Intrinsic Stabilizer Reduction and Generalized Donaldson-Thomas Invariants

Let $\sigma$ be a stability condition on the bounded derived category $D^b({\mathop{\rm Coh}\nolimits} W)$ of a Calabi-Yau threefold $W$ and $\mathcal{M}$ a moduli stack parametrizing $\sigma$-semistable objects of fixed topological type. We define generalized Donaldson-Thomas invariants which act as virtual counts of objects in $\mathcal{M}$, fully generalizing the approach introduced by Kiem, Li and the author in the case of semistable sheaves. We construct an associated proper Deligne-Mumford stack $\widetilde{\mathcal{M}}^{\mathbb{C}^\ast}$, called the $\mathbb{C}^\ast$-rigidified intrinsic stabilizer reduction of $\mathcal{M}$, with an induced semi-perfect obstruction theory of virtual dimension zero, and define the generalized Donaldson-Thomas invariant via Kirwan blowups to be the degree of the associated virtual cycle $[\widetilde{\mathcal{M}}^{\mathbb{C}^\ast}]^{\mathrm{vir}} \in A_0 (\widetilde{\mathcal{M}}^{\mathbb{C}^\ast})$. This stays invariant under deformations of the complex structure of $W$. Examples of applications include Bridgeland stability, polynomial stability, Gieseker and slope stability.

1. Introduction 1.1. Brief historical background. Donaldson-Thomas (abbreviated as DT from now on) invariants constitute one of the main approaches for curve counting on Calabi-Yau threefolds. They naturally appear in many enumerative problems of interest in algebraic geometry and string theory and are conjecturally equivalent with other counting invariants, such as Gromov-Witten invariants, Stable Pair invariants [PT09] and Gopakumar-Vafa invariants [MT18]. These relations have now been proven in many cases (for example, in [MNOP06a,MNOP06b,Tod10]).
Let W be a smooth, projective Calabi-Yau threefold and γ ∈ H * (W, Q). Classical DT theory was introduced in [Tho00] in order to obtain virtual counts of stable sheaves on W of Chern character γ. More precisely, Thomas considered the (coarse) moduli space M := M ss L (γ) parameterizing Gieseker semistable sheaves on W of positive rank, fixed determinant L and Chern character γ. Assuming that every semistable sheaf is stable, M is proper and admits a perfect obstruction theory in the sense of Li-Tian [LT98]  Another important feature of DT invariants is their motivic nature. Behrend [Beh09] showed that the obstruction theory of M is symmetric in a certain sense and established an equality where ν M : M → Z is a canonical constructible function on M and the right-hand side a weighted Euler characteristic.
However, when stability and semistability of sheaves do not coincide, the above methods do not suffice and new techniques are required in order to define generalized DT invariants counting semistable sheaves. A main obstacle is that strictly semistable sheaves can have more automorphisms beyond C * -scaling and, as a result, the stacks M ss (γ) parameterizing semistable sheaves are generally Artin and no longer Deligne-Mumford (after rigidifying C * -scaling, cf. Subsection 6.3). This is necessary for the standard machinery of perfect obstruction theory and virtual cycles to apply.
In [JS12], Joyce and Song constructed generalized DT invariants of moduli stacks M ss (γ) taking advantage of the above motivic behaviour and using motivic Hall algebras to obtain a generalization of the right-hand side of ( †). Their DT invariant is easy to work with and amenable to computation. However, the proof of its deformation invariance is indirect and proceeds via wall-crossing to a stable pairs theory where semistability and stability coincide and thus a virtual cycle exists. Kontsevich and Soibelman [KS10] have also defined motivic generalized DT invariants using similar ideas. We also mention the related work of Behrend and Ronagh [BR19,BR16].
In [KLS17], the authors develop a new direct approach towards defining generalized DT invariants of such a moduli stack M ss (γ). Their method The author was partially supported by a Stanford Graduate Fellowship, an Alexander S. Onassis Foundation Graduate Scholarship and an A.G. Leventis Foundation Grant during the course of this work. adapts Kirwan's partial desingularization procedure [Kir85] to define an invariant as the degree of a zero-dimensional virtual cycle in an associated Deligne-Mumford stack M. The constructed invariant is called the generalized DT invariant via Kirwan blowups (or DTK invariant for short) and is a direct generalization of the left-hand side of ( †). By the usual properties of virtual cycles, they establish the deformation invariance of DTK invariants.
We make a final comment on the need for orientation data in the above approaches. Since motivic DT invariants are more refined, Kontsevich and Soibelman do need to assume the existence of such data to define their invariants. This assumption has been recently proved to hold in [JU21]. Orientation data are not necessary for the definition of numerical generalized DT invariants given in [JS12] and [KLS17]. This will be the case for the DTK invariant defined in this paper as well.
1.2. Statement of results. This paper serves as a sequel to [KLS17]. We generalize the construction of DTK invariants to the case of moduli stacks M := M σ−ss (γ) parametrizing σ-semistable objects of Chern character γ in the derived category D b (Coh W ) of coherent sheaves on W , where σ is one of the following stability conditions: (1) A Bridgeland stability condition [Bri07], as considered by Piyaratne-Toda [PT19] and [Li19].
Our main results are summarized in the following theorem, giving a definition of DTK invariants counting σ-semistable complexes.
Theorem. Let W be a smooth, projective Calabi-Yau threefold, σ a stability condition on D b (Coh W ) as in Definition 6.2 and M := M σ−ss (γ) be the (C * -rigidified) moduli stack parametrizing σ-semistable complexes of Chern character γ. Then there exist: (2) [Theorem 5.18, Theorem-Definition 6.10] A natural semi-perfect obstruction theory of virtual dimension zero on M C * , which extends the symmetric obstruction theory of the Deligne-Mumford stack M σ−s (γ).
We thus have a virtual fundamental cycle and the generalized Donaldson-Thomas invariant via Kirwan blowups of M is defined as DTK(M σ−ss (γ)) := deg [ M C * ] vir ∈ Q. By Theorem 6.14, this is invariant under deforming the complex structure of W .
1.3. Brief review of the case of sheaves and sketch of construction. We first give a brief account of the results of [KLS17] and then explain the necessary adjustments in order to generalize their approach.
Let M be an Artin stack parametrizing Gieseker semistable sheaves of a fixed Chern character on a smooth, projective Calabi-Yau threefold. By construction (see [HL10]), M is obtained by Geometric Invariant Theory (GIT) (see [MFK94]), meaning that it is a quotient stack of the form where G is a reductive group acting on a projective space P N via a homomorphism G → GL(N + 1, C) and X is an invariant closed subscheme of the GIT semistable locus X ⊆ (P N ) ss .
In order to define an invariant out of M, the first step in [KLS17] is the construction of a proper DM quotient stack M := [ X/G] over M, called the intrinsic stabilizer reduction of M. This is produced by an iterative blowup procedure, which successively resolves the loci of closed G-orbits in X with stabilizer groups of maximum dimension. This procedure is an adaptation of Kirwan partial desingularization construction [Kir85] to singular GIT quotient stacks, using the notion of intrinsic blowup introduced in [KL13] and suitably generalized in [KLS17].
The second step is to endow M with a semi-perfect obstruction theory [CL11] of virtual dimension zero. This is a usual perfect obstruction theory in the sense of Behrend-Fantechi [BF97] defined on anétale cover of M together with compatibility data which play the role of suitable descent data. It is at this step where the existence of a virtual structure on M plays a crucial role. More precisely, by [PTVV13], M is the truncation of a (−1)-shifted symplectic derived Artin stack and it then follows that M is a d-critical stack [Joy15]. In particular, applying Luna'sétale slice theorem [Dré04] and using the theory of d-critical loci developed in [Joy15], we can obtain an explicit local description of M: for every closed point x ∈ M with (reductive) stabilizer H, there exists a smooth affine scheme V with an H-action, an invariant function f : V → A 1 and anétale morphism critical locus of f . Any two such local presentations can be compared as they give the same d-critical structure on M in the sense of [Joy15].
To obtain the semi-perfect obstruction theory of M, consider the following H-equivariant 4-term complex on U : Around u ∈ U with finite stabilizer group, this complex is quasi-isomorphic to a 2-term complex which gives a symmetric perfect obstruction theory of [U/H] and thus of M near u.
The intrinsic stabilizer reduction algorithm produces lifts of theétale morphisms (1.2) to give anétale cover where T = {ω S = 0} ⊆ S for S a smooth affine H-scheme and ω S ∈ H 0 (S, F S ) an invariant section of an H-equivariant vector bundle F S on S. Moreover, there exists an effective invariant divisor D S such that (1.3) lifts to a 4-term complex whose first arrow is injective with locally free cokernel and last arrow is surjective. Therefore, (1.5) is quasi-isomorphic to a 2-term complex, whose dual can be shown to be a perfect obstruction theory on [T /H].
In [KLS17], it is shown that these perfect obstruction theories satisfy the axioms of a semi-perfect obstruction theory on M of virtual dimension zero.
Since, as usual, semi-perfect obstruction theories produce virtual cycles, M admits a virtual cycle [ M] vir ∈ A 0 ( M) whose degree defines the DTK invariant counting Gieseker semistable sheaves parametrized by M. A relative version of the above construction, using derived symplectic geometry, implies its deformation invariance.
In the present paper, we work with moduli stacks M of semistable perfect complexes. These are truncations of (−1)-shifted symplectic derived Artin stacks as before, however they are not global quotient stacks obtained by GIT of the form (1.1).
Our main contribution is to remove this assumption. To do this, we use several recent major technical results on theétale local structure of stacks [AHR19, AHR20], moduli spaces of objects in abelian categories [AHH18] and stability conditions in families [BLM + 21].
We generalize the requirement of a GIT presentation to the condition that M admit a good moduli space morphism [Alp13]. We show that such stacks (under additional reasonable assumptions) admit an intrinsic stabilizer reduction M, which is a Deligne-Mumford stack and of independent interest in its own right (see Subsection 1.4 below).
We then proceed to show that if in addition M is the truncation of a (−1)shifted symplectic derived Artin stack, then the above statements about local models of M and their comparison data carry over, using the main structural result of [AHR20] in place of Luna'sétale slice theorem. This allows us to define a semi-perfect obstruction theory on M and the associated DTK invariant.
Our approach thus works for a wide class of Artin stacks, which we call stacks of DT type (see Definition 3.8). Based on our discussion here, their two main characteristics can be summarized as the existence of a good moduli space and a derived enhancement that is (−1)-shifted symplectic.
Moduli stacks of semistable complexes fit into this context. This is the case by the recent results of [AHH18], which combine the theory of good moduli spaces with the notion of Θ-reductivity introduced in [Hal14], and apply to the stacks considered in the present paper.
Finally, the deformation invariance of the DTK invariant follows from the compatibility of the intrinsic stabilizer reduction with base change and the recent results on stability conditions in [BLM + 21].
1.4. Relation to derived algebraic geometry. At the time of writing of this paper, intrinsic blowups and the intrinsic stabilizer reduction procedure were expected to be the classical shadow of a corresponding construction in derived algebraic geometry.
Subsequently, derived blowups were first defined in full generality in the work of Hekking [Hek21] and this expectation has recently been materialized by the results of [HRS22], also using some of our results. Namely, the intrinsic blowup notion used in the present paper coincides with the classical truncation of an appropriate derived blowup of a derived scheme with a G-action along its derived fixed locus by the group G. Therefore, the intrinsic stabilizer reduction construction is indeed the classical truncation of a derived version of the Kirwan partial desingularization algorithm in GIT [Kir85] and its generalization to stacks with good moduli spaces by Edidin-Rydh [ER21], called derived stabilizer reduction. Moreover, the semi-perfect obstruction theory constructed here is closely related to the derived cotangent complex of the stabilizer reduction of a (−1)-shifted symplectic derived stack.
1.5. Layout of the paper. In §2 we review background material on dcritical loci and (−1)-shifted symplectic structures. In §3, we define the notion of stacks of DT type and establish their main properties that will be used throughout. §4 reviews the intrinsic stabilizer reduction procedure for GIT quotient stacks and generalizes it to stacks with good moduli spaces. In §5, we explain how to construct a semi-perfect obstruction theory on stacks of DT type by following the arguments of [KLS17]. Finally, in §6, we construct DTK invariants of semistable complexes by combining the above with the results of [AHH18] and G m -rigidification for Artin stacks and discuss their deformation invariance using recent work of [BLM + 21].
1.6. Acknowledgements. The author would like to thank his advisor Jun Li for introducing him to the subject, his constant encouragement and many enlightening discussions during the course of the completion of this work. He also benefitted greatly by conversations with Jarod Alper, Jack Hall, Daniel Halpern-Leistner, Young-Hoon Kiem, Alex Perry, David Rydh and Ravi Vakil. 1.7. Notation and conventions. Here are the various notations and other conventions that we use throughout the paper: -All schemes and stacks are defined over the field of complex numbers C or a smooth C-scheme C, unless stated otherwise. For this reason, reductive group schemes will be linearly reductive automatically.
-M typically denotes an Artin stack, of finite type, with affine stabilizers and separated diagonal unless stated otherwise. -W denotes a smooth, projective Calabi-Yau threefold over C and D b (Coh W ) its bounded derived category of coherent sheaves. -C denotes a smooth quasi-projective scheme over C.
-G, H denote complex reductive groups. Usually, H will be a subgroup of G. T denotes the torus C * . -If x ∈ M, G x denotes the automorphism group/stabilizer of x. We will only consider stabilizers of closed points x ∈ M. These will be reductive for most stacks of interest in this paper. -If U → V is a closed embedding, I U ⊆V or I U (when V is clear from context) denotes the ideal sheaf of U in V . -For a morphism ρ : U → V and a sheaf E on V , we systematically use E| U to denote ρ * E, suppressing the pullback from the notation. -If V is a G-scheme, V G is used to denote the fixed point locus of G in V . -If U is a scheme with a G-action, then U is used to denote the Kirwan blowup of U with respect to G. U denotes the intrinsic stabilizer reduction of U . -The abbreviations DT, DM, GIT, whenever used, stand for Donaldson-Thomas, Deligne-Mumford and Geometric Invariant Theory respectively.

D-Critical Loci and (−1)-Shifted Symplectic Derived Stacks
This section collects background material and terminology that will be used throughout the rest of the paper.
We first briefly recall Joyce's theory of d-critical loci, as developed in [Joy15], and establish some notation, and then proceed to quickly review shifted symplectic structures on derived stacks.
2.1. d-critical schemes. We begin by defining the notion of a d-critical chart.
Definition 2.1. (d-critical chart) A d-critical chart for a scheme X is the data of (U, V, f, i) such that: Joyce defines a canonical sheaf S X of C-vector spaces with the property that for any Zariski open U ⊆ X and an embedding U → V into a smooth scheme V with ideal I, S X fits into an exact sequence gives a section of S X | U .
Definition 2.2. (d-critical scheme) A d-critical structure on a scheme X is a section s ∈ Γ(X, S X ) such that X admits a cover by d-critical charts (U, V, f, i) and s| U is given by f + I 2 as above on each such chart. We refer to the pair (X, s) as a d-critical scheme.
2.2. Equivariant d-critical loci. For our purposes, we need equivariant analogues of the results of Subsection 2.1. The theory works in parallel as before (cf. [Joy15, Section 2.6]).
Definition 2.3. (Good action) Let G be an algebraic group acting on a scheme X. We say that the action is good if X has a cover {U α } α∈A where every U α ⊆ X is a G-invariant Zariski open affine subscheme.
Remark 2.4. If X is affine, then trivially every action of G on M is good.
Suppose now that a scheme X admits a good action by a reductive group G.  Remark 2.6. If G is a torus (C * ) k , then Proposition 2.5 is true without the assumption that x is a fixed point of G.
Remark 2.7. One may replace Zariski open morphisms byétale morphisms without any difference to the essence of the theory. Another option is to work in the complex analytic topology.
2.3. d-critical Artin stacks. The theory of d-critical loci extends naturally to Artin stacks. For more details, we point the interested reader to Section 2.8 of [Joy15]. We mention the following definition and basic properties which we will need in the form of remarks.  If M ′ → M is a smooth morphism of Artin stacks and M is d-critical, then one may pull back the d-critical structure on M making M ′ d-critical as well.
2.4. Shifted symplectic structures. Let cdga ≤0 C be the category of nonpositively graded commutative differential graded C-algebras.
There is a spectrum functor Spec : cdga ≤0 C → dSt C to the category of derived stacks (see [ [Toë09] or [TV04] for its definition).
Shifted symplectic structures on derived Artin stacks were introduced by Pantev-Toën-Vaquié-Vezzosi in [PTVV13]. The definition is given in the affine case first, and then generalized by showing the local notion satisfies smooth descent. We recall the local definition: let us set where the differential d is induced by the differential of the algebra A. For a fixed k ∈ Z, define a k-shifted p-form on M to be an element ω 0 ∈ (Λ p L M ) k such that dω 0 = 0. To define the notion of closedness, consider the de Rham differential d dR : in L qcoh (M ), and we say that ω 0 is non-degenerate if this morphism is an isomorphism in L qcoh (M ).
When k = −1, it is shown in [BBBBJ15] that if M is the classical truncation of a (−1)-shifted symplectic derived Artin stack, then M admits an induced d-critical structure. This defines a 'truncation functor' τ from the ∞-category of (−1)-shifted symplectic derived Artin stacks to the 2-category of d-critical Artin stacks.

Stacks of DT Type
In this section, we first give an account of results regarding theétale local structure of Artin stacks and the theory of good moduli spaces. We then proceed to define stacks of DT type, develop standard local models for them and describe how to compare these models.
3.1. Local structure of Artin stacks. The following theorem is anétale slice theorem for stacks, which generalizes Luna'sétale slice theorem [Dré04]. It states that Artin stacks areétale locally quotient stacks.
Theorem 3.1. [AHR20, Theorem 1.2] Let M be a quasi-separated Artin stack, locally of finite type over C with affine stabilizers. Let x ∈ M and H ⊆ G x a maximal reductive subgroup of the stabilizer of x. Then there exists an affine scheme U with an action of H, a point u ∈ U fixed by H, and a smooth morphism which maps u → x and induces the inclusion H → G x of stabilizers at u. Moreover, if G x is reductive, the morphism isétale. If M has affine diagonal, then Φ x can be taken to be affine.

3.2.
Good moduli spaces. We now collect some useful results about the structure of a certain class of Artin stacks, namely those with affine diagonal admitting a good moduli space, following the theory developed by Alperét al. All the material of the section can be found in [Alp13] and [AHR20].
We have the following definition of a good moduli space. (1) q is quasi-compact and q * : The intuition behind the introduction of the notion of good moduli space is that stacks M that admit good moduli spaces behave like quotient stacks [X ss /G] obtained from Geometric Invariant Theory (GIT) with good moduli space morphism given by the map [X ss /G] → X/ /G. In this sense, it is a generalization of GIT quotients for stacks.
We state the following properties of stacks with good moduli space.
The following notions will be useful for us.
If U is saturated, the morphism U → U := q(U) gives a good moduli space for U fitting into a Cartesian diagram Remark 3.6. In the sequel, we will be using the fact that stronglyétale morphisms are stabilizer-preserving.
We have the following theorem regarding theétale local structure of good moduli space morphisms for stacks with affine diagonal.
Theorem-Definition 3.7. (Quotient chart) [AHR20, Theorem 4.12] Let M be a locally noetherian Artin stack with a good moduli space q : M → M such that q is of finite type with affine diagonal. If x ∈ M is a closed point with (reductive) stabilizer G x , then there exists an affine scheme U with an action of G x and a Cartesian diagram such that Φ x has the same properties as in Theorem 3.1, is affine and We refer to any choice of data affine and stabilizer-preserving as a quotient chart for M centered at x. We say that the quotient chart is stronglyétale if the morphism Φ x is stronglyétale.
3.3. Stacks of DT type. We start with the following definition.
Definition 3.8. (Stack of DT type) Let M be an Artin stack. We say that M is of DT type if the following are true: (1) M is quasi-separated and finite type over C.
(2) There exists a good moduli space q : M → M , where q is of finite type and has affine diagonal.
A variant of the following proposition first appeared and was used in [Tod16].
Proposition 3.11. Let M be a stack of DT type and x ∈ M a closed point. Then there exists a d-critical quotient chart for M centered at x, which can be taken to be stronglyétale.
Proof. By Theorem-Definition 3.7, we have a quotient chart Φ x : [U ′′ x /G x ] → M. By Proposition 2.5, Definition 2.8 and Remark 2.9, there exists a This concludes the proof. □ We now obtain the following key lemma, which gives a way to compare two choices of d-critical quotient charts.
Lemma 3.12. Let M be a stack of DT type. Let be the two projections of z. Then the following hold: (1) We have G-equivariant commutative diagrams for λ = α, β where the vertical arrows are unramified morphisms, the horizontal arrows are embeddings, t ∈ T αβ maps to x λ ∈ U λ , and T αβ , S αβ are affine.
Proof. It follows by the second condition in Definition 3.8 that M must have affine diagonal. In particular, the Cartesian diagram there exists a smooth affine scheme V αβ and a G αβ -equivariant embedding fitting into a commutative diagram Concretely, we may choose any G αβ -equivariant embedding U αβ → A N for big enough N and then take V αβ = V α ×V β ×A N with the obvious choices of morphisms that make the above diagram commute.
(we are liberally abusing notation at this point). Using Luna'ś etale slice theorem, take S αβ ⊆ V αβ to be an affineétale slice for t in V αβ and T αβ := S αβ ∩ U αβ the induced affineétale slice for t ∈ U αβ .
It is clear that conditions (1) and (2) are now satisfied and remain so even after G-invariant shrinking of S αβ around t.
For conditions (3) and (4), notice that by diagrams (3.2) and (3.3) and the properties of d-critical structures, [T αβ /G] is a d-critical stack and thus, up to possibly shrinking S αβ G-invariantly around t, there exists an induced d-critical G-invariant quotient chart (T αβ , S αβ , f αβ , Φ αβ ) for M centered at t ∈ T αβ . Finally, since by construction S αβ , T αβ areétale slices of V λ , U λ at t, we obtain that (df αβ ) = (θ * λ df λ ) as ideals and θ * λ f λ + I 2 , f αβ + I 2 both represent the pullback of the d-critical structure of M to [T αβ /G], which completes the proof. □ Remark 3.13. The above results only use the d-critical structure of M and not the existence of its (−1)-shifted symplectic derived enhancement. In fact, using the property that M is the truncation of a (−1)-shifted symplectic derived Artin stack M, Proposition 3.11 may be strengthened. Since we won't need this strengthening in this paper, we are content with just making this remark.
Remark 3.14. We finally make a remark on the conditions in Definition 3.8. The first condition is there to ensure boundedness. The second condition implies the existence of quotient charts as in diagram (3.1), which will be necessary in order to construct the intrinsic stabilizer reduction M and its good moduli space. The third condition is necessary to induce a d-critical structure on M which is the crucial component in obtaining a semi-perfect obstruction theory for M.

Intrinsic Stabilizer Reduction
In this section, we review the notions of intrinsic and Kirwan blowups and then generalize the construction of the intrinsic stabilizer reduction of a quotient stack obtained by GIT given in [KLS17] to the more general setting of stacks with good moduli space. 4.1. Kirwan blowups for affine schemes. We recall the notion of Kirwan blowup developed in [KLS17].
Suppose that U is an affine scheme with an action of a reductive group G. For now, let us assume that G is connected, as this will be the case when we take blowups throughout the paper.
Suppose that we have an equivariant closed embedding U → V into a smooth affine G-scheme V and let I be the ideal defining U . Since U ⊆ V is G-invariant, G acts on I and we have a decomposition I = I fix ⊕ I mv into the fixed part of I and its complement as G-representations.
Let V G be the fixed point locus of G inside V , defined by the ideal generated by O mv V , and π : bl , given by multiplication by ξ.
We define I intr ⊆ O bl G (V ) to be I intr = ideal sheaf generated by π −1 (I fix ) and ξ −1 π −1 (I mv ). (4.1) If U G = ∅, then U = U and there is nothing to show. So suppose this is not the case and u ∈ U G .
Let U → V 1 , U → V 2 be two equivariant embeddings of U into smooth G-schemes. By composing with the equivariant inclusions V 1 ×{u} ⊆ V 1 ×V 2 and {u}×V 2 ⊆ V 1 ×V 2 , we may reduce to the case of a sequence of equivariant Applying the above construction using the two embeddings of U , we obtain G-intrinsic blowups U V , induced by the embedding U → V , and U W , induced by the composite embedding U → W .
Then an identical argument as in the proof of [KL13, Lemma 3.1], shows that Φ gives rise to an isomorphism U V ≃ U W , which moreover is independent of the choice of Φ interpolating between the two embeddings U → V and U → W . This proves that U is indeed canonical. □ Suppose U is an affine G-scheme with an equivariant embedding into a smooth affine G-scheme V as above. We can make sense of the notion of semistability of points in U intr without ambiguity as follows.
We first work on the ambient scheme V . As it is affine, we can think of all points of V as being semistable (in the usual sense of GIT). Definition 4.3. We say that v ∈ V is stable if its G-orbit is closed in V and its stabilizer finite. A point v ∈ bl G (V ) is unstable if its G-orbit closure meets the unstable locus of E. If v is not unstable, we say that it is semistable.
Thus for any smooth affine G-scheme V , we can define its Kirwan blowup V = (bl G (V )) ss . By [Kir85], it satisfies V G = ∅. Now, if we have an equivariant embedding V → W between smooth Gschemes, then (W intr ) ss ∩ V intr = (V intr ) ss based on our description. Hence we may define (U intr ) ss := U intr ∩ (V intr ) ss for any equivariant embedding U → V into a smooth scheme V , which is independent of the choice of U → V . The notion of semistability above is exactly motivated by the corresponding notion in GIT in Kirwan's original blowup procedure in [Kir85]. This can be seen by the following theorem of Reichstein, which asserts that the locus of unstable points on bl G (V ) is exactly the locus of unstable points in the sense of GIT for the ample line bundle O bl G (V ) (−E).
We obtain the following corollary. Example 4.7. Suppose that G = C * is the one-dimensional torus acting on the affine plane V = C 2 x,y with weights 1 and −1 on the coordinates x and y respectively and U ⊆ V is the closed G-invariant subscheme cut out by the ideal I = (x 2 y, xy 2 ).
V intr is the blowup of V along the fixed locus V G = {0}. The unstable points are the punctured x-axis and y-axis together with the points 0, ∞ of the exceptional divisor P 1 . Thus we have that V = Spec[u, v, v −1 ] where G acts on u, v with weights 1, −2 respectively and the blowdown map V → V is given on coordinates by x → u, y → uv.
U ⊆ V is the closed subscheme cut out by the ideal (u 2 ), so that We now explain how one can proceed if G is not connected. Suppose that U → V is a G-equivariant embedding into a smooth Gscheme V . Let, as before, I be the ideal of U in V . Let G 0 be the connected component of the identity. This is a normal, connected subgroup of G of finite index. Let I = I fix ⊕ I mv be the decomposition of I into fixed and moving parts with respect to the action of G 0 . Using the normality of G 0 , we see that the fixed locus V G 0 is a closed, smooth G-invariant subscheme of V and also I fix , I mv are G-invariant.
Let π : bl V G 0 V → V be the blowup of V along V G 0 with exceptional divisor E and local defining equation ξ. Then, as before, take I intr to be the ideal generated by π −1 (I fix ) and ξ −1 π −1 (I mv ). Everything is G-equivariant and we define U intr as the subscheme of bl V G 0 V defined by the ideal I intr .
Finally, we need to delete unstable points. By the Hilbert-Mumford criterion (cf. [MFK94, Theorem 2.1]) it follows that semistability on E with respect to the action of G is the same as semistability with respect to the action of G 0 , since every 1-parameter subgroup of G factors through G 0 , and hence we may delete unstable points exactly as before and define the Kirwan blowup U .
One may check in a straightforwardly analogous way that this has the same properties (and intrinsic nature). It is obvious that if G is connected we recover Definition 4.4.

4.2.
Kirwan blowups for GIT schemes. So far, we have defined Kirwan blowups for affine G-schemes (with non-empty fixed locus of the G-action). We now generalize the construction to any G-scheme with a G-action coming from GIT (see Definition 4.8 below) and the associated quotient stack, as performed in [KLS17, Subsection 2.4].
We begin with an affine G-scheme whose G-fixed locus can now be possibly empty. Let M = [X/G] be a quotient Artin stack, where X is an affine Gscheme.
Fix a G-equivariant closed embedding X → Y , where Y is a smooth affine G-scheme, and write Y = [Y /G]. Let d be the maximum dimension of the stabilizers of closed points x ∈ X. By possibly equivariantly shrinking Y , we may assume that the maximum dimension of stabilizers of closed points of Y is also equal to d.
Kirwan's partial desingularization, combined with Theorem 4.5, yields a smooth G-scheme Y by blowing up the locus Y d ⊆ Y consisting of points satisfying dim G y = d and deleting unstable points. Y satisfies the crucial property that its maximum stabilizer dimension is strictly less than d.
Now, for any point x ∈ X with closed G-orbit and dim G x = d, let S x ⊆ Y be anétale slice for the G-orbit of x in Y . The Kirwan blowups S x (cf. Definition 4.4) of the slices S x associated with G x provide natural stronglyétale morphisms Ψ Then T x is anétale slice for the G-orbit of x in X and we have the Kirwan blowup T x using the closed G x -equivariant embedding T x ⊆ S x . In [KLS17, Proposition 2.12], it is shown that the closed embedding satisfiesétale descent with respect to the morphism Ψ and thus defines a closed subscheme Lemma 4.2 and the above discussion on stability imply that these are canonical and independent of any choices made regarding the embedding X → Y andétale slices S x .
If X is not affine, but is the GIT semistable locus of a G-scheme with a G-linearized ample line bundle, then it admits a cover by G-invariant Zariski open, saturated, affine subschemes U 1 , ..., U n ⊆ X. Here, as in Definition 3.4, saturated means that the U i fit in Cartesian squares We may thus apply the Kirwan blowup to each U i . The Kirwan blowups U i glue to the Kirwan blowup X of X and we are led to the following definition. Remark 4.9. When X is not affine, we could also argue globally as in the affine case using an equivariant closed embedding X → Y into a smooth G-scheme Y , afforded by GIT. However, the more local argument using a cover by saturated, affine, G-invariant open subschemes is closer in spirit to the proof of Theorem-Construction 4.16.

4.3.
Intrinsic stabilizer reduction for GIT quotient stacks. We quickly recall how the construction of the intrinsic stabilizer reduction works for GIT quotient stacks and some useful properties in that setting.
Let X be a G-scheme coming from GIT and M = [X/G] the associated quotient stack.
Write M 0 = M and X 0 = X. Taking Kirwan blowups, we obtain X 1 = X and M 1 = [ X/G]. The G-action on X is still G-linearized, so M 1 is a GIT quotient stack of lower maximum stabilizer dimension compared to M 0 . Thus we may iterate the procedure to produce a canonical sequence of GIT quotient stacks where M ℓ is a Deligne-Mumford stack, since at each step the maximum stabilizer dimension strictly decreases.

Intrinsic stabilizer reduction for stacks with good moduli spaces.
We now generalize the procedure of intrinsic stabilizer reduction to the case of stacks with good moduli spaces.
The following properties of Kirwan blowups will be useful, so we record them separately in a proposition before our main construction.
Suppose that f : X 1 → X 2 is a smooth morphism which is equivariant with respect to φ and such that the induced morphism f : [X 1 /G 1 ] → [X 2 /G 2 ] on quotient stacks is stronglyétale (cf. Definition 3.5).
Then there is a canonically induced smooth, affine morphism f : X 1 → X 2 , equivariant with respect to φ, such that the associated quotient morphism f : [ X 1 /G 1 ] → [ X 2 /G 2 ] is stronglyétale and fits in a Cartesian square Proof. Let x 1 ∈ X 1 a point with closed G 1 -orbit and of maximum stabilizer dimension and x 2 = f (x 1 ) ∈ X 2 . Since f is stronglyétale, the G 2 -orbit of x 2 is also closed, by Proposition 3.3(5), and x 2 is of maximum stabilizer dimension.
Up to G 1 -equivariant shrinking around x 1 , we can find smooth, affine G ischemes Y i , i = 1, 2, and a smooth morphism g : Y 1 → Y 2 , equivariant with respect to φ, fitting in a commutative diagram where the horizontal arrows are equivariant closed embeddings. We may additionally assume that the induced morphism g : stronglyétale. This is because we can initially take g to map x 1 to x 2 , be stabilizer-preserving at x 1 and g to beétale at x 1 , so using the Fundamental Lemma [Alp10, Theorem 6.10] at x 1 , after possible further shrinking around x 1 , we may take g to be stronglyétale. By the intrinsic nature of intrinsic blowups, using an identical argument as the one outlined in the proof of Lemma 4.2, one may consider diagram (4.2) as x 1 varies to obtain a canonical morphism f intr : X intr 1 → X intr 2 . By construction, f intr is stabilizer-preserving and maps closed G 1 -orbits to closed G 2 -orbits, as this is the case for g intr . It is moreover smooth and the associated quotient morphism f intr : [X intr 1 /G 1 ] → [X intr 2 /G 2 ] isétale. This can be seen as follows: Let T be an affine,étale slice for the G 1 -orbit of x 1 in X 1 , whose stabilizer we denote by H. By the assumptions on the morphism f , we have a commutative diagram / / X 2 and the corresponding commutative diagram at the level of quotient stacks has allétale arrows. By definition, these give rise to identical diagrams with T, X 1 , X 2 replaced by their intrinsic blowups. Varying x 1 , this immediately implies the smoothness and affineness of f intr andétaleness of f intr . Moreover, a similar, straightforward local computation, using the definition of intrinsic blowup and the arguments of the proof of Lemma 4.2 and [KL13, Lemma 3.1], shows that the natural map X intr 1 → X 1 × X 2 X intr 2 is an isomorphism. Now Corollary 4.6 lets us pass from intrinsic blowups to Kirwan blowups. Finally, since f isétale, stabilizer-preserving and maps closed points to closed points, applying the Fundamental Lemma [Alp10, Theorem 6.10] at all closed points of [ X 1 /G 1 ], we obtain that it is stronglyétale. The existence of the isomorphism [  This agrees with Definition 4.3 by the following proposition, which also explains why the stable locus is open and saturated, since this is a property that can be checked stronglyétale locally. Proof. Any stronglyétale morphism induces an isomorphism of fibers q −1 U (q U (u)) ≃ q −1 (q(x)) whenever Φ(u) = x and we identify u ∈ U with its G-orbit as a point u ∈ [U/G]. The conclusion is immediate by Proposition 3.3(5) and the fact that Φ is stabilizerpreserving. □ We can now construct the Kirwan blowup of an Artin stack with good moduli space.
Theorem-Construction 4.14. Let M be an Artin stack of finite type over C with affine diagonal, which is not Deligne-Mumford. Moreover, suppose that q : M → M is a good moduli space morphism with q of finite type and with affine diagonal.
Then there exists a canonical Artin stack M, called the Kirwan blowup of M, together with a morphism π : M → M, such that: (1) M is of finite type over C, has affine diagonal and admits a good moduli space morphism q : M → M with affine diagonal.
The morphisms Φ x are affine, stronglyétale and cover the locus M max . We may take the Kirwan blowup of each quotient stack [U x /G x ] to obtain good moduli space morphisms We need to check that these glue to give a stack M with a universally closed projection M → M and a good moduli space M → M satisfying the same conditions as M and its good moduli space morphism. By the properties of the Kirwan blowup, the maximum stabilizer dimension of M will be lower than that of M.
Suppose x, y are two closed points of M such that G x , G y are of maximum dimension. We obtain a Cartesian diagram of stacks where U xy := U x × M U y is an affine scheme. This is due to the Cartesian diagram and the fact that M has affine diagonal. Using Proposition 4.11, we obtain a diagram with affine, stronglyétale arrows and we have, moreover, canonical associ- Using the charts [ U x /G x ] together with a cover of M \ q −1 (M max ), we therefore obtain an atlas for a stack M with a map to M. By the canonical isomorphisms of the previous paragraph, M is independent of the particular choices of charts for M.
Since the arrows in diagram (4.5) are stronglyétale, we obtain a corresponding diagram ofétale arrows at the level of good moduli spaces of the Kirwan blowups together with an atlas of M \ q −1 (M max ) glue to give a morphism M → M . By Proposition 3.3 again, this is a good moduli space morphism. M has affine diagonal since we have a cartesian diagram Finally, the last assertion about the intrinsic blowup M intr can be shown using the same arguments. Since we won't be needing M intr , we leave the details to the reader. □ Remark 4.15. The fact that M and q : M → M have affine diagonal is not crucial for the above proof to go through. In fact, by [AHR19, Theorem 13.1] and [AHR19, Corollary 13.11], it is enough to assume that M has affine stabilizers and separated diagonal. Since the stacks we are interested in will have affine diagonal, we make this assumption for convenience of presentation.
As before, repeatedly applying the operation of Kirwan blowup we obtain the instrisic stabilizer reduction. By construction of the Kirwan blowup, the stable locus is preserved at each step and hence stays invariant under the whole procedure. In general, if at some step of the procedure the stack M i is a gerbe over a Deligne-Mumford stack (i.e. M i has constant stabilizer dimension and every point is prestable), then M i is empty. We could elect to terminate the procedure at this step instead and define M := M i . We choose not to do so as for our purpose of defining Donaldson-Thomas invariants, setting M = ∅ in this case seems more geometrically motivated and better aligned with the BPS invariants considered in [JS12] and [DM20].
Remark 4.18. Edidin-Rydh have also developed a blowup procedure for stacks with good moduli spaces in [ER21]. For smooth stacks, our stabilizer reduction is the same as theirs. For singular stacks, Kirwan blowups can be phrased in their language of saturated blowups, however the resolution they obtain is a closed substack of the one here. Nevertheless, our construction is closely related to the Edidin-Rydh resolution of stabilizers performed at the level of derived stacks, using the recently developed notion of derived blowups in [Hek21] and the results of [HRS22].

Obstruction Theory
In this section, we first recall the basic principles of semi-perfect obstruction theories and then explain how to construct such a gadget on the intrinsic stabilizer reduction of a stack of DT type. Our discussion goes through the notion of a local model and its obstruction theory, as used in [KLS17].
5.1. Semi-perfect obstruction theory. This subsection contains background material about semi-perfect obstruction theories and their induced virtual cycles, as developed in [CL11].
Let U → C be a morphism, where U is a scheme of finite type and C denotes a smooth quasi-projective scheme, which will typically be either a point or a smooth quasi-projective curve. We first recall the definition of perfect obstruction theory [BF97,LT98].
We refer to Ob ϕ := H 1 (E ∨ ) as the obstruction sheaf of ϕ. such that the image of g contains a point p ∈ U , the problem of findinḡ g :∆ → U making the diagram commutative is the "infinitesimal lifting problem of U/C at p". Definition 5.3. (Obstruction space) For a point p ∈ U , the intrinsic obstruction space to deforming p is T 1 p,U/C := H 1 (L ≥−1 U/C ) ∨ | p . The obstruction space with respect to a perfect obstruction theory ϕ is Ob(ϕ, p) := H 1 (E ∨ | p ).
Given an infinitesimal lifting problem of U/C at a point p, there exists by the standard theory of the cotangent complex a canonical element ω g, ∆,∆ ∈ Ext 1 g * L ≥−1 U/C | p , I = T 1 p,U/C ⊗ C I (5.2) whose vanishing is necessary and sufficient for the liftḡ to exist.
Definition 5.4. (Obstruction assignment) For an infinitesimal lifting problem of U/C at p and a perfect obstruction theory ϕ the obstruction assignment at p is the element Definition 5.5. Let ϕ : E → L ≥−1 U/C and ϕ ′ : E ′ → L ≥−1 U/C be two perfect obstruction theories and ψ : Ob ϕ → Ob ϕ ′ be an isomorphism. We say that the obstruction theories give the same obstruction assignment via ψ if for any infinitesimal lifting problem of U/C at p ψ ob U (ϕ, g, ∆,∆) = ob U (ϕ ′ , g, ∆,∆) ∈ Ob(ϕ ′ , p) ⊗ C I.
We are now ready to give the definition of a semi-perfect obstruction theory.
Definition 5.6. (Semi-perfect obstruction theory [CL11, Definition 3.1]) Let M → C be a morphism, where M is a DM stack, proper over C, of finite presentation and C is a smooth quasi-projective scheme. A semiperfect obstruction theory ϕ consists of anétale covering {U α } α∈A of M and perfect obstruction theories ϕ α : E α → L ≥−1 Uα/C such that (1) For each pair of indices α, β, there exists an isomorphism so that the collection {Ob ϕα , ψ αβ } gives descent data of a coherent sheaf on M.
(2) For each pair of indices α, β, the obstruction theories E α | U αβ and E β | U αβ give the same obstruction assignment via ψ αβ (as in Definition 5.5).
Remark 5.7. The obstruction sheaves {Ob ϕα } α∈A glue to define a sheaf Ob ϕ on M. This is the obstruction sheaf of the semi-perfect obstruction theory ϕ.
Suppose now that M → C is as above and admits a semi-perfect obstruction theory. Then, for each α ∈ A, we have , where C Uα/C and N Uα/C denote the intrinsic normal cone stack and intrinsic normal sheaf stack respectively, where by abuse of notation we identify a sheaf F on M with its sheaf stack.
We therefore obtain a cycle class [c ϕα ] ∈ Z * Ob ϕα by taking the pushforward of the cycle [C Uα/C ] ∈ Z * N Uα/C . Remark 5.9. Observe that in Definition 5.6 it is not strictly necessary to take the U α to be schemes. It is straightforward to generalize the definition and the construction of the associated virtual fundamental cycle to includé etale covers by Deligne-Mumford stacks.

5.2.
Local models, standard forms and their blowups. Let V be a smooth affine G-scheme. The action of G on V induces a morphism g⊗O V → T V and its dual σ V : (3) Let R be the identity component of the stabilizer group of a closed point in V with closed orbit. Let V R denote the fixed point locus of is zero, where r is the Lie algebra of R. We say that the data give a local model structure for U . Thinking of the quotient stack, we say that [U/G] also has a local model structure and denote the data by Λ [U/G] .
Remark 5.11. Note that if f : V → A 1 is a G-invariant function on V , then (U, V, G, Ω V , df, 0, σ V ) give a local model for U , being equivalent to an invariant d-critical chart (U, V, f, i) for U . Therefore, an invariant d-critical locus is a particular case of a local model. Now, let Λ U = (U, V, G, F V , ω V , D V , ϕ V ) define a local model structure on U . Since G is reductive, we have a splitting By equivariance, π * ω V maps to zero under the second map and hence induces an invariant section ω V of F V . A local computation shows the following.
Proposition 5.12. The zero locus of ω V is the Kirwan blowup U of U with respect to G.
Proof. This is Proposition 2.11 in [KLS17]. Proof. This is [KLS17,Lemma 5.3]. The fact that F V is locally free is discussed shortly after Definition 2.10 in [KLS17]. Denoting the Kirwan blowup map V → V by π, the divisor D V is equal to π * D V + 2E, where E is the exceptional divisor of V . Finally, ϕ V is induced by ϕ V through a diagram chase and the definition of F V . The reasoning forétale slices is similar. □

5.3.
Obstruction theory of local model. Let be the data of a local model structure. Consider the sequence given by the diagram where I denotes the ideal sheaf of U in V . Observe that the diagram commutes, since by Setup-Definition 5.10(2), the composition ϕ V • ω V is identically zero. Let U s , V s denote the stable loci of U and V respectively (cf. Definition 4.3). We thus have that σ V | V s : Ω V | V s → g ∨ is surjective. By Setup-Definition 5.10, it follows that ϕ V | U s is surjective and hence its dual is injective with locally free cokernel.
Therefore, K [U s /G] := K [U/G] | [U s /G] is a two-term perfect complex, whose dual E [U s /G] gives a perfect obstruction theory on the DM stack [U s /G] via the diagrams (5.7) and (5.8).
Definition 5.14. Let (U, V, G, F V , ω V , D V , ϕ V ) be the data of a local model structure. Then E [U s /G] is the induced perfect obstruction theory on the stable locus [U s /G].

5.4.
Semi-perfect obstruction theory of the intrinsic stabilizer reduction of a stack of DT type. Let M be an Artin stack of DT type. We now explain how to construct a semi-perfect obstruction theory on its intrinsic stabilizer reduction M, one of the main results of this paper.
We begin by recording and abstracting some of the data associated to a stack of DT type.
Setup 5.15. For any closed point x ∈ M, by Proposition 3.11, there exists a stronglyétale d-critical quotient chart for M centered at x. In particular, we obtain a stronglyétale cover where, by Remark 5.11, each quotient stack [U x /G x ] is equipped with data of a local model By Proposition 3.12, for any two closed points x, y ∈ M we have anétale cover For any morphism [T z /G z ] → [U x /G x ] in (5.11), Proposition 3.12 also produces the following data: (1) A commutative G z -equivariant diagram where the horizontal arrows are closed embeddings and θ zx is unramified, inducing theétale morphism ble with the morphisms ϕ Vx | Sz and ϕ Sz , such that ω ′ z := η zx (ω Vx | Sz ) and ω Sz are Ω-equivalent (cf. [KLS17, Definition 5.9]).
(3) An induced isomorphism We abstract this situation by saying that the data of a stronglyétale cover (5.9) equipped with a local model structure (5.10), anétale cover (5.11) with a local model structure (5.12) and compatibility data described by items (1)-(4) above constitute a VC package for an Artin stack M satisfying the conditions of Theorem-Construction 4.14. Note that if M is an Artin stack of DT type then, by our discussion, we may take the local model structures to correspond to invariant d-critical quotient charts, however in general we only require local models that satisfy the compatibilities (1)-(4).
For a detailed account of Ω-equivalence, we refer the reader to [KLS17,Section 5]. For the purposes of our discussion, it suffices to note that ω ′ z being Ω-equivalent to ω Sz essentially means that we have another local model Proof. By the discussion in Setup 5.15, M admits a VC package where moreover all local models corresponds to d-critical quotient charts.
By the construction of M, it suffices to show that if a stack N (satisfying the conditions of Theorem-Construction 4.14) admits a VC package, then its Kirwan blowup N admits a canonically induced VC package.
By the construction in the proof of Proposition 3.12, Proposition 4.11 and Theorem-Construction 4.14, we may obtain newétale covers 5.9 and 5.11 for N by replacing every stack by its Kirwan blowup.
By Proposition 5.13, there are canonically induced local model structures on theseétale covers. It is moreover clear that condition (1) is satisfied.
Conditions (2)-(4) follow from [KLS17, Lemma 6.1]. This concludes the proof. □ Now, a VC package on a Deligne-Mumford stack naturally provides data of a semi-perfect obstruction theory. Thus, by Definition 5.14, the local model structures Λ [Ux/Gx] induce perfect obstruction theories Conditions (3) and (4) of a VC package give descent data for the obstruction sheaves h 1 (E ∨ x ), while conditions (1) and (2) and the properties of Ω-equivalence imply that these descent data are compatible with obstruction assignments of infinitesimal lifting problems.
Using Remark 5.9 and Theorem-Construction 5.8, we obtain a semiperfect obstruction theory on M and a virtual fundamental cycle [M] vir ∈ A * (M). □ Combining the above two theorems, we arrive at the main result of this section. Proof. The existence of the semi-perfect obstruction theory follows immediately from Theorems 5.16 and 5.17.
The virtual dimension is zero, since for a d-critical chart the rank of F V = Ω V is equal to dim V and the virtual dimension dim V − rk F V is preserved under Kirwan blowups.
Finally, any two differentétale covers (5.9) that are part of a VC package of M can be refined to a commonétale cover. It is routine to check that the virtual fundamental cycle thus obtained is the same using this common refinement. □ Remark 5.19. The existence of the intrinsic stabilizer reduction of M and its obstruction theory only uses its (−1)-shifted symplectic derived enhancement to deduce the existence of a d-critical structure on M. We could have thus used a weaker notion of stacks of DT type, replacing condition (3) in Definition 3.8 by the requirement that M admits a d-critical structure. However, we will be interested in replicating our arguments in the relative case where M is a stack over a base smooth scheme S. In that context, there is no well-developed theory of d-critical stacks to the author's knowledge and the existence of a derived enhancement will allow us to still perform our constructions.

Donaldson-Thomas Invariants of Derived Objects
In this section, W denotes a smooth, projective Calabi-Yau threefold over C. We first describe the stability conditions σ on D b (Coh W ) that we will be interested in. We then quote results in [AHH18] which imply that moduli stacks of σ-semistable complexes are stacks of DT type and explain how to rigidify the C * -scaling automorphisms of objects. Finally, we define generalized DT invariants via Kirwan blowups and show their deformation invariance.
6.1. Stability conditions. By [Lie06], there is an Artin stack P := Perf(W ) of (universally gluable) perfect complexes on W , which is locally of finite type and has separated diagonal. Following [AHH18], we will consider the following type of stability condition. Definition 6.1. (Stability condition) A stability condition σ on D b (Coh W ) consists of the following data: denote the stack of perfect complexes in A.
(2) A vector γ ∈ H * (W, Q). Let P γ A denote the stack of perfect complexes in A with Chern character γ.
(3) A locally constant function where V is a totally ordered abelian group, p γ (E) = 0 for E ∈ P γ A and p γ is additive so that p γ (E ⊕ F ) = p γ (E) + p γ (F ).
We say that E ∈ P γ A is semistable if for any subobject F ⊆ E we have p γ (F ) ≤ 0 and stable if p γ (F ) < 0. If E is not semistable, we say it is unstable.
In order for the stack of semistable objects to be of DT type, we will need to consider stability conditions satisfying certain properties.
Definition 6.2. (Nice stability condition) Given a stability condition σ on D b (Coh W ), let M be the stack of σ-semistable objects in P γ A . We say that σ is nice if the following hold: (1) M is an Artin stack of finite type.
(3) M satisfies the existence part of the valuative criterion of properness.
We then say that M is quasi-proper or universally closed.  To obtain a meaningful nonzero DT invariant, it will be necessary to rigidify the C * -scaling automorphisms of objects in M.
For each family of complexes E S ∈ M(S) there exists an embedding which is compatible with pullbacks and moreover G m (S) is central. In the terminology used in [AGV08], we say that M has a G m -2-structure.
Using the results of [AOV08] or [AGV08], we may take the G m -rigification M G m of M. From the properties of rigification, for any point x ∈ M, one has Aut M Gm (x) = Aut M (x)/T . In particular, if x ∈ M s is stable, then Even though M G m is not a stack of DT type, we show in the next two propositions that we can construct its intrinsic stabilizer reduction and equip it with a semi-perfect obstruction theory of virtual dimension zero. Proof. M is a stack of DT type and hence admits a VC package by Setup 5.15. By the existence of the G m -2-structure, T naturally embeds in each stabilizer group G x . Replacing all such groups by their quotients G x /T , we obtain a VC package for M G m . □ Remark 6.7. In the case of semistable sheaves treated in [KLS17], rigidification is much simpler, since the moduli stack is a global GIT quotient stack M = [X/G] where G = GL(N, C), and then one may work with [X/PGL (N, C)] as the G m -rigidication.
We thus obtain the following.
Theorem-Definition 6.8. M C * is called the C * -rigidified intrinsic stabilizer reduction of M. It is a proper DM stack with a semi-perfect obstruction theory of virtual dimension zero.
Proof. For properness, the good moduli space q : M C * → M C * is proper. Since M is proper and M C * is proper over M , M C * is proper. Everything else follows from Theorems 5.16, 5.17 as applied in the proof of Theorem 5.18. □ Remark 6.9. By identical reasoning, all of the above hold in greater generality when M is an Artin stack of DT type with a G m -2-structure.
6.4. Generalized DT invariants via Kirwan blowups. Suppose as before that M is an Artin stack of DT type parametrizing σ-semistable objects in a heart A of a t-structure on D b (Coh W ) with Chern character γ, for a nice stability condition σ. Then, by Theorem-Definition 6.8, there is an induced C * -rigidified intrinsic stabilizer reduction M C * with a good moduli space M C * and an induced semi-perfect obstruction theory and virtual cycle of dimension zero. We can now state the main theorem of this paper.
Theorem-Definition 6.10. Let W be a smooth, projective Calabi-Yau threefold, σ a nice stability condition on a heart A ⊂ D b (Coh W ) of a tstructure, as in Definition 6.1, γ ∈ H * (W ) and let M denote the stack parametrizing σ-semistable complexes with Chern character γ. Remark 6.11. In [KS21], the results of [KLS17] and the present paper are refined to define a virtual structure sheaf [O vir M ] ∈ K 0 ( M C * ) and a corresponding K-theoretic generalized DTK invariant.
6.5. Relative theory and deformation invariance. Let C be a smooth quasi-projective scheme over C and W → C a smooth, projective family of Calabi-Yau threefolds. Without loss of generality, we assume that H * (W t , Q) stays constant for t ∈ C and identify it with H * (W 0 , Q) where W 0 is the fiber of the family over a point 0 ∈ C. Let γ ∈ H * (W 0 , Q). Moreover, let Perf(W/S) denote the stack of (universally gluable) perfect complexes on the morphism W → C as in [Lie06].
We consider families σ of stability conditions σ t , where, for each t ∈ C, σ t is a stability condition on D b (Coh W t ) as in Definition 6.2 with Chern character γ ∈ H * (W t , Q) = H * (W 0 , Q). Let M → C be the stack parametrizing relatively semistable objects in D b (Coh W ), i.e. perfect complexes E such that the derived restriction E t := E| Wt is σ t -semistable for all t ∈ C. We require that the conditions characterizing a nice stability condition hold relative to the base C as follows.
Setup 6.12. We say that the family of stability conditions σ t is nice if the following hold.
(1) M is an Artin stack of finite type.
(3) M → C admits a good moduli space M → C, proper over C.
Remark 6.13. When we have a GIT description of M → C, then the above conditions are satisfied. This is the case for Gieseker and slope stability of coherent sheaves.
When we are in the situation of the above setup, all of our results extend to the relative case.
Theorem 6.14. Let W → C be a smooth, projective family of Calabi-Yau threefolds over a smooth, quasi-projective scheme C, {σ t } t∈C a nice family of stability conditions on D b (Coh W ), γ ∈ H * (W 0 , Q) and M → C the stack of fiberwise σ t -semistable objects of Chern character γ in D b (Coh W ).
Then there exists an induced C * -rigidified intrinsic stabilizer reduction M → C, a DM stack proper over C, endowed with a semi-perfect obstruction theory and a virtual fundamental cycle [ M] vir ∈ A 0 ( M).
Moreover, the fiber M t over t ∈ C is the C * -rigidified intrinsic stabilizer reduction of M t and the obstruction theory pulls back to the one constructed in the absolute case, so that if i t : M t → M is the inclusion, we have Proof. This is the generalization of [KLS17, Theorem 7.17] adapted to our context. We briefly explain the steps and necessary changes.
M admits a G m -2-structure. By conditions (1) and (3) of Setup 6.12, M G m satisfies the conditions of Theorem 4.16, so using Proposition 6.5 we obtain the C * -rigidified intrinsic stabilizer reduction M C * → M → C.
To see that the fiber over t ∈ C is M C * t , observe that by definition M C * is obtained by taking iterated Kirwan blowups of a cover by stronglyétale quotient charts (where T = C * ) Taking fibers over t ∈ C, we obtain a cover of M t G m by stronglyétale quotient charts, so it suffices to show that ( N x ) t = (N x ) t . But this is true by the same argument used in the proof of [KLS17,Proposition 7.4].
Constructing the obstruction theory of M C * is slightly more subtle. First, we observe that there is an obvious generalization of Setup-Definition 5.2 to define local model structures on G-schemes U over the base scheme C, cf. [KLS17,Definition 7.11].
Using the fact that the morphism M → C is the truncation of a (−1)shifted symplectic derived stack over C, we can show that M G m admits a VC package, as in Setup 5.15. The only difference is that now in (5.10), we have F Vx = Ω Vx/C and ω Vx is a G x -invariant 1-form, and similarly in (5.12), F Sz = Ω Sz/C and ω Sz is a G z -invariant 1-form. The construction of the VC package follows verbatim the reasoning of Subsections 7.2 and 7.3 in [KLS17]. We refer the reader there for details.
Finally, to see that the restriction of the semi-perfect obstruction theory of M C * to M C * t agrees with the absolute semi-perfect obstruction theory constructed using the d-critical structure of M t G m , we use the fact that for any choice of relative local models in the VC package of M G m , the sections ω Vx can be taken so that ω Vx | (Vx)t is Ω-equivalent to an exact 1form df x induced by the d-critical structure of M t G m . As a consequence of properties of Ω-equivalence (cf. [KLS17, Lemma 5.14]), the construction of the semi-perfect obstruction theory of M C * t can be performed equivalently using ω Vx | (Vx)t instead of df x in the VC package of M t G m . The necessary arguments are carried out in detail in [KLS17,Subsection 7.4]. □ In the case of Bridgeland stability conditions constructed in [PT19] and [Li19], the results of [BLM + 21] imply that we get nice families of Bridgeland stability conditions.
As an immediate corollary, we have the following theorem.
Theorem 6.15. The generalized DT invariant via Kirwan blowups for σsemistable objects on Calabi-Yau threefolds, where σ is a Bridgeland stability condition as in [PT19] and [Li19], is invariant under deformations of the complex structure of the Calabi-Yau threefold.
Remark 6.16. In the case of Gieseker stability and slope stability of coherent sheaves, deformation invariance follows directly from the results of [KLS17].