Semiorthogonal decompositions of stable pair moduli spaces via d-critical flips

We show the existence of semiorthogonal decompositions (SOD) of Pandharipande-Thomas (PT) stable pair moduli spaces on Calabi-Yau 3-folds with irreducible curve classes, assuming relevant moduli spaces are non-singular. The above result is motivated by categorifications of wall-crossing formula of PT invariants in the derived category, and also a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture. We also give SOD of stable pair moduli spaces on K3 surfaces, which categorifies Kawai-Yoshioka's formula proving Katz-Klemm-Vafa formula for PT invariants on K3 surfaces with irreducible curve classes.


Introduction
The purpose of this paper is to give applications of d-critical birational geometry proposed in [Tod] to the study of derived categories of coherent sheaves on moduli spaces of stable objects on Calabi-Yau (CY for short) 3-folds. The main result is that Pandharipande-Thomas (PT for short) stable pair moduli spaces [PT09] on CY 3-folds with irreducible curve classes admit certain semiorthogonal decompositions (SOD for short), assuming relevant moduli spaces are non-singular. Our results are motivated by categorifications of the wall-crossing formula for Donaldson-Thomas invariants on CY 3-folds [JS12,KS]  the derived category, and also by a d-critical analogue of the D/K equivalence conjecture of Bondal-Orlov and Kawamata [BO,Kaw02].

SOD of stable pair moduli spaces
Let X be a smooth projective CY 3-fold over C. By definition, a stable pair on X consists of data [PT09] (F, s), s : O X → F, (1.1) where F is a pure one-dimensional coherent sheaf on X and s is surjective in dimension 1. For β ∈ H 2 (X, Z) and n ∈ Z, we denote by The study of PT invariants is one of the central topics in curve counting on CY 3-folds (see [PT14]). Suppose that n ≥ 0 and β is an irreducible curve class, i.e. β cannot be written as β 1 + β 2 for effective curve classes β i . Then we have the following diagram: P n (X, β) Here U n (X, β) is the moduli space of one-dimensional Gieseker stable sheaves F on X with [F ] = β and χ (F ) = n. The maps π ± are defined by π + (F, s) := F, π − (F , s ) := Ext 2 X (F , O X ).
For a variety Y , we denote by D b (Y ) the bounded derived category of coherent sheaves on Y . The following is the main result in this paper: Theorem 1.1 (Theorem 5.7). Suppose that U n (X, β) is fine and non-singular. Then P ±n (X, β) are also non-singular, and we have the following: (i) The Fourier-Mukai functor P : D b (P −n (X, β)) → D b (P n (X, β)) with kernel the structure sheaf of the fiber product of (1.3) is fully faithful.
(ii) There is a π + -ample line bundle O P (1) on P n (X, β) such that if n ≥ 1 then the functor ϒ i P : D b (U n (X, β)) → D b (P n (X, β)) defined by Lπ + * (−) ⊗ O P (i) is fully faithful. (iii) We have the SOD D b (P n (X, β)) = Im ϒ −n+1 P , . . . , Im ϒ 0 P , Im P . (1.4) Here U n (X, β) is called fine if it admits a universal sheaf on X × U n (X, β), which is guaranteed if (D · β, n) is coprime for some divisor D. The result of Theorem 1.1 will also be applied to some non-compact CY 3-folds. We apply Theorem 1.1 in the case of where S is a smooth projective surface. The assumption of Theorem 1.1 is satisfied when −K S · β > 0, and we obtain the SOD of derived categories of relative Hilbert schemes of points on the universal curve over a complete linear system on S (see Corollary 5.10).
We also apply Theorem 1.1 in the case of where C is a smooth projective curve. Here n ∈ Z ≥0 , C [k] is the k-th symmetric product of C, g is the genus of C, and J C is the Jacobian of C. The above SOD seems to give a new result on the properties of symmetric products of curves and the associated Abel-Jacobi maps.

Motivations behind Theorem 1.1
We have two motivations behind the result of Theorem 1.1. The first one is to give a categorification of the formula (see [PT10,Tod12a]) P n,β − P −n,β = (−1) n−1 nN n,β . (1.5) Here N n,β ∈ Z is obtained by the integration of the virtual class on U n (X, β). The identity (1.5) is the key ingredient to show the rationality of the generating series of PT invariants P β (X) = n∈Z P n,β q n when β is irreducible (see [PT10]). As observed in [Tod12a], the diagram (1.3) is a wallcrossing diagram in D b (X), and (1.5) is the associated wall-crossing formula. Under the assumption of Theorem 5.7, the invariants in (1.5) are given by P ±n,β = (−1) n+d−1 e(P ±n (X, β)), N n,β = (−1) d e(U n (X, β)) where d is the dimension of U n (X, β). Therefore the SOD in (1.4) categorifies the formula (1.5), as it recovers the formula (1.5) by taking the Euler characteristics of the Hochschild homology of both sides of (1.4). The second motivation is to give an evidence for a d-critical analogue of Bondal-Orlov's and Kawamata's D/K equivalence conjecture [BO,Kaw02]. The original D/K equivalence conjecture asserts that for a flip of smooth varieties Y + Y − there exists a fully faithful functor On the other hand, the diagram (1.3) is an example of a d-critical flip introduced in [Tod]. Therefore Theorem 1.1(i) gives evidence for a d-critical analogue of the D/K equivalence conjecture. We will come back to this point of view in Subsection 1.4.

Categorification of Kawai-Yoshioka's formula
We will apply the argument as in the proof of Theorem 1.1 to show the existence of SOD on relative Hilbert schemes of points associated with linear systems on K3 surfaces. Let S be a smooth projective K3 surface such that Pic(S) is generated by O S (H ) for an ample divisor H with H 2 = 2g − 2. Let π : C → |H | = P g (1.6) be the universal curve. Below we fix n > 0, and define to be the π -relative Hilbert scheme of n + g − 1 points. The moduli space (1.7) is known to be isomorphic to the moduli space of PT stable pairs P n (S, [H ]) on S. For each k ≥ 0, let U k be the moduli space of H -Gieseker stable sheaves E on S such that v(E) = (k, H, k + n) ∈ H 2 * (S, Z) where v(−) is the Mukai vector. The moduli space U k is an irreducible holomorphic symplectic manifold. We assume that U k is a fine moduli space for all k ≥ 0, e.g. it is satisfied if (2g − 2, n) is coprime. Let N ≥ 0 be defined to be the largest k ≥ 0 such that U k = ∅. In this situation, we have the following: Theorem 1.2 (Corollary 6.3). We have the following SOD: k , . . . , A (n+2k) k such that each A (i) k is equivalent to D b (U k ). Theorem 1.2 is proved by using the zigzag diagram constructed by Kawai-Yoshioka [KY00]. We show that each step of the above diagram is described in terms of a d-critical simple flip, by investigating wall-crossing diagrams on a CY 3-fold X = S × C for an elliptic curve C. Then Theorem 1.2 is proved by applying the argument used for Theorem 1.1 to each step of the diagram. The SOD in Theorem 1.2 is interpreted as a categorification of Kawai-Yoshioka's formula [KY00] for PT invariants on K3 surfaces with irreducible curve classes, defined by P n,g := (−1) n−1 e(C [n+g−1] ). Indeed, the following formula is proved in [KY00]: (1.8) The SOD in Theorem 1.2 recovers the formula (1.8) by taking the Euler characteristics of Hochschild homologies of both sides of (1.2). In [KY00], the formula (1.8) led to the Katz-Klemm-Vafa (KKV) formula for PT invariants with irreducible curve classes (see Remark 6.4).

D-critical analogue of the D/K equivalence conjecture
Here we explain the notion of d-critical flips for Joyce's d-critical loci [Joy15], and an analogue of the D/K equivalence conjecture mentioned earlier. By definition, a d-critical locus consists of data where M is a C-scheme or an analytic space and S 0 M is a certain sheaf of C-vector spaces on M. The section s is called a d-critical structure of M. Roughly speaking, if M admits a d-critical structure s, this means that M is locally written as a critical locus of some function on a smooth space, and the section s remembers how M is locally written as a critical locus. If M is a truncation of a derived scheme with a (−1)-shifted symplectic structure [PT + 13], then it has a canonical d-critical structure [BB + 15].
Let (M ± , s ± ) be two d-critical loci and consider a diagram of morphisms of schemes or analytic spaces The above diagram is called a d-critical flip if for any p ∈ U , there is a commutative diagram We expect that an analogue of the D/K equivalence conjecture may hold for d-critical loci. Namely for a d-critical locus (M, s) 1 there may exist a certain triangulated category D(M, s) such that if the diagram (1.9) is a d-critical flip, we have a fully faithful functor (1.11) The category D(M − , s − ) may be constructed as a gluing of Z/2Z-periodic triangulated categories of matrix factorizations defined locally on each d-critical chart, though its construction seems to be a hard problem at this moment (see [Joy, (J)], [Toë14, Section 6.1]). For a flip Y + Y − in the diagram (1.10), suppose that the D/K equivalence conjecture holds, i.e. we have a fully faithful functor Then it induces the fully faithful functor (see Theorem 2.1) where D(Y ± , w ± ) are the derived factorization categories associated with pairs (Y ± , w ± ). If the desired categories D(M ± , s ± ) are gluings of D(Y ± , w ± ) defined locally on U , then we may try to globalize the functor (1.12) to give a fully faithful functor (1.11). If this is possible, then the numerical realization of a semiorthogonal complement of the embedding (1.11) may recover the wall-crossing formula for DT invariants [JS12,KS]. 2 For a d-critical flip (1.9), suppose that M ± are smooth, so in particular s ± = 0. In this case, we can use usual derived categories of coherent sheaves to ask an analogue of the above question. Namely for a d-critical flip (1.9) with M ± smooth, we can ask whether we have a fully faithful functor The results of Theorems 1.1 and 1.2 are proved by establishing such a result in the case of d-critical simple flips (see Theorem 4.5).

Relation to other works
There exist some recent works studying wall-crossing behavior of derived categories of moduli spaces of stable objects on algebraic surfaces. In [Bal17], Ballard showed the existence of SOD under wall-crossing of Gieseker moduli spaces of stable sheaves on rational surfaces. Also Halpern-Leistner [HL] announces that, under wall-crossing of Bridgeland moduli spaces of stable objects on K3 surfaces, their derived categories are equivalent. The results in our paper can be regarded as a CY 3-fold version of these works. One of the crucial differences is that, although the moduli spaces considered in [Bal17,HL] are birational under wall-crossing, the moduli spaces in our paper are not necessarily birational under wall-crossing. For example the moduli spaces P ±n (X, β) in the diagram (1.3) have different dimensions if n > 0. Instead, the fact that they are birational in d-critical birational geometry plays an important role for the existence of SOD in Theorems 1.1 and 1.2.

Outline of the paper
The outline of this paper is as follows. In Section 2, we review basics on derived factorization categories which we will use in later sections. In Section 3, we show the existence of SOD of gauged LG models on simple flips over a complete local base, and describe the relevant kernel objects. In Section 4, we globalize the result in Section 3 and show the SOD for formal d-critical simple flips. In Section 5, we use the result in Section 4 to show Theorem 1.1. In Section 6, we prove Theorem 1.2.

Notation and conventions
In this paper, all the varieties and schemes are defined over C. For C-schemes U , S, T and a morphism f : S → T , we set For a variety Y , we denote by D b (Y ) the bounded derived category of coherent sheaves on Y . If an algebraic group G acts on Y , we also denote by D b G (Y ) the bounded derived category of G-equivariant coherent sheaves on Y . For smooth varieties Y 1 , Y 2 with projective morphisms Y i → T , and an object P ∈ D b (Y 1 × Y 2 ) supported on Y 1 × T Y 2 , we denote by P the Fourier-Mukai functor Here p i : Y 1 × Y 2 → Y i are the projections. The object P is called the kernel of the functor P .
Recall that a semiorthogonal decomposition of a triangulated category D is a collection C 1 , . . . , C n of full triangulated subcategories such that Hom(C i , C j ) = 0 for all i > j and the smallest triangulated subcategory of D containing C 1 , . . . , C n coincides with D. In this case, we write D = C 1 , . . . , C n . If each C i is equivalent to D b (M i ) for a variety M i , we also write D = D b (M 1 ), . . . , D b (M n ) for simplicity.

Review of derived factorization categories
In this section, we recall the notion of gauged Landau-Ginzburg (LG) models, and the associated derived factorization categories introduced by Positselski. We refer to the articles [EP15] for basics on these notions.

Definitions of derived factorization categories
Let us consider data (called gauged LG model) where Y is a C-scheme, G is a reductive algebraic group which acts on Y , χ : G → C * is a character and w ∈ (O Y ) satisfies g * w = χ (g)w for any g ∈ G. Given data as above, the derived factorization category is defined as a triangulated category whose objects consist of factorizations of w, i.e. sequences of G-equivariant morphisms of G-equivariant coherent sheaves The category (2.2) is defined to be the localization of the homotopy category of the factorizations (2.3) by its subcategory of acyclic factorizations. When Y is a smooth affine scheme and G = {1}, the category (2.2) is equivalent to the triangulated category of matrix factorizations of w (see [Orl09]). In the case of G = C * and χ = id, we simply write For a character χ : G → C * , let χ be defined by We have the functor When G = {1}, the functor (2.4) gives the equivalence (see [Isi13,Shi12,Hir17b])

Derived functors between derived factorization categories
Let (Y, G, χ , w) be a gauged LG model (2.1), and W be another variety with a G-action. For a G-equivariant projective morphism f : W → Y , we have another gauged LG model Similarly to the usual derived functors between derived categories, if Y is smooth we have the derived functors Also for another object P ∈ D G (Y, χ , w ), we have the derived tensor product Below we omit the subscripts R, L when the relevant functors are exact functors of coherent sheaves, e.g. write Lf * as f * when f is flat. Let Y 1 , Y 2 be regular C-schemes with G-actions. Let T be a C-scheme with a G-action and consider G-equivariant projective morphisms Y i → T . Let us take w ∈ (O T ) and a character χ : G → C * satisfying g * w = χ (g)w for any g ∈ G. We consider the commutative diagram Y 1 be the projections, and suppose G acts on Y 1 × Y 2 diagonally. For any object Here is given in (2.4) and forg G , forg C * are forgetting the G-action and the C * -action respectively. For P ∈ D b (Y 1 × T Y 2 ) and (P) ∈ D C * (Y 1 × T Y 2 , 0), the following diagram commutes: Moreover we have the following: is fully faithful (resp. an equivalence). Then for the object is fully faithful (resp. an equivalence).

Knörrer periodicity
Let E → Y be an algebraic vector bundle on a regular C-scheme Y , and s : Y → E be a regular section of it, i.e. its zero locus has codimension equal to the rank of E. The section s naturally defines a morphism Q s : E ∨ → A 1 sending (y, v) for y ∈ Y and v ∈ E| ∨ y to s(y), v . We have the diagram Let C * act on Z trivially, and on E ∨ with weight 1 on the fibers of the projection E ∨ → Y . The following is the version of Knörrer periodicity used in this paper: is an equivalence of triangulated categories. By composing it with the equivalence (2.5), we obtain the equivalence

SOD via simple flips
Let U be the formal completion of an affine space at the origin. In this section, we show that for a simple d-critical flip for smooth schemes M ± satisfying some conditions, we have the SOD The result is proved by combining derived factorization analogue of Bondal-Orlov's SOD associated with simple flips [BO] (see Theorem 3.3) with the Knörrer periodicity of derived factorization categories (see Theorem 2.2). The main ingredient in this section is to show that the kernel object of the fully faithful functor D b ( M − ) → D b ( M + ) is given by the structure sheaf of the fiber product M + × U M − . This explicit description of the kernel will be important in the next section to globalize the result of this section.

Simple toric flips
Let V + , V − be C-vector spaces of dimensions a, b respectively. We assume that a ≥ b, and set n := a − b ≥ 0. Let C * act on V + , V − with weight 1, −1 respectively. We fix bases of V ± and denote the coordinates of V + , V − by x = (x 1 , . . . , x a ), y = (y 1 , . . . , y b ) respectively. Let V ± * := V ± \ {0}, and define We have a toric flip diagram, called a simple flip (see [Rei]): We also have the projections and closed embeddings where i ± are the zero sections of pr ± . By setting W := Y + × Z Y − , we have the diagram where p ± are the projections. Note that p ± are the blow-ups of Y ± at the smooth loci i ± (P(V ± )). The fiber product W is also described as (3.5) Under the description of W in (3.5), the projections p ± : W → Y ± are induced by the maps respectively. Then the diagram (3.4) is equivariant with respect to the above C * -actions.

Critical loci
Let U be a smooth C-scheme of dimension g, given by Let us take an element w ∈ (O Z U ) written as for some w ij ( u) ∈ (O U ). We consider the following commutative diagram: Then the composition can be written as in the description of W by (3.5). We define M ± to be Lemma 3.1. Suppose that M ± are smooth and irreducible of dimension respectively. Then M ± are contained in the images of i ± U : (3.11) Proof. Let N + be the scheme defined by the RHS of (3.11). Note that we obviously have the closed embedding Therefore the assumption on M + implies that the embedding (3.12) is an isomorphism. The claim for M − is proved similarly. Under the assumption of Lemma 3.1, we have f ± Moreover for each c ∈ Z, we have the line bundles

SOD of derived factorization categories under simple flips
Let us consider the diagram (3.8). Since w + , w − and w are of weight 1 with respect to the C * -actions (3.6), we have the associated derived factorization categories respectively. Since the diagram (3.4) is C * -equivariant, we have the functors . By composing them, we obtain the functor (3.14) Let be the projection and the inclusion into the zero section (3.3) respectively. We also have the functor Here C * acts on U , P(V + ) U trivially. The following result should be well-known, but we include a proof here as we cannot find a reference. Here Proof. (i) The functor Y is written as O W in the notation of Theorem 2.1. On the other hand, the functor is fully faithful by [BO]. Therefore (i) follows from Theorem 2.1. The proof of (ii) is similar. We prove (iii). Let us recall that we have a similar SOD using windows [HL15, are C * -equivariant, where the C * -action on the LHS is given by (3.6) and that on the RHS is given by the above T 2 -action. Let O • (i) be the T 1 -equivariant line bundle on Spec C, given by a one-dimensional T 1 -representation with weight i. We denote by Let χ : T 1 × T 2 → C * be the second projection, and take w ∈ (O (V + ×V − ) U ) as in (3.7). For a subset I ⊂ R, the window subcategory is defined to be the thick triangulated subcategory generated by the factorizations (2.3) where F 0 , F 1 are of the form as T 1 -equivariant sheaves. Here we regard the (T 1 × T 2 )-equivariant sheaves F j as T 1equivariant sheaves by the inclusion T 1 → T 1 × T 2 , t 1 → (t 1 , 1). By [BFK19, Theorem 3.5.2], there exists a fully faithful functor Here the horizontal arrows are equivalences of triangulated categories, defined by pullbacks via open immersions (3.17) restricted to W I , and the left vertical arrow is a natural inclusion. Moreover by loc. cit., we have the SOD It is enough to show that Im Y = Im Y . Note that By the diagram (3.19), it follows that Im Y is generated by factorizations (2.3) such that F 0 , F 1 are of the form On the other hand, an easy calculation shows that where O W is the functor (3.16). Together with the equivalence of the top horizontal arrow of (3.19), it follows that Im Y is also generated by the objects of the form (3.20). Therefore Im Y = Im Y .

SOD in the complete local setting
We return to the situation of Subsection 3.2. Under the assumption of Lemma 3.1, we have the diagram Here A ± are defined by the above Cartesian square. By Theorem 2.2, the above diagram induces the equivalence Here C * acts on M ± trivially, and on Y ± U with weight 1 on fibers of pr ± U . Set where the right vertical arrow is the projection and E is defined by the above Cartesian square. Again by Theorem 2.2 and the description of w in (3.9), the above diagram induces the equivalence Here C * acts on B trivially, and on W U with weight 1 on fibers of pr U . Set Let ± be the functor defined by The following diagram is commutative: Here the vertical arrows are the equivalences (3.21), (3.22).
Proof. Let F ± ⊂ W U be defined by the Cartesian square Here the right vertical arrow is the projection. We have two diagrams Since all Cartesians in the above diagrams are derived Cartesians, the base change shows that Therefore the lemma holds.
Lemma 3.5. The following diagram is commutative: Here ± R are the right adjoint functors of , i.e.
± R := Rr ± * • (k ± ) ! , where (k ± ) ! are the right adjoint functors of k ± * . They are written as Here Proof. The commutativity of (3.24) follows from that of (3.23) together with the fact that Rp ± U * , ± R are the right adjoint functors of Lp ± * U , ± respectively. As for the formula for (k ± ) ! , note that By the exact sequences Together with the dimension computations we obtain (3.25).
Proposition 3.6. Suppose that Then the following diagram is commutative: Here the vertical arrows are the equivalences (3.21), Y is given by (3.14) and M is defined by where is the equivalence in (2.5): :

Proof. By Lemmas 3.4 and 3.5, it is enough to check that
. Let us consider the composition Lk + * • k − * in the above formula. We have the Cartesian diagram By the definition of F ± , we have Then the assumption (3.27) on the dimension of the fiber product implies dim(F + ∩ F − ) = g − 1 and (3.28) is a derived Cartesian diagram. Therefore by base change, By substituting into the above formula for + R • − , and again noting (3.29), we have Therefore the proposition holds.
Lemma 3.7. The following diagram is commutative: Proof. The inverse of the equivalence of the right vertical arrow in (3.30) is given by Rq + * • j +! . Therefore it is enough to check that (3.32) We use the commutative diagram Since the left Cartesian diagram above is a derived Cartesian, by base change we have as expected.
By putting all the arguments in this subsection together, we have the following: Proposition 3.8. In the setting of Subsection 3.2, suppose that the assumptions of Lemma 3.1 and the dimension condition (3.27) hold. Then we have the following: : is fully faithful. Therefore (i) follows by the commutative diagram (2.6). Similarly, (ii) follows from Theorem 3.3(ii), Lemma 3.7 and the commutative diagram (2.6). As for (iii), by Theorem 3.3(iii), Proposition 3.6 and Lemma 3.7 we have the SOD , we obtain the desired SOD.

SOD via d-critical simple flips
In this section, we show that for a d-critical simple flip M + → U ← M − satisfying some conditions, we have an associated SOD of D b (M + ). The SOD in this section is obtained by globalizing the SOD in Proposition 3.8.

D-critical simple flips
Let U be a smooth variety with g := dim U . Let (M ± , s ± ) be two d-critical loci, and suppose that we have projective morphisms For each p ∈ U , we set Definition 4.1. A diagram (4.1) is called a formal d-critical simple flip if for any p ∈ U , there exist finite-dimensional vector spaces V ± with dim V + ≥ dim V − such that, with Y ± , Z defined as in (3.2), and there exist w ∈ O Z U and a commutative diagram where the horizontal arrows are closed immersions, w ± are defined by the above commutative diagram, j sends p to (0, p) and ι ± induce the isomorphisms of d-critical loci For a formal d-critical simple flip (4.1) and p ∈ U , let V ± be vector spaces as in Definition 4.1. Below we use the notation in Subsection 3.1, e.g. a = dim V + , b = dim V − , n := a − b ≥ 0, the coordinates x, y of V + , V − , etc. Note that (a, b) may depend on the choice of p ∈ U . We assume the following on the diagram (4.1): Assumption 4.2. (i) The diagram (4.1) is a formal d-critical simple flip.
(ii) For any p ∈ U , the formal function w in (4.4) is of the form x i x i y j y j w ii jj ( u) + · · · (4.6) for some w for some a ij k ∈ C. Moreover the bilinear map is injective on each factor, i.e. for any non-zero α ∈ C a and β ∈ C b , the maps (4.10) Proof. For p ∈ U , let us consider the diagram (4.4). The subscheme {d w + = 0} ⊂ Y + U is contained in the closed subscheme of Y + U defined by the equations ii jj ( u) ∂u k + · · · = 0 (4.11) for all 1 ≤ k ≤ g. Note that Then by the assumption on the map (4.8), the subscheme Since the higher order terms in (4.11) have degrees greater than or equal to 2 in y, by the Nakayama lemma the zero locus defined by (4.11) equals y = 0 on Y + U , i.e. the zero section P(V + ) U p ⊂ Y + U . Therefore {d w + = 0} ⊂ Y + U is as in (4.10).
Let g j for 1 ≤ j ≤ b be the defining equations in the RHS of (4.10). Again the property of the map (4.8) implies that the Jacobian matrix ∂g j ∂x i , ∂g j ∂u k 1≤i≤a,1≤j ≤b,1≤k≤g is of maximal rank b at any point in the RHS of (4.10). Therefore {d w + = 0} is smooth of dimension a − 1 + g − b = n + g − 1. By the isomorphism (4.5), M + p is smooth of dimension of n + g − 1 for any p ∈ U , hence M + is smooth of dimension n + g − 1. The claim for M − is similarly proved. (4.12) Proof. Let us take p ∈ U , and vector spaces V ± as in Definition 4.1 with a = dim V + , b = dim V − as before. For each k ≥ 0, let U (k) be the locally closed subset U defined by Then p ∈ U (b) as (π − ) −1 (p) = P(V − ), and the descriptions of {d w ± = 0} in (4.10) and the isomorphisms (4.5) show that It follows from the description of w Therefore the dimension of T U (b) | p is the dimension of the cokernel of ψ in (4.8). By the assumption on the map (4.8), the Hopf lemma (see [Gin,Lemma 2 By applying the above argument for all p ∈ U , we see that for any k ≥ 0 we have Since U is a disjoint union of strata U (k) , the condition (4.12) holds.

SOD under d-critical simple flips
The following is the main result in this section.
Theorem 4.5. Suppose that the diagram (4.1) satisfies Assumption 4.2, so that M ± are smooth of dimension ±n + g − 1 for some n ∈ Z ≥0 by Lemma 4.3. Then: is fully faithful. The proof of Theorem 4.5 is in three steps.
Step 1. For each p ∈ U , we may assume that the formal function (4.6) satisfies w (k) * ( u) = 0 for k ≥ 2. Proof. In the notation of Assumption 4.2(ii), let w ∈ O Z ⊗ O U,p be defined by We set w ± : Y ± U p → A 1 as in the diagram (3.8) for U = U p . Then the argument used for Lemma 4.3 shows that {dw ± = 0} ⊂ Y ± U p are described as in the RHS of (4.10) and the isomorphisms (4.5) give Therefore we may replace w with w and assume that w (k) * ( u) = 0 for k ≥ 2.
Proof. Let M,R be the right adjoint functor of M , and let P ∈ D b (M − × M − ) be the kernel object for the composition functor Then there is a canonical morphism (4.14) corresponding to the adjunction id M − → M,R • M . Let Q be the cone of the morphism (4.14). In order to show that M is fully faithful, it is enough to show that Q = 0. Indeed, if this is the case, then the adjunction id M − → M,R • M is an isomorphism hence M is fully faithful. Note that Q is supported on the fiber product M − × U M − by construction. Since U p → U is faithfully-flat, the vanishing Q = 0 is equivalent to for all p ∈ U . Now by Step 1 and Lemmas 4.3 and 4.4, the diagram satisfies the assumptions in Proposition 3.8. Then Proposition 3.8(i) shows that the morphism (4.14) is an isomorphism after pulling it back by M − p × M − p → M − × M − . Therefore (4.15) holds for any p ∈ U , and Theorem 4.5(i) is proved. The proof of (ii) is similar.
Proof. We first show the semiorthogonality of the RHS of (4.13), i.e. the vanishings where M,R , ϒ i M,R are the right adjoint functors of M , ϒ i M respectively. Again it is enough to check these vanishings formally locally at every p ∈ U , and Proposition 3.8(iii) implies that they indeed hold.
Let E ∈ D b (M + ) be an object in the right orthogonal complement of the RHS of (4.13). Then Proposition 3.8(iii) implies that E = 0 on M + p for any p ∈ U . Therefore E = 0, and the RHS of (4.13) generates the LHS.

SOD for stable pair moduli spaces
In this section, we apply Theorem 4.5 to prove Theorem 1.1, i.e. the existence of certain SOD on moduli spaces of Pandharipande-Thomas stable pairs on CY 3-folds.

Stable pairs and stable sheaves
Let X be a smooth quasi-projective variety. By definition, a stable pair in the sense of Pandharipande-Thomas [PT09] consists of data where F is a pure one-dimensional coherent sheaf on X with compact support, and s is surjective in dimension 1. For β ∈ H 2 (X, Z) and n ∈ Z, the moduli space of stable pairs (F, s) satisfying the condition is denoted by P n (X, β). Here [F ] is the homology class of the fundamental one-cycle associated with F . The moduli space P n (X, β) is a quasi-projective scheme (see [PT09]).
We define the open subscheme P • n (X, β) ⊂ P n (X, β) to consist of stable pairs (F, s) such that the fundamental one-cycle associated with F is irreducible. We denote by U n (X, β) the moduli space of compactly supported one-dimensional Gieseker stable sheaves F on X with respect to a fixed polarization, satisfying the condition (5.1). The moduli space U n (X, β) is a quasi-projective scheme (see [HL97]). We define the open subscheme U • n (X, β) ⊂ U n (X, β) consisting of one-dimensional stable sheaves whose fundamental one-cycles are irreducible. Note that U • n (X, β) is the moduli space of pure one-dimensional sheaves F with irreducible fundamental one-cycles satisfying (5.1). In particular, U • n (X, β) is independent of the choice of a polarization. Below we always assume that U • n (X, β) is a fine moduli space, i.e. it admits a universal sheaf F ∈ Coh(X × U • n (X, β)).
Remark 5.2. The existence of the universal sheaf (5.2) is guaranteed if (D · β, n) is coprime for some divisor D on X. See [HL97, Corollary 4.6.6].

Wall-crossing diagram of stable pair moduli spaces
Suppose that X is a smooth projective CY 3-fold, i.e. dim X = 3, K X = 0.
Let us take β ∈ H 2 (X, Z) and n ∈ Z ≥0 . Then as in [PT10], we have the diagram Here π ± are defined by If furthermore H 1 (O X ) = 0, then the diagram (5.3) gives an example of an analytic (in particular formal) d-critical simple flip (see [Tod,Theorem 6.18]). Here we recall some more details. Let us take a point p ∈ U • n (X, β) corresponding to a pure one-dimensional sheaf F on X. We write U n (X, β) p := Spec O U n (X,β),p , P n (X, β) p := P • n (X, β) × U • n (X,β) U n (X, β) p . We take a collection of objects in D b (X) (5.4) We define vector spaces V + , V − and U as follows: Below we use the notation and conventions of Subsections 3.1 and 3.2, e.g. C * -actions on V ± , the GIT quotients Y ± , Z, coordinates x, y, u on We also take the formal completion Z U of Z U at (0, 0), and define f ± U : Y ± U → Z U as in (4.3). The following result is obtained in [Tod]: Theorem 5.3 ([Tod, Theorem 6.18]). In the above situation, there exist an element w ∈ O Z U ,(0,0) and the commutative diagram Here w ± are defined by the above commutative diagram, the bottom left arrow sends p to (0, 0), and the map j is the composition of the inclusion {dw (0) = 0} ⊂ U with the inclusion U → Z U given by u → (0, u).
Remark 5.4. In [Tod14b, Theorem 6.18], it is stated that we can take w as an analytic function on an analytic open neighborhood of 0 ∈ Z U , and the diagram (5.6) can be extended to analytic neighborhoods of 0 ∈ Z U and p ∈ U • n (X, β). The formal version in Theorem 5.3 is weaker than the analytic version in [Tod14b, Theorem 6.18], but enough for the purpose of this paper.
Let us write the formal function w in Theorem 5.3 as ii jj ( u) + · · · (5.7) for w (k) * ( u) ∈ O U,0 . The function (5.7) is constructed using the minimal cyclic A ∞ -structure on the subcategory of D b (X) generated by E 1 and E 2 (see [Tod,Subsection 5.1]). In particular, the linear term of w (1) ij ( u) is given as follows. Let us consider the triple product given by composition, where the last isomorphism is given by the Serre duality. For be the dual basis of x i , y j , u k respectively. Then using the triple product (5.8), we have given by composition is injective on each factor.
Proof. Recall that E 1 , E 2 are as in (5.4), i.e. E 1 = O X and E 2 = F [−1] for a pure onedimensional sheaf F on X with irreducible fundamental one-cycle. So F can be written as j * E where j : C → X is an irreducible Cohen-Macaulay curve and E is a rank-one torsion free sheaf on C. Therefore the map (5.10) is where ω C is the dualizing sheaf on C. Also H 1 (C, End(E)) ⊂ Ext 1 X (j * E, j * E), and the Serre duality gives the surjection Ext 2 X (j * E, j * E) Hom(End(E), ω C ).
By composing it with (5.11) we obtain the map H 0 (C, E) ⊗ Hom(E, ω C ) → Hom(End(E), ω C ). (5.12) The above bilinear map is given by the natural composition map. Since E and End(E) are torsion free on C, and ω C is also torsion free on C as C is Cohen-Macaulay, the bilinear map (5.12) is injective on each factor. Therefore the lemma holds.
We also have the following lemma: Lemma 5.6. There is a π + -ample line bundle O P (1) on P • n (X, β) such that for any p ∈ U • n (X, β), the isomorphism ι + in the diagram (5.3) satisfies Proof. Let H be a sufficiently ample divisor on X such that for any [F ] ∈ U • n (X, β), the sheaf F (H ) := F ⊗ O X (H ) satisfies H 1 (X, F (H )) = 0, and the natural map F → F (H ) defined by taking the tensor product with O X ⊂ O X (H ) is injective. Such an ample divisor H exists as U • n (X, β) is of finite type. By setting d = H · β, we have the commutative diagram Here the top arrow is given by and the bottom arrow sends a stable sheaf F to F (H ). By the condition H 1 (X, F (H )) = 0, the right arrow is a projective bundle with fiber P(H 0 (X, F (H ))). Indeed, using the universal sheaf (5.2), we have an isomorphism over U • n (X, β) P • n+d (X, β) ∼ = P(p U * (F ⊗ p * X O X (H ))). Here p X , p U are the projections from X × U • n (X, β) to X, U • n (X, β) respectively. By restricting the tautological line bundle on P • n+d (X, β) to P • n (X, β) by the top arrow of (5.13), we obtain the desired O P (1).

SOD for stable pair moduli spaces
We keep the situation of the previous subsections. For the diagram (5.3), let W • be the fiber product W • := P • n (X, β) × U • n (X,β) P • −n (X, β).
The following is the main result in this section: Theorem 5.7. For n ≥ 0 and β ∈ H 2 (X, Z), suppose that U • n (X, β) is fine and nonsingular of dimension g. Then P • ±n (X, β) are also non-singular of dimension ±n + g − 1, and we have the following: , . . . , Im ϒ 0 P , Im P . Proof. We show that the diagram (5.3) satisfies Assumption 4.2. Let us take p ∈ U • n (X, β) corresponding to a pure one-dimensional sheaf F . The assumption that U • n (X, β) is smooth and the bottom left isomorphism in the diagram (5.6) indicate that, for the formal function w written as in (5.7), we may assume that w (0) ( u) = 0. Then (i) of Assumption 4.2 follows from Theorem 5.3, (ii) follows from Lemma 5.5, and (iii) follows from Lemma 5.6. Therefore the theorem follows from Theorem 4.5.
Remark 5.8. When X is a non-compact CY 3-fold, suppose that X has a smooth compactification X such that H i (O X ) = 0 for i = 1, 2. Then the result of Theorem 5.7 also holds without any modification with X replaced by X. This is because for E 1 = O X and E 2 = F [−1] where the support of F is contained in X, we have the perfect pairing by the CY3 condition for X and since H i (O X ) = 0 for i = 1, 2.

Stable pairs on local surfaces
We apply Theorem 5.7 to some local surfaces. Let S be a smooth projective surface satisfying H i (O S ) = 0 for i = 1, 2. We consider the non-compact CY 3-fold X = Tot S (K S ).
We will apply Theorem 5.7 to show the existence of SOD of relative Hilbert schemes of points on the universal curve over a complete linear system. Let us take β ∈ H 2 (S, Z) = H 2 (S, Z) such that −K S · β > 0. By the assumption H i (O S ) = 0 for i = 1, 2, there is a unique L ∈ Pic(S) such that c 1 (L) = β. Let the universal curve. Note that any member C ∈ |L| • has arithmetic genus g = 1 + 1 2 (β 2 + K S · β).
We have the diagram Here π [n] is the π -relative Hilbert scheme of n points, and π J is the π -relative rank-one torsion free sheaf on the fibers of π with Euler characteristic n. Let i : S → X be the zero section. We have the following lemma: and they are non-singular.
and they are non-singular.
Proof. As for (i), the isomorphism C [n+g−1] ∼ = → P • n (S, β) and the smoothness of P • n (S, β) follow by applying the arguments used for [PT10, Propositions B.8, C.2]. The assumption −K S · β > 0 implies that any compactly supported irreducible curve on X with homology class i * β must lie on the zero section S ⊂ X. Therefore we have the settheoretic bijection P • n (S, β) → P • n (X, i * β), and these schemes have the same scheme structures by [KT14,Proposition 3.4].
As for (ii), the smoothness of U • n (S, β) follows from → U • n (X, i * β) follows similarly to (i). By Lemma 5.9, the diagram (5.3) in this case is Again we assume that n ≥ 0 and U • n (S, β) is fine, which is guaranteed if gcd(β·D, n) = 1 for some divisor D on S (see Remark 5.2). Applying Theorem 5.7 to X = Tot S (K S ) and noting Remark 5.8, we have the following: Corollary 5.10. In the above situation, we have the SOD

SOD of symmetric product of curves
Let C be a smooth projective curve over C of genus g. Its k-fold symmetric product C [k] is defined by where the action of the symmetric group S k is by permutation. The variety C [k] is a smooth projective variety of dimension k, and identified with the Hilbert scheme of k-points on C.
Let Pic k (C) be the moduli space of degree k line bundles on C, which is a gdimensional complex torus. Once we fix a point c ∈ C, we have the isomorphism Remark 5.11. For k > 2g − 2, the map (5.16) is a projective bundle. In general, the map (5.16) is a stratified projective bundle, where strata on Pic k (C) are given by Brill-Noether loci. The geometry of Brill-Noether loci is complicated and depends on the complex structure of C, whose study is a classical subject in the study of symmetric products of curves (see [Fla, Section 5], [Kas13, Examples 1.0.7-1.0.10]).
For n ≥ 0, we consider the diagram Here AJ ∨ sends Z ⊂ C to ω C (−Z). Applying Theorem 5.7 and using the isomorphism (5.15), we obtain the following corollary: Corollary 5.12. For each n ≥ 0, we have the SOD Proof. Let X be the non-compact CY 3-fold where L 1 , L 2 are general line bundles of degree g − 1 satisfying L 1 ⊗ L 2 ∼ = ω C . Then the diagram (5.3) in this case coincides with the diagram (5.17). As mentioned in [Tod, Example 9.22, Remark 9.23], Theorem 5.3 applies to the non-compact CY 3-fold X. Therefore the result follows by the same argument as for Theorem 5.7 and isomorphisms (5.15).
For n = 0, the images of AJ and AJ ∨ coincide with the theta divisor which is singular in general, but has only rational singularities [Kem73]. So we have the diagram which gives a (possibly non-isomorphic) resolutions of . Let W be the fiber product of the above diagram. Applying Theorem 5.7 as in the proof of Corollary 5.12 for n = 0, we have the following: Corollary 5.13. We have the autoequivalence (5.18) Below we give some examples related to Corollaries 5.12 and 5.13.
Example 5.14. Suppose that n > g − 1. Then C [−n+g−1] = ∅ and is a projective bundle whose fibers are P n−1 . Then the SOD in Theorem 5.12 is which is nothing other than Orlov's SOD for projective bundles [Orl92].
Example 5.15. Suppose that n = g − 1. Then C [−n+g−1] = Spec C and is a projective bundle outside the point [ω C ] ∈ Pic 2g−2 (C). For the fiber F = AJ −1 ([ω C ]), its structure sheaf O F is exceptional, and the SOD in Theorem 5.12 is Example 5.16. Suppose that n = g − 2. Then C [−n+g−1] = C and is a projective bundle outside AJ ∨ (C) ⊂ Pic 2g−3 (C). In this case, the SOD in Theorem 5.12 is . The SOD in Theorem 5.12 becomes ) .
If C is not hyperelliptic, the above SOD seems to be the blow-up formula of derived categories obtained in [Orl92].
Example 5.19. Suppose that g = 4 and n = 0. Then the birational map is a crepant resolution of which is a divisorial contraction if C is hyperelliptic, and a small resolution which contracts one or two smooth rational curves if C is not hyperelliptic (see [Kas13, Example 1.0.10]). In the latter case, the equivalence (5.18) seems to be the derived equivalence under flops [BO01,Bri02].

Categorification of Kawai-Yoshioka's formula
In this section, we prove Theorem 1.2 as another application of Theorem 4.5. We use Kawai-Yoshioka's diagram [KY00] relating moduli spaces of stable pairs on K3 surfaces to moduli spaces of stable sheaves on them. The key ingredient, which was essentially observed in [Tod12b], is to interpret Kawai-Yoshioka's diagram in terms of a wall-crossing diagram in a CY 3-fold defined by the product of the K3 surface and an elliptic curve.

SOD of relative Hilbert schemes of points
Let S be a smooth projective K3 surface such that for an ample divisor H on S. Let g ∈ Z be defined by H 2 = 2g −2. We have the complete linear system |H | and the universal curve In what follows, we fix n > 0. Let → P g (6.1) be the π -relative Hilbert scheme of n+g −1 points on C. As in Lemma 5.9, the π -relative Hilbert scheme (6.1) is isomorphic to the moduli space of Pandharipande-Thomas stable pair moduli space P n (S, For elements (r i , β i , m i ) ∈ S with i = 1, 2, the Mukai pairing is defined by ((r 1 , β 1 , m 1 ), (r 2 , β 2 , m 2 )) := β 1 β 2 − r 2 m 1 − r 1 m 2 .
For each k ∈ Z ≥0 , we define U k to be the moduli space of H -Gieseker stable sheaves E on S satisfying Here we refer to [HL97] for basics on moduli spaces of stable sheaves and their properties. The moduli space U k is known to be a projective irreducible holomorphic symplectic manifold with Below we assume that U k is a fine moduli space, i.e. there is a universal sheaf on S × U k .
Let P k be the moduli space of pairs where [E] ∈ U k and s is a non-zero morphism. By [KY00, Lemma 5.117], the moduli space P k is a smooth projective variety with We have the diagram (see [KY00, Lemma 5.113]) As an application of Theorem 4.5, we have the following result whose proof will be given in Subsection 6.5: Theorem 6.2. For k ≥ 0, we have the following SOD: Applying the above theorem from k = 0 to k = N, and noting that where the latter is due to (6.2), we have the following result: Corollary 6.3. For n > 0, we have the SOD where each A k has the SOD Remark 6.4. As mentioned in Subsection 1.3, the SOD (6.5) recovers Kawai-Yoshioka's formula (1.8), P n,g = (−1) n−1 N k=0 (n + 2k)e(U k ).
The formula (6.6) is the KKV formula mentioned above.

Tilting on S × C
Let S be a K3 surface as in the previous subsection. We fix a smooth elliptic curve C and consider a compact CY 3-fold X := S × C with projections p S , p C , In what follows, we will interpret the diagram (6.4) in terms of wall-crossing diagrams in D b (X). We define the triangulated subcategory consisting of objects whose cohomology is supported on fibers of p C . The triangulated category D 0 is the derived category of the abelian subcategory consisting of sheaves supported on fibers of p C . For c ∈ C, let The category Coh 0 (X) is the extension closure of objects of the form i c * F for some c ∈ C and F ∈ Coh(S).
for v i (F ) ∈ H 2i (S, Z). We define the following slope function on Coh 0 (X): Here F ∈ Coh 0 (X) and we set µ(F ) = ∞ if v 0 (F ) = 0. The slope function defines µstability on Coh 0 (X) in the usual way: a non-zero object F ∈ Coh 0 (X) is µ-(semi)stable if for any non-zero subsheaf F F , Let T , F be the subcategories of Coh 0 (X) defined by Here − ex means extension closure. The pair of subcategories (T , F) is a torsion pair on Coh 0 (X). We have the associated tilting For t ∈ R >0 , let be the group homomorphism defined by Then the pair is a Bridgeland stability condition on D 0 (see [Tod12b,Lemma 3.3]). In particular it defines Z t -(semi)stable objects: a non-zero object E ∈ B is Z t -(semi)stable if for any non-zero subobject 0 = E E in B, we have the inequality in (0, π]: arg Z t (E ) < (≤) arg Z t (E).
We have the following lemma: for some c ∈ C and some H -Gieseker stable sheaf F ∈ Coh(S).
Proof. The lemma is well-known (for example see [Bay18, proof of Lemma 6.1]). Let C ⊂ B be the subcategory defined by the object E is Z t -stable if and only if Hom(C, E) = 0. First suppose that E is Z t -stable, so Hom(C, E) = 0. Then H 0 (E) = 0, and H 1 (E) is either a µ-stable two-dimensional sheaf or a one-dimensional H -Gieseker stable sheaf. It follows that E ∼ = i c * F [−1] for some c ∈ C, where F is a µ-stable sheaf on S or an H -Gieseker stable one-dimensional sheaf on S. In the former case, since the Mukai vector of F is primitive, its µ-stability is equivalent to its H -Gieseker stability. Conversely, if E ∼ = i c * F [−1] as in the statement, then it is obvious that Hom(C, E) = 0. Therefore the lemma is proved.
We define the following subcategory of D b (X): The category A is the heart of a bounded t-structure on the triangulated subcategory of D b (X) generated by p * C Pic(C) and objects in D 0 (see [Tod12b, Proposition 2.9]). In particular, A is an abelian category. Note that E ∈ A satisfies rank(E) = 0 if and only if E ∈ B. We will use the following property of A: Lemma 6.6. For any object E ∈ A, there is an exact sequence in A 0 → E → E → E → 0 (6.10) such that E ∈ B and E ∈ p * C Pic(C) ex .
Proof. This is proved in [Tod12b,Lemma 7.5] in the case of S × P 1 , and the same argument works here. For simplicity, we prove the lemma when E fits into a non-split extension in A for L ∈ Pic(C), [F ] ∈ U k , and c ∈ C. The full details are in [Tod12b, Lemma 7.5]. Let ξ be the extension class of (6.11). Then since i ! c p * Therefore ξ gives rise to the non-trivial extension of sheaves on S It is easy to see that F is H -Gieseker stable so [F ] ∈ U k+1 . We have the commutative diagram E Here horizontal and vertical sequences are distinguished triangles. By the above diagram, we obtain the exact sequence in A The above exact sequence is the desired sequence (6.10).

Weak stability conditions on A
Let A be the abelian category given in (6.9). For t ∈ R >0 and E ∈ A, we define µ t (E) ∈ R ∪ {∞} by Here if rank(E) = 0, then E ∈ B and Z t (E) ∈ C is given in (6.8). The following stability condition on A appeared in [Tod12b] in the framework of weak stability conditions: Definition 6.7. A non-zero object E ∈ A is µ t -(semi)stable if for any exact sequence 0 → E → E → E → 0 in A with non-zero E , E , we have µ t (E ) < (≤) µ t (E ).
Below we fix n ∈ Z >0 and characterize µ t -semistable objects E ∈ A satisfying ch(E) = (1, 0, −i c * [H ], −n) ∈ H 0 (X) ⊕ H 2 (X) ⊕ H 4 (X) ⊕ H 6 (X). (6.12) Proposition 6.8. For k ∈ Z >0 , suppose that t ∈ R >0 satisfies t k < t < t k−1 , t k := n + k (g − 1)k , t 0 := ∞. (6.13) Then an object E ∈ A satisfying (6.12) is µ t -semistable if and only if E is isomorphic to a two-term complex for some c ∈ C, [F ] ∈ U k , L ∈ Pic k (C) and s is a non-zero morphism. Here Pic k (C) ⊂ Pic(C) is the subset of degree k line bundles, and p * C L is located in degree zero. Moreover in this case, E is µ t -stable. Proof.
Step 1. The 'only if' direction. Let us take t ∈ (t k , t k−1 ) and a µ t -semistable object E ∈ A satisfying (6.12). By [Tod12b,Lemma 7.5], there is an exact sequence in A for some A ∈ B, L ∈ Pic r (C) for some r ∈ Z. The condition (6.12) imply v(A) = −v r . The above exact sequence and the µ t -semistability of E yield Z t (A) = −r − n + r(g − 1)t 2 ≥ 0.
As t < t k−1 and n > 0, the above inequality yields r ≥ k > 0. The µ t -semistability of E implies that Hom(C, E) = 0, where C ⊂ B is defined in the proof of Lemma 6.5. By the exact sequence (6.15) we have Hom(C, A) = 0, and Lemma 6.5 shows that A ∼ = i c * F [−1] for some c ∈ C and [F ] ∈ U r . Therefore E is isomorphic to a two-term complex where s must be non-zero due to the µ t -semistability of E. Let us show that r = k. By taking the cohomology of E, we obtain the exact sequence in A where G is the cokernel of s . Since v(G) = v r−1 , the µ t -semistability of E yields Z t (G[−1]) = −r + 1 − n + (r − 1)(g − 1)t 2 ≤ 0. (6.17) As t > t k , the above inequality implies that k ≥ r. As we already proved r ≥ k, it follows that r = k. Therefore we have proved the 'only if' direction of the proposition.
Step 2. The 'if' direction. Conversely, let us take an object E ∈ A of the form (6.14). We show that E is µ t -stable if t ∈ (t k , t k−1 ). Let us take an exact sequence in A such that A, B are non-zero. We will show that Since rank(E) = 1, we have (rank(A), rank(B)) = (0, 1) or (1, 0). We will show (6.19) in each case. Therefore Hom(A, p * C L) = 0. By the exact sequences (6.18) and (6.20), we have an injection A → i c * F [−1] in B. By Lemma 6.5, the object i c * F [−1] is Z t -stable in B. Therefore where µ t (−v k ) < 0 is due to t k < t. Hence (6.19) holds.
Next suppose that rank  Similarly to (6.16), we have the exact sequence in A where G is the cokernel of s in (6.14). By the exact sequences (6.18), (6.22), and the vanishing (6.21), there is a surjection G[−1] B in B. By Lemma 6.5, the object G[−1] ∈ B is Z t -stable. Therefore where µ t (−v k−1 ) > 0 due to t < t k−1 . Hence (6.19) holds.
When t lies on a wall, the µ t -semistable objects are characterized by the following lemma.
Lemma 6.9. An object E ∈ A satisfying (6.12) is µ t k -semistable if and only if E is S-equivalent to a µ t k -polystable object of the form for some c ∈ C, [F ] ∈ U k and L ∈ Pic k (C).
Proof. The 'if' direction is obvious as both E 1 , E 2 are µ t k -semistable with µ t k (E 1 ) = µ t k (E 2 ) = 0. The 'only if' direction is proved similarly to Step 1 in the proof of Proposition 6.8. If we apply the proof above for t = t k , the only point to notice is that, just after (6.17) we only have k ≥ r − 1 as we take t = t k . Therefore either r = k or r = k + 1. In the latter case, the exact sequence (6.16) shows that E is S-equivalent to an object of the form (6.23).

Moduli stacks of semistable objects
Let M be the 2-functor M : Sch/C → Groupoid sending a C-scheme S to the groupoid of relatively perfect objects E ∈ D b (X × S) such that for each point s ∈ S, the object E s := Li * s E for the inclusion i s : X × {s} → X × S satisfies Ext <0 (E s , E s ) = 0. The stack M is known to be an Artin stack locally of finite type [Lie06]. For a fixed n ∈ Z ≥0 and t ∈ R >0 , we define the substack to be the stack whose S-valued points consist of E ∈ M(S) such that for each s ∈ S, the object E s is a µ t -semistable object in A satisfying (6.12). Using Proposition 6.8 and Lemma 6.9, we show the following: Proposition 6.10. The stack M t is an Artin stack of finite type such that (6.24) is an open immersion. Moreover if t ∈ (t k , t k−1 ), the stack M t is smooth.
Proof. By [Tod12b, Lemma 4.13(ii)], M t ⊂ M is constructible. Therefore for the first statement, it is enough to show that M t ⊂ M is open in the analytic topology. By Lemma 6.9, for t = t k an object corresponding to a C-valued point of M t is a small deformation of an object of the form (6.23). Set Then the analytic local deformation space of E 1 ⊕ E 2 is given by the critical locus of some analytic function w defined in an analytic neighborhood of 0 ∈ V + × V − × U . Similarly to the case of stable pairs in (5.7), the function w is invariant under the conjugate Aut(E 1 ⊕ E 2 ) = (C * ) 2 -action on V + × V − × U , so it is of the form x i x i y j y j w ii jj ( u) + · · · where x, y and u are coordinates of V + , V − and U respectively. As in [Tod, Subsection 5.1], the function w is constructed using the minimal A ∞ -structure on D b (X). By the construction in loc. cit., the function w (0) ( u) can be written as such that the critical locus of w (0) i ( u i ) in Ext 1 X (E i , E i ) gives the local deformation space of E i . Since the deformation space of E i is smooth, we may assume that w (0) ( u) = 0.
Similarly to Subsection 5.2, the function w (1) ij ( u) can be written as in (4.7) such that the coefficients of the linear terms a ij k are determined by the triple product given by composition and the Serre duality. Then by Lemma 6.11 below, the coefficients a ij k satisfy the condition in Assumption 4.2(ii). Therefore Lemma 4.3 shows that (6.25) This implies that any small deformation E of E 1 ⊕ E 2 fits into one of the following exact sequences in A: where (F , L , c ) is a small deformation of (F, L, c), so that [F ] ∈ U k and L ∈ Pic k (C). Therefore E is µ t k -semistable, and M t k ⊂ M is open. Suppose that t ∈ (t k , t k−1 ), and take an object E as in (6.14) which corresponds to a C-valued point of M t . Then E is isomorphic to a small deformation of the object E 1 ⊕E 2 as above, which lies in the LHS of (6.25). Then any small deformation E of E fits into a non-split sequence (6.26). Therefore E is again µ t -semistable by Proposition 6.8, and M t ⊂ M is open. Moreover the argument used for Lemma 4.3 implies that the LHS of (6.25) is smooth, hence M t is smooth.
We have used the following lemma, which is an analogue of Lemma 5.5.
Lemma 6.11. For the objects E 1 , E 2 in (6.23), the composition map is injective.
Proof. Note that We also have the surjection Ext 2 X (E 2 , E 2 ) = Ext 2 X (i c * F, i c * F ) Ext 1 S (F, F ) which is Serre dual to the natural map Ext 1 S (F, F ) → Ext 1 X (i c * F, i c * F ). By composing it with (6.11), we obtain the composition map Then it is well-known that U is a µ-stable sheaf (see [Yos99,Tod14a]). Applying Hom(−, F ) to the above exact sequence, we obtain the exact sequence 0 → C → Hom(U, F ) → H 0 (S, F ) ⊗ Ext 1 S (F, O S ) → Ext 1 S (F, F ). (6.30) Since (6.29) is the universal extension, applying Hom(−, O S ) to (6.29) we obtain Ext 1 S (U, O S ) = 0. Then applying Hom(U, −) to (6.29) and using the stability of U, we get Hom(U, F ) = Hom(U, U) = C.
Therefore by the exact sequence (6.30), we see that (6.28) is injective.
For t ∈ R >0 , let M t → M t (6.31) be the good moduli space for the stack M t , which exists by [AHLH]. The good moduli space M t is an algebraic space of finite type which parametrizes µ t -polystable objects in A satisfying (6.12), i.e. direct sums of µ t -stable objects with µ t (−) = 0. By Proposition 6.8, the moduli space M t is constant if t ∈ (t k , t k−1 ) for some k. So we can write M k := M t , t ∈ (t k , t k−1 ).
By Proposition 6.8, M k consists of µ t -stable objects for t ∈ (t k , t k−1 ) and is also smooth by Proposition 6.10.
Recall that J C := Pic 0 (C) is defined to be the moduli space of degree zero line bundles on C, which is isomorphic to C itself as C is an elliptic curve. In the k = 1 case, we can describe M 1 by the stable pair moduli space: Proof. First we need to show that the map (6.33) is well-defined, i.e. the object p * C L ⊗ D(O X → i c * F ) ∈ A (6.34) on the RHS of (6.33) corresponds to a point in M 1 . By [Tod,Remark 9.8], an object in E ∈ A is of the form (6.34) if and only if E fits into an exact sequence in A where F is a pure one-dimensional sheaf on S such that Hom(T [−1], E) = 0 for any one-dimensional sheaf T on X. Moreover in this case we have i c * F = Ext 2 X (i c * F , O X ). The proof of Lemma 6.6 shows that E fits into an exact sequence 0 → i c * F [−1] → E → p * C L(c) → 0 (6.36) for [F ] ∈ U 1 . Therefore E is isomorphic to (p * C L(c) s → i c * F ) for a non-zero s, hence gives a point in M 1 by Proposition 6.8.
Conversely, by Proposition 6.8, any object [E] ∈ M 1 fits into an exact sequence of the form (6.36). By taking the cohomology of E, it also fits into a non-split exact sequence of the form (6.35). On the other hand, by the exact sequence (6.36) we see that Hom(T [−1], E) = 0 for any one-dimensional sheaf T on X. Therefore E is of the form (6.34), and the map (6.33) is bijective on closed points. Since both sides of (6.32) are smooth, it is an isomorphism.
In general for k > 0, we can describe M k in terms of pair moduli spaces P k on S: Lemma 6.13. For k > 0, we have an isomorphism Proof. The map (6.38) is a morphism of smooth algebraic spaces which is bijective on closed points by Proposition 6.8. Hence it is an isomorphism.
We also set By the open immersions M t k +ε ⊂ M t k ⊃ M t k −ε for 0 < ε 1, and noting that M k is smooth, we have the induced morphisms (6.40) (ii) Under the isomorphisms (6.37), (6.40), the diagram (6.39) is identified with the diagram (6.4) × id C×J C .
Proof. (i) By Lemma 6.9, a point in M t k corresponds to a µ t k -polystable object of the form (6.23). Therefore we have the morphism The morphism (6.41) is a bijection on closed points. Moreover the proof of [Tod, Lemma 9.21] shows that (6.41) is a closed immersion. Therefore we have the isomorphism (6.40) by taking the reduced parts of (6.41).
(ii) The statement π + k = π k × id C×J C is obvious from the descriptions of the maps (6.38), (6.41). As for π − k , let us take a point ((O S s → F ), c, L ) ∈ P k+1 × (C × J C ).
Under the map (6.37), it corresponds to a point in M k+1 of the form E = (p * C L → i c * F ) ∈ A, L = O C ((k + 1)[c]) ⊗ L ∈ Pic k+1 (C).
By taking the cohomology of E , we have an exact sequence in A where G is the cokernel of s . Then the map π − k is given by As L (−c) = O C (k[c]) ⊗ L , it comes from (G, c, L ) ∈ U k × (C × J C ) under the map (6.41). Therefore the identity π − k = π − k × id C×J C also holds.
Proposition 6.15. The diagram (6.39) satisfies Assumption 4.2 by setting M + = M k , M − = M k+1 , U = U k , π ± = π ± k . Proof. Note that the diagram (6.39) is a wall-crossing diagram in the CY 3-fold X. Together with the fact that a point in U k corresponds to a µ t -polystable object (6.23), it is a d-critical simple flip by [Tod,Example 6.3] (see also [Tod, proof of Theorem 9.22]). Therefore Assumption 4.2(i) holds. By Lemma 6.11 Assumption 4.2(ii) also holds, and Assumption 4.2(iii) holds by the same argument as used for Lemma 5.6. 6.5. Proof of Theorem 6.2 We first prove Theorem 6.2 for k > 0. Let W k be the fiber product of the diagram (6.39), and O M k (1) be a π + k -ample line bundle on M k satisfying Assumption 4.2(iii) for the diagram (6.39). By Theorem 4.5 and Proposition 6.15, we have the fully faithful functors Then by Lemma 6.14(ii), the functors (6.42) are linear over C × J C under the isomorphisms (6.37), (6.40), so Theorem 6.2 for k > 0 follows by restricting the SOD (6.42) to by taking the product of the diagram (5.3) with J C via the isomorphism (6.32). The diagram (6.43) satisfies Assumption 4.2 as in the proof of Theorem 5.7. On the other hand, similarly to Lemma 6.13 and Lemma 6.14, we have isomorphisms Under the above isomorphisms, the diagram (6.43) is identified with the diagram (6.4) × id C×J C for k = 0. Therefore the argument for k > 0 also implies Theorem 6.2 for k = 0.