − ) ⊕ O P ( − ) and0 −→ K −→ ( O P ( − )) ⊕ −→ O P −→ n >
2, we obtain the following short exact sequence in Coh ( Y n − ) −→ K n −→ ( O Y n − ( − ) ⊗ π ∗ n − ( R − s )) ⊕ m −→ E ∨ n − −→ R a very ample line bundle on Y n − and m , s ∈ Z > . As a consequence, we obtain a tower ofprojective bundles.We then define the following functors v n , i : D b ( X ) , → D b ( Y X , n ) E η n ∗ ( E ) ⊗ e L n , i .It also follows from Remark 3.6 that e L n , i = η ∗ n ( e L n − i − ) for i =
2, . . ., n and e L n ,1 = O Y X , n ( − ) . Remark 3.8. By [
27, Lemma 2.1 ] , we have that the functor η ∗ : D b ( X ) −→ D b ( Y X , n ) is fully faithful.The functor v n , i is the composition of two fully faithful functors; η ∗ and an autoequivalence givenby tensoring by a line bundle. Therefore, the functor v n , i is also fully faithful. By [
27, Assertion 2.4 ] we have that D b ( X ) is saturated, therefore v n , i ( D b ( X )) is an admissible triangulated subcategory.Moreover, using the adjunction ( η n ∗ , R η n ∗ ) , we obtain that the right adjoint of v n , i is given by v ! n , i : D b ( Y X , n ) , → D b ( X ) F R η n ∗ ( F ⊗ e L ∨ n , i ) . Theorem 3.9.
There is a fully faithful functor v n : T X , n , → D b ( Y X , n ) . Moreover, for the standard se-miorthogonal decomposition T X , n = 〈 D n , . . ., D nn 〉 , the image of D ni under v n is given by v n , i ( D b ( X )) and the following diagram commutes (4) D n − i − η ∗ n / / v n − i − (cid:15) (cid:15) D ni v n , i (cid:15) (cid:15) D b ( Y X , n − ) η ∗ n / / D b ( Y X , n ) for i =
2, . . . , n.Proof.
We proceed by induction on n . For n =
1, we have that T X ,1 = D b ( X ) and Y X ,1 = X × P = P X ( O X ⊕ O X ) . As in the proof of Theorem 3.5, it holds that g L = O P and by [
27, Lemma 2.1 ] thestatement follows.We now assume the assertion for T X , n − . First note that we have already a fully faithful functor η ∗ n ◦ v n − : T X , n − −→ D b ( Y X , n ) .Abusing notation, we identify image of this functor with T X , n − itself. Our strategy is to show that v n ,1 ( D b ( X )) ⊆ T ⊥ X , n − and that the triangulated category 〈 v n ,1 ( D b ( X )) , T X , n − 〉 ⊆ D b ( Y X , n ) is equivalent to T X , n .Let us consider the standard semiorthogonal decomposition T X , n − = 〈 D n − , . . ., D n − n − 〉 . In or-der to prove that v n ,1 ( D b ( X )) ⊆ T ⊥ X , n − , it is enough to show thatHom D b ( Y X , n ) ( η n ∗ ( E ) ⊗ e L n , j , η n ∗ ( F ) ⊗ e L n ,1 ) = j =
2, . . . , n and E , F ∈ D b ( X ) . ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 15
Note that e L n ,1 = O Y X , n ( − ) . Applying the adjunction ( η n ∗ , R η n ∗ ) and the projection formula wehave Hom D b ( Y X , n ) ( η n ∗ ( E ) ⊗ e L n , j , η n ∗ ( F ) ⊗ O Y X , n ( − ))= Hom D b ( Y X , n ) ( η ∗ n ( η n − ∗ ( E ) ⊗ ( e L n − j − )) , η n ∗ ( F ) ⊗ O Y X , n ( − ))= Hom D b ( Y X , n ) ( η ∗ n ( η n − ∗ ( E ) ⊗ ( e L n − j − )) , η ∗ n η n − ∗ ( F ) ⊗ O Y X , n ( − ))= Hom D b ( Y X , n − ) ( η n − ∗ ( E ) ⊗ e L n − j − , η n − ∗ ( F ) ⊗ R η n ∗ ( O Y X , n ( − )))= R η n ∗ ( O Y X , n ( − )) =
0, since η n ∗ is a projective bundle.Consider the two triangulated categories T X , n = 〈 D n , T X , n − 〉 and 〈 v n ,1 ( D b ( X )) , T X , n − 〉 ⊆ D b ( Y X , n ) and note that 〈 v n ,1 ( D b ( X )) , T X , n − 〉 , as a full triangulated subcategory of D b ( Y X , n ) , has a dg-enha-ncement. By Lemma 3.2 both T X , n − and D b ( X ) have unique dg-enhancements. Moreover,Lemma 3.3 implies that it is now enough to show that the gluing functor v ! n ,1 v n ,2 : D b ( X ) −→ D b ( X ) between D and D in D b ( Y X , n ) is given precisely by v ! n ,1 v n ,2 ( E ) = E [ − ] .Indeed, this is the case. Consider: v ! n ,1 v n ,2 ( E ) = v ! n ,1 ( η n ∗ ( E ) ⊗ e L n ,2 )= R η n ∗ ( η n ∗ ( E ) ⊗ e L n ,2 ⊗ e L ∨ n ,1 )= E ⊗ R η n − ∗ R η n ∗ ( e L n ,2 ⊗ O Y X , n ( ))= E ⊗ R η n − ∗ R η n ∗ ( η ∗ n ( e L n − ) ⊗ O Y n ( ))= E ⊗ R η n − ∗ ( e L n − ⊗ R η n ∗ ( O Y n ( )))= E ⊗ R η n − ∗ ( q ∗ n − ( L n − ⊗ K n )) .For n =
2, we have that L = O P , therefore E ⊗ R η ∗ ( q ∗ ( L ⊗ K )) = E ⊗ R η ∗ ( q ∗ ( K ))= E ⊗ R η ∗ ( O Y X ,2 ( − ) ⊕ O Y X ,2 ( − ) ⊕ ) = E ⊗ O X [ − ] = E [ − ] .For n >
2, we have that E ⊗ R η n − ∗ ( q ∗ n − ( L n − ⊗ K n )) = E ⊗ R η n − ∗ ( q ∗ n − R π n − ∗ ( O Y n − ( − ) ⊗ K n )) ,by (3) and the fact that L n − = O Y n − ( − ) . Using the exact sequences from Remark 3.6 and since rk ( K n ) >
2, we obtain two short exactsequences 0 −→ O Y n − −→ E ∨ n − ⊗ O Y n − ( − ) −→ π ∗ n ( E ∨ n − ) ⊗ O Y n − ( − ) −→ −→ K n ⊗ O Y n − ( − ) −→ ( O Y n − ( − ) ⊗ π ∗ n ( R s )) m ′ −→ E ∨ n − ⊗ O Y n − ( − ) −→ R π n − ∗ ( O Y n − ( − )) =
0, we get that R π n − ∗ ( E ∨ n − ⊗ O Y n − ( − )) ∼ = O Y n − . Moreover, as rk ( K n − ) > R π n − ∗ (( O Y n − ( − ) ⊗ π ∗ n − ( R ⊗ s )) m ′ ) =
0, and therefore R π n − ∗ ( K n ⊗ O Y n − ( − )) = R π n − ∗ ( E ∨ n − ⊗ O Y n − ( − ))[ − ] = O Y n − [ − ] .As a consequence, v ! n ,1 v n ,2 ( E ) = E ⊗ R η n − ∗ ( q ∗ n − R π n − ∗ ( O Y n − ( − ) ⊗ K n )) = E ⊗ R η n − ∗ ( O Y X , n − )[ − ] = E [ − ] . (cid:3) We conclude this section with two technical lemmas which will require.
Lemma 3.10.
There is a semiorthogonal decomposition of v n ( T X , n ) ⊥ = 〈 B , . . ., B m 〉 ⊆ D b ( Y X , n ) , such that B i ∼ = D b ( X ) for i =
1, . . ., m for some m ∈ N .Proof. The proof goes by induction on n . Let n = T X ,1 ∼ = D b ( X ) and Y X ,1 = X × P = P X ( O X ⊕ O X ) . Consider the embedding w : D b ( X ) −→ D b ( Y X ,1 ) given by E η ∗ ( E ) ⊗ O Y X ,1 ( − ) .By [
27, Theorem 2.6 ] , there is a semiorthogonal decomposition D b ( Y X ,1 ) = 〈 w ( D b ( X )) , v ( D b ( X ) 〉 and consequently v ( T X ,1 ) ⊥ ∼ = w ( D b ( X )) .Now assume the statement for n −
1. As Y X , n is a projective bundle over Y X , n − , by [
27, Theorem2.6 ] there is r ∈ N such that D b ( Y X , n ) = 〈 C − r , . . ., C 〉 where C i ∼ = D b ( Y X , n − ) . Moreover, the cate-gory C is the image under the functor η ∗ n : D b ( Y X , n − ) → D b ( Y X , n ) and C − is the image under thefunctor η ∗ n ( − ) ⊗ O Y X , n ( − ) . Also note that v n ( T X , n ) = 〈 D n − n − − , ( T X , n − ) 〉 , where ( T X , n − ) is the im-age of the embedding T X , n − , → C and T X , n − = 〈 D n − , . . ., D n − n − 〉 is the standard semiorthogonaldecomposition.We also have that D b ( Y X , n − ) = 〈 v n − ( T X , n − ) ⊥ , v n − ( T X , n − ) 〉 and hence by the induction hy-pothesis it holds that T ⊥ X , n − = 〈 B , . . ., B s 〉 with B i ∼ = D b ( X ) for all i =
1, . . . , s . As a consequence,we have that C j = 〈 B j , . . ., B s j , D n − j , . . . D n − n − j 〉 . Therefore we have that D b ( Y X , n ) splits up into: 〈 B − r , . . ., B s − r , D n − − r , . . . D n − n − − r , . . ., B − , . . ., B s − , D n − − , . . ., D n − n − − , B , . . ., B s , D n − , . . ., D n − n − 〉 .where D n − n − − is precisely the image of v n ,1 : D b ( X ) → D b ( Y X , n ) . ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 17
We can then mutate the semiorthogonal decomposition given above. Indeed, consider thefunctor L D n − n − − : D b ( Y X , n ) −→ D b ( Y X , n ) F C ( i ∗ i ! ( F )) ,where i : D n − n − − , → D b ( Y X , n ) is the inclusion and we mutate as in [
20, Section 2.4 ] .We obtain D b ( Y X , n ) = 〈 B − r , . . ., B s − r , D n − − r , . . . D n − n − − , . . ., B − , . . ., L D n − n − − ( B ) , D n − n − − , . . ., B s , D n − , . . . D n − n − 〉 .In addition, the functor L D n − n − − induces an equivalence ⊥ D n − n − − D n − n − − ⊥ , as a consequence L D n − n − − ( B ) ∼ = D b ( X ) .By mutating repeatedly as above, we obtain D b ( Y X , n ) = 〈 B − r , . . . , B s − r , D n − − r , . . . D n − n − − , . . . , B − , . . ., B ′ , . . ., B ′ s , D n − n − − , D n − , . . . D n − n − 〉 .with B ′ i ∼ = D b ( X ) .Since v n ( T X , n ) = 〈 D n − n − − , D n − , . . ., D n − n − 〉 we have that v n ( T X , n ) ⊥ = 〈 B − r , . . . , B s − r , D n − − r , . . . D n − n − − , . . . , B − , . . ., B ′ , . . ., B ′ s 〉 as desired. (cid:3) Remark 3.11.
Let Y X , n as in Theorem 3.9, then T X , n is an admissible subcategory of D b ( Y X , n ) .Indeed, note that T X , n is saturated by [
8, Proposition 2.10 ] and therefore T X , n is admissible by [ ] .4. M ODULI OF B RIDGELAND SEMISTABLE HOLOMORPHIC CHAINS
We begin by introducing a moduli stack analogous to the moduli stack defined and studied byLieblich [ ] . These moduli stacks were first studied by Bayer et. al [
5, Section 9 ] and our notationfollows theirs.Let Y be a smooth projective variety. From this point on T ⊆ D b ( Y ) is always an admissiblesubcategory. See Remark 2.18. Definition 4.1.
Let S be a C -scheme. We say that an S -perfect (in the sense of [
5, Definition 8.1 ] )object E ∈ D ( Y × S ) is universally gluable if Ext i ( E s , E s ) = i < C -point s ∈ S .We write D pug ( Y × S ) ⊂ D ( Y × S ) for the subcateogry consisting of universally gluable objects in D ( Y × S ) . Remark 4.2.
Note that by [
5, Lemma 8.3 ] , we have that D pug ( Y × S ) ⊆ D b ( Y × S ) . Moreover, if S issmooth then if E ∈ D b ( Y × S ) implies that E is S -perfect. Definition 4.3.
We denote M pug ( T ) : Sch / C −→ Gpdsthe functor whose value on a C -scheme S is the set of all E ∈ D pug ( Y × S ) such that E s ∈ T forall s ∈ S . The groupoid structure is given by the standard notion of equivalence: E ∼ E ′ if thereexists a line bundle L ∈ Pic ( S ) such that E ∼ = E ′ ⊗ q ∗ L , where q : Y × S → S .The following is a version of [
23, Theorem 4.2.1 ] . Proposition 4.4. [
5, Proposition 9.2 ] The functor M pug ( T ) is an algebraic stack locally of finitetype over C .We now define the moduli stacks which are of our primary interest. Definition 4.5.
Let σ ∈ Stab Λ ( T ) . We define M β , φ ( σ ) to be the substack of M pug ( T ) parameter-ising the set of σ -semistable objects of phase φ and class β . In particular M β , φ ( σ )( S ) = (cid:8) E ∈ M pug ( S ) | E s σ -semistable, class β ∈ Λ , phase φ (cid:9) .The groupoid structure is given by the standard notion of equivalence.The remainder of this section will be concerned with studying these moduli stacks in the case T = T X , n , where X is a smooth projective variety. The ultimate aim is prove that the modulispaces M β , φ ( σ ) are algebraic stacks of finite type. The algebracity will follow from the open heartproperty and generic flatness which are addressed in respective sections below. That the stack isof finite type will follow from boundedness .Let C be a curve. For certain stability conditions on T C , n we can prove all three of the aboveproperties and in the case of holomorphic triples T C for the entire stability manifold. Further-more, we can prove some partial results for T X , n .4.1. Boundedness.
As before, let Y be a smooth projective variety and T ⊆ D b ( Y ) be an admis-sible subcategory. Definition 4.6.
A set of objects B ⊆ T is called bounded if there is a C -scheme of finite type S ,and an object E ∈ T S such that any object in B is isomorphic to E s for some C -point s ∈ S . If thisrather holds for some E ∈ D b ( Y × S ) we say that B is bounded in D b ( Y ) .The following lemma states that if boundedness holds for one stability condition on the sta-bility manifold, it must hold for the entire stability manifold. The result appears in [
30, Theorem4.2 ] . We provide a proof for the sake of completeness. ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 19
Lemma 4.7.
Let
Stab ◦ Λ ( T ) ⊆ Stab Λ ( T ) be a connected component. If M β , φ ( σ ) is bounded for σ ∈ Stab ◦ Λ ( T ) an algebraic stability condition. Then, for any τ ∈ Stab ◦ Λ ( T ) , we have that M β , φ ( τ ) isalso bounded for all β ∈ Λ and φ ∈ R . Proof.
Let us consider the slicings P and Q of σ and τ respectively. By connecting σ to τ via apath, we can assume that d ( P , Q ) = inf { δ ∈ R | Q ( φ ) ⊆ P ([ φ − δ , φ + δ ]) for all φ ∈ R } = ε <
18 ,where d is the metric on the stability manifold, see [ ] . Hence Q ( φ ) ⊆ P (( φ − ε , φ + ε )) .We now show that M β , φ ( τ ) is bounded. Let E ∈ Q ( φ ) with [ E ] ∈ Λ . The semistable factors F i of E , with 1 ≤ i ≤ n E , satisfy φ − ε < φ σ ( F i ) < φ + ε . Since σ is algebraic and ε small enough, wehave that the map M β , φ ( τ ) → N , given by E n ( E ) is bounded. As a consequence, the set { Z ( F i ) | ≤ i ≤ n E , with E ∈ M β , φ ( τ ) } is finite.Let φ i : = φ ( F i ) for i =
1, . . ., n . Since M β , φ i ( σ ) is bounded and applying [
5, Lemma 9.8 ] , we getthat M β , φ ( τ ) is bounded. (cid:3) Lemma 4.8.
Let C be a curve, if E ∈ P σ α ( φ ) for σ α = ( Z α , Q C , n ) ∈ Stab ( T C , n ) as in Examples 2.13and < φ < . Then E i is torsion free for i =
1, . . . , n.Proof.
Let E ∈ Q C , n be a σ α -semistable holomorphic triple. We can decompose E n = T ( E n ) ⊕ F ( E n ) , where T ( E n ) is the torsion part of E n and F ( E n ) the torsion-free part. Note that i n ( T ( E n )) is a subchain of E and φ σ α ( i n ( T ( E n ))) =
1, where i n is as in Section 2.2.1. This contradicts thesemistability of E . Therefore E n is torsion free.Note that the chain T n − = −→ · · · −→ T ( E n − ) −→ E with φ ( T n − ) =
1. Again contradicting semistability and hence T ( E n − ) = E i is torsion free for all i =
1, . . . , n − (cid:3) Lemma 4.9.
Let C be curve and let σ α ∈ Stab ( T C , n ) , as in Example 2.13 with α ∈ Q n . Then M β , φ ( σ α ) is bounded for all β ∈ Z n and < φ < .Proof. The moduli space of σ α -semistable holomorphic chains of vector bundles was constructedin [
32, Theorem 1.6 ] . Therefore, by Lemma 4.8 we obtain that M β , φ ( σ α ) is bounded. (cid:3) Remark 4.10.
Under the assumptions of Lemma 4.9, if E ∈ P σ α ( ) , we have that P ni = r i = r i ≥ i , we get that r i =
0. Therefore E i is a torsion sheaf for each i . After fixing β =( i , d i ) i = n we get M β ,1 ( σ α ) ∼ = Sym n ( C ) and is bounded.The following corollary holds under the assumption that the support property is satisfied for σ α . Let Stab ◦ ( T C , n ) be the connected component containing σ α . Corollary 4.11.
Let C be a curve and
Stab ◦ ( T C , n ) the connected component containing σ α . Thenfor every σ ∈ Stab ◦ ( σ α ) , the set M β , φ ( σ ) is bounded for every β ∈ Z n and φ ∈ R . Corollary 4.12.
For σ ∈ Stab ( T C ) , then M β , φ ( σ ) is bounded for all β ∈ Z and all φ ∈ R . Proof.
It is proven in [
26, Theorem 1.1 ] that Stab ( T C ) is connected. The statement follows fromLemmas 4.9 and 4.7. (cid:3) The open heart property.
We begin by stating the open heart property.
Definition 4.13.
Let S be a C -scheme of finite type and A ⊂ T a noetherian heart. We say that A satisfies the open heart property if for every E ∈ T S and smooth C -point s ∈ S with E s ∈ A ,there exists an open neighbourhood s ∈ U ⊂ S such that E U ∈ A U .In the case of D b ( X ) the open heart property was proven by Abramovich and Polishchuk. Theorem 4.14. [
1, Proposition 3.3.2 ] Let
A ⊆ D b ( X ) be a noetherian heart. Then A satisfies theopen heart property. The rest of the section will be concerned with proving the analogous result in our setting:
Theorem 4.15.
Any noetherian heart
A ⊂ T X , n satisfies the open heart property. Lemma 4.16.
Suppose that T = 〈 D , D 〉 is a semiorthogonal decomposition with D ∼ = D b ( X ) with X a smooth projective variety. If D admits a noetherian heart, then T also admits a noe-therian heart.Proof. Let
A ⊂ D be a noetherian heart. We claim that A ′ = gl ( Coh ( X )[ m ] , A ) for m >> A ′ is noetherian and hence would conclude the proof.Let us prove the claim. It suffices to showHom ≤ T (cid:0) i ( Coh ( X ))[ m ] , j ( A ) (cid:1) = ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 21 where i : D → T and j : D → T are the inclusions. Consider E ∈ Coh ( X ) and F ∈ A , thenHom T (cid:0) i ( E ) , j ( F ) (cid:1) = Hom D (cid:0) j ∗ i ( E ) , F (cid:1) .Consider the composition j ∗ i : D → D . By [
19, Proposition 2.5 ] , there are a , b ∈ Z such that forall E ∈ Coh ( X ) we have that j ∗ i ( E ) ∈ D [ a , b ] , where ( D ≤ , D ≥ ) is the t-structure given by A on D .Now we choose m >> ≤ D ( j ∗ i ( E )[ m ] , F ) = (cid:3) The next step towards proving the open heart property is to prove that the recollement oftwo hearts is stable under base change. By this we mean that one can either base change twohearts and then consider the recollement or first take the recollement of two hearts and thenbase change; the resulting heart is the same.
Proposition 4.17. [ Recollement is stable under base change ] Let j : T , → D b ( Y ) be an admissabletriangulated subcategory, where Y is a smooth projective variety, with T = 〈 D , D 〉 and S be aprojective variety. Let A ⊆ D and A ⊆ D be hearts of bounded t-structures. Then rec ( A S , A S ) = rec ( A , A ) S . Moreover, if A and A satisfy gluing conditions, for E ∈ rec ( A , A ) S we have that i ( E ) ∈ A , where i : D , → T is the inclusion functor.Proof. It suffices to show that rec ( A S , A S ) ⊆ rec ( A , A ) S , since inclusion of hearts impliesequality. For brevity we write φ n = R p ∗ ( − ⊗ ( q ∗ L ) n ) where L is an ample line bundle on S .We first note that F ∈ rec ( A , A ) S if and only if φ n F ∈ rec ( A , A ) for all n ≫ F → φ n F → F → F [ ] F ′ → φ n F → F → F ′ [ ] ,for some F ∈ A , F ∈ D ≥ , F ′ ∈ D ≤ , where ( D ≤ j , D ≥ j ) is the t-structure with heart A j for j =
1, 2, and F ′ ∈ ⊥ D . Note that these triangles are unique.Suppose now that E ∈ rec ( A S , A S ) . Thus we have the exact triangle E → E → E → E [ ] ,where E = j ! E ∈ ( D ≤ ) S and E ∈ A S . Applying the functor φ n we get φ n E → φ n E → φ n E → φ n E [ ] .Choosing n large enough allows us to conclude that φ n E ∈ A and φ n E ∈ D ≤ . This recoversthe first exact sequence from (5) for φ n E .Similarly, consider the triangle E → E → E ′ → E [ ] with respect to the decomposition T S = 〈 ( D ) S , ( ⊥ D ) S 〉 . Again, E ′ ∈ ( D ≤ ) S and hence, for n ≫ φ n E ′ ∈ D ≤ .To argue that φ n E ∈ ⊥ D we note that by [
1, Proposition 2.1.3 ] we have E ⊗ q ∗ L ∈ ( ⊥ D ) S forany L . Given that S is projective, R p ∗ (( ⊥ D ) S ) ⊂ ⊥ D by Proposition 2.19. We now assume that rec ( A , A ) = gl ( A , A ) and E ∈ rec ( A , A ) S , inthis case, for n ≫ i ( φ n ( E )) = φ n E ∈ A , which is precisely the definition of i E = E ∈ A S . (cid:3) Corollary 4.18.
Let
B ⊆ D b ( X ) be the heart of a bounded t-structure. If A = gl ( B , . . . , B ) ⊆ T X , n , then A S = gl ( B S , . . ., B S ) .Proof. Note that if we take the same heart in each component the gluing conditions are automat-ically satisfied. The result follows from the recursive definition of A , Proposition 4.17 and [ ] . (cid:3) Corollary 4.19.
Suppose that T = 〈 D , D 〉 and A i ⊂ D i are hearts of bounded t-structures suchthat f A = rec ( A , A ) . Then ( A ) S = ( D ) S ∩ f A S . Proof.
It follows directly from the definition of recollement A = D ∩ rec ( A , A ) , then the state-ment follows directly from Proposition 4.17. (cid:3) Let
A ⊆ T X , n be a noetherian heart and consider the realisation Y X , n as in Theorem 3.9. ByLemma 3.10, there is a decomposition T ⊥ X , n = 〈B , . . ., B m 〉 with B i ∼ = D b ( X ) . Recursively apply-ing Lemma 4.16 we construct a noetherian heart f A = gl ( B , A ) ⊆ D b ( Y X , n ) with B = gl ( Coh ( X )[ n ] , . . ., Coh ( X )[ n k ]) ⊆ T X , n ⊥ . Corollary 4.20.
Let
A ⊆ T X , n be the noetherian heart of the t-structure ( D ≤ , D ≥ ) on T X , n . Thenthe functor L i ∗ s is right t-exact with respect to ( D ≤ S , D ≥ S ) and ( D ≤ , D ≥ ) .Proof. Let E ∈ D ≤ S . By the definition in Theorem 2.14 we have that E ∈ e D ≤ S , where ( e D ≤ , e D ≥ ) is the t-structure with heart f A . It follows from [
1, Lemma 2.5.3 ] that L i ∗ s ( e D ≤ S ) ⊆ e D ≤ . Note that L i ∗ s (( T X , n ) S ) ⊆ T X , n . Consequently L i ∗ s ( E ) ∈ e D ≤ ∩ T X , n = D ≤ . (cid:3) We also get a result analogous to [
1, Lemma 2.6.2 ] . Corollary 4.21.
Let E ∈ A S , then there is H ∈ A and n ∈ Z such that p ∗ ( H ) ⊗ q ∗ ( L ) n ։ E in A S . ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 23
Proof.
By the definition of recollement E ∈ f A , after applying [
1, Lemma 2.6.2 ] there is G ∈ f A and n ∈ Z such that p ∗ ( G ) ⊗ L n ։ E in f A S . By applying j ! , we obtain p ∗ ( j ! ( G )) ⊗ q ∗ ( L ) n ։ E in A S . Indeed, it holds that j ! ( f A ) ⊆ A and j ! ( p ∗ ( G ) ⊗ L n ) = p ∗ ( j ! ( G )) ⊗ q ∗ ( L ) n . Therefore, take H = j ! ( G ) ∈ A . (cid:3) Now we are in a position to prove Theorem 4.15.
Proof of Theorem 4.15.
Let E ∈ ( T X , n ) S , such that E s ∈ A ⊆ f A . Theorem 4.14 ensures the openheart property is satisfied for the heart f A S . Thus there is an open set s ∈ U ⊆ S such that E U ∈ f A U and by definition E U ∈ ( T X , n ) U . Since hearts constructed via gluing coincide with those con-structed via recollement as in Remark 2.15, by Corollary 4.19 we then have that E U ∈ A U . (cid:3) Remark 4.22.
Using the techniques above, one can prove the open heart property for noether-ian hearts for a wider class of triangulated categories T . That is, where T admits a realisation D b ( Y ) and T ⊥ is geometric i.e. admit semiorthogonal decompositions with semiorthogonal com-ponents given by D b ( X i ) for various smooth projective X i .4.3. Generic flatness.
Let Y be a smooth projective variety and T ⊆ D b ( Y ) an admissible sub-category. Definition 4.23.
Let
A ⊆ T be the heart of a bounded t-structure. We say that E ∈ A S is t-flat iffor every s ∈ S , we have that E s ∈ A . Definition 4.24.
Let
A ⊆ T be a heart of a bounded t-structure, we say that A satisfies the genericflatness property if for all E ∈ A S , where S is a projective variety, there is an open set U ⊆ S suchthat for all s ∈ U , we have that E s ∈ A .As in [
1, Proposition 3.5.3 ] , using the techniques of the proof of the open heart property, wecan immediately provide a partial result of generic flatness of a heart A ⊂ T X , n . Proposition 4.25.
For E ∈ A S , then there is a dense set Z ⊆ S such that E s ∈ A for every s ∈ Z . Proof.
We consider
A ⊆ f A as in the proof of Theorem 4.15. Let E ∈ A S ⊆ f A S . By [
1, Proposition3.5.3 ] , there is a dense set Z ⊆ S , such that for all s ∈ Z we have that E s ∈ f A . Due to the fact that E s ∈ T X , n and A = T X , n ∩ f A , it follows that E s ∈ A for all s ∈ Z . (cid:3) We start by proving the generic flatness property in the case of curves. The following lemmafollows from the same arguments of [
33, Lemma 4.7 ] , after replacing the K Lemma 4.26.
Let C be a curve and σ = ( Z , A ) ∈ Stab ( C ) . Then A satisfies generic flatness. Proof.
First note that by [
24, Theorem 2.7 ] , we have A = P σ µ (( r , 1 + r ]) where r = m + θ with θ ∈ [
0, 1 ) and m ∈ Z , where σ µ is given by slope stability and T θ = P σ µ ( θ , 1 ] and F θ = P σ µ ( θ ] .Therefore, we get A = 〈F θ [ m + ] , T θ [ m ] 〉 . It is enough to prove the statement for m =
0. Let
E ∈A S , we have that R p ∗ ( E ⊗ q ∗ ( L ) n ) ∈ A for n >>
0. Note that the cohomology of E is concentratedin degree −
1, 0.The spectral sequence E i , j = R i p ∗ ( H i ( E ) ⊗ q ∗ ( L ) n ) ⇒ R i + j ( E ⊗ q ∗ ( L ) n ) ∈ A degenerates for n >>
0, from which it follows that H i ( E ) = i = −
1, 0. Then by [
17, Theo-rem 2.3.2 ] , there is an open set U ⊆ S and a filtration0 = F ⊆ F ⊆ · · · F l = H − ( E ) U such that F i + / F i are U -flat for i =
1, . . . , l . Moreover, for s ∈ S , the filtration0 = F s ⊆ F s ⊆ · · · ⊆ F ls = H − ( E ) s is precisely the HN-filtration of H − ( E ) s with respect to µ -stability. By [
1, Proposition 3.5.3 ] , wehave that there is dense set S ′ ⊆ S , such that E s ∈ A for s ∈ S ′ . This implies that for every s ∈ S ′ we get that H − ( E ) s ∈ F θ . Since F i + / F i are U -flat, we have that [ F i + s / F is ] = [ F i + s ′ / F is ′ ] for s ∈ U and s ′ ∈ S ′ and that µ ( F i + s / F is ) = µ ( F i + s ′ / F is ′ ) ≤ − cot ( πθ ) . It implies that for all s ∈ U , we get that H − ( E ) s ∈ F θ . Analogously for H ( E ) . (cid:3) Lemma 4.27.
Let T be a triangulated category and T = 〈 D , D 〉 a semiorthogonal decomposi-tion. Suppose there are hearts A j ⊆ D j for j =
1, 2 , satisfying generic flatness, then the heart A = rec ( A , A ) also satisfies generic flatness.Proof. Let E ∈ A S . By Proposition 4.17, we have that A S = rec ( A S , A S ) and that i ∗ ( E ) ∈ A S , i !2 ( E ) ∈ D ≥ S and i ∗ ( E ) ∈ D ≤ , where ( D ≤ , D ≥ ) is the t-structure with heart A . From Lemma4.26, it follows that there are open sets U and U , such that for all s ∈ U we have i ∗ ( E ) s ∈ A andfor all s ∈ U we have i !2 ( E ) s ∈ D ≥ and i ∗ ( E ) ∈ D ≤ .Finally we define U : = U ∩ U and note that it follows directly from the definition of recollementthat E s ∈ A for all s ∈ U . (cid:3) Corollary 4.28.
Let A be a gluing heart with respect to the standard semiorthogonal decomposi-tion T C , n = 〈 D , . . ., D n 〉 . Then A satisfies the generic flatness property.Proof. Since A is a gluing heart, there exist hearts A j ⊆ D j ∼ = D b ( C ) for j =
1, . . ., n such that A = gl ( A , . . . , A n ) ⊆ T C , n . By Lemma 4.26, we have that A j satisfies the generic flatness property.Moreover, it follows from Remark 2.15 that if T = 〈 D , D 〉 and two hearts B ⊆ D and B ⊆ D satisfy gluing conditions, then gl ( B , B ) = rec ( B , B ) . By the recursive construction of A = gl ( A , . . ., A n ) and Lemma 4.27, we obtain that A satisfies the generic flatness property. (cid:3) We will need the following result which follows from the same arguments given in [
33, Lemma3.15 ] . Lemma 4.29.
Let σ = ( Z , A ) ∈ Stab ( T C , n ) be an algebraic stability condition. Assume that M β , φ ( σ ) is bounded for all φ ∈ R and β ∈ Z n . Then for φ ∈ (
0, 1 ) and G ∈ A the following set ofQ ( G , φ ) = { E ∈ A | there exists a surjection G ։ E ∈ A and φ ( E ) ≤ φ } is bounded in D b ( Y C , n ) . We now assume that for an algebraic stability condition σ = ( Z , A ) ∈ Stab ( T C , n ) the set M β , φ ( σ ) is bounded for all φ ∈ R and β ∈ Z n and that A satisfies generic flatness. Let E ∈ A S be t-flatand take φ ∈ (
0, 1 ) . We consider the following functorsQuot ( E , φ ) , (cid:0) Sub ( E , φ ) (cid:1) : ( Sch / S ) → Setsdefined as follows: A scheme T over S is mapped to pairs of the form ( F , E T → F ) (respectively ( F , F → E T ) ) where F ∈ M pug ( T C , n )( T ) such that:(1) For each t ∈ T , we have that F t ∈ A and φ ( F t ) ≤ φ (respectively φ ( F t ) ≥ φ ).(2) For each closed point t ∈ T , the induced morphism E t → F t is surjective (respectively F t → E t injective) in A . Remark 4.30.
These functors are a subspaces of the quot spaces defined in [
5, Definition 11.3 ] .It follows from [
5, Proposition 11.6 ] that they are algebraic spaces.We prove the following proposition [
33, Proposition 3.17 ] . The proofs are essentially the samewith some minor modifications. In particular, we incorporate the techniques of [ ] . Proposition 4.31.
For any φ ∈ (
0, 1 ) there exist S -schemes Q ( E , φ ) , S ( E , φ ) , of finite type over S ,and S -morphisms Q ( E , φ ) → Quot ( E , φ ) , S ( E , φ ) → Sub ( E , φ ) , which are surjective on C -valued points of Quot ( E , φ ) and Sub ( E , φ ) .Proof. Let E ∈ A S . By Corollary 4.21 we have that there is an object H ∈ A , some integer n ∈ Z and a surjection H S ⊗ L − n ։ E in A S , for i =
1, . . . , n . Corollary 4.20 states that the functor L i ∗ : ( T C , n ) S → T C , n is right t-exact with respect to A and therefore the kernel of H ։ E is alsot-flat. As a consequence, we obtain a morphism from H s ։ E s for each s ∈ S .By Lemma 4.29 there is a C -scheme of finite type Q and F ∈ D b ( Y C , n × S ) , such that any objectin Q ( H , φ ) is isomorphic to F q for some q ∈ Q . As A satisfies both generic flatness and the openheart property by Theorem 4.15, the set Q = { q ∈ Q | F q ∈ A } is open.Define Q = Q × S . By [
5, Lemma 8.9 ] there is an open U ⊆ Q such thatHom U ( E , F ) : ( Sch / U ) op −→ ( Sets ) T Hom D ( X T ) ( E T , F T ) is representable by an affine scheme Z U .Once again there is an open set V ⊆ Q \ U , such that the functor above over V is representableby Z V . Recursively we construct a scheme Q whose C -points are in bijection with the C -pointsof Q and where the functor above over Q is representable by Z , an affine scheme of finite pre-sentation over Q . Moreover, note that since Q , and therefore also Q , is of finite type over S , wehave that Z is of finite type over S .Let us consider the universal family E Z → F Z and the triangle K → E Z → F Z in D b ( Y C , n × Z ) .For q ∈ Z , we have that F q ∈ A . As a consequence, the morphism E q → F q is surjective in A ifand only if K q ∈ A . Then, we define Q ( E , φ ) : = { q ∈ Z | K q ∈ A } which induces a morphism to Quot ( E , φ ) that is surjective on C -valued points. We get that Q ( E , φ ) is an open subscheme of Z . Indeed, we apply the open heart property and generic flatness of A .The arguments for Sub ( E , φ ) are the same as in [
33, Proposition 3.17 ] . (cid:3) Proposition 4.32. [
33, Proposition 3.18 ] Let σ = ( Z , A ) ∈ Stab ( T C , n ) be as in Proposition 4.31. Ifthere is g ∈ Ý GL + ( R ) such that σ ′ = ( Z ′ , A ′ ) = σ · g and σ ′ is algebraic then A ′ also satisfies thegeneric flatness property. Corollary 4.33.
Let σ = ( Z , A ) ∈ Stab ( T C ) an algebraic stability condition, then A satisfies thegeneric flatness property.Proof. By Theorem 2.16, we have that either σ is a gluing stability condition or σ satisfies ( ) or ( ) . If σ is a gluing stability condition, our statement follows from Corollary 4.28.If σ satisfies ( ) , then there is a g ∈ Ý GL + ( R ) such that σ · g is a gluing stability condition, inthis case the result then follows from Proposition 4.32. ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 27 If σ satisfies ( ) , then there is g ∈ G , such that σ · g = ( Z ′ , A ′ ) has the following property: Thereis a non-gluing stability condition σ ′′ = ( Z ′′ , A ′′ ) satisfying ( ) with such that A ′ = A ′′ . Sincegeneric flatness is a property of the heart, we apply the same argument as before. (cid:3) Algebraic moduli stacks.
Following the strategy laid out in [ ] , we prove, using the openheart property, that generic flatness is sufficient for the algebraicity of the substack of Bridgelandsemistable chains.First we state the following lemma. The proof is exactly the same as in [ ] . Lemma 4.34. [
33, Lemma 3.13 ] Let σ = ( Z , A ) ∈ Stab ( T C ) be an algebraic stability condition. Fora smooth quasi-projective variety S and E ∈ M pug ( T C )( S ) , assume that the locusS ◦ = (cid:8) s ∈ S | E s is of numerical type β ∈ Z and E s ∈ P σ ( φ ) (cid:9) for φ ∈ R is not empty. Then there is an open subset U ⊆ S which is contained in S ◦ .We now state and prove the main result. Theorem 4.35.
Let σ ∈ Stab ( T C ) be a stability condition, then M β , φ ( σ ) is an algebraic stack offinite type over C for all β ∈ Z and φ ∈ R . Proof.
First assume that σ is an algebraic stability condition. Following standard arguments (seefor example [
33, Lemma 3.6 ] ), Lemma 4.34 implies that M β , φ ( σ ) is an open substack of M pug ( T C ) .By Corollary 4.12, we obtain that M β , φ ( σ ) is bounded. Therefore M β , φ ( σ ) is an algebraic stackof finite type over C . See [
5, Lemma 9.7 ] .The result for a non-algebraic stability condition follows from the algebraic case proved above.We omit the proof since it is exactly the same as in [
33, Proposition 3.20, Step 3 ] which relies onthe well-behaved wall and chamber decomposition, see [
6, Proposition 3.3 ] . (cid:3) Corollary 4.36.
For every σ = ( Z , A ) ∈ Stab ( T C ) , we have that A satisfies generic flatness.Proof. An adapted version of the arguments of [
30, Proposition 4.12 ] to our set up follows fromTheorem 4.35, Corollary 4.20 and Corollary 4.21. (cid:3) Under the assumption that σ α ∈ Stab ◦ ( T C , n ) satisfies the support property we obtain the fol-lowing proposition. Let Stab ◦ ( T C , n ) be the connected of σ α . Proposition 4.37.
Let σ = ( Z , A ) ∈ Stab ◦ ( T C , n ) be a gluing algebraic stability condition. Then M β , φ ( σ ) is an algebraic stack of finite type over C for all β ∈ Z n and φ ∈ R . Proof.
By Corollary 4.11, we have that M β , φ ( σ ) is bounded. Subsequently, A satisfies the openheart property (Theorem 4.15). Moreover, Corollary 4.28 tells us that A satisfies the generic flat-ness property. By Proposition 4.32, for every g ∈ Ý GL + ( R ) , we also get that if σ · g = ( Z ′ , A ′ ) isalgebraic then A ′ also satisfies the generic flatness property. As a consequence, we can prove theanalogous to Lemma 4.34 for this case. Therefore, the statement follows from [
5, Lemma 9.7 ] . (cid:3) Remark 4.38.
Let X be a smooth projective variety of dim ( X ) > σ = ( Z , A ) ∈ Stab ( X ) . We then have that there is an algebraic gluing pre-stability condition τ = ( W , B ) = gl ( σ , . . . , σ ) on T X , n . Moreover, if A satisfies generic flatness, byCorollary 4.28 we get that B also satisfies generic flatness. By Theorem 4.15, the heart B alsosatisfies the open heart property. Consequently, it is enough to prove the boundedness of themoduli stack M β , φ ( τ ) , which is expected, in order to conclude that it is algebraic of finite typeover C . See [ ] for K [ ] for 3-folds.4.5. Good moduli spaces.
We will now apply the groundbreaking result [ ] to show the existenceof the good moduli spaces as defined by Alper [ ] .The main theorem of the section is the following. Theorem 4.39.
Consider the moduli stack M β , φ ( σ ) as in Theorem 4.35 or as in Proposition 4.37.Then M β , φ ( σ ) admits a good moduli space M β , φ ( σ ) which is an algebraic space over C . Moreover,M β , φ ( σ ) is proper.Proof. Since M β , φ ( σ ) is an algebraic stack of finite type over C , we can follow the same steps asin [
3, Theorem 7.25 ] . We obtain that M β , φ ( σ ) admits a separated good moduli space M β , φ ( σ ) .To prove that M β , φ ( σ ) is proper it suffices to prove the existence part of the valuative criteria forproperness [
3, Theorem A ] , which follows from Proposition 4.40 below. (cid:3) The following is an analogous result to [
1, Proposition 4.1.1 ] in our setting. Combining thefollowing proposition with [
3, Theorem A ] we deduce the properness claimed in Theorem 4.39. Proposition 4.40.
Let σ = ( Z , A ) ∈ Stab ( T X , n ) be an algebraic stability condition. Let S be a curveand U : = S \ { p } where p ∈ S is a closed point and j : U , → S . Let E U ∈ A U such that E s ∈ P σ ( ) forall s ∈ U . Then there is an E ∈ A S such that j ∗ ( E ) = E U and E s ∈ P σ ( ) for all s ∈ S .
Remark 4.41.
Proposition 4.40 also appears in the more general relative setting in [
5, Lemma21.22 ] . We include a proof as a pleasant application of our methodology. Proof.
Consider the realisation T X , n ⊂ D b ( Y X , n ) and the heart f A ⊂ D b ( Y X , n ) as constructed inSection 4.2 such that A ⊂ f A . Then A U ⊆ f A U . By [
1, Lemma 3.2.1 ] , there is an object E ∈ ODULI OF BRIDGELAND SEMISTABLE HOLOMORPHIC TRIPLES 29 f A S , such that j ∗ ( E ) = E U . We have a triangle E → E → E induced by the semiorthogonaldecomposition D b ( Y X , n × S ) = 〈 ( T X , n ) ⊥ S , ( T X , n ) S 〉 .By Proposition 4.17, we get that E ∈ A S and moreover j ! ( E ) = j ! ( E ) = E U . As A is Noetherianthere is a maximal S -torsion subobject F ⊆ E in A S , with support { p } . See [
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ECHNISCHE U NIVERSITÄT B ERLIN , S
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