Dimensional reduction in cohomological Donaldson-Thomas theory
aa r X i v : . [ m a t h . AG ] F e b DIMENSIONAL REDUCTION IN COHOMOLOGICALDONALDSON–THOMAS THEORY
TASUKI KINJO
Abstract.
For oriented − − Contents
1. Introduction 12. Shifted symplectic structures and vanishing cycles 53. Dimensional reduction for schemes 164. Dimensional reduction for stacks 215. Applications 36Appendix A. Remarks on the determinant functor 39References 441.
Introduction
Motivations.
For a Calabi–Yau 3-fold X , Thomas introduced enumerativeinvariants in [Tho00] which is now called the Donaldson–Thomas invariants (DTinvariants for short) which virtually count stable sheaves on X . Later severalvariations and generalizations of DT invariants were introduced. One such exampleis the DT invariants for quivers with potentials first introduced in [Sze08], which isnow understood as the local version of the original DT invariants. Another exampleis cohomological Donaldson–Thomas theory (CoDT theory for short), which studiesthe sheaf theoretic categorification of DT invariants.CoDT theory was initiated by the work of Kontsevich–Soibelman [KoSo11] inthe case of quivers with potentials. It studies the vanishing cycle cohomologies ofthe moduli stacks of representations over Jacobi algebras associated with quiverswith potentials. Later [BBBBJ15, BBDJS15, JU20] opened the door to CoDTtheory for CY 3-folds by defining a natural perverse sheaf ϕ M H -ss X (resp. ϕ M H -st X ) on the moduli stack M H -ss X of compactly supported H -semistable sheaves (resp. themoduli scheme M H -st X of H -stable sheaves) on a CY 3-fold X with a fixed ampledivisor H , which can be regarded as a categorification of the original DT invariantin the following sense: for a compact component N ⊂ M H -st X we have Z [ N ] vir X i ( − i dim H i ( N ; ϕ M H -st X | N ) . where [ N ] vir denotes the virtual fundamental class of N .CoDT theory for quivers with potentials is well-developed and shown to havea rich theory. For example, Kontsevich–Soibelman in [KoSo11] constructed alge-bra structures called critical cohomological Hall algebras (critical CoHAs for short)on the CoDT invariants for quivers with potentials. Later Davison–Meinhardt in[DM20] proved the wall crossing formulas for CoDT invariants of quivers with po-tentials, and realized it as natural maps induced by the CoHA multiplications. Incontrast to these developments, almost nothing is known concerning CoDT the-ory for CY 3-folds though it is expected that the local theory as above can beextended to the global settings. The aim of this paper is to make the first steptowards the development of CoDT theory for local surfaces (i.e. CY 3-folds of theform Tot S ( ω S ) where S is a smooth surface) by proving a global version of thedimensional reduction theorem [Dav17, Theorem A.1].1.2. Dimensional reduction.
In this paper, we always use the term “dimensionalreduction” as a statement that relates three-dimensional things to two-dimensionalthings. A dimensional reduction in DT theory was first observed by the work of[BBS13] in the motivic context. They computed the motivic refinement of theDT invariant of zero-dimensional closed subschemes of C by relating it with themotive of the moduli stack of zero-dimensional sheaves on C . Later Davison proveda categorified version of the dimensional reduction theorem in [Dav17], which webriefly recall now. Let U be a scheme, E be a vector bundle on U , s ∈ Γ( U, E ) bea section, and ¯ s be the regular function on Tot U ( E ∨ ) corresponding to s . Denoteby π : Tot U ( E ∨ ) → U the projection. Write Z = s − (0). Then [Dav17, TheoremA.1] states that we have an isomorphism π ! ϕ p ¯ s ( Q Tot U ( E ∨ ) ) | Z ∼ = Q Z [ − E ](1.1)where ϕ p ¯ s denotes the vanishing cycle functor. This statement has many applica-tions in CoDT theory for quiver with potentials. For example, Davison in [Dav16]computed the vanishing cycle cohomology for the Hilbert scheme of points in C byusing the quiver description and applying (1.1). Another interesting application ofthe isomorphism (1.1) also discussed in [Dav16] is the study of compactly supportedcohomologies of the moduli stacks of representations of preprojective algebras, e.g.the purity of their Hodge structures. Therefore the isomorphism (1.1) not only al-lows us to compute the vanishing cycle cohomology but also can be used to importthree-dimensional techniques to the study of two-dimensional objects.The main theme of this paper is to globalize the isomorphism (1.1) so that itcan be applied to the study of CoDT theory for local surfaces. Before statingthe main theorem of this paper, we briefly recall the construction of the perversesheaves introduced in [BBBBJ15, BBDJS15]. Let ( X , ω ) be a − IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 3
It is shown in [BBBBJ15] that X is locally (in the smooth topology) written as acritical locus. If we are given a square root of the line bundle det( L X | X red ) o : M ⊗ ∼ = det( L X | X red ) , which is called an orientation for X , we can construct a natural perverse sheaf ϕ p X = ϕ p X ,ω,o on t ( X ) locally isomorphic to the vanishing cycle complex twisted by a certainlocal system. Our main theorem in this paper is as follows: Theorem. (Theorem 4.14)
Let Y be a quasi-smooth derived Artin stack, and T ∗ [ − Y := Spec Y (Sym( T Y [1])) be the − -shifted cotangent stack. Equip T ∗ [ − Y with the canonical − -shifted symplectic structure ω T ∗ [ − Y and the canonical orien-tation o T ∗ [ − Y (see Example 2.6 and Example 2.15). If we write π : t ( T ∗ [ − Y ) → t ( Y ) the projection, we have a natural isomorphism π ! ϕ p T ∗ [ − Y ∼ = Q t ( Y ) [vdim Y ] . (1.2)Now we return back to the story of the dimensional reduction in CoDT theory.Consider a smooth surface S and write X = Tot S ( ω S ). Denote by M S (resp. M X )the derived moduli stack of compactly supported coherent sheaves on S (resp. X ),and we write M S = t ( M S ) and M X = t ( M X ). By applying the work of [BCS20]and [IQ18], we can show that there is a natural equivalence between M X and T ∗ [ − M S over M S preserving the − Corollary. (Corollary 5.2)
Let M X and M S be as above, and equip M X withthe orientation induced by the canonical orientation on T ∗ [ − M S . If we write π : M X → M S the canonical projection, we have an isomorphism π ! ϕ p M X ∼ = Q M S [vdim M S ] . (1.3)By the Verdier self-duality of ϕ p M X , the isomorphism (1.3) impliesH ∗ ( M X ; ϕ p M X ) ∼ = H BMvdim M S −∗ ( M S )(1.4)where H BM denotes the Borel–Moore homology. Since it is shown in [KV19] thatH BM ∗ ( M S ) carries a convolution product, the isomorphism (1.4) induces an algebrastructure on H ∗ ( M X ; ϕ p M X ). We expect that this is isomorphic to the conjecturalcritical CoHA for X and it is useful in the study of wall-crossing formulas for CoDTinvariants of X as in the local case. Further, as the local dimensional reductionisomorphism (1.1) plays an important role in the cohomological study of modulistacks of representations of preprojective algebras in [Dav16], we expect that itsglobal variant (1.3) is useful in the cohomological study of moduli stacks of coherentsheaves on K3 surfaces or Higgs sheaves on curves. These directions will be pursuedin future work. In [JU20], natural orientation data for a wide class of CY 3-folds including all projective onesand local surfaces are constructed using gauge theoretic techniques. In [JU20, Remark 4.12] it isconjectured that our choice coincides with theirs for local surfaces.
TASUKI KINJO
Thom isomorphism.
For a quasi-smooth derived scheme Y , the dimensionalreduction isomorphism (1.2) has another interpretation: a version of the Thomisomorphism for the dual obstruction cone. By imitating the construction of theEuler class, we construct a class e ( T ∗ [ − Y ) ∈ H BM2 vdim Y ( Y ) where we write Y = t ( Y ) as follows. Consider the natural morphism π ! ϕ T ∗ [ − Y → π ∗ ϕ T ∗ [ − Y . (1.5)By taking the Verdier dual of the isomorphism (1.2), we have π ∗ ϕ T ∗ [ − Y ∼ = ω Y [ − vdim Y ] . Therefore the map (1.5) defines an element in H
BM2 vdim Y ( Y ), which we name e ( T ∗ [ − Y ).Since the virtural fundamental class is a generalization of the Euler class, it is nat-ural to compare e ( T ∗ [ − Y ) with the virtual fundamental class [ Y ] vir . Concerningthis we have obtained the following claim, which will be proved in a subsequentpaper: Theorem ([Kin]) . Assume Y is quasi-projective. Then we have e ( T ∗ [ − Y ) = ( − vdim Y · (vdim Y − / [ Y ] vir . In other words, this theorem gives a new construction of the virtual fundamentalclass (at least for quasi-projective cases). It is an interesting problem to constructother enumerative invariants (e.g. Donaldson–Thomas type invariants for Calabi–Yau 4-folds constructed in [CL14, BJ17, OT20]) based on the isomorphism (1.2) orits variant. This direction will be investigated in future work.1.4.
Plan of the paper.
This paper is organized as follows. In Section 2 we re-call some basic facts used in CoDT theory. In Section 3 we prove the dimensionalreduction theorem for quasi-smooth derived schemes, by gluing the local dimen-sional reduction isomorphisms in [Dav17, Theorem A.1]. In Section 4 we extendthe dimensional reduction theorem to quasi-smooth derived Artin stacks. The keypoint of the proof is the observation that the canonical − − Acknowledgments .I am deeply grateful to my supervisor Yukinobu Toda for introducing cohomo-logical Donaldson–Thomas theory to me, for sharing many ideas, and for usefulcomments on the manuscript of this paper. I also thank Ben Davison for answeringseveral questions related to this work. I was partly supported by WINGS-FMSPprogram at the Graduate School of Mathematical Science, the University of Tokyo.
Notation and convention . • All commutative dg algebras (cdga for short) sit in non-positive degree withrespect to the cohomological grading. • All derived or underived Artin stacks are assumed to be 1-Artin, and tohave quasi-compact and quasi-separated diagonals. • All cdgas and derived Artin stacks are assumed to be locally of finite pre-sentation over the complex number field C . • Denote by S , dSch , and dSt the ∞ -categories of spaces, derived schemes,and derived stacks respectively. IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 5 • For a derived scheme or a derived Artin stack X , t ( X ) denotes the classicaltruncation. Denote by X red = t ( X ) red the reduction of X . • For a morphism of derived Artin stacks f : X → Y , L f denotes the relativecotangent complex. We write L X for the absolute cotangent complex of X ,and T X := L ∨ X for the tangent complex. • For a derived Artin stack X , vdim X denotes the locally constant functionon t ( X ) whose value at p ∈ t ( X ) is P ( − i H i ( L X | p ). We define vdim f for a morphism locally of finite presentation between derived Artin stacks f in a similar manner. • For a derived Artin stack X and a perfect complex E on X , we define c det( E ) := det( E | X red ). • For a complex analytic space or a scheme X , we will only consider (analyti-cally) constructible sheaves or perverse sheaves which are of Q -coefficients.Denote by D bc ( X, Q ) the full subcategory of the bounded derived categoryof sheaves of Q -vector spaces D b ( X, Q ) spanned by the complexes with(analytically) constructible cohomology sheaves. • Concerning sign conventions for derived categories, we always follow [Stacks,Tag 0FNG]. • If there is no confusion, we use expressions such as f ∗ , f ! , and H om for thederived functors Rf ∗ , Rf ! , and R H om .2. Shifted symplectic structures and vanishing cycles
In this section, we briefly recall the notion of shifted symplectic structures intro-duced by [PTVV13], and some facts about vanishing cycle functors which will beneeded later.2.1.
Shifted symplectic geometry.
Here we briefly recall some notions in de-rived algebraic geometry and shifted symplectic geometry.
Definition 2.1.
A derived Artin stack X is called quasi-smooth if the cotangentcomplex L X is perfect of amplitude [ − , U be a smooth scheme, and s ∈ Γ( U, E ) be a section of a vector bundle E on U . Write Z ( s ) for the derived zero locus of s . We have the following Cartesiandiagram in dSch Z ( s ) ❴✤ f / / g (cid:15) (cid:15) U s (cid:15) (cid:15) U / / E. Since L g [ −
1] and g ∗ L U are locally free sheaves concentrated in degree zero, weconclude that Z ( s ) is quasi-smooth. Definition 2.2.
For a quasi-smooth derived scheme X , a Kuranishi chart is a tuple(
Z, U, E, s, ι ) where Z is an open subscheme of t ( X ), U is a smooth scheme, E isa vector bundle on U , s is a section of E , and ι : Z ( s ) → X is an open immersionwhose image is Z . A Kuranishi chart is said to be minimal at p = ι ( q ) ∈ Z if thedifferential ( ds ) q : T q U → E q is zero. A Kuranishi chart ( Z, U, E, s, ι ) is called good if U is affine and has global ´etale coordinate i.e. regular functions x , x , . . . , x n such that d dR x , d dR x , . . . , d dR x n form a basis of Ω U , and E is a trivial vectorbundle of a constant rank. TASUKI KINJO
Proposition 2.3.
Let X be a quasi-smooth derived scheme, and p ∈ X be a point. (i) [BBJ19, Theorem 4.1] There exists a Kuranishi chart ( Z, U, E, s, ι ) of X minimal at p ∈ Z . (ii) [BBJ19, Theorem 4.2] For i = 1 , , let ( Z i , U i , E i , s i , ι i ) be a Kuranishichart on X minimal at p = ι i ( q i ) . Then there exists a third Kuranishi chart ( Z ′ , U ′ , E ′ , s ′ , ι ′ ) of X minimal at p = ι ′ ( q ′ ) , ´etale morphisms η i : U ′ → U i ,and isomorphisms τ i : E ′ → η ∗ i E i with the following properties: – τ i ( s ′ ) = η ∗ i s i . – The composition Z ( s ′ ) → Z ( s i ) ι i −→ X is equivalent to ι ′ where thefirst map is induced by η i and τ i .Proof. (i) is a direct consequence of [BBJ19, Theorem 4.1].(ii) follows from [BBJ19, Theorem 4.2] except for η i being ´etale and τ i beingisomorphism. Since ( Z i , U i , E i , s i , f i ) is minimal at q i , we haveH ( L Z ( s i ) | q i ) ∼ = Ω U i | q i , H − ( L Z ( s i ) | q i ) ∼ = E ∨ i | q i . Similarly, we haveH ( L Z ( s ′ ) | q ′ ) ∼ = Ω U ′ | q ′ , H − ( L Z ( s ′ ) | q ′ ) ∼ = E ′∨ | q ′ . Since the open immersion Z ( s ′ ) → Z ( s i ) induces a quasi-isomorphism L Z ( s i ) | q i ≃ L Z ( s ′ ) | q ′ , we see that η i is ´etale at q ′ and τ i is isomorphic at q ′ . Thus by shrinking U ′ around q ′ if necessary, we obtain the required properties. (cid:3) Let A be a cdga, and take a semi-free resolution A ′ → A . We define the spaceof n -shifted p -forms and the space of n -shifted closed p -forms by A p ( A, n ) := | ( ∧ p Ω A ′ [ n ] , d ) | , A p,cl ( A, n ) := | ( Y i ≥ ∧ p + i Ω A ′ [ − i + n ] , d + d dR ) | respectively where d is the internal differential induced by the differential of Ω A ′ ,and d dR is the de Rham differential. Here for a dg module E , | E | denotes the simpli-cial set corresponding to the truncation τ ≤ ( E ) by the Dold–Kan correspondence.These constructions can be made ∞ -functorial, and they satisfy the sheaf conditionwith respect to the ´etale topology (see [PTVV13, Proposition 1.11]). Therefore weobtain ∞ -functors A p ( − , n ) , A p,cl ( − , n ) : dSt op → S . We write f ⋆ for A p ( f , n ) and also for A p,cl ( f , n ). For a derived Artin stack X , itis shown in [PTVV13, Proposition 1.14] that we have an equivalence A p ( X , n ) ≃ Map( O X , ∧ p L X [ n ] ) . (2.1)We have natural morphisms π : A p,cl ( − , n ) → A p ( − , n )d dR : A p ( − , n ) → A p +1 ,cl ( − , n )where π is induced by the projection Q i ≥ ∧ p + i Ω A ′ [ − i + n ] → ∧ p Ω A ′ [ p ], and d dR is induced by the map A p ( A, n ) ∋ ω ( d dR ω , , , . . . ) ∈ A p +1 ,cl ( A, n ) . IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 7
Definition 2.4.
Let X be a derived Artin stack. A closed n -shifted 2-form ω ∈A ,cl ( X , n ) is called an n -shifted symplectic structure if π ( ω ) is non-degenerate (i.e.the morphism T X → L X [ n ] induced by π ( ω ) using the identification (2.1) is anequivalence). An n -shifted symplectic derived Artin stack is a derived Artin stackequipped with an n -shifted symplectic structure on it.We say that two shifted symplectic derived Artin stacks ( X , ω X ) and ( X , ω X )are equivalent if there exists an equivalence Φ : X ∼ −→ X as derived stacks and anequivalence Φ ⋆ ω X ∼ ω X . Example 2.5.
Let U = Spec A be a smooth affine scheme and f : U → C be aregular function on it. Denote by X the derived critical locus Crit ( f ). Since X is the derived zero locus of the section d dR f : U → Ω U , the derived scheme X isequivalent to Spec B where B is a cdga defined by the Koszul complex B := ( · · · → ∧ Ω ∨ A · d dR f −−−−→ Ω ∨ A · d dR f −−−−→ A ) . Assume there exists a global ´etale coordinate ( x , . . . , x n ) on U . We write the dualbasis of d dR x , . . . , d dR x n by ∂∂x , . . . , ∂∂x n , and y i ∈ B − denotes the element ofdegree − ∂∂x i for each i = 1 , . . . , n . Although B is not semi-free in general, one can see that Ω B gives a model for L B . Define an element ω ′ X ∈ ( ∧ Ω B ) − of degree − ω ′ X := d dR x ∧ d dR y + · · · + d dR x n ∧ d dR y n . This defines a − d dR ω ′ X = 0, the closed form ω X := ( ω ′ X , , . . . ) defines a − X . Example 2.6.
Let Y be a derived Artin stack, and n be an integer. Definethe n -shifted cotangent stack of Y by T ∗ [ n ] Y := Spec Y (Sym( T Y [ − n ])). Let π : T ∗ [ n ] Y → Y be the projection. We have a tautological n -shifted 1-form λ T ∗ [ n ] Y on T ∗ [ n ] Y defined by the image of the tautological section of π ∗ L Y [ n ] under thecanonical map π ∗ L Y [ n ] → L T ∗ [ n ] Y [ n ]. In [Cal19, Theorem 2.2] it is shown that ω T ∗ [ n ] Y := d dR λ T ∗ [ n ] Y is non-degenerate, and we obtain the canonical n -shiftedsymplectic structure on T ∗ [ n ] Y .It is proved in [BBJ19, Theorem 5.18] that any − − T ∗ [ − Y for some quasi-smooth derivedscheme Y , its local model as derived critical locus can be described by combiningProposition 2.3 and the following lemma. Lemma 2.7.
Let U = Spec A be a smooth affine scheme admitting a global ´etalecoordinate, E be a trivial vector bundle on U , and s ∈ Γ( U, E ) be a section. De-note by ¯ s the regular function on Tot U ( E ∨ ) corresponding to s . Then we have anequivalence of − -shifted symplectic derived schemes ( Crit (¯ s ) , ω Crit (¯ s ) ) ≃ ( T ∗ [ − Z ( s ) , ω T ∗ [ − Z ( s ) )(2.2) equipped with the − -shifted symplectic structures constructed in Example 2.5 andExample 2.6 respectively.Proof. Fix a global ´etale coordinate x , . . . , x n on U and a basis e , . . . , e m of M :=Γ( U, E ). Write s = a e + · · · + a m e m . If we write α i := d dR x i , then α , . . . , α n TASUKI KINJO defines a basis of Ω U . Denote by e ∨ , . . . , e ∨ m and α ∨ , . . . , α ∨ n the dual bases of e , . . . , e m and α , . . . , α n respectively. Define a cdga C by the Koszul complex C := ( · · · → (Ω ∨ A ⊕ M ∨ ) ⊗ A A [ z , . . . , z m ] α ∨ i ⊗ P ( ∂a j /∂x i ) z j ,e ∨ j ⊗ a j −−−−−−−−−−−−−−−−−−−−−→ A [ z , . . . , z m ]) . Now it is clear that
Spec C gives models for both Crit (¯ s ) and T ∗ [ − Z ( s ). Thetautologial 1-form λ T ∗ [ − Z ( s ) is represented by n X i =1 α ∨ i ( d dR x i ) + m X j =1 z j ( d dR e ∨ j ) ∈ A ( Spec C, − . A direct computation shows that d dR λ T ∗ [ − Z ( s ) ∼ ω Crit (¯ s ) , which implies thelemma. (cid:3) D-critical schemes.
In this section, we briefly recall the notion of d-criticalstructures introduced in [Joy15] which is a classical model for − − X , Joyce in [Joy15, Theorem 2.1] introduced asheaf S X ∈ Mod( C X )(2.3)of C -vector space on X with the following property: for any open subset R ⊂ X and any closed embedding i : R ֒ → U where U is a complex manifold, we have anexact sequence of sheaves on R → S X | R → i − O U /I R,U d dR −−→ i − Ω U / ( I R,U · i − Ω U ) . Here I R,U is the ideal sheaf of i − O U corresponding to R . The following composition S X | R → i − O U /I R,U → i − O U /I R,U ∼ = O R glue to define a morphism β X : S X → O X , and we define a subsheaf S X of S X bythe kernel of the composition S X β X −−→ O X → O X red . It can be shown that we have a decomposition S X = C X ⊕ S X where C X is theconstant sheaf and C X ֒ → S X is induced by the inclusion C U ֒ → O U identifying C U with the sheaf of locally constant functions on U . If X is the critical locus of somefunction f on a complex manifold U such that f | X red = 0, then f + I X,U defines anelement of Γ( X, S X ) since d dR f | X = 0. Definition 2.8. [Joy15, Definition 2.5] Let X be a complex analytic space. Asection s ∈ Γ( X, S X ) is called an (analytic) d-critical structure if for any closedpoint x ∈ X there exist an open neighborhood x ∈ R ⊂ X , a complex manifold U , a regular function f on U with f | R red = 0, and a closed embedding i : R → U such that i ( R ) = Crit( f ) and f + I R,U = s | R . The tuple ( R, U, f, i ) as above iscalled a d-critical chart for (
X, s ). A d-critical scheme is a scheme equipped witha d-critical structure on its analytification.
Remark 2.9.
Joyce [Joy15, Definition 2.5] also introduced the algebraic versionof the d-critical structure, and some authors define a d-critical scheme as a schemeequipped with an algebraic d-critical structure. We always work with analyticd-critical structures since they are enough for our purposes.
IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 9
For a d-critical chart (
R, U, f, i ) of a d-critical scheme (
X, s ) and an open subset U ′ ⊂ U , define R ′ = i − ( U ′ ), f ′ = f | U ′ , and i ′ = i | R ′ : R ′ ֒ → U ′ . Then ( R ′ , U ′ , f ′ , i ′ )defines a d-critical chart on ( X, s ). We call ( R ′ , U ′ , f ′ , i ′ ) an open subchart of( R, U, f, i ).In order to compare two d-critical charts, Joyce introduced the notion of em-bedding : for a d-critical scheme (
X, s ) and its d-critical charts ( R , U , f , i ) and( R , U , f , i ) such that R ⊂ R , an embedding ( R , U , f , i ) ֒ → ( R , U , f , i )is defined by a locally closed embedding Φ : U ֒ → U such that f = f ◦ Φ andΦ ◦ i = i | R . The following theorem is useful when comparing two d-criticalcharts: Theorem 2.10.
Let ( X, s ) be a d-critical scheme. (i) [Joy15, Theorem 2.20] Let ( R , U , f , i ) and ( R , U , f , i ) be d-criticalcharts, and x ∈ R ∩ R be a point. Then by shrinking these d-critical chartsaround x if necessary, we can find a third d-critical chart ( R , U , f , i ) with x ∈ R and embeddings ( R , U , f , i ) ֒ → ( R , U , f , i ) and ( R , U , f , i ) ֒ → ( R , U , f , i ) . (ii) [Joy15, Theorem 2.22] Let
Φ : ( R , U , f , i ) ֒ → ( R , U , f , i ) be an em-bedding of d-critical charts, and x ∈ R be a point. Then by shrinkingthese d-critical charts around x keeping Φ( U ) ⊂ U if necessary and re-placing Φ by its restriction, we can find holomorphic maps α : U → U and β : U → C n for n = dim U − dim U , such that ( α, β ) : U → U × C n is biholomorphic onto its image, and we have α ◦ Φ = id , β ◦ Φ = 0 , and f = f ◦ α + ( z + · · · + z n ) ◦ β where z i is the i -th coordinate of C n . For an embedding of d-critical charts Φ : ( R , U , f , i ) ֒ → ( R , U , f , i ) of ad-critical scheme ( X, s ), Joyce defined in [Joy15, Definition 2.26] a natural isomor-phism J Φ : i ∗ ( K ⊗ U ) | R red1 ∼ = i ∗ ( K ⊗ U ) | R red1 . If there exist α, β as in Theorem 2.10 (ii), J Φ is defined as follows. Firstly, we haveisomorphisms K U ∼ = ( α, β ) ∗ ( K U × C n ) ∼ = α ∗ K U ⊗ β ∗ K C n ∼ = α ∗ K U where the final isomorphism is defined by the trivialization K C n ∼ = O C n · (d z ∧ · · · ∧ d z n ) . Then by taking the square of this composition and pulling back to R , we obtainthe desired isomorphism.Using this preparation, we can construct a natural line bundle K X,s on X red ,which is a d-critical version of the canonical line bundle as follows: Theorem 2.11. [Joy15, Theorem 2.28]
For a d-critical scheme ( X, s ) , one candefine a line bundle K X,s on X red which we call the virtual canonical bundle of ( X, s ) characterized by the following properties: (i) For a d-critical chart R = ( R, U, f, i ) of ( X, s ) , we have an isomorphism ι R : K X,s | R red ∼ = ( i ∗ K U ) ⊗ | R red . (ii) For an embedding of d-critical charts
Φ : R = ( R , U , f , i ) ֒ → R = ( R , U , f , i ) , we have ι R | R red1 = J Φ ◦ ι R . Definition 2.12. [Joy15, Definition 2.31] An orientation o of a d-critical scheme( X, s ) is a choice of a line bundle L on X red and an isomorphism o : L ⊗ ∼ = −→ K X,s . As we have seen in the previous section, − − Theorem 2.13. [BBJ19, Theorem 6.6]
Let ( X , ω X ) be a − -shifted symplecticderived scheme. Then its underlying scheme X = t ( X ) carries a canonical d-critical structure s X with the following property: for any − shifted symplecticderived scheme ( R , ω R ) of the form R = Crit ( f ) where f is a regular function on asmooth scheme U , ω R the − -shifted symplectic form on R constructed in Example2.5, and an open inclusion ι : R ֒ → X such that ι ∗ ω X ∼ ω R , the tuple ( R, U, f, i ) gives a d-critical chart for ( X, s X ) where we write R = t ( R ) and i : R ֒ → U thenatural closed embedding. Furthermore there exists a canonical isomorphism of linebundles Λ X : c det( L X ) ∼ = K X,s X . where c det( L X ) = det( L X | X red ) by definition. We define the notion of orientations for − X in the above theorem for X = Crit ( f ) where f is a regular function on a smooth scheme U . In this case, L X | X is represented by the following two-term complex( T U | X Hess( f ) −−−−−→ Ω U | X )where Hess( f ) denotes the Hessian of f . We define Λ X by the following composi-tion: c det( L X ) ∼ = det(Ω U | X red ) ⊗ det( T U | X red ) − ∼ = ( i ∗ K U ) | ⊗ X red · ( ) dim U −−−−−−→ ( i ∗ K U ) | ⊗ X red ∼ = K X,s . (2.4)where det( T U ) − ∼ = K U is locally defined by( ∂/∂z ∧ · · · ∧ ∂/∂z n ) ∨ d z ∧ · · · ∧ d z n . (2.5)The constant ( ) dim U is just a convention, and it corresponds to the fact thatHess( z ) = 2(d z ) ⊗ .Now we discuss the canonical orientation for − Lemma 2.14.
Let X be a derived Artin stack. (i) For a perfect complex E on X and an integer m , we have natural isomor-phisms ˆ η E : c det( E ∨ ) ∼ = c det( E ) − , ˆ χ ( m ) E : c det ( E [ m ]) ∼ = c det( E ) ( − m . IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 11 (ii)
For a distinguished triangle ∆ : E → F → G → E [1] of perfect complexeson X , we have a natural isomorphism ˆ ı (∆) : c det( E ) ⊗ c det( G ) ∼ = c det( F ) . We write ˆ χ E = ˆ χ (1) E . The following example defines the canonical orientation for − Example 2.15. [Tod19, Lemma 4.3] Let Y be a quasi-smooth derived scheme, π : T ∗ [ − Y → Y be the projection and write s T ∗ [ − Y the d-critical structureassociated with the canonical − π : π ∗ L Y → L T ∗ [ − Y → π ∗ T Y [1] → L Y [1] . (2.6)Define an isomorphism o ′ T ∗ [ − Y : ( π red ) ∗ c det( L Y ) ⊗ ∼ = c det( L T ∗ [ − Y )by the composition of the isomorphisms( π red ) ∗ c det( L Y ) ⊗ id ⊗ ˆ η π ∗ T Y −−−−−−−→ ( π red ) ∗ c det( L Y ) ⊗ ( π red ) ∗ c det( T Y ) − ⊗ ˆ χ − π ∗ T Y −−−−−−−→ ( π red ) ∗ c det( L Y ) ⊗ ( π red ) ∗ c det( T Y [1]) ˆ ı (∆ π ) −−−−→ c det( L T ∗ [ − Y )where we write π = t ( π ). Define o T ∗ [ − Y : ( π red ) ∗ c det( L Y ) ⊗ ∼ = K T ∗ [ − Y ,s T ∗ [ − Y (2.7)by the composition Λ T ∗ [ − Y ◦ o ′ T ∗ [ − Y and we call this the canonical orientation for T ∗ [ − Y .2.3. Vanishing cycle complexes.
Let f be a holomorphic function on a com-plex manifold U , and set U = f − (0) and U ≤ = f − ( { z ∈ C | Re( z ) ≤ } ).The (shifted) vanishing cycle functor ϕ pf : D bc ( U, Q ) → D b ( U , Q ) is defined by thecomposition of the functors ϕ pf := ( U ֒ → U ≤ ) ∗ ( U ≤ ֒ → U ) ! . The functor ϕ pf preserves the constructibility (see e.g. [Dim04, Definition 4.2.4,Proposition 4.2.9]). The canonical morphism ( U ≤ ֒ → U ) ! → ( U ≤ ֒ → U ) ∗ inducesa natural transform γ f : ϕ pf → ( U ֒ → U ) ∗ . (2.8)Here we list basic properties of the functor ϕ pf we use later: Proposition 2.16.
Let U be a complex manifold and f be a holomorphic functionon it. Write U = f − (0) . (i) If F is a perverse sheaf on U , then ϕ pf ( F ) is also a perverse sheaf. (ii) The support of ϕ pf ( Q U ) is contained in Crit( f ) . (iii) Let q : V → U be a holomorphic map where V is a complex manifold,and q : V → U denotes the restriction of q where V = ( f ◦ q ) − (0) .Then we have a canonical morphism Θ q,f : q ∗ ϕ pf ( F ) → ϕ pf ◦ q ( q ∗ F ) for each F ∈ D bc ( U, Q ) , which is an isomorphism if q is a submersion. Further, thefollowing diagram commutes: q ∗ ϕ pf ( F ) Θ q,f / / q ∗ γ f ( F ) % % ▲▲▲▲▲▲▲▲▲▲ ϕ pf ◦ q ( q ∗ F ) γ f ◦ q ( q ∗ F ) x x qqqqqqqqqq q ∗ ( F | U ) . (2.9)(iv) (Thom–Sebastiani) Let V be a complex manifold, g be a holomorphic func-tion on it, and f ⊞ g be the function on U × V defined by ( f ⊞ g )( u, v ) = f ( u ) + g ( v ) . For F ∈ D bc ( U, Q ) and G ∈ D bc ( V, Q ) , we have a canonicalisomorphism T S f,g,F,G : ϕ pf ⊞ g ( F ⊠ G ) | U × V ∼ = ϕ pf ( F ) ⊠ ϕ pg ( G ) where V = g − (0) . Further, the following diagram commutes: ϕ pf ⊞ g ( F ⊠ G ) | U × V γ f ⊞ g ( F ⊠ G ) | U × V / / T S f,g,F,G ∼ (cid:15) (cid:15) ( F ⊠ G ) | U × V ∼ (cid:15) (cid:15) ϕ pf ( F ) ⊠ ϕ pg ( G ) γ f ( F ) ⊠ γ g ( G ) / / F | U ⊠ G | U (2.10)(v) (Verdier duality) For F ∈ D bc ( U, Q ) , there exists a canonical isomorphism D U ( ϕ pf ( F )) ∼ = ϕ pf ( D U ( F )) where D U and D U denote the Verdier duality functors on U and U respec-tively.Proof. (i) is proved in [KaSc90, Corollary 10.3.13]. (ii) easily follows from thedefinition. (iii) follows from the smooth base change theorem. (iv) is proved in[Sch03, Corollary 1.3.4]. (v) is proved in [Mas16]. (cid:3) By abuse of notation, we write ϕ pf = ϕ pf ( Q U [dim U ]) if there is no confusion. Weidentify i -th cohomology of the stalk of ϕ pf at some point with the ( i + dim U )-threlative cohomology of a ball modulo the Milnor fiber at a small positive value. Weregard ϕ pf as a perverse sheaf on U , U , or Crit( f ) depending on each situation.If we write z : C → C to be the identity map, we have natural isomorphisms( ϕ pz ) ∼ = H ( C , { z ∈ C | Re( z ) > } ; Q ) ∼ = H ( R , R \ Q ) . (2.11)The latter isomorphism is induced by the inclusion( R , R \ ֒ → ( C , { z ∈ C | Re( z ) > } ) . The orientation of R given by the positive direction defines a cohomology class a + ∈ H ( R , R \ Q ), hence a trivialization h ,z : ϕ pz ∼ = Q . (2.12)Let ( z , . . . , z n ) be the standard coordinate of C n . Then Thom–Sebastiani theoremand (2.12) gives an isomorphism h n, ( z ,...,z n ) : ϕ pz + ··· + z n | (0 ,..., ∼ = Q (0 ,..., . (2.13) IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 13
We now recall the construction of the globalization of the vanishing cycle com-plexes associated with oriented d-critical schemes introduced in [BBDJS15]. To dothis we introduce the following notation. For a scheme X , a principal Z / Z -bundle P on X , and F ∈ D bc ( X, Q ), one defines F ⊗ Z / Z P := F ⊗ Q X ( Q X ⊗ Z X / Z X P )where Z X / Z X -module structure on Q X is defined by the multiplication by −
1. Fora d-critical scheme (
X, s ) with a fixed orientation o : L ⊗ ∼ −→ K X,s and its d-criticalchart R = ( R, U, f, i ), we define a principal Z / Z -bundle Q o R (2.14)over R whose sections are local isomorphisms a : L → ( i ∗ K U ) | R red such that a ⊗ = ι R,U,f,i ◦ o . For an embedding of d-critical chartsΦ : R = ( R, U , f , i ) ֒ → R = ( R, U , f , i )such that α : U → U and β : U → C n as in Theorem 2.10 (ii) exist,( Q o R ) − ⊗ Q o R parameterizes square roots of i ∗ β ∗ (d z ∧ · · · ∧ d z n ) | ⊗ R red . Thus the choice i ∗ β ∗ (d z ∧ · · · ∧ d z n ) | R red gives an isomorphism Q o R ∼ = Q o R . (2.15)On the other hand, we have isomorphisms i ∗ ϕ pf ∼ = i ∗ α ∗ ϕ pf ⊗ β ∗ ϕ pz + ··· + z n ∼ = i ∗ ϕ pf (2.16)where the first one is defined using (2.12) and the second one is Thom–Sebastianiisomorphism.Under these notations, the globalized vanishing cycle complex is defined by thefollowing theorem: Theorem 2.17. [BBDJS15, Theorem 6.9]
Let ( X, s, o ) be an oriented d-criticalscheme, R = ( R , U , f , i ) and R = ( R , U , f , i ) be any d-critical charts with R ⊂ R . Then there exists a natural isomorphism Υ R , R : i ∗ ϕ pf ⊗ Z / Z Q o R → i ∗ ϕ pf ⊗ Z / Z Q o R | R with the following properties: (i) If we are given another d-critical chart R = ( R , U , f , i ) with R ⊂ R ,we have Υ R , R = Υ R , R | R ◦ Υ R , R . (ii) If R is an open subchart of R , then Υ R , R is defined by the canonicalisomorphisms ϕ pf ∼ = ϕ pf | U , Q o R ∼ = Q o R | U . (iii) For an embedding of d-critical charts R = ( R, U , f , i ) ֒ → R = ( R, U , f , i ) such that α : U → U and β : U → C n as in Theorem 2.10 (ii) exist, Υ R , R is defined by isomorphisms (2.15) and (2.16) .Using (i), we can define a perverse sheaf ϕ pX,s,o on X such that for a given d-criticalchart R = ( R, U, f, i ) there exists a natural isomorphism ω R : ϕ pX,s,o | R ∼ = i ∗ ϕ pf ⊗ Z / Z Q o R . Moreover, there exists an isomorphism σ X,s,o : D X ( ϕ pX,s,o ) ∼ = ϕ pX,s,o . For later use, we recall the construction of σ X,s,o . For a d-critical chart R =( R, U, f, i ), the Verdier self-duality of ϕ pf induces an isomorphism σ ′ R : D X ( ϕ pX,s,o ) | R ∼ = ϕ pX,s,o | R . (2.17)If we define σ R = ( − dim U · (dim U − / σ ′ R , one can show that it glues to define an isomorphism σ X,s,o (the necessity of thesign intervention is due to the fact that the first diagram in [BBDJS15, Theorem2.13] commutes up to the sign ( − dim U · dim V ). If there is no confusion, we write σ X = σ X,s,o .For an oriented − X , ω X , o ), define ϕ X ,ω X ,o to be the perverse sheaf ϕ pX,s X ,o on X = t ( X ) where s X is the d-critical structureassociated with ω X . If there is no confusion, we simply write ϕ X ,o or ϕ X insteadof ϕ X ,ω X ,o .2.4. Dimensional reduction.
Let U be a smooth variety of dimension n , and s be a section of a trivial vector bundle E of rank r on U . Denote ¯ s : Tot U ( E ∨ ) → A the regular function corresponding to s . We have a canonical morphism γ ¯ s ( Q Tot U ( E ∨ ) [ n + r ]) : ϕ p ¯ s → Q ¯ s − (0) [ n + r ] . (2.18)Define Z := Z ( s ) to be the zero locus of s , and e Z := ( π E ∨ ) − ( Z ) where π E ∨ : Tot U ( E ∨ ) → U is the projection. By restricting (2.18) to e Z , we obtain γ ¯ s := γ ¯ s ( Q Tot U ( E ∨ ) [ n + r ]) | e Z : ϕ p ¯ s → Q e Z [ n + r ] . (2.19)Here we identify ϕ p ¯ s and ϕ p ¯ s | e Z since the support of ϕ p ¯ s is contained in e Z . Theorem 2.18. [Dav17, Theorem A.1]
The natural map ¯ γ ¯ s := ( π E ∨ ) ! γ ¯ s : ( π E ∨ ) ! ϕ p ¯ s → ( π E ∨ ) ! Q e Z [ n + r ] ∼ = Q Z [ n − r ](2.20) is an isomorphism. We want to globalize this statement for the − Z / Z -bundleintroduced in (2.14) associated with the canonical orientations for shifted cotangentschemes and certain d-critical charts: IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 15
Lemma 2.19.
Let
U, s be as above, and Z := Z ( s ) be the derived zero locus of s .Assume U is affine and carries a global ´etale coordinate. Denote by o T ∗ [ − Z thecanonical orientation constructed in Example 2.15 and f Z = (Crit(¯ s ) , Tot U ( E ∨ ) , ¯ s, i ) the d-critical chart induced by the equivalence (2.2) . Then Q o T ∗ [ − Z f Z is a trivial Z / Z -bundle.Proof. The distinguished triangle (2.6) for Z restricted to Crit(¯ s ) is represented bythe following short exact sequence of two term complexes:0 / / π ∗ E ∨ E ∨ | Crit(¯ s ) / / (d s ) ∨ (cid:15) (cid:15) T Tot U ( E ∨ ) | Crit(¯ s ) / / Hess(¯s) (cid:15) (cid:15) π ∗ E ∨ T U | Crit(¯ s ) / / d s (cid:15) (cid:15) / / π ∗ E ∨ Ω U | Crit(¯ s ) / / Ω Tot U ( E ∨ ) | Crit(¯ s ) / / π ∗ E ∨ E | Crit(¯ s ) / / . (2.21)Thus the canonical orientation (2.7) for Crit (¯ s ) is identified with µ : ( c det( π ∗ E ∨ Ω U ) ⊗ c det( π ∗ E ∨ E ∨ ) − ) ⊗ ∼ = c det(Ω Tot U ( E ∨ ) ) ⊗ . (2.22)where c det( π ∗ E ∨ E ) ⊗ c det( π ∗ E ∨ T U ) − ∼ = c det( π ∗ E ∨ Ω U ) ⊗ c det( π ∗ E ∨ E ∨ ) − and det( T U ) − ∼ = det(Ω U )are defined in the same manner as (2.5). On the other hand, we have an isomor-phism ν : c det( π ∗ E ∨ Ω U ) ⊗ c det( π ∗ E ∨ E ∨ ) − ∼ = c det( π ∗ E ∨ Ω U ) ⊗ c det( π ∗ E ∨ E ) ∼ = c det(Ω Tot U ( E ∨ ) ) . (2.23)where the first isomorphism is defined as (2.5) and the second isomorphism isinduced by the lower short exact sequence in (2.21). By the definition of o T ∗ [ − Z and (2.4) we see that µ = ( − ( n + r )( n + r − / (1 / n + r · ν ⊗ (2.24)where we write n = dim U and r = rank E . The appearance of the sign ( − ( n + r )( n + r − / is caused by the difference of the maps (2.5) and (A.2), and the difference of thesymmetric monoidal structure for the category of graded line bundles (A.1) and thestandard symmetric monoidal structure for the category of ungraded line bundles.The equality (2.24) implies the triviality of Q o T ∗ [ − Z f Z . (cid:3) For later use, we explicitly choose a trivialization of Q o T ∗ [ − Z f Z . For each ( a, b ) ∈ Z ≥ , take ǫ a,b ∈ { , − , √− , −√− } so that • ǫ , = 1. • ǫ a,b = ( − b √− ǫ a − ,b − • ǫ a +1 ,b = ( −√− a − b ǫ a,b . Then ǫ n,r (1 / √ n + r · ν (2.25)gives a square root of µ , hence a trivialization of Q o T ∗ [ − Z f Z . Corollary 2.20.
Let Y be a quasi-smooth derived scheme , π Y : T ∗ [ − Y → Y the projection and write π Y = t ( π Y ) . Then ( π Y ) ! ϕ p T ∗ [ − Y is a rank one localsystem shifted by vdim Y .Proof. Since the statement is local, we may assume Y is a derived zero locus Z ( s )as in the previous lemma. The conclusion of the lemma implies that ϕ p T ∗ [ − Y isisomorphic ϕ p ¯ s hence the statement follows from Theorem 2.18. (cid:3) Dimensional reduction for schemes
In this section, we will prove that the local system appeared in Corollary 2.20 isin fact trivial, by showing that the local dimensional reduction isomorphism (2.20)is independent of the choice of the Kuranishi chart.Let Y be a quasi-smooth derived scheme and π Y : T ∗ [ − Y → Y be the projec-tion. We always equip T ∗ [ − Y with the − o = o T ∗ [ − Y . Write π Y = t ( π Y ), e Y = t ( T ∗ [ − Y ), and Y = t ( Y ). Take a good Kuranishichart Z = ( Z, U, E, s, ι )of Y . The map ι induces an open immersion e ι : T ∗ [ − Z ( s ) ֒ → T ∗ [ − Y withthe image e Z := π − Y ( Z ). Lemma 2.7 shows that there exists a natural embedding˜ i : e Z ֒ → Tot U ( E ∨ ) such that f Z = ( e Z, Tot U ( E ∨ ) , ¯ s, ˜ i )gives a d-critical chart on e Y .Now we have an isomorphism( π Y ) ! ϕ p T ∗ [ − Y | Z ∼ = i ∗ ( π E ∨ ) ! ˜ i ∗ ( ϕ p T ∗ [ − Y | e Z )(3.1)where π E ∨ : Tot U ( E ∨ ) → U is the projection and i : Z ֒ → U is the natural embed-ding. By the definition of ϕ p T ∗ [ − Y in Theorem 2.17 we also have an isomorphism ω f Z : ˜ i ∗ ( ϕ p T ∗ [ − Y | e Z ⊗ Z / Z ( Q o f Z ) − ) ∼ = ϕ p ¯ s . (3.2)By combining isomorphisms (3.1) and (3.2), the trivialization of Q o f Z in (2.25), andTheorem 2.18, we obtain the following isomorphism:¯ γ Z : ( π Y ) ! ϕ p T ∗ [ − Y | Z ∼ = Q Z [vdim Y ] . (3.3) Theorem 3.1.
For i = 1 , , let Z i = ( Z i , U i , E i , s i , ι i ) be good Kuranishi chartson Y . Then we have ¯ γ Z | Z ∩ Z = ¯ γ Z | Z ∩ Z . Therefore there exists a naturalisomorphism ¯ γ Y : ( π Y ) ! ϕ p T ∗ [ − Y ∼ = Q Y [vdim Y ] . IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 17
We say that two Kuranishi charts Z = ( Z , U , E , s , ι ) and Z = ( Z , U , E , s , ι )have compatible dimensional reductions at p ∈ Z ∩ Z if there exists an analyticopen neighborhood p ∈ W ⊂ Z ∩ Z such that ¯ γ Z | W = ¯ γ Z | W . Lemma 3.2.
Let Z = ( Z , U , E , s , ι ) and Z = ( Z , U , E , s , ι ) be goodKuranishi charts on Y . Assume that these Kuranishi charts are minimal at p ∈ Z ∩ Z . Then they have compatible dimensional reductions at p .Proof. Denote by i : Z ֒ → U and i : Z ֒ → U the natural embeddings, and e Z = ( f Z , Tot U ( E ∨ ) , ¯ s , ˜ i ) , e Z = ( f Z , Tot U ( E ∨ ) , ¯ s , ˜ i )be d-critical charts associated with Z and Z respectively. Using Proposition 2.3(ii), we may assume that we have the following commutative diagramTot U ( E ∨ ) τ ∨ ∼ / / ¯ s * * ' ' PPPPPPPPPPPPP
Tot U ( η ∗ E ∨ ) ˜ η / / ❴✤ (cid:15) (cid:15) Tot U ( E ∨ ) ¯ s / / (cid:15) (cid:15) A U η / / U such that η is ´etale and η ( i ( p )) = i ( p ). The natural isomorphism(˜ η ◦ τ ∨ ) ∗ K Tot U ( E ∨ ) ∼ = K Tot U ( E ∨ ) induces an isomorphism Q o f Z ∼ = Q o f Z | f Z that identifies the trivializations (2.25). Then Threorem 2.17 (ii) implies that thefollowing composition˜ i ∗ ϕ p ¯ s ∼ = ˜ i ∗ ( ϕ p ¯ s ⊗ Z / Z Q o f Z ) ∼ = ˜ i ∗ ( ϕ p ¯ s ⊗ Z / Z Q o f Z ) | f Z ∼ = (˜ i ∗ ϕ p ¯ s ) | f Z (3.4)is the natural isomorphism ϕ p ¯ s ∼ = η ∗ ϕ p ¯ s given in Proposition 2.16 (iii) pulled back to f Z . Hence the commutativity of thediagram (2.9) implies the lemma. (cid:3) Proposition 3.3.
Let Z = ( Z, U, E, s, ι ) be a good Kuranishi chart on Y , whichis not minimal at p ∈ Z . Then there exists another good Kuranishi chart Z ′ =( Z ′ , U ′ , E ′ , s ′ , ι ′ ) with p ∈ Z ′ and dim U ′ < dim U such that Z and Z ′ have com-patible dimensional reductions at p .Proof. Take a trivialization E = O U · e ⊕ · · · ⊕ O U · e r and write s = f e + · · · f r e r . By the non-minimality assumption, we may assume that f = 0 and the zero locus Z ( f ) is smooth at p . Take a smooth affine open neighborhood p ∈ U ′ ⊂ Z ( f ) anddefine a vector bundle E ′ on U ′ by E ′ := ( O U · e ⊕ · · · ⊕ O U · e r ) | U ′ . Let s ′ ∈ Γ( U ′ , E ′ ) be the section induced by s | U ′ . Then we obtain a natural openimmersion of the derived zero loci Z ( s ′ ) ֒ → Z ( s ). Define ι ′ : Z ( s ′ ) → Y by the composition Z ( s ′ ) ֒ → Z ( s ) ι −→ Y , and denote its image by Z ′ . By shrinking around p if necessary, we may assume that Z ′ = ( Z ′ , U ′ , E ′ , s ′ , ι ′ )is a good Kuranishi chart. We prove that this Kuranishi chart has the desiredproperty.Firstly take a local coordinate x , . . . , x n of U around p U = ι − ( p ) with x = f and an analytic open neighborhood p U ∈ V in U which maps biholomorphicallyto a polydisc B nǫ ⊂ C n under ( x , . . . , x n ) Write V ′ = V ∩ U ′ . By shrinking V ifnecessary, we can write f i | V = x h i + r i ◦ ¯ α for each i ∈ { , . . . , n } where h i is a holomorphic function on V , r i is a holomorphicfunction on V ′ , and ¯ α : V → V ′ is the map identified with the projection B nǫ → B n − ǫ .Let f Z = ( e Z, Tot U ( E ∨ ) , ¯ s, ˜ i ) , f Z ′ = ( f Z ′ , Tot U ′ (( E ′ ) ∨ ) , ¯ s ′ , ˜ i ′ )be d-critical charts on e Y associated with Z and Z ′ respectively. Write g Z V = ( f W , e V , ¯ s | e V , ˜ i | f W ) , g Z ′ V ′ = ( f W , e V ′ , ¯ s ′ | e V ′ , ˜ i ′ | f W )the restrictions of f Z and f Z ′ , where we define W := i − ( V ) , f W := π − Y ( W ) , e V := Tot V ( E | V ) , e V ′ := Tot V ′ ( E ′ | V ′ ) . To simplify the notation, we write e i = e i | V , ¯ s = ¯ s | e V , ¯ s ′ = ¯ s ′ | f V ′ , ˜ i = ˜ i | f W , ˜ i ′ = ˜ i ′ | f W . Define a closed immersion linear over V ′ ֒ → V Φ : e V ′ ֒ → e V by e ∨ i | V ′
7→ − h i e ∨ + e ∨ i where e ∨ , . . . , e ∨ r is the dual basis of e , . . . , e r . A direct computation shows that Φdefines an embedding of d-critical charts g Z V ֒ → g Z ′ V ′ , in other words, we have the following commutative diagram e V ′ ¯ s ′ ' ' Φ / / e V ¯ s / / A f W ?(cid:31) ˜ i ′ O O f W ?(cid:31) ˜ i O O such that Crit(¯ s ′ ) = Im(˜ i ′ ) , Crit(¯ s ) = Im(˜ i ) . Now define a map linear over the projection ¯ α : V ։ V ′ α : e V ։ e V ′ by e ∨ i e ∨ i | V ′ IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 19 and a map linear over the projection x : V ։ B ǫ β = ( x , y ) : e V ։ B ǫ × C by e ∨ , e ∨ i h i ( i > . It is clear by the construction that α ◦ Φ = id e V ′ , β ◦ Φ = 0 , the map ( α, β ) : e V → e V ′ × ( B ǫ × C ) is isomorphic, and the following diagramcommutes: f W π Y | f W (cid:15) (cid:15) (cid:31) (cid:127) ˜ i / / e V ¯ s ) ) ( α,β ) ∼ / / π V (cid:15) (cid:15) e V ′ × ( B ǫ × C ) ¯ s ′ ⊞ x y / / π V ′ × x (cid:15) (cid:15) A W (cid:31) (cid:127) i / / V (¯ α,x ) ∼ / / V ′ × B ǫ Here π V and π V ′ are natural projections.Consider the following composition of morphisms of perverse sheaves on f W ˜ i ∗ ϕ p ¯ s ∼ = ˜ i ∗ ϕ p ¯ s ⊗ Z / Z Q o g Z V ∼ = ϕ T ∗ [ − Y ∼ = (˜ i ′ ) ∗ ϕ p ¯ s ′ ⊗ Z / Z Q o g Z ′ V ′ ∼ = (˜ i ′ ) ∗ ϕ p ¯ s ′ (3.5)where the first and final isomorphisms are induced by (2.25), the second and thirdisomorphisms are ω − f Z and ω f Z ′ defined in Theorem 2.17 respectively. Now we showthat this is equal to the following composition˜ i ∗ ϕ p ¯ s ∼ = i ∗ ( α, β ) ∗ ( ϕ p ¯ s ′ ⊠ ϕ px y ) ∼ = (˜ i ′ ) ∗ ϕ p ¯ s ′ (3.6)where the first map is the Thom–Sebastiani isomorphism, and the second map isconstructed by substituting z = ( x − y ) / √− z = ( x + y ) / h , ( z ,z ) in(2.13). To do this, it suffices to prove the commutativity of the following diagram,thanks to Theorem 2.17 (iii):( Z / Z ) f W ǫ n,r (1 / √ n + r · ν / / Q o f Z (cid:15) (cid:15) ( Z / Z ) f W ǫ n − ,r − (1 / √ n + r − · ν ′ / / Q o f Z ′ . Here the right vertical map is (2.15) and ν : ˜ i ∗ K e V ∼ = ( π red Y ) ∗ c det( L Y ) | f W , ν ′ : (˜ i ′ ) ∗ K e V ′ ∼ = ( π red Y ) ∗ c det( L Y ) | f W are constructed in the same manner as (2.23). The commutativity of the diagramabove is equivalent to the commutativity of the following diagram( π red Y ) ∗ c det( L Y ) | f W ǫ n,r (1 / √ n + r · ν / / ǫ n − ,r − (1 / √ n + r − · ν ′ (cid:15) (cid:15) (˜ i ) ∗ K e V ∼ (cid:15) (cid:15) (˜ i ′ ) ∗ K f V ′ a a ∧ d z ∧ d z / / (˜ i ) ∗ ( α, β ) ∗ K V ′ × C . The commutativity of this diagram follows by the definitions of ν and ν ′ , and theequations ǫ n,r /ǫ n − ,r − = ( − r √− z ∧ d z = ( −√− / x ∧ d y . Thereforewe have obtained the equality of isomorphisms (3.5) and (3.6).Now consider the following commutative diagram ϕ p ¯ s γ ¯ s / / ∼ (cid:15) (cid:15) Q e V [ n + r ] ∼ (cid:15) (cid:15) ( α, β ) ∗ ( ϕ p ¯ s ′ ⊠ ϕ px y ) ( α,β ) ∗ ( γ ¯ s ′ ⊠ γ x y ) / / ( α, β ) ∗ ( Q e V ′ [ n + r − ⊠ Q (0 , [2])where the left vertical map is induced by the Thom–Sebastiani isomorphism. Thecommutativity follows from the commutativity of the diagram (2.10). By applyingthe functor ( π V ) ! , we obtain the following commutative diagram( π V ) ! ϕ p ¯ s ∼ / / ∼ (cid:15) (cid:15) Q V [ n − r ] ∼ (cid:15) (cid:15) ( π V ) ! ( α, β ) ∗ ( ϕ p ¯ s ′ ⊠ ϕ px y ) ∼ (cid:15) (cid:15) (¯ α, x ) ∗ (( π V ′ ) ! ϕ p ¯ s ′ ⊠ ( x ) ! ϕ px y ) ∼ / / (¯ α, x ) ∗ ( Q e V ′ [ n − r ] ⊠ Q (0 , )By combining the commutativity of the diagram above and the equality of isomor-phisms (3.5) and (3.6), the proposition follows from the next lemma. (cid:3) Lemma 3.4.
The following diagram commutes ( x ) ! ϕ px y ( x ) ! h , ( z ,z / / ∼ ¯ γ y (cid:15) (cid:15) ( x ) ! Q (0 , ∼ (cid:15) (cid:15) Q Q (3.7) where we use z = ( x − y ) / √− and z = ( x + y ) / .Proof. Since x y is homogenous and Crit( x y ) has compact support, we have nat-ural isomorphismsH ( C , ( x ) ! ϕ px y ) ∼ = H ( C , { ( x , y ) ∈ C | Re( x y ) > } ; Q ) ∼ = H ( C , C \ C × { } ; Q ) . (3.8)The left vertical map in (3.7) is given by the Thom class of H ( C , C \ C × { } ; Q )and (3.8). Now consider the following composition of isomorphismsH ( C , ( x ) ! ϕ px y ) ∼ = H ( C , { ( x , y ) ∈ C | Re( x y ) > } ; Q )= H ( C , { ( z , z ) ∈ C | Re( z + z ) > } ; Q ) ∼ = H ( C , { ( z , z ) ∈ C | Re( z ) > , or Re( z ) > } ; Q ) ∼ = H ( C , { z ∈ C | Re( z ) > } ; Q ) ⊗ H ( C , { z ∈ C | Re( z ) > } ; Q ) ∼ = Q ⊗ Q ∼ = Q . (3.9) IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 21
The third isomorphism is the relative Kunneth isomorphism and the fourth isomor-phism is given by (2.12). Since the Thom–Sebastiani isomorphism is induced bythe relative Kunneth isomorphism (see [Sch03, p.62]), this composition correspondsto h , ( z ,z ) . Therefore we only need to show that Q → H ( C , C \ C × { } ; Q )constructed by combining (3.8) and (3.9) gives the Thom class. Consider the com-position R z ,z ) −−−−→ C y −→ C . (3.10)If we equip R with the product orientation of the positive directions, this compo-sition preserves the orientation. This proves the claim. (cid:3) By repeatedly using Proposition 3.3, we obtain the following corollary.
Corollary 3.5.
Under the assumption of Proposition 3.3, there exists a good Ku-ranishi chart Z ′ = ( Z ′ , U ′ , E ′ , s ′ , ι ′ ) containing and minimal at p , such that Z and Z ′ have compatible dimensional reductions at p .Proof of Theorem 3.1. By the sheaf property, it suffices to show that Z and Z have compatible dimensional reductions at each p ∈ Z ∩ Z . By Corollary 3.5, wemay assume these Kuranishi charts are minimal at p , and then the claim followsfrom Lemma 3.2. (cid:3) Remark 3.6.
For a d-critical scheme (
X, s ), it is shown in [BBDJS15, § ϕ X,s has a natural extension to a mixed Hodge module. We can extend Theorem3.1 to an isomorphism of mixed Hodge modules with the same proof as above.4.
Dimensional reduction for stacks
The aim of this section is to extend Theorem 3.1 to quasi-smooth derived Artinstacks.4.1.
Lisse-analytic topology.
We briefly recall the theory of lisse-analytic toposintroduced in [Sun17], which is a complex analytic analogue of the lisse-´etale topos.All statements in this section can be deduced in the same manner as in [Ols07] or[LO08], so we do not give detailed proofs.Let
AnSp denotes the site of complex analytic spaces equipped with the analytictopology. A stack in groupoid X over AnSp is called complex analytic stack if thefollowing conditions hold:(i) The diagonal morphism
X → X × X is representable by complex analyticspaces.(ii) There exists a smooth surjection U → X from a complex analytic space U . Definition 4.1.
Let X be a complex analytic stack. The lisse-analytic site Lis-An( X )is the site defined as follows: • The underlying category of Lis-An( X ) is the full subcategory of complexanalytic spaces over X spanned by ones smooth over X . • A family of morphisms { ( U i → X ) → ( U → X ) } i ∈ I is a covering if { U i → U } i ∈ I is an open covering.The topos X lis − an associated with Lis-An( X ) is called the lisse-analytic topos of X . It can be easily seen that a sheaf F ∈ X lis − an is given by the following data: • a sheaf F ( U,u ) on U for each ( u : U → X ) ∈ Lis-An( X ) and • a morphism c f : f − F ( V,v ) → F ( U,u ) for each f : ( u : U → X ) → ( v : V → X )in Lis-An( X )such that the following conditions hold:(1) c f is an isomorphism if f is an open immersion, and(2) if we are given a composition( u : U → X ) f −→ ( v : V → X ) g −→ ( w : W → X ) , we have c g ◦ f = c f ◦ f − c g .Denote by Mod( X lis − an , Q ) the category of sheaves of Q -vector spaces over X and by D ( X lis − an , Q ) the derived category of Mod( X lis − an , Q ). Definition 4.2.
A sheaf F ∈ Mod( X lis − an , Q ) is called Cartesian if for any mor-phism f : ( U → X ) → ( V → X ) in Lis-An( X ), c f is an isomorphism. A Cartesiansheaf F ∈ Mod( X lis − an , Q ) is called (analytically) constructible if for any U → X in Lis-An( X ) the restriction F | U an to the analytic topos of U is (analytically) con-structible.Denote by D cart ( X lis − an , Q ) (resp. D c ( X lis − an , Q )) the full subcategory of D ( X lis − an , Q )spanned by complexes whose cohomologies are Cartesian sheaves (resp. constructiblesheaves).For an Artin stack X , one can define its associated complex analytic stack X an as in [Sun17, 3.2.2]. By abuse of notation, we write Lis-An( X ) (resp. X lis − an )instead of Lis-An( X an ) (resp. X anlis − an ). For ∗ ∈ { b, + , −} , D ( ∗ ) ( X lis − an , Q ) de-notes the full subcategory of D ( X lis − an , Q ) consists of complexes K such that K | U ∈ D ∗ ( U lis − an , Q ) for any quasi-compact Zariski open subset U ⊂ X . Define D ( ∗ )cart ( X lis − an , Q ) and D ( ∗ ) c ( X lis − an , Q ) in a similar manner.Arguing as in [LO08], if we are given a morphism f : X → Y of finite typebetween Artin stacks, one can construct six functors: Rf ∗ : D (+) c ( X lis − an , Q ) → D (+) c ( Y lis − an , Q ) , f ∗ : D c ( Y lis − an , Q ) → D c ( X lis − an , Q ) ,Rf ! : D ( − ) c ( X lis − an , Q ) → D ( − ) c ( Y lis − an , Q ) , f ! : D c ( Y lis − an , Q ) → D c ( X lis − an , Q ) , ( − ) ⊗ ( − ) : D ( − ) c ( X lis − an , Q ) × D ( − ) c ( X lis − an , Q ) → D ( − ) c ( X lis − an , Q ) and R H om : D ( − ) c ( X lis − an , Q ) op × D (+) c ( X lis − an , Q ) → D (+) c ( X lis − an , Q ) . We briefly recall the construction of Rf ∗ , f ∗ , Rf ! and f ! . Firstly define f ∗ : Mod( X lis − an , Q ) → Mod( Y lis − an , Q )by the rule that f ∗ F ( U ) = F ( U × Y X ). By taking the derived functor of f ∗ we obtain Rf ∗ . When f is a smooth morphism, f ∗ is nothing but the restrictionfunctor. In general f ∗ is constructed by taking simplicial covers, but we use pullbackfunctors only for smooth morphisms in this paper, so we do not need this generalconstruction. To define Rf ! and f ! we use the Verdier duality functor. Arguing asin [LO08, § ω X ∈ D ( b ) c ( X lis − an , Q )and define the Verdier duality functor D X := R H om ( − , ω X ) : D ( − ) c ( X lis − an , Q ) op → D (+) c ( X lis − an , Q ) . IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 23
Now define Rf ! := D Y ◦ f ∗ ◦ D X and f ! := D X ◦ f ∗ ◦ D Y . If f is a smooth morphismof relative dimension d , we have natural isomorphisms f ∗ D X ( F ) = f ∗ R H om ( F, ω Y ) ∼ −→ R H om ( f ∗ F, f ∗ ω Y ) ∼ −→ R H om ( f ∗ F, ω X [ − d ]) ∼ = D Y ( f ∗ F )[ − d ](4.1)for F ∈ D c ( Y , Q ). Therefore we have f ! ∼ = f ∗ [2 d ].If we are given a 2-morphism ξ : f ⇒ g between morphisms of finite type of Artinstacks, we have a natural isomorphism ξ ∗ : Rf ∗ ⇒ Rg ∗ compatible with the verticaland horizontal compositions. The same statement also holds for Rf ! , f ∗ and f ! .Now we discuss the base change isomorphisms. Consider the following 2-Cartesiandiagram of Artin stacks: X ′ ❴✤ g ′ / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) η v ~ ✈✈✈✈ Y ′ g / / Y . By adjunction, we have the base change map bc η : g ∗ Rf ∗ → Rf ′∗ g ′∗ (4.2)which is isomorphic if g is smooth. Now assume that f is of finite type and g issmooth with relative dimension d , and take F ∈ D ( − ) c ( X , Q ). Then we can constructthe proper base change map pbc η : g ∗ Rf ! F ∼ −→ Rf ′ ! g ′∗ F by the following composition g ∗ Rf ! F = g ∗ D Y Rf ∗ D X F ∼ = D Y ′ g ∗ Rf ∗ D X F [2 d ] ∼ = D Y ′ Rf ′∗ g ′∗ D X F [2 d ] ∼ = D Y ′ Rf ′∗ D X ′ g ′∗ F = Rf ′ ! g ′∗ F where the first and third isomorphism is defined by using (4.1) and the secondisomorphism is the base change map (4.2). Now consider the following compositionof 2-Cartesian diagrams η (cid:11) (cid:19) X ′′ ❴✤ h ′ / / f ′′ (cid:15) (cid:15) k ′ X ′ ❴✤ g ′ / / f ′ (cid:15) (cid:15) X f (cid:15) (cid:15) η t | qqqq η t | qqqq Y ′′ k < < h / / Y ′ η (cid:11) (cid:19) g / / Y where f is of finite type, and g and h are smooth. We define η : f ◦ k ′ ⇒ k ◦ f ′′ bycomposing 2-morphisms in the diagram. Arguing as [LO08, Lemma 5.1.2], we can show the commutativity of the following diagram: k ∗ f ! pbc η / / η ∗ (cid:15) (cid:15) ( f ′′ ) ! ( k ′ ) ∗ η ∗ (cid:15) (cid:15) h ∗ g ∗ f ! h ∗ pbc η / / h ∗ ( f ′ ) ! ( g ′ ) ∗ pbc η ( g ′ ) ∗ / / ( f ′′ ) ! ( h ′ ) ∗ ( g ′ ) ∗ . (4.3)For an Artin stack X , we define a full subcategory Perv( X ) ֒ → D c ( X , Q ) con-sists of objects K such that for any smooth morphism f : U → X from a scheme, f ∗ K [dim f ] is a perverse sheaf on U . An object in Perv( X ) is called a perversesheaf on X . Arguing as [LO09, Proposition 7.1], we see that U Perv( U ) definesa stack on Lis-An( X ) whose global section category is Perv( X ).4.2. D-critical stacks.
In this section we first recall the notion of d-critical stacksintroduced in [Joy15, § − Proposition 4.3. [Joy15, Proposition 2.3, 2.8]
Let f : X → Y be a morphism ofcomplex analytic spaces, and S X (resp. S Y ) be the sheaf on X (resp. Y ) definedin (2.3) . Then there exists a natural map θ f : f − S Y → S X with the followingproperty: If R ⊂ X and S ⊂ Y are open subsets with f ( R ) ⊂ S , i : R ֒ → U and j : S ֒ → V are closed embeddings into complex manifolds, and ˜ f : U → V is aholomorphic map with j ◦ f | R = ˜ f ◦ i , then the following diagram commutes: f − S Y | R / / θ f | R (cid:15) (cid:15) ( f | R ) − (( j − O V ) /I S,V ) (cid:15) (cid:15) S X | R / / ( i − O U ) /I R,U . Here horizontal maps are induced by the natural inclusions, and right vertical mapis induced by ˜ f ♯ : ˜ f − O V → O U . The map θ f induces natural map f − S Y → S X (also written as θ f ). If f is smooth and s ∈ Γ( Y, S Y ) is a d-critical structure, f ⋆ s := θ f ( f − s ) is also a d-critical structure. Now we explain that the definition of the sheaf S X can be extended to complexanalytic stacks: Proposition 4.4. [Joy15, Corollary 2.52]
Let X be a complex analytic stack. Thenthere exists a sheaf of complex vector spaces S X in Lis-an( X ) with the followingproperties. • For ( u : U → X ) ∈ Lis-an( X ) , we have an isomorphism θ u : S X | U an ∼ = S U . IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 25 • For a morphism f : ( u : U → X ) → ( v : V → X ) in Lis-an( X ) , the followingdiagram commutes f − ( S X | V an ) f − ( θ v ) / / c f (cid:15) (cid:15) f − S Vθ f (cid:15) (cid:15) S X | U an θ u / / S U . Definition 4.5.
Let X be a complex analytic stack. A section s ∈ Γ( X lis-an , S X )is called a d-critical structure if for any ( u : U → X ) ∈ Lis-An( X ), u ⋆ s := θ u ( s | U an )is a d-critical structure on U . A d-critical Artin stack is an Artin stack X with ad-critical structure on its analytification.We have a stacky version of Theorem 2.13: Theorem 4.6. [BBBBJ15, Theorem 3.18(a)]
Let ( X , ω X ) be a − -shifted sym-plectic derived Artin stack. Then there exists a natural d-critical structure s X on X := t ( X ) uniquely characterized by the following property:Assume we are given derived schemes X and c X , morphisms g : X → X and τ : X → c X such that g is smooth. Further assume that there exists a − -shiftedsymplectic structure ω b X and an equivalence g ⋆ ω X ∼ τ ⋆ ω b X . If we write g = t ( g ) , τ = t ( τ ) , and s b X the d-critical structure on b X = t ( c X ) associated with ω b X , wehave g ⋆ s X = τ ⋆ s b X .Proof. The uniqueness part is proved in [BBBBJ15, Theorem 3.18(a)]. We nowverify that the d-critical structure constructed in loc. cit. satisfies the propertyas above. Using [BBBBJ15, Theorem 2.10], we have derived schemes U and b U ,a smooth surjection u : U → X , a morphism τ U : U → b U , and a − ω b U on b U such that u ⋆ ω X ∼ τ ⋆ U ω b U . Further, if we write s b U thed-critical structure associated with ω b U , we may assume t ( u ) ⋆ s X = t ( τ U ) ⋆ s b U . Wehave the following diagram of derived stacks: U × X X ❴✤ u ′ / / g ′ (cid:15) (cid:15) X g (cid:15) (cid:15) τ / / c X b U U τ U o o u / / X . Now take any point x ∈ t ( X ) and an ´etale morphism from a derived scheme η : W → U × X X such that the image of t ( u ′ ◦ η ) contains x . Since t ( u ′ ◦ η ) isa smooth morphism, it suffices to show that t ( τ U ◦ g ′ ◦ η ) ⋆ s b U = t ( τ ◦ u ′ ◦ η ) ⋆ s b X . This follows by arguing as [BBJ19, Example 5.22] since we have ( τ U ◦ g ′ ◦ η ) ⋆ ω b U ∼ ( τ ◦ u ′ ◦ η ) ⋆ ω b X . (cid:3) Now we discuss the behavior of the d-critical structure associated with the canon-ical − − f : Y → Y be a smooth morphismfrom a derived scheme Y to a quasi-smooth derived Artin stack Y . Consider the following diagram: f ∗ T ∗ [ − Y τ / / e f ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ π Y , f ( ( T ∗ [ − Y π Y / / Y f (cid:15) (cid:15) T ∗ [ − Y π Y / / Y . (4.4)Here f ∗ T ∗ [ − Y is the total space Tot Y ( L f ∗ Y [ − π Y , π Y , and π Y , f are the pro-jections, τ is induced by the canonical map f ∗ L Y [ − → L Y [ − e f : f ∗ T ∗ [ − Y → T ∗ [ − Y is the base change of f . The smoothness of f implies that τ induces anisomorphism on underlying schemes, so we use the identification t ( f ∗ T ∗ [ − Y ) = t ( T ∗ [ − Y )(4.5)throughout the paper. Proposition 4.7.
Consider the situation as above. Denote by s T ∗ [ − Y (resp. s T ∗ [ − Y ) the d-critical structure associated with the canonical − -shifted symplec-tic form ω T ∗ [ − Y (resp. ω T ∗ [ − Y ) constructed in Example 2.6. Then we have s T ∗ [ − Y = ˜ f ⋆ s T ∗ [ − Y where we write ˜ f = t ( e f ) and use the identification (4.5) .Proof. By Theorem 4.6, we only need to show that τ ⋆ ω T ∗ [ − Y ∼ e f ⋆ ω T ∗ [ − Y . If wewrite λ T ∗ [ − Y and λ T ∗ [ − Y the tautological 1-forms on T ∗ [ − Y and T ∗ [ − Y respectively, we have d dR λ T ∗ [ − Y = ω T ∗ [ − Y and d dR λ T ∗ [ − Y = ω T ∗ [ − Y bydefinition. Therefore we only need to prove τ ⋆ λ T ∗ [ − Y ∼ e f ⋆ λ T ∗ [ − Y . By the functoriality of the cotangent complex, we have the following homotopycommutative diagram:( π Y , f ) ∗ f ∗ L Y [ − ∼ a / / ( π Y , f ) ∗ θ f (cid:15) (cid:15) e f ∗ π ∗ Y L Y [ − e f ∗ θ π Y / / e f ∗ L T ∗ [ − Y [ − θ e f ∗ (cid:15) (cid:15) L f ∗ T ∗ [ − Y [ − π Y , f ) ∗ L Y [ − ∼ b / / τ ∗ π ∗ Y L Y [ − τ ∗ θ πY / / τ ∗ L T ∗ [ − Y [ − . θ τ O O Here a and b are defined by using f ◦ π Y , f ≃ π Y ◦ e f and π Y , f ≃ π Y ◦ τ re-spectively, and other morphisms are induced by the functoriality of the cotangentcomplex. Now write γ f ∗ T ∗ [ − Y , γ T ∗ [ − Y and γ T ∗ [ − Y the tautological sections of( π Y , f ) ∗ f ∗ L Y [ − π ∗ Y L Y [ −
1] and π ∗ Y L Y [ −
1] respectively. By definition, we have e f ⋆ λ T ∗ [ − Y ∼ θ e f ( e f ∗ λ T ∗ [ − Y ) ∼ θ e f ◦ e f ∗ θ π Y ( e f ∗ γ T ∗ [ − Y ) , τ ⋆ λ T ∗ [ − Y ∼ θ τ ( τ ∗ λ T ∗ [ − Y ) ∼ θ τ ◦ τ ∗ θ π Y ( τ ∗ γ T ∗ [ − Y ) . Since we have the following tautological relations e f ∗ γ T ∗ [ − Y ∼ a ( γ f ∗ T ∗ [ − Y ) , τ ∗ γ T ∗ [ − Y ∼ b ◦ ( π Y , f ) ∗ θ f ( γ f ∗ T ∗ [ − Y ) , IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 27 the proposition follows. (cid:3)
We now discuss the notion of the virtual canonical bundles and the orientationsfor d-critical stacks. Before doing this we recall a property of the virtual canonicalbundle of d-critical schemes. For a d-critical chart (
R, U, f, i ) of a d-critical scheme(
X, s ) and a point x ∈ R , consider the following complex concentrated in degree − L x := (T U | x Hess( f ) | x −−−−−−→ Ω U | x ) . Since H ( L x ) ∼ = Ω X | x and H − ( L x ) ∼ = (Ω X | x ) ∨ , we can define an isomorphism κ x : K X,s | x ∼ = det( L x ) ∼ = det(Ω X | x ) ⊗ det((Ω X | x ) ∨ ) − ∼ = det(Ω X | x ) ⊗ . (4.6)Here the final isomorphism is defined in the same manner as (2.5). It is proved in[Joy15, Theorem 2.28] that κ x does not depend on the choice of a d-critical chart.Now the virtual canonical bundle for a d-critical stack is defined by the followingproposition: Proposition 4.8. [Joy15, Theorem 2.56]
Let ( X , s ) be a d-critical stack. Thenthere exists a line bundle K X ,s on X red , which we call the virtual canonical bundle of ( X , s ) , characterized uniquely up to unique isomorphism by the following properties: (i) For x ∈ X , there exists an isomorphism κ x : K X ,s | x ∼ = det( τ ≥ ( L X ) | x ) ⊗ . (4.7)(ii) For a smooth morphism u : U → X from a scheme U , there exists an iso-morphism Γ U,u : ( u red ) ∗ K X ,s ∼ = K U,u ⋆ s ⊗ c det(Ω U/ X ) ⊗ − . (4.8)(iii) In the situation of (ii), take any p ∈ U . The following distinguished triangle ∆ : u ∗ τ ≥ ( L X ) → Ω U → Ω U/ X → u ∗ τ ≥ ( L X )[1] induces an isomorphism ˆ ı (∆) p : det( τ ≥ ( L X ) | u ( p ) ) ⊗ det(Ω U/ X | p ) ∼ = det(Ω U | p ) where ˆ ı is defined in Lemma 2.14. Then the following diagram commutes: K X ,s | u ( p ) Γ U,u | p / / κ u ( p ) (cid:15) (cid:15) K U,u ⋆ s | p ⊗ det(Ω U/ X | p ) ⊗ − κ p ⊗ id (cid:15) (cid:15) det( τ ≥ ( L X ) | u ( p ) ) ⊗ / / det(Ω U | p ) ⊗ ⊗ det(Ω U/ X | p ) ⊗ − Here the bottom horizontal map is defined by using ˆ ı (∆) p . An orientation o of a d-critical stack ( X , s ) is the choice of a line bundle L on X red and an isomorphism o : L ⊗ ∼ = K X ,s . An isomorphism between orientations o : L ⊗ ∼ = K X ,s and o : L ⊗ ∼ = K X ,s is defined by an isomorphism φ : L ∼ = L such that o = o ◦ φ ⊗ . If there exists a smooth morphism u : U → X , we definean orientation u ⋆ o for ( U, u ⋆ s ) by the following composition: u ⋆ o : (( u red ) ∗ L ⊗ c det(Ω U/X )) ⊗ ∼ = ( u red ) ∗ K X ,s ⊗ c det(Ω U/X ) ⊗ Γ U,u −−−→ K U,u ⋆ s . If we are given a smooth morphism q : ( u : U → X ) → ( v : V → X ) in Lis-An( X ),define an isomorphism u ⋆ o ∼ = q ⋆ v ⋆ o (4.9) by using the natural isomorphism c det(Ω U/ X ) ∼ = ( f red ) ∗ c det(Ω V/ X ) ⊗ c det(Ω U/V ) . We now discuss the relation of the cotangent complex of a − Theorem 4.9. [BBBBJ15, Theorem 3.18(b)]
Let ( X , ω X ) be a − -shifted symplec-tic derived Artin stack, and ( X , s X ) the associated d-critical stack. Then there existsa natural isomorphism Λ X : c det( L X ) ∼ = K X ,s X (4.10) characterized by the following property:Assume we are given derived schemes X and c X , morphisms g : X → X and τ : X → c X such that g is smooth and L τ | x is concentrated in degree − for each x ∈ X . Note that it automatically follows that τ = t ( τ ) is ´etale. Further as-sume that there exist a − -shifted symplectic structure ω b X on c X with associatedd-critical locus ( b X, s b X ) and an equivalence g ⋆ ω X ∼ τ ⋆ ω b X . This equivalence inducesa homotopy between the composition T g → T X → τ ∗ T b X τ ⋆ ω c X −−−−→ τ ∗ L b X [ − → L X [ − and , hence an isomorphism ℓ : T g ∼ −→ L τ [ − . (4.11) Then the composition c det( g ∗ L X ) ∼ = c det( L X ) ⊗ c det( L g ) − ∼ = c det( τ ∗ L b X ) ⊗ c det( L τ ) ⊗ c det( L g ) − ∼ = c det( τ ∗ L b X ) ⊗ c det( L g ) ⊗ − ∼ = ( τ red ) ∗ K b X,s c X ⊗ c det( L g ) ⊗ − ∼ = K X,τ ⋆ s c X ⊗ c det( L g ) ⊗ − ∼ = ( g red ) ∗ K X ,s X is equal to ( − rank(Ω g ) ( g red ) ∗ Λ X where we write g = t ( g ) . Here the first andsecond isomorphisms are defined by using ˆ ı (∆ g ) and ˆ ı (∆ τ ) respectively, where ∆ g and ∆ τ are distinguished triangles ∆ g : g ∗ L X → L X → L g → g ∗ L X [1] , ∆ τ : τ ∗ L b X → L X → L τ → τ ∗ L b X [1] . The third isomorphism is defined in the same manner as (2.5) using ℓ (withoutany sign intervention used in Appendix A), the fourth isomorphism is Λ b X in The-orem 2.13, and the fifth isomorphism is Γ X,τ defined in Proposition 4.8. The finalisomorphism is Γ X,g , where we use the fact that τ ⋆ s b X = g ⋆ s X proved in Theorem4.6.Proof. The proof is essentially same as one in [BBBBJ15], but we include this forreader’s convenience and to fix the sign. Suppose we are given X , c X , g and τ asabove. Define Λ X , b X , g , τ : c det( g ∗ L X ) → ( g red ) ∗ K X ,s X IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 29 by the composition as above multiplied by ( − rank(Ω g ) . Write pr , pr : R = X × X X ⇒ X the first and second projections. We have a natural 2-morphism ξ : g ◦ pr ⇒ g ◦ pr . Now we prove the commutativity of the following diagram:(pr red1 ) ∗ ( c det( g ∗ L X )) (pr red1 ) ∗ Λ X , c X , g , τ / / ξ ∗ (cid:15) (cid:15) (pr red1 ) ∗ ( g red ) ∗ K X ,s X ξ ∗ (cid:15) (cid:15) (pr red2 ) ∗ ( c det( g ∗ L X )) (pr red2 ) ∗ Λ X , c X , g , τ / / (pr red2 ) ∗ ( g red ) ∗ K X ,s X . (4.12)By the reducedness of R red , we only need to prove the commutativity at each point r ∈ R . Write pr ( r ) = x and pr ( r ) = x . Now consider the following diagram: c det( g ∗ L X ) | x Λ X , c X , g , τ | x / / ξ ∗ | r (cid:15) (cid:15) (A) ' ' PPPPPPPPPPPPP ( g red ) ∗ K X ,s X | x ξ ∗ | r (cid:15) (cid:15) (B) u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ det(H ∗ ( g ∗ L X | x )) (C) / / ξ ∗ | r (cid:15) (cid:15) det(H ∗ ( τ ≥ ( g ∗ L X ) | x )) ⊗ ξ ∗ | r (cid:15) (cid:15) c det(H ∗ ( g ∗ L X | x )) (C) / / det(H ∗ ( τ ≥ ( g ∗ L X ) | x )) ⊗ det( g ∗ L X ) | x Λ X , c X , g , τ | x / / (A) ♥♥♥♥♥♥♥♥♥♥♥♥♥ ( g red ) ∗ K X ,s X | x . (B) i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Here (A) i is defined by the quasi-isomorphism g ∗ L X | x i ≃ H ∗ ( g ∗ L X | x i ) and (B) i is defined by κ g ( x i ) in Proposition 4.8 and the quasi-isomorphism τ ≥ ( g ∗ L X ) | x i ≃ H ∗ ( τ ≥ ( g ∗ L X ) | x i ). The map (C) i is defined in the same manner as (2.5) using theisomorphismsH n ( g ∗ L X | x i ) ∼ = ( H n ( τ ≥ ( g ∗ L X ) | x i ) n = 0 , − n − ( τ ≥ ( g ∗ L X ) | x i ) ∨ n = − , − . The commutativity of the left trapezoid and middle square is obvious, and thecommutativity of the right trapezoid follows from the proof of [Joy15, Theorem2.56]. It is easy to see that the upper and lower trapezoids commute up to the sign( − rank(H ( L X | g ( xi ) )) by using the equality (A.3). These commutativity propertiesimply the commutativity of the outer square, and hence the commutativity of thediagram (4.12).By Darboux theorem [BBBBJ15, Theorem 2.10], we can take X , c X , g and τ in the proposition so that g is surjective. By the commutativity of the diagram(4.12), Λ X , b X , g , τ descends to Λ X : c det( L X ) ∼ = K X ,s X satisfying the property in theproposition. The uniqueness of Λ X as in the theorem is clear from the construction. (cid:3) The notion of orientation for − Y be a quasi-smooth derived Artinstack. The argument in Example 2.15 works also for the stacky case and defines a natural isomorphism o ′ T ∗ [ − Y : c det( π ∗ Y L Y ) ⊗ ∼ = c det( L T ∗ [ − Y ) . (4.13)We define the canonical orientation o T ∗ [ − Y for T ∗ [ − Y by the composition o T ∗ [ − Y := Λ Y ◦ o ′ T ∗ [ − Y . Proposition 4.10.
Under the notation as in Proposition 4.7, we have an isomor-phism Ξ f : ˜ f ⋆ o T ∗ [ − Y ∼ = o T ∗ [ − Y . (4.14) Proof.
Throughout the proof we use the following notation: for a morphism h : Z → W of derived Artin stacks, we write∆ h : h ∗ L W θ h −→ L Z ζ h −→ L h δ h −→ h ∗ L W [1]the natural distinguished triangle of cotangent complexes.Define Ξ ′ f : ( ˜ f red ) ∗ c det( π ∗ Y L Y ) ⊗ c det( L e f ) ∼ = c det( π ∗ Y L Y )by using ˆ ı (∆ f ) and the identification ( π Y , f ) ∗ L f ∼ = L e f . WriteΞ f := √− vdim f · (vdim f − / Y · vdim f · Ξ ′ f . (4.15)Now it is enough to prove the commutativity of the following diagram of line bundleson T ∗ [ − Y red :(( ˜ f red ) ∗ c det( π ∗ Y L Y ) ⊗ c det( L e f )) ⊗ Ξ ⊗ f / / o ′ T ∗ [ − Y ⊗ id (cid:15) (cid:15) c det( π ∗ Y L Y ) ⊗ o ′ T ∗ [ − Y (cid:15) (cid:15) ( ˜ f red ) ∗ c det( L T ∗ [ − Y ) ⊗ c det( L e f ) ⊗ ( ˜ f red ) ∗ Λ T ∗ [ − Y ⊗ id (cid:15) (cid:15) c det( L T ∗ [ − Y ) Λ T ∗ [ − Y (cid:15) (cid:15) ( ˜ f red ) ∗ K t ( T ∗ [ − Y ) ,s T ∗ [ − Y ⊗ c det( L e f ) ⊗ / / K t ( T ∗ [ − Y ) ,s T ∗ [ − Y . (4.16)Here Λ T ∗ [ − Y and Λ T ∗ [ − Y are defined in Theorem 2.13 and Theorem 4.9 respec-tively, and the bottom arrow is defined by using Γ t ( T ∗ [ − Y ) , ˜ f in (4.8) and theidentification L e f | T ∗ [ − Y red ∼ = Ω ˜ f | T ∗ [ − Y red . IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 31
Consider the following diagram in Perf( f ∗ T ∗ [ − Y )( π Y , f ) ∗ L Y τ ∗ θ πY / / τ ∗ L T ∗ [ − Y τ ∗ ζ πY / / θ τ (cid:15) (cid:15) ( π Y , f ) ∗ T Y [1] ( π Y , f ) ∗ θ ∨ f [1] (cid:15) (cid:15) ( π Y , f ) ∗ L Y θ π Y , f / / L f ∗ T ∗ [ − Y ζ π Y , f / / ζ τ (cid:15) (cid:15) ( π Y , f ) ∗ f ∗ T Y [1] − ( π Y , f ) ∗ δ ∨ f [2] (cid:15) (cid:15) L τ k ∼ / / ❴❴❴❴❴❴❴❴ δ τ (cid:15) (cid:15) ( π Y , f ) ∗ T f [2] ( π Y , f ) ∗ ζ ∨ f [2] (cid:15) (cid:15) τ ∗ L T ∗ [ − Y [1] τ ∗ ζ πY [1] / / ( π Y , f ) ∗ T Y [2](4.17)where the top vertical arrows are identified with a part of the natural morphismbetween distinguished triangles τ ∗ ∆ π Y → ∆ π Y , f , by the natural isomorphisms τ ∗ π ∗ Y L Y ∼ = ( π Y , f ) ∗ L Y , τ ∗ L π Y ∼ = ( π Y , f ) ∗ T Y [1] , L π Y , f ∼ = ( π Y , f ) ∗ f ∗ T Y [1]and k is taken so that the right horizontal arrows define a morphism betweendistinguished triangles ∆ τ → ∆ ∨ f , rot . Here ∆ ∨ f , rot denotes the right vertical distinguished triangle in the diagram above.Now we claim that c det( k ) ◦ c det( ℓ [2]) = det( φ [2])(4.18)where ℓ : T e f ∼ −→ L τ [ −
2] is defined in (4.11), and φ : ( π Y , f ) ∗ T e f ∼ −→ T f is the naturalisomorphism. To see this consider the following commutative diagram: T f ∗ T ∗ [ − Y [1] ( · τ ∗ ω T ∗ [ − Y )[1] ◦ θ ∨ τ [1] / / θ ∨ e f [1] (cid:15) (cid:15) τ ∗ L T ∗ [ − Y τ ∗ ζ πY / / θ τ (cid:15) (cid:15) ( π Y , f ) ∗ T Y [1] ( π Y , f ) ∗ θ ∨ f [1] (cid:15) (cid:15) e f ∗ T T ∗ [ − Y [1] − δ ∨ e f [2] (cid:15) (cid:15) θ e f ◦ ( · e f ∗ ω T ∗ [ − Y )[1] / / L f ∗ T ∗ [ − Y ζ τ (cid:15) (cid:15) ζ π Y , f / / ( π Y , f ) ∗ f ∗ T Y [1] − ( π Y , f ) ∗ δ ∨ f [2] (cid:15) (cid:15) T e f [2] ℓ [2] / / L τ k / / ( π Y , f ) ∗ T f [2] . By using [KM76, Proposition 6], it is enough to prove the following equalities τ ∗ ζ π Y ◦ ( · τ ∗ ω T ∗ [ − Y )[1] ◦ θ ∨ τ [1] = θ ∨ π Y , f [1] ζ π Y , f ◦ θ e f ◦ ( · e f ∗ ω T ∗ [ − Y )[1] = e f ∗ θ ∨ π Y [1]but these are consequences of [Cal19, Remark 2.5]. Now consider the following diagram of line bundles on f ∗ T ∗ [ − Y red , in whichwe omit the pullback functors τ ∗ , π ∗ Y , f , and ( f ◦ π Y , f ) ∗ to simplify the notation: c det( L f ∗ T ∗ [ − Y ) ˆ ı (∆ π Y , f ) − (cid:15) (cid:15) ˆ ı (∆ τ ) − / / c det( L T ∗ [ − Y ) ⊗ c det( L τ ) ˆ ı (∆ πY ) − ⊗ c det( ℓ − [2]) (cid:15) (cid:15) c det( L Y ) ⊗ c det( T Y [1]) id ⊗ ˆ ı (∆ ∨ f , rot ) − / / id ⊗ ˆ χ T Y (cid:15) (cid:15) c det( L Y ) ⊗ c det( T Y [1]) ⊗ c det( T e f [2]) id ⊗ ˆ χ T Y ⊗ ˆ χ (2) T e f (cid:15) (cid:15) c det( L Y ) ⊗ c det( T Y ) ⊗ − id ⊗ (ˆ η ⊗− L Y ) − (cid:15) (cid:15) c det( L Y ) ⊗ c det( T Y ) ⊗ − ⊗ c det( T e f ) id ⊗ (ˆ η ⊗− L Y ) − ⊗ (ˆ η L e f ) − (cid:15) (cid:15) c det( L Y ) ⊗ c det( L Y ) id ⊗ ˆ ı (∆ f ) ⊗ id d det( L e f ) ⊗− / / c det( L Y ) ⊗ ⊗ c det( L e f ) ⊗ − . (4.19)Here ˆ η, ˆ χ are defined in Lemma 2.14. The commutativity of the diagram (4.17),the equality (4.18), and [KM76, Theorem 1] implies the commutativity of the up-per square. By applying Proposition A.1 and Proposition A.4 we see that thelower square also commutes. Next consider the following commutative diagram inPerf( f ∗ T ∗ [ − Y ): e f ∗ π ∗ Y L Y e f ∗ θ π Y / / ( π Y , f ) ∗ θ f (cid:15) (cid:15) e f ∗ L T ∗ [ − Y e f ∗ ζ π Y / / θ e f (cid:15) (cid:15) e f ∗ π ∗ Y T Y [1]( π Y , f ) ∗ L Y θ π Y , f / / ( π Y , f ) ∗ ζ f (cid:15) (cid:15) L f ∗ T ∗ [ − Y ζ π Y , f / / ζ e f (cid:15) (cid:15) e f ∗ π ∗ Y T Y [1]( π Y , f ) ∗ L f ∼ / / L e f . (4.20)The upper vertical arrows are identified with a part of the natural morphism ofdistinguished triangles e f ∗ ∆ π Y → ∆ π Y , f by the natural isomorphisms e f ∗ L π Y ∼ = e f ∗ π ∗ Y T Y [1] , L π Y , f ∼ = e f ∗ π ∗ Y T Y [1]and the left horizontal arrows are identified with a part of the natural morphism( π Y , f ) ∗ ∆ f → ∆ e f . by the natural isomorphism ( π Y , f ) ∗ L Y ∼ = e f ∗ π ∗ Y L Y . IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 33
Now consider the following diagram of line bundles on f ∗ T ∗ [ − Y red , in which weomit pullback functors as previous: c det( L f ∗ T ∗ [ − Y ) ˆ ı (∆ π Y , f ) − (cid:15) (cid:15) ˆ ı (∆ e f ) − / / c det( L f ) ⊗ c det( L T ∗ [ − Y ) ( − vdim Y · vdim f · id ⊗ ˆ ı (∆ π Y ) − (cid:15) (cid:15) c det( L Y ) ⊗ c det( T Y [1]) ˆ ı (∆ f ) − ⊗ id / / id ⊗ ˆ χ T Y (cid:15) (cid:15) c det( L f ) ⊗ c det( L Y ) ⊗ c det( T Y [1]) id ⊗ id ⊗ ˆ χ T Y (cid:15) (cid:15) c det( L Y ) ⊗ c det( T Y ) ⊗ − id ⊗ (ˆ η ⊗− L Y ) − (cid:15) (cid:15) c det( L f ) ⊗ c det( L Y ) ⊗ c det( T Y ) ⊗ − id ⊗ id ⊗ (ˆ η ⊗− L Y ) − (cid:15) (cid:15) c det( L Y ) ⊗ c det( L Y ) ˆ ı (∆ f ) − ⊗ id / / c det( L f ) ⊗ c det( L Y ) ⊗ . (4.21)The commutativity of the diagram (4.20) and [KM76, Theorem 1] implies the com-mutativity of the upper square, and the commutativity of the lower square is obvi-ous. By combining the commutativity of the diagrams (4.19) and (4.21), we obtainthe commutativity of the diagram (4.16) (the sign ( − vdim f · (vdim f − / appearsdue to the difference of the maps (2.5) and (A.2)). (cid:3) Remark 4.11.
Under the situation of the proposition above, assume further thatthere exists a smooth morphism q : Y ′ → Y , and write ˜ q : t ( T ∗ [ − Y ′ ) → t ( T ∗ [ − Y )the base change of q = t ( q ). Then it is clear that the following composition( ˜ f ◦ ˜ q ) ⋆ o T ∗ [ − Y ∼ (4.9) / / ˜ q ⋆ ˜ f ⋆ o T ∗ [ − Y ∼ ˜ q ∗ Ξ f / / ˜ q ⋆ o T ∗ [ − Y ∼ Ξ q / / o T ∗ [ − Y ′ is equal to Ξ f ◦ q .4.3. Dimensional reduction for Artin stacks.
We first recall the definition ofthe vanishing cycle complexes associated with d-critical stacks. To do this, wediscuss the functorial behavior of the vanishing cycle complexes associated withd-critical schemes with respect to smooth morphisms.
Proposition 4.12. [BBBBJ15, Proposition 4.5]
Let ( Y, s, o ) be an oriented d-critical scheme, and q : X → Y be a smooth morphism. Then there exists a naturalisomorphism Θ q = Θ q,s,o : ϕ pX,q ⋆ s,q ⋆ o ∼ = q ∗ ϕ pY,s,o [dim q ] characterized by the following property: for a d-critical chart R = ( R, U, f, i ) of ( X, s ) , a d-critical chart S = ( S, V, g, j ) of ( Y, s ) such that q ( R ) ⊂ S , and asmooth morphism ˜ q : U → V such that f = g ◦ ˜ q and j ◦ q = ˜ q ◦ i , the followingdiagram commutes ϕ pX,q ⋆ s,q ⋆ o | R ω R / / Θ q | R (cid:15) (cid:15) i ∗ ϕ pf ⊗ Z / Z Q q ⋆ o R | R Θ ˜ q,f ⊗ ρ q (cid:15) (cid:15) q ∗ ϕ pY,s,o [dim q ] | R q ∗ ω S [dim q ] / / j ∗ ˜ q ∗ ϕ pg [dim q ] ⊗ Z / Z ( q | R ) ∗ ( Q o S | S ) where Θ ˜ q,f is defined in Proposition 2.16(iii), and ρ q is defined by using the naturalisomorphism K U ∼ = ˜ q ∗ K V ⊗ det(Ω U/V ) . Theorem 4.13. [BBBBJ15, Theorem 4.8]
Let ( X , s, o ) be an oriented d-criticalstack. Then there exists a natural perverse sheaf ϕ X ,s,o with the following property:for each ( u : U → X ) ∈ Lis-An( X ) there exists an isomorphism Θ u = Θ u,s,o : ϕ pU,u ⋆ s,u ⋆ o ∼ = u ∗ ϕ p X ,s,o [dim u ] satisfying Θ u,s,o = q ∗ Θ v,s,o [dim q ] ◦ Θ q,v ⋆ s,v ⋆ o for any smooth morphism q : ( u : U → X ) → ( v : V → X ) in Lis-An( X ) . Here we identify q ⋆ v ⋆ o and u ⋆ o by using (4.9) . Let ( X , s ) be a d-critical stack, and Ξ : o ∼ = o be an isomorphism betweenorientations on ( X , s ). We write a Ξ : ϕ X ,s,o ∼ = ϕ X ,s,o the isomorphism induced by Ξ.Now we state our main theorem. Theorem 4.14.
Let Y be a quasi-smooth derived Artin stack, and equip T ∗ [ − Y with the canonical − -shifted sympelectic structure and the canonical orientation.Then we have a natural isomorphism ¯ γ Y : ( π Y ) ! ϕ p T ∗ [ − Y ∼ = Q Y [vdim Y ] where we write Y = t ( Y ) and π Y = t ( π Y ) .Proof. Take a smooth surjective morphism v : V → Y and an ´etale morphism η : U → V × Y V where V and U are derived schemes. q , q : U → V denotethe composition of η and the first and second projections respectively. Write U = t ( U ), e U = t ( T ∗ [ − U ), V = t ( V ), e V = t ( T ∗ [ − V ), e Y = t ( T ∗ [ − Y ), v = t ( v ), and q i = t ( q i ) for i = 1 ,
2. Denote by π Y : e Y → Y , π U : e U → U ,and π V : e V → V the projections, and by ˜ v : e V → e Y (resp. ˜ q i : e U → e V ) the basechange of v (resp. q i ). Denote by s the d-critical structure on e Y associated withthe canonical − ω T ∗ [ − Y .Define ¯ γ Y , v : v ∗ ( π Y ) ! ϕ p T ∗ [ − Y ∼ = v ∗ Q Y [vdim Y ]by the following composition: v ∗ ( π Y ) ! ϕ p T ∗ [ − Y ∼ = ( π V ) ! ˜ v ∗ ϕ p T ∗ [ − Y ∼ = ( π V ) ! ϕ p T ∗ [ − V [ − dim v ] ∼ = v ∗ Q Y [vdim Y ] . where the first isomorphism is the proper base change map, the second isomorphismis ( π V ) ! ( a Ξ v ◦ Θ − v )[ − dim v ], and the third isomorphism is ¯ γ V [ − dim v ]. By the sheafproperty, it is enough to prove the commutativity of the following diagram: q ∗ v ∗ ( π Y ) ! ϕ p T ∗ [ − Y ξ ∗ (cid:15) (cid:15) q ∗ ¯ γ Y , v / / q ∗ v ∗ Q Y [vdim Y ] ξ ∗ (cid:15) (cid:15) q ∗ v ∗ ( π Y ) ! ϕ p T ∗ [ − Y q ∗ ¯ γ Y , v / / q ∗ v ∗ Q Y [vdim Y ](4.22) IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 35 where ξ : v ◦ q ⇒ v ◦ q is the natural 2-morphisim. We define¯ γ Y , v ◦ q i : ( v ◦ q i ) ∗ ( π Y ) ! ϕ p T ∗ [ − Y ∼ = ( v ◦ q i ) ∗ Q Y [vdim Y ]for i = 1 , γ Y , v . The commutativity of the diagram (4.3)implies the commutativity of the following diagram:( v ◦ q ) ∗ ( π Y ) ! ϕ p T ∗ [ − Y ξ ∗ (cid:15) (cid:15) ¯ γ Y , v ◦ q / / ( v ◦ q ) ∗ Q Y [vdim Y ] ξ ∗ (cid:15) (cid:15) ( v ◦ q ) ∗ ( π Y ) ! ϕ p T ∗ [ − Y ¯ γ Y , v ◦ q / / ( v ◦ q ) ∗ Q Y [vdim Y ] . Therefore the commutativity of the diagram (4.22) follows once we prove the com-mutativity of the following diagram( v ◦ q i ) ∗ ( π Y ) ! ϕ p T ∗ [ − Y ∼ (cid:15) (cid:15) ¯ γ Y , v ◦ q i / / ( v ◦ q i ) ∗ Q Y [vdim Y ] ∼ (cid:15) (cid:15) q ∗ i v ∗ ( π Y ) ! ϕ p T ∗ [ − Y q ∗ i ¯ γ Y , v / / q ∗ i v ∗ Q Y [vdim Y ](4.23)for each i = 1 ,
2. We drop i from the notation, and write q = q i and ˜ q = ˜ q i . ByRemark 4.11, the following diagram commutes: ϕ p T ∗ [ − U a − v ◦ Θ ˜ v w w ♦♦♦♦♦♦♦♦♦♦♦♦ a − v ◦ q ◦ Θ ˜ q ◦ ˜ v ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ ˜ q ∗ ϕ p T ∗ [ − V [dim q ] a − q ◦ Θ ˜ q / / ˜ q ∗ ˜ v ∗ ϕ p T ∗ [ − Y [dim v ◦ q ] . Using this and the commutativity of the diagram (4.3) again, the commutativityof the diagram (4.23) is implied by the commutativity of the following diagram( π U ) ! ϕ p T ∗ [ − U ¯ γ U / / pbc q ◦ ( π V ) ! ( a − q ◦ Θ ˜ q ) (cid:15) (cid:15) Q U [vdim U ] ∼ (cid:15) (cid:15) q ∗ ( π V ) ! ϕ p T ∗ [ − V [dim q ] q ∗ ¯ γ V [dim q ] / / q ∗ Q V [vdim U ](4.24)where pbc q denotes the base change map. Arguing as the proof of [BBBBJ15,Theorem 2.9] and by shrinking if necessary, we may assume that there exist asmooth morphism F : M → N with a constant relative dimension between smoothschemes, a vector bundle E of rank r on N , and its section e ∈ Γ( N, E ) such that V = Z ( e ), U = Z ( F ∗ e ), and q : Z ( F ∗ e ) → Z ( e ) is the base change of F . Write e F : Tot M ( F ∗ E ∨ ) → Tot N ( E ∨ ) the base change of F , and ¯ e : Tot N ( E ∨ ) → A denotes the regular function corresponding to e . Then U = ( e U ,
Tot M ( F ∗ E ∨ ) , ¯ e ◦ e F , i ) , V = ( e V ,
Tot N ( E ∨ ) , ¯ e, j )define d-critical charts on e U and e V respectively, where i and j denote the naturalembeddings. Consider the following composition ρ ′ ˜ q : Q o T ∗ [ − U U ∼ = Q q ⋆ o T ∗ [ − V U ρ ˜ q −→ ˜ q ∗ Q o T ∗ [ − V V where the first map is induced by Ξ q . Recall that we have chosen trivializations of Q o T ∗ [ − U U and Q o T ∗ [ − V V in (2.25). Since we have ǫ dim N,r /ǫ dim M,r = √− vdim q · (vdim q − / V · vdim q , these trivializations are identified by ρ ′ ˜ q . This shows the commutativity of thefollowing diagram ϕ p T ∗ [ − U ω U / / a − q ◦ Θ ˜ q (cid:15) (cid:15) i ∗ ϕ p ¯ e ◦ e F ⊗ Z / Z Q o T ∗ [ − U U id ⊗ triv / / i ∗ ϕ p ¯ e ◦ e Fi ∗ Θ e F, ¯ e (cid:15) (cid:15) ˜ q ∗ ϕ p T ∗ [ − V ˜ q ∗ ω U / / ˜ q ∗ ( j ∗ ϕ p ¯ e [dim q ] ⊗ Z / Z Q o T ∗ [ − V V ) ˜ q ∗ (id ⊗ triv) / / ˜ q ∗ j ∗ ϕ p ¯ e [dim q ]where two triv in the right horizontal arrows denote the trivialization as above.Then the commutativity of the diagram (4.24) follows from the commutativity ofthe diagram (2.9). (cid:3) Applications
In this section, we will discuss two applications of Theorem 3.1 and its stackygeneralization Theorem 4.14. Firstly, we will apply it to prove the dimensional re-duction theorem for the vanishing cycle cohomology of the moduli stacks of sheaveson local surfaces. Secondly, we will propose a sheaf theoretic construction of virtualfundamental classes of quasi-smooth derived schemes by regarding Theorem 3.1 asa version of Thom isomorphism for − Cohomological Donaldson–Thomas theory for local surfaces.
Let S be a smooth quasi-projective surface and denote by p : X = Tot S ( ω S ) → S theprojection from the total space of the canonical bundle. Denote by M S (resp. M X ) the derived moduli stack of coherent sheaves on S (resp. X ) with propersupports, and π p : M X → M S the projection defined by p ∗ . By applying the maintheorem of [BD19], M X carries a canonical − ω M S . Theorem 5.1.
There exists an equivalence of − -shifted symplectic derived Artinstacks Ψ : ( M X , ω M X ) ≃ ( T ∗ [ − M S , ω T ∗ [ − M S )(5.1) such that π p ≃ π M S ◦ Φ .Proof. Let G be a compact generator of D (Qcoh( S )), and A = R Hom(
G, G ) and B = R Hom( p ∗ G, p ∗ G ) be the derived endomorphism algebras. It is clear that p ∗ G is a compact generator of D (Qcoh( X )), and we have quasi-equivalences: R Hom( G, − ) : L coh ( S ) ∼ −→ per dg A,R
Hom( p ∗ G, − ) : L coh ( X ) ∼ −→ per dg B where L coh ( S ) (resp. L coh ( X )) denotes the derived dg-category of Coh( S ) (resp.Coh( X )), and per dg A (resp. per dg B ) denotes the derived dg-category of perfect A -modules (resp. B -modules). It is proved in [IQ18, § B is equivalent tothe 3-Calabi–Yau completion of A . Therefore, by applying [BCS20, Theorem 6.17],we only need to prove the coincidence of two left Calabi–Yau structure c , c ∈ HC − ( L coh ( X )) ∼ = H ( X, ω X ) IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 37 where c is induced by the Calabi–Yau completion description and c correspondsto the canonical Calabi–Yau form on X . Since the statement is local on S , wemay assume S is affine. By the discussion after [BCS20, Theorem 5.8], we see that c = δc where δ denotes the mixed differential and c ∈ HH ( L coh ( X )) ∼ = H ( X, ∧ Ω X )corresponds to the tautological 2-form on X under the Hochschild-Kostant-Rosenbergisomoprhism. Since the Hochschild-Kostant-Rosenberg isomorphism identifies themixed differential on the Hochschild homology and the de Rham differential (see[TV11]), the theorem is proved. (cid:3) We always equip M X with the canonical − t ( Ψ ) ⋆ o T ∗ [ − M S . The following statement is a directconsequence of Theorem 4.14: Corollary 5.2.
We have an isomorphism ( π p ) ! ϕ p M X ∼ = Q M S [vdim M S ] where we write M X = t ( M X ) , M S = t ( M S ) , and π p = t ( π p ) . Now assume S is quasi-projective and ω S is trivial. Take an ample divisor H on S and denote by M H -ss S ⊂ M S (resp. M p ∗ H -ss X ⊂ M X ) the moduli stackof H -semistable sheaves on S (resp. p ∗ H -semistable sheaves on X ) with propersupports. By the triviality of ω S , we have an equality π − p ( M H -ss S ) = M p ∗ H -ss X . Thisobservation and the Verdier self-duality of ϕ p M X implies the following corollary: Corollary 5.3.
Write ϕ p M p ∗ H -ss X := ϕ p M X | M p ∗ H -ss X . Then we have following isomor-phisms: H ∗ c ( M p ∗ H -ss X ; ϕ p M p ∗ H -ss X ) ∼ = H ∗ +vdim M H -ss S c ( M H -ss S ) , H ∗ ( M p ∗ H -ss X ; ϕ p M p ∗ H -ss X ) ∼ = H BMvdim M H -ss S −∗ ( M H -ss S ) . Here H c denotes the cohomology with compact support, and H BM denotes the Borel–Moore homology. Thom isomorphism.
Let Y be a quasi-smooth derived scheme, and write Y = t ( Y ) and e Y = t ( T ∗ [ − Y ). Thanks to Theorem 3.1, we have the followingisomorphism: H ∗ ( e Y ; ϕ p T ∗ [ − Y ) ∼ = H BMvdim Y −∗ ( Y ) . (5.2)Since ϕ p T ∗ [ − Y is conical, by using Theorem 3.1 and [KaSc90, Proposition 3.7.5],we also have the following isomorphism:H ∗ ( e Y , e Y \ Y ; ϕ p T ∗ [ − Y ) ∼ = H ∗ +vdim Y ( Y ) . (5.3)This isomorphism can be regarded as a version of the Thom isomorphism. Indeed,if Y = M × R E ,E, E M where M is a smooth scheme, E is a vector bundle on M , and0 E is the zero section of E , the isomorphism (5.3) is the usual Thom isomorphism.By imitating the construction of the Euler class, we construct a class e ( T ∗ [ − Y ) ∈ H BM2 vdim Y ( Y ) by the image of 1 ∈ H ( Y ) under the following composition:H ( Y ) (5.3) ∼ / / H − vdim Y ( e Y , e Y \ Y ; ϕ p T ∗ [ − Y ) / / H − vdim Y ( e Y ; ϕ p T ∗ [ − Y ) ∼ (5.2) / / H BM2vdim Y ( Y ) . Denote by [ Y ] vir ∈ H BM2 vdim Y ( Y ) the virtual fundamental class of Y constructed byBehrend–Fantechi in [BF97]. We have the following conjecture: Conjecture 5.4. e ( T ∗ [ − Y ) = ( − vdim Y · (vdim Y − / [ Y ] vir . In other words, we expect that this gives a new construction of the virtualfundamental class.
Example 5.5.
Assume Y = M × R E ,E, E M . In this case, we have e Y ∼ = Tot N ( E ∨ )and ϕ p T ∗ [ − Y ∼ = Q e Y [dim e Y ], and the construction of the Verdier duality isomor-phism σ e Y in (2.17) implies that the following diagram commutes: D e Y ( ϕ p T ∗ [ − Y ) ( − dim e Y · (dim e Y − / σ e Y / / ∼ (cid:15) (cid:15) ϕ p T ∗ [ − Y ∼ (cid:15) (cid:15) D e Y ( Q e Y [dim e Y ]) ∼ / / Q e Y [dim e Y ] . Therefore we have e ( T ∗ [ − Y ) = ( − dim e Y · (dim e Y − / e ( E ∨ ) ∩ [ M ]= ( − (dim e Y · (dim e Y − / E e ( E ) ∩ [ M ]= ( − vdim Y · (vdim Y − / [ Y ] vir . The author has verified Conjecture 5.4 under a certain assumption:
Theorem 5.6. [Kin] If L Y | Y is represented by a two-term complex of vector bun-dles, then Conjecture 5.4 is true. In particular, Conjecture 5.4 holds when Y isquasi-projective. Remark 5.7.
We can extend the above construction for stacky cases as follows.Let Y be a quasi-smooth derived Artin stack and write Y = t ( Y ) and e Y = t ( T ∗ [ − Y ). Denote by π : e Y → Y the projection and by i : Y → e Y the zerosection. Then [KaSc90, Proposition 3.7.5] and the smooth base change theoremimplies isomorphisms of functors π ! ∼ = i ! , π ∗ ∼ = i ∗ . Therefore Theorem 4.14 implies isomorphisms i ! ϕ T ∗ [ − Y ∼ = Q Y [vdim Y ] ,i ∗ ϕ T ∗ [ − Y ∼ = ω Y [ − vdim Y ] . Consider the following composition of natural transforms: i ! ϕ p T ∗ [ − Y → i ! i ∗ i ∗ ϕ p T ∗ [ − Y ∼ = i ! i ! i ∗ ϕ p T ∗ [ − Y ∼ = i ∗ ϕ p T ∗ [ − Y IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 39 where the first map is the ∗ -unit, the second map defined in the same manner as[LO08, Proposition 4.6.2], and the third map is the inverse of the !-unit. Thiscomposition defines an element e ( T ∗ [ − Y ) ∈ H BM2 vdim Y ( Y ) . We conjecture that this is equal (up to the sign ( − vdim Y · (vdim Y − / ) to thestacky virtual fundamental class recently constructed by [AP19] and [Kha19]. Remark 5.8.
By [Cal19, Theorem 2.2], the zero section Y ֒ → T ∗ [ − Y carries acanonical Lagrangian structure. Further, arguing as Example 2.15, we see that thisLagrangian structure admits a canonical orientation with respect to o T ∗ [ − Y . Theisomorphism (5.3) can be regarded as [ABB17, Conjecture 5.18] for this orientedLagrangian structure. Appendix A. Remarks on the determinant functor
In this appendix, we prove some results on the determinant of perfect complexes.All results follow easily from [KM76], but we include this for completeness and tofix the sign conventions.Let X be a scheme. Denote by P is X the category of invertible sheaves on X withthe isomorphisms, and P gr is X by the category of locally Z / Z -graded invertiblesheaves with the isomorphisms defined as follows: • Objects of P gr is X are pairs ( L, α ) where L is an invertible sheaf on X and α is a locally constant Z / Z -valued function. • A morphism from (
L, α ) to (
M, β ) is an isomorphism form L to M when α = β , and otherwise there is no morphism between them.If there is no confusion, we omit the local grading. We define a monoidal structureon P gr is X by ( L, α ) ⊗ ( M, β ) := ( L ⊗ M, α + β ) , with the monoidal unit ( O X ,
0) and the obvious associator. By the Koszul sign rulewith respect to the local grading, we define the symmetrizer s ♭ ( L,α ) , ( M,β ) : ( L, α ) ⊗ ( M, β ) ∼ = ( M, β ) ⊗ ( L, α ) . (A.1)This makes P gr is X a symmetric monoidal category. In this paper we do not equip P gr is X with any other symmetric monoidal structure. Note that the forgetfulfunctor P gr is X → P is X is monoidal but not symmetric monoidal with respect tothe standard symmetric monoidal structure on P is X . For ( L, α ) ∈ P gr is X , defineits (right) inverse by ( L, α ) − := ( L − , − α ), and define morphisms δ ♭ ( L,α ) , ( δ ′ ( L,α ) ) ♭ as follows: δ ♭ ( L,α ) : ( L, α ) ⊗ ( L, α ) − ∼ = ( L ⊗ L − , ∼ = ( O X , δ ′ ( L,α ) ) ♭ : ( L, α ) − ⊗ ( L, α ) s ♭ ( L,α ) − , ( L,α ) −−−−−−−−−→ ( L, α ) ⊗ ( L, α ) − δ ♭ ( L,α ) −−−−→ ( O X , . Define µ ♭ ( L,α ) : (( L, α ) − ) − → ( L, α ) so that the following diagram commutes:(( L, α ) − ) − ⊗ ( L, α ) − µ ♭ ( L,α ) ⊗ id / / ( δ ′ ( L,α ) − ) ♭ ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ ( L, α ) ⊗ ( L, α ) − δ ♭ ( L,α ) w w ♥♥♥♥♥♥♥♥♥♥♥♥ ( O X , . Note that the map µ ♭ ( L,α ) differs from the natural isomorphism of the ungraded linebundles ( L − ) − → L by the sign ( − α . For L, M ∈ P gr is X , define θ ♭L,M : ( L ⊗ M ) − → L − ⊗ M − so that the following diagram commutes:( L ⊗ M ) ⊗ ( L ⊗ M ) − δ ♭L ⊗ M / / id L ⊗ M ⊗ θ ♭L,M (cid:15) (cid:15) O X ( L ⊗ M ) ⊗ ( L − ⊗ M − ) id L ⊗ s ♭M,L − ⊗ id M − / / ( L ⊗ L − ) ⊗ ( M ⊗ M − ) . δ ♭L ⊗ δ ♭M O O Note that the map θ ♭ ( L,α ) , ( M,β ) differs from tha natural isomorphism ( L ⊗ M ) − → L − ⊗ M − defined by using the standard symmetric monoidal structure on thecategory of ungraded line bundles by ( − α · β .Write C • X the category of bounded complexes of finite locally free O X -modules,and C is • X the subcategory of C • X with the same objects and the morphisms are thequasi-isomorphisms. For a locally free O X -module F , define a graded line bundledet ♭ ( F ) ∈ P gr is X by det ♭ ( F ) := ( ∧ rank( F ) F, rank( F ) mod 2) . Clearly, det ♭ is functorial with respect to isomorphisms. For F • ∈ C is • X , define agraded line bundle det ♭ ( F • ) ∈ P gr is X bydet ♭ ( F • ) := ( · · · ⊗ det ♭ ( F i ) ( − i ⊗ det ♭ ( F i − ) ( − i − ⊗ · · · ) . In [KM76, Theorem 1], it is shown that det ♭ extends naturally to a functor C is • X → P gr is X , which we also write as det ♭ . Define a functor det : C is • X → P is X by thecomposition C is • X det ♭ −−→ P gr is X → P is X where the latter functor is the forgetful one. For a short exact sequence 0 → E • u • −→ F • v • −→ G • → C • X , define i ♭ ( u • , v • ) : det ♭ ( E • ) ⊗ det ♭ ( G • ) ∼ = det ♭ ( F • ) IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 41 by the following composition:det ♭ ( E • ) ⊗ det ♭ ( G • ) = ( O i det ♭ ( E i ) ( − i ) ⊗ ( O i det ♭ ( G i ) ( − i ) (i) ∼ = O i (det ♭ ( E i ) ( − i ⊗ det ♭ ( G i ) ( − i ) (ii) ∼ = O i (det ♭ ( E i ) ⊗ det ♭ ( G i )) ( − i (iii) ∼ = O i det ♭ ( F i ) ( − i = det ♭ ( F • ) . Here (i) is defined by the symmetric monoidal structure on P gr is X , (ii) is definedusing θ ♭ det( E i ) , det( G i ) , and (iii) is defined by the natural isomorphisms det( E i ) ⊗ det( G i ) ∼ = det( F i ). We define i ( u • , v • ) : det( E • ) ⊗ det( G • ) ∼ = det( F • ) by forgettingthe local grading from i ♭ ( u • , v • ).A.1. Compatibility with the derived dual functor.
For a free O X -module E with a fixed basis e , . . . , e n , define η E : det( E ∨ ) ∼ = −→ det( E ) − by the rule e ∨ ∧ · · · ∧ e ∨ n ( e n ∧ · · · ∧ e ) ∨ (A.2)where e ∨ , . . . , e ∨ n denotes the dual basis of e , . . . , e n and ( e n ∧ · · · ∧ e ) ∨ denotesthe dual of e n ∧ · · · ∧ e . Clearly η E is independent of the choice of the basis, andwe can define η E for any locally free O X -module. For E • ∈ C • X , define( η ♭E • ) ′ : det ♭ ( E •∨ ) ∼ = det ♭ ( E • ) − by the following composition:det ♭ ( E •∨ ) = O i det ♭ (( E − i ) ∨ ) ( − i (i) ∼ = O i (det ♭ ( E − i ) ( − i ) − ∼ = (det ♭ ( E • )) − . Here (i) is defined by η E i and (ii) is defined by iterating θ ♭ . Write ǫ ( E • ) := P i ≡ , rank( E i ) and define η ♭E • := ( − ǫ ( E • ) ( η ♭E • ) ′ . Proposition A.1. (i)
For a short exact sequence → E • u • −→ F • v • −→ G • → in C • X , the following diagram commutes: det ♭ ( G •∨ ) ⊗ det ♭ ( E •∨ ) η ♭G • ⊗ η ♭E • / / i (( v • ) ∨ , ( u • ) ∨ ) (cid:15) (cid:15) det ♭ ( G • ) − ⊗ det ♭ ( E • ) − s ♭ det ♭ ( G • ) − , det ♭ ( E • ) − / / det ♭ ( E • ) − ⊗ det ♭ ( G • ) − det ♭ ( F •∨ ) η ♭F • / / det ♭ ( F • ) − i ♭ ( u • ,v • )) ⊗− / / (det ♭ ( E • ) ⊗ det ♭ ( G • )) − . θ ♭ det ♭ ( E • ) , det ♭ ( E • ) O O (ii) For a quasi-isomorphism u • : E • → F • in C is X , the following diagramcommutes: det ♭ ( F •∨ ) η ♭F • / / det ♭ (( u • ) ∨ ) (cid:15) (cid:15) det ♭ ( F • ) − ♭ ( u • ) ⊗− (cid:15) (cid:15) det ♭ ( E •∨ ) η ♭E • / / det ♭ ( E • ) − . Proof. (i) Clearly we may assume that these three complexes are concentratedin a single degree i . Then the claim follows from a direct computation.(ii) Arguing as the proof of [KM76, Lemma 2] and using (i), we may assumethat E • is an acyclic complex and F • = 0. Further, by localizing X ifnecessary and using (i), we may assume that E • has length two, but thenthe claim follows from a direct computation (note the sign convention ofthe dual complex). (cid:3) In [KM76, Theorem 2], the determinant functor is defined for the category ofperfect complexes with quasi-isomorphisms. By using the proposition above, wecan define η ♭E : det ♭ ( E ∨ ) ∼ = −→ det ♭ ( E ) − for any perfect complex E . We define η E : det( E ∨ ) ∼ = −→ det( E ) − by forgetting the local grading from η ♭E .A.2. Compatibility with the shift functor.
For E • ∈ C is X , define χ ♭E • : det ♭ ( E • [1]) ∼ = det ♭ ( E • ) − by the following composition:det ♭ ( E • [1]) = O i det ♭ ( E i +1 ) ( − i (i) ∼ = O i (det ♭ ( E i +1 ) ( − i +1 ) − ∼ = det ♭ ( E • ) − . Here (i) is defined by using µ ♭ det ♭ ( E i ) and (ii) is defined by iterating θ ♭ . The followingproposition can be shown similarly to Proposition A.1: Proposition A.2. (i)
For a short exact sequence → E • u • −→ F • v • −→ G • → in C • X , the following diagram commutes: det ♭ ( E • [1]) ⊗ det ♭ ( G • [1]) χ ♭E • ⊗ χ ♭G • / / i ♭ ( u • [1] ,v • [1]) (cid:15) (cid:15) det ♭ ( E • ) − ⊗ det ♭ ( G • ) − det ♭ ( F • [1]) χ ♭F • / / det ♭ ( F • ) − i ♭ ( u • ,v • ) ⊗− / / (det ♭ ( E • ) ⊗ det ♭ ( G • )) − . θ ♭ det ♭ ( E • ) , det ♭ ( E • ) O O (ii) For a quasi-isomorphism u • : E • → F • in C is X , the following diagramcommutes: det ♭ ( E • [1]) χ ♭E • / / det ♭ ( u • [1]) (cid:15) (cid:15) det ♭ ( E • ) − ♭ ( u • ) ⊗− ) − (cid:15) (cid:15) det ♭ ( F • [1]) χ ♭F • / / det ♭ ( F • ) − . IMENSIONAL REDUCTION IN COHOMOLOGICAL DONALDSON–THOMAS THEORY 43
This proposition implies that we can define χ ♭E : det ♭ ( E [1]) ∼ = det ♭ ( E ) − for anyperfect complex E . We define χ E : det( E [1]) ∼ = det( E ) − by forgetting the local grading from χ ♭E . We also define χ ( n ) E : det( E [ n ]) ∼ = det( E ) ( − n for each n ∈ Z so that χ (1) E = χ E and χ ( n + m ) E = ( χ ( m ) E [ n ] ) ( − ⊗ n ◦ χ ( n ) E holds foreach n, m ∈ Z , where we identify (det( E ) ( − n ) ( − m and det( E ) ( − n + m by using µ ♭ det( E ) ♭ if both n and m are odd.A.3. Compatibility with distinguished triangles.
Consider a distinguishedtriangle E → F → G → E [1] of perfect complexes on X . In [KM76] it is observedthat there exists an isomorphism det ♭ ( E ) ⊗ det ♭ ( G ) ∼ = det ♭ ( F ), though there is nonatural choice in general . However, it is also observed in [KM76] that there is acanonical choice when X is reduced: Proposition A.3. [KM76, Proposition 7]
For each distinguished triangle of perfectcomplexes ∆ : E u −→ F v −→ G w −→ E [1] on a reduced scheme X , there exists a uniqueisomorphism i ♭ (∆) = i ♭ ( u, v, w ) : det ♭ ( E ) ⊗ det ♭ ( G ) ∼ = det ♭ ( F ) characterized by the following properties: • If E u −→ F v −→ G w −→ E [1] is represented by a short exact sequence ofcomplexes of locally free sheaves → E • u • −→ F • v • −→ G • → , then i ♭ ( u, v, w ) = i ♭ ( u • , v • ) . • If there exists a morphism of reduced schemes f : Y → X , then f ∗ i ♭ ( u, v, w ) = i ( f ∗ u, f ∗ v, f ∗ w ) . We define i (∆) = i ( u, v, w ) : det( E ) ⊗ det( G ) ∼ = det( F ) by forgetting the localgrading from i ♭ ( u, v, w ). The following statement follows by a direct computation: Proposition A.4.
Let X be a reduced scheme, ∆ : E u −→ F v −→ G w −→ E [1] adistinguished triangle of perfect complexes over X , and ∆ ′ : F v −→ G w −→ E [1] − u [1] −−−→ F [1] the rotated triangle. Then the following diagram commutes: det( E ) ⊗ det( F ) ⊗ det( E [1]) id ⊗ i (∆ ′ ) / / id ⊗ χ E (cid:15) (cid:15) det( E ) ⊗ det( G ) i (∆) / / det( F ) (cid:15) (cid:15) det( E ) ⊗ det( F ) ⊗ det( E ) − · ( − rank( E ) rank( F ) / / det( E ) ⊗ det( F ) ⊗ det( E ) − . Here the right vertical map is defined by unit map for the standard symmetricmonoidal structure on P is X . The essential reason is the non-functoriality of the mapping cone. See [BS17, §
5] for anapproach via the ∞ -categorical enhancement. Let k be a field, and E be a perfect complex over Spec k . Write H ∗ ( E ) := L i H i ( E )[ − i ], considered as a complex with zero differential. The natural isomor-phism H ∗ ( E ) ≃ E in D ( k ) induces an isomorphism j E : det(H ∗ ( E )) ∼ = det( E ) . Let E u −→ F v −→ G w −→ E [1] be a distinguished triangle of perfect complexes overSpec k . By decomposing the long exact sequence induced by the above distinguishedtriangle into short exact sequences, we can construct i ′ H ( u, v, w ) : det(H ∗ ( E )) ⊗ det(H ∗ ( G )) → det(H ∗ ( F )) . Define i ′ ( u, v, w ) := j F ◦ i ′ H ( u, v, w ) ◦ ( j − E ⊗ j − G ) : det( E ) ⊗ det( G ) ∼ = det( F ) . The maps i ( u, v, w ) and i ′ ( u, v, w ) do not coincide in general because we used thesymmetric monoidal structure on P gr is X to construct i ( u, v, w ). However we canexplicitly write down the difference between i ( u, v, w ) and i ′ ( u, v, w ) as follows:write a i := rank ker H i ( u ), b i := rank ker H i ( v ), and c i := rank ker H i ( w ). Then wehave i ( u, v, w ) = ( − T i ′ ( u, v, w ) , where T = X i : even ( X j ≤ i − a i a j + X j ≤ i a i b j + X j ≤ i +1 c i a j + X j ≤ i − c i b j )+ X i : odd ( X j ≤ i a i a j + X j ≤ i − a i b j + X j ≤ i c i a j + X j ≤ i c i b j ) . (A.3)A.4. Extension to Artin stacks.
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