Motivic Semiorthogonal Decompositions for Abelian Varieties
aa r X i v : . [ m a t h . AG ] F e b MOTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIANVARIETIES
BRONSON LIM AND FRANCO ROTAAbstract.
Suppose G is a finite group acting on an Abelian variety A such that thecoarse moduli A/G is smooth. Using the recent classification result due to Auffarth, Luc-chini, and Quezada, we construct a motivic semiorthogonal decomposition for D [ A/G ] provided G = T ⋊ H with T a subgroup of translations and H is a subgroup of groupautomorphisms. Introduction
Motivic semiorthogonal decompositions.
Suppose G is a finite group acting ef-fectively on a smooth quasi-projective variety X . If each of the quotients ¯ X λ = X λ /C ( λ ) are smooth, where X λ is the fixed loci of λ ∈ G and C ( λ ) is the centralizer of λ ∈ G ,then there is an orthogonal decomposition HH ∗ [ X/G ] = M λ ∈ G/ ∼ HH ∗ ( ¯ X λ ) where G/ ∼ is the set of conjugacy classes. This decomposition holds for any additiveinvariant, see [TVdB18]. In [PVdB19], the authors conjecture that this remains true onthe derived (or dg) level. Since the derived category decomposes orthogonally if and onlyif the variety is not connected, we should not expect a fully orthogonal decompositionbut only a semiorthogonal decomposition , see Definition 1. Motivic Semiorthogonal Decomposition Conjecture.
Suppose G is a a finite groupacting effectively on a smooth variety X and that for all λ ∈ G/ ∼ the coarse moduli ¯ X λ = X λ /C ( λ ) is smooth. Then there is a semiorthogonal decomposition of D [ X/G ] where the components C [ λ ] are in bijection with conjugacy classes and C [ λ ] ∼ = D ( ¯ X λ ) . A semiorthogonal decomposition satisfying the conditions of the Motivic Semiorthog-onal Decomposition Conjecture is called a motivic semiorthogonal decomposition , see Def-inition 2. It is motivic in the sense of [TVdB18] and [Toe00].
Remark . In [BGLL17] it is shown that an effective action is necessary. Roughly, theytake an ineffective group action on an elliptic curve without complex multiplication and,assuming the conjecture is true, construct an endomorphisms with cyclic kernel.There are several known cases of the conjecture which we list now. • The natural actions of complex reflection groups of types
A, B, G , F , and G ( m, , n ) acting on A n , [PVdB19, Theorem C]. • Semidirect product actions of type G ⋊ S n on C n where C is a curve and G is aneffective group acting on C , [PVdB19, Theorem B]. Mathematics Subject Classification.
Primary 14F05; Secondary 13J70.
Key words and phrases.
Derived Categories, Fourier-Mukai Functors. • Dihedral group actions on A , [Pot17, Corollary 6.5.5]. • Curves, [Pol06, Theorem 1.2]. • Cyclic group quotients, [KP17, Theorem 4.1].1.2.
Main result.
Our main result is an affirmative answer to the Motivic Semiorthogo-nal Decomposition Conjecture when A is an Abelian variety and G = T ⋊ H with T atranslation subgroup and H a subgroup of group homomorphisms. Theorem 1.
Let A be an Abelian variety and G = T ⋊ H a finite group of automorphismsof A such that the coarse quotient A/G is smooth. Then there exists a motivic semiorthogonaldecomposition for D [ A/G ] .Remark . We did not have to add the hypothesis that A λ /C G ( λ ) is smooth for all λ .This is because it is automatically satisfied provided A/G is smooth by the classificationresults in [AL17, ALQ18] which we review in Section 2.As an application, we get another proof of the orthogonal decomposition of the non-commutative motive for [ A/G ] , see [TVdB18, Theorem 1.24, Theorem 1.27]. Let U be theuniversal additive invariant. Corollary 1.
Let A be an Abelian variety and G = T ⋊ H a finite group of automorphismsof A such that the coarse quotient A/G is smooth. Then the motive U ([ A/G ]) decomposesorthogonally: U ([ A/G ]) ∼ = M λ ∈ G/ ∼ U ( ¯ A λ ) , where ¯ A λ = A λ /C ( λ ) . Outline of Proof.
We rely on the recent work in [AL17, ALQ18] characterizingsmooth quotients of Abelian varieties A by finite groups. In particular, there are onlyfinitely many indecomposable types with A/G smooth. With this classification, we proveTheorem 1 by using the known motivic semiorthogonal decomposition for products ofcurves, using the results of [LP17], and by constructing a new one for the stacky surface [ E × E/G (4 , , where E = C / Z [ i ] is an elliptic curve with automorphism group µ .1.4. Outline of Paper.
Section 2 is a preliminary section. Section 3 is devoted to theconstruction of a motivic semiorthogonal decomposition for D [ A /G (4 , , . This is alocal model of the exceptional case in the classification of smooth quotients of Abelianvarieties. The proof of the global exceptional case is Section 4. In Section 5, we com-plete the proof of the main result. Lastly, in Section 6 we explicitly compute motivicsemiorthogonal decompositions for Abelian surfaces in the indecomposable cases.1.5. Conventions and notation.
We work over C . Unless otherwise stated, all functorsare assumed to be derived. For X a scheme or stack, by D ( X ) we mean the boundedderived category of coherent sheaves on X .1.6. Acknowledgements.
The first author would like to thank Robert Auffarth, AaronBertram, Giancarlo Lucchini Arteche, and Alexander Polishchuk for helpful conversa-tions. This research was part of the RTG grant at the University of Utah.
OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 3 Preliminaries
We recall preliminaries on equivariant derived categories, motivic semiorthogonal de-compositions, and smooth quotients of Abelian varieties. Throughout X is a smoothquasi-projective variety and G is a finite group acting on X . In particular, ¯ X = X/G is aquasi-projective variety. We refer to [BKR01, Section 4] for a review of G -equivariant de-rived categories and [Kuz14] for a survey of semiorthogonal decompositions in algebraicgeometry.2.1. Equivariant derived categories.
For computational purposes it is often conve-nient to work with the G -equivariant derived category of X instead of the derived cate-gory of [ X/G ] . By [ ? ] These are equivalent: D [ X/G ] ∼ = D ( X ) G .The structure map π : X → pt is trivially G -equivariant and therefore descends tothe stack quotients: π : [ X/G ] → BG . Here BG = [pt /G ] is the classifying space of G -bundles. Any finite dimensional representation V of G determines a coherent sheafon BG and hence an object in D ( BG ) . For any object F ∈ D ( X ) G we therefore have anoperation of tensoring by finite dimensional representations: F ⊗ V := F ⊗ π ∗ ( V ) . For any two G -equivariant sheaves on X , say E , F ∈ D ( X ) G , the set of morphismsis a complex of G -representations R Hom( E , F ) ∈ D ( BG ) and we have Ext ∗ [ X/G ] ( E , F ) ∼ = ( H ∗ R Hom( E , F )) G . Motivic semiorthogonal decompositions.Definition 1.
Let T be a triangulated category. A semiorthogonal decomposition of T is a pair A , B of full triangulated subcategories of T such that • Hom T ( b, a ) = 0 for all a ∈ A , b ∈ B ; • for all t ∈ T , there exists a t ∈ A and b t ∈ B such that b t → t → a t → b t [1] is an exact triangle.In this case, we write T = hA , Bi .We can iterate Definition 1 to get semiorthogonal decompositions with an arbitraryfinite number of subcategories: T = hA , . . . , A r i . As suggested in the introduction, the following definition is motivated by a motivicorigin of the conjecture.
Definition 2.
Suppose a finite group G acts effectively on a smooth variety X . A motivicsemiorthogonal decomposition of D [ X/G ] is the data of • a total order λ , . . . , λ r on conjugacy classes G/ ∼ ; • embedding functors Φ λ i : D ( X λ i /C ( λ i )) ֒ → D [ X/G ] for each λ i ∈ G/ ∼ which are linear over D ( X/G ) ; B. LIM AND F. ROTA • a semiorthogonal decomposition D [ X/G ] = h C , . . . , C r i ; where C i is the image of Φ λ i .Given semiorthogonal decompositions T = hA , Bi = C , Ai , we can produce a newsemiorthogonal decomposition by mutating A either to the left through C or to the rightthrough B . That is, there exists mutation functors L C , R B : T → T such that T = hB , R B ( A ) i = h L C ( A ) , Ci . In the case T is a derived category of a smooth projective variety or Deligne-Mumfordstack, the mutation functors are given by the Serre functor and its inverse, see [BK89]. Example 1.
The easiest example is when µ acts on a smooth variety X fixing a smoothdivisor Y pointwise. Then ¯ X = X/µ is smooth. Set ι : Y ֒ → X to be the inclusion,which is µ -equivariant, and π : X → ¯ X the quotient mapping. Then it is straightforwardto check that there is a semiorthogonal decomposition D [ X/µ ] = h π ∗ D ( ¯ X ) , ι ∗ D ( Y ) i . This semiorthogonal decomposition is D ( ¯ X ) -linear and hence defines a motivic semiorthog-onal decomposition.Since µ acts trivially on ι ∗ D ( Y ) , we can tensor by irreducible representations to getdifferent objects. Let χ : µ → C ∗ be the unique nontrivial representation. Then forany object ι ∗ F ∈ ι ∗ D ( Y ) , we can consider the object ι ∗ F ⊗ χ . This is still an object in D [ X/µ ] but no longer in ι ∗ D ( Y ) . More precisely, the G -equivariant adjunction formulashows the Serre functor comes with an additional tensoring by χ . Thus, by mutating, wehave another motivic semiorthogonal decomposition D [ X/µ ] = h ι ∗ D ( Y ) ⊗ χ, π ∗ D ( ¯ X ) i . This example indicates one of the central problems in solving this conjecture. A canon-ical ordering on embeddings doesn’t exist: one could have fully-faithful embedding func-tors but they (or any reordering) are not necessarily semiorthogonal.2.3.
Smooth quotients of Abelian varieties.
We recall the main results of [AL17, ALQ18].Let G be a finite group of automorphisms of an Abelian variety A such that G = T ⋊ H with T a group of translations and H a group of group homomorphisms. Theorem 2.
Suppose G fixes the identity and dim( A G ) = 0 . If G doesn’t act irreduciblyon T e ( A ) , then there exists direct product decompositions A ∼ = A × · · · × A k and G ∼ = G × · · · × G k where the action is diagonal and G i acts irreducibly on T e ( A i ) . Since D [ A × · · · × A k /G × · · · × G k ] ∼ = D [ A /G ] ⊗ · · · ⊗ D [ A k /G k ] we can assume that G acts irreducibly. Theorem 3.
Suppose G fixes the identity, dim( A G ) = 0 , G acts irreducibly on T e ( A ) , and A/G is smooth. If dim( A ) ≥ , then there exists an elliptic curve E such that A ∼ = E n andeither(A) G ∼ = C n ⋊ S n where C is a finite cyclic group acting on E so that C n acts diagonallyand S n permutes the factors; or We need A and B to be admissible but in our setting this will be satisfied. OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 5 (B) G ∼ = S n +1 and acts on A = { ( x , . . . , x n +1 ) ∈ E n +1 | x + · · · + x n +1 = e } bypermutations.If dim( A ) = 2 , then both (a) and (b) can happen and there is a third case:(C) E ∼ = C / Z [ i ] and G = G (4 , , ∼ = H ⋊ S where H = { ( ξ a , ξ b ) | a + b ≡ } . Here ξ is a primitive fourth root of unity. We will refer to these as irreducible quotients of Types (A), (B), and (C). Motivic semiorthog-onal decompositions of Type (A) have been constructed in [PVdB19]. Types (B) and (C)have not yet been considered. We will construct one for Type (C) in Section 4. We willshow they exist for type (B) in Theorem 6.3.
Motivic semiorthogonal decomposition for [ A /G (4 , , Before constructing a motivic semiorthogonal decomposition for type (C), we start bystudying the local case of G (4 , , acting naturally on A .3.1. Representation Theory of G (4 , , . We will need the representation theory of G (4 , , . We recall this information now. The complex reflection group G (4 , , isconstructed as follows. Fix ξ = √− to be a primitive fourth root of unity, let µ = h ( ξ, , (1 , ξ ) i and K the subgroup given by K = { ( ξ a , ξ b ) ∈ µ | a + b ≡ } . The symmetric group S = h σ i acts on K by σ ( ξ a , ξ b ) = ( ξ b , ξ a ) and we set G := G (4 , ,
2) = K ⋊ S . The natural action of G induced by that of µ ⋊ S on X := A is an irreducible twodimensional representation that we call V . Note that V is not self-dual, as χ V ( ξ, ξ,
1) =2 ξ is purely complex. We have X = A ∼ = Spec(Sym( V ∨ )) . Proposition 1.
There are ten irreducible representations of G . Eight of the irreducible rep-resentations are one dimensional and the remaining two are two dimensional. Moreover, theone dimensional irreducible representations are 2-torsion.Proof. The center of G is generated by ( − , − , and it is easy to see that the quotientgroup is an elementary Abelian 2-group of order 8: G/ h ( − , − i = µ . Indeed, define a homomorphism ρ : G → µ by ρ ( ξ a , ξ b , σ ) = ( ξ a + b , ξ a − b , sgn( σ )) . It is a homomorphism as ξ a − b = ξ b − a in this case. The kernel is ( − , − , and it isevidently surjective. It follows that there are eight irreducible one dimensional represen-tations. Since the sum of the squares of the dimensions of the irreducible representationsmust be | G | = 16 we have V ) + · + dim( V k ) = 16 and thus k = 2 and dim( V ) = dim( V ) = 2 . These two dimensional representations are V and V ∨ . (cid:3) B. LIM AND F. ROTA
Let ρ : G → µ to be the surjective homomorphism from the proof of Proposition 1and π i +1 : µ → C ∗ the projection onto the i th factor character. Then set χ i = π i ◦ ρ : G → C ∗ . Explicitly χ ( ξ a , ξ b , σ c ) = ξ a + b χ ( ξ a , ξ b , σ c ) = ξ a − b χ ( ξ a , ξ b , σ c ) = sgn( σ c ) . The characters χ , χ , and χ generate the character group of G . We also have Λ V = χ χ . Proposition 2.
There are isomorphisms of representations V ⊗ V ∼ = Λ V ⊕ χ χ ⊕ χ χ χ ⊕ χ V ∨ ∼ = V ⊗ Λ V ∼ = V ⊗ χ χ ∼ = V ⊗ χ χ χ ∼ = V ⊗ χ V ⊗ V ∨ ∼ = χ χ ⊕ ⊕ χ ⊕ χ . Proof.
For the first isomorphipsm we exhibit an explicit decomposition. Pick the standardbasis e , e for V , then we have the correspondence Λ V = Span { e ⊗ e − e ⊗ e } χ χ = Span { e ⊗ e + e ⊗ e } χ χ χ = Span { e ⊗ e − e ⊗ e } χ = Span { e ⊗ e + e ⊗ e } The second isomorphism follows since if f , f are the bases dual to e and e we havethat σ ( f ) = f and converseley. Moreover, ( ξ a , ξ b )( f ) = ξ − a f and ( ξ a , ξ b )( f ) = ξ − b f . The last isomorphism follows from the first two. (cid:3)
The following is straightforward. We include it for labeling purposes.
Proposition 3.
The conjugacy classes of G are: D = { Id } D = { (1 , , σ ) , ( − , − , σ ) } D = { (1 , − , , ( − , , } D = { ( ξ, − ξ, σ ) , ( − ξ, ξ, σ ) } D = { ( − , − , } D = { ( ξ, ξ, } D = { ( ξ, ξ, σ ) , ( − ξ, − ξ, σ ) } D = { ( − ξ, − ξ, } D = { (1 , − , σ ) , ( − , , σ ) } D = { ( ξ, − ξ, , ( − ξ, ξ, } . OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 7
The centralizers of each conjugacy class are: H = G H = h ( ξ, ξ, , (1 , , σ ) i ∼ = µ × µ H = K H = h ( ξ, ξ, , ( ξ , , σ ) i ∼ = µ × µ H = G H = GH = h ( ξ, ξ, , (1 , , σ ) i ∼ = µ × µ H = GH = h ( ξ, ξ, , (1 , ξ , σ ) i ∼ = µ × µ H = K. For all i = 1 , ..., , let g i denote a representative of D i . Then, the fixed loci X i := X g i are (1) X = X, X = ... = X = { } , and X , X , X are lines intersecting at the origin. Notice that the centralizers H , H , H , H are index two and hence normal. Conse-quently the quotients by these normal subgroups define characters: ρ , ρ , ρ . Explicitly, ρ = χ , ρ = χ , ρ = χ χ . Fourier-Mukai kernels.
For i = 1 , . . . , , we define Fourier-Mukai kernels byconstructing subschemes H i × G -invariant subschemes Z i of X i × X as follows. Define Z i = [ g ∈ G g ∗ Γ i where Γ i is the graph of the inclusion X i ֒ → X , and we give Z i its reduced schemestructure. Then Z i is an H i × G - invariant subscheme and so O Z i can be taken as akernel for the Fourier-Mukai functor Φ i = Φ O Zi ◦ π ∗ i : D ( ¯ X i ) → D [ X/G (4 , , , where π i : X i → ¯ X i is the quotient mapping. Equivalently, let ¯ Z i be the image of Z i under the quotient ( π i , Id) : X i × X → ¯ X i × X , then Φ i = Φ O ¯ Zi . We note that if X i is a point, then Z i is also a point and the corresponding Φ i is simply thepushforward along the inclusion of the origin which is G -invariant. If X i is a hyperplane,then Z i is the union of two lines in X i × X ∼ = A as the centralizer is index two. If X i = X , then Φ = π ∗ , the pullback functor from the coarse space. Proposition 4. (1) We have Φ ( O ) = C [ x, y ] / ( x y , x + y ) and for p = 0 every other point in X we set L p to be its orbit. If p has trivialstabilizer, then L p consists of 16 distinct points and Φ ( O p ) = O L p . If p has a µ stabilizer. In this case the orbit L p of p contains 8 points, and Φ ( O p ) is the sheaf supported on L p with a nilpotent of order 2 at every point. B. LIM AND F. ROTA (2) For each i = 2 , , , we have Φ i ( O ) = C [ x, y ] / ( x y , x − y ) i = 2 C [ x, y ] / ( xy, ( x + y ) ) i = 3 C [ x, y ] / ( x y , x + y ) i = 4 and for p = 0 any other point in ¯ X i we set L p to be the orbit of the preimage of p under the inclusion X i ֒ → X . That is, let p , p , p , p be the four distinct points inthe preimage of p , then L p = { p , p , p , p , γp , γp , γp , γp } where γ is a nontrivial coset representative. Then Φ i, ( O p ) ∼ = O L p . Proof. (1) The functor Φ is pullback π ∗ from the coarse moduli. The maximal idealof the origin pulls back to the ideal of G -invariant polynomials ( x y , x + y ) ,and a point p = 0 to its orbit.If p has trivial stabilizer, then the pull back of its image under X → ¯ X con-sists of 16 points, all belonging to the locus where the quotient map is a localisomorphism.If p has a µ stabilizer, then there are 8 other points in its orbit, and then thepull-back π ∗ ( O π ( p ) ) is non-reduced of length 2: locally around p the quotient map X → ¯ X is given by C [ u, v ] → C [ u, v ] and π ( p ) pulls back to C [ u, v ] / ( u, v ) .There are no other possibilities for the stabilizer of a point p ∈ A .(2) The second statement follows since the pullup of O p under the quotient map X i → ¯ X i is π ∗ i ( O p ) ∼ = M i =1 O p i . Then since p = 0 , the support of π ∗ i ( O p ) is contained in the open locus where theprojection is an isomorphism onto its image. Finally, if γ is a coset representativeof the only nontrivial coset, then when we equivariantize it just translates eachof the four points under γ .The first statement is more involved. We will only do the computation for i = 3 as the other computations are analagous under a linear change of coordinates. Inthis case λ = (1 , − , . We can take { , σ } as a transversal for C ( λ ) in G . Thus Z λ = Γ λ ∪ σ ∗ Γ λ with its reduced scheme structure. As modules we have H ( O Z λ ) ∼ = C [ z, x, y ] / ( z − x − y, xy ) . Let X → ¯ X denote the quotient map. Note that C ( λ ) / ( λ ) ∼ = µ and actsin the natural way on X . In particular if x is the coordinate on X , then thequotient map is x x . It follows that for ∈ ¯ X ∼ = A , we have π ∗ ( O ) = O where H ( O ) ∼ = C [ z ] / ( z ) is a non-reduced zero-dimensional scheme of length4 concentrated at the origin. Then we pull-up the resolution O X z −→ O X to X × X ∼ = A z × A x,y . Tensoring with O Z λ yields O Z λ z −→ O Z λ . OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 9
As this is injective we have that this complex is quasi-isomorphic to C [ z, x, y ] / ( z − x − y, xy, z ) This pushes forward to C [ x, y ] / (( x + y ) , xy ) . (cid:3) Corollary 2.
Let R denote the regular representation of G . Then: Φ ( O ) ∼ = R. Moreover we have the following isomorphisms of representations: Φ ( O ) ∼ = ⊕ V ∨ ⊕ V ⊕ χ ⊕ χ χ ⊕ χ Φ ( O ) ∼ = ⊕ V ∨ ⊕ V ⊕ χ χ ⊕ χ χ χ ⊕ χ Φ ( O ) ∼ = ⊕ V ∨ ⊕ V ⊕ χ ⊕ χ χ χ ⊕ χ χ Proof.
This is straightforward. For example, a basis for Φ ( O ) as a C -vector space is , x, y, xy, x , xy, y ,x , x y, xy , y , x ,x y, xy , x , x y, x y. We have isomorphisms C { } ∼ = , C { x, y, x , x y } ∼ = ( V ∨ ) ⊕ , C { x , x y, xy , y } ∼ = V ⊕ , C { x , y } ∼ = χ χ ⊕ χ χ χ , C { x y, xy } ∼ = χ ⊕ χ χ , C { xy } ∼ = χ , C { x } ∼ = χ , C { x y } ∼ = χ χ . (cid:3) Proposition 5.
For each i = 1 , . . . , , the Fourier-Mukai functors Φ i : D ( ¯ X i ) → D [ X/G (4 , , are fully-faithful.Proof. In the case i = 1 , this is pullback from X/G (4 , , which is fully-faithful. If i = 5 , . . . , , this is pushforward of the origin, which is fully-faithful since the structuresheaf of the origin is exceptional:(2) Ext ∗ ( O , O ) G = (cid:0) [0] ⊕ V [ − ⊕ Λ V [ − (cid:1) G = [0] . The computations for i = 2 , , are all similar. For notational convenience, we willcompute the case of i = 3 . In this case, we have two types of points. If p = 0 , then welook at the orbit of a preimage under the quotient map. The orbit is a reduced subschemeof length 8 L p . Moreover, we have Ext ∗ ( O L p , O L p ) G = (cid:0) ⊕ i =1 Ext ∗ ( O p i , O p i ) (cid:1) G . Clearly when ∗ = 0 the G -action permutes the hom-sets and so the 8-dimensional hom-space has only one copy of the trivial representation. If ∗ = 2 , then we have (cid:0) ⊕ i =1 T p i X i ⊗ N p i X i (cid:1) G and since X is codimension 1 it must be that λ acts nontrivially on the normal bundle.Thus there are no invariants.For p = 0 , we recall that Γ(Φ ( O )) ∼ = C [ x, y ] / ( xy, ( x + y ) ) . Hence, Hom(Φ ( O ) , Φ ( O )) G ∼ = C as this is determined by the image of and Φ ( O ) G ∼ = . For vanishing of the secondext group we use duality: Ext (Φ ( O ) , Φ ( O )) ∼ = Hom(Φ ( O ) , Φ ( O ) ⊗ Λ V ) and by Corollary 2 Φ ( O ) ∼ = ⊕ V ⊕ V ∨ ⊕ χ χ ⊕ χ χ χ ⊕ χ and so (Φ ( O ) ⊗ Λ V ) G = 0 . This implies vanishing of the second ext group andcompletes the proof. (cid:3) Assembling the Motivic Semiorthogonal Decomposition.
In this section weconstruct a motivic semiorthogonal decomposition of [ X/G ] using the functors Φ i . Westart by observing that the positive dimensional loci have semiorthogonal images: Lemma 1. (1) Let i = 2 , , and p ∈ X i . We have Ext ∗ (Φ i ( O p ) , Φ ( O p )) G = 0 (2) For j = 2 , , , j = i , Ext ∗ (Φ i ( O ) , Φ j ( O )) G = 0 . Proof.
Recall that Φ = π ∗ , and write Ext ∗ (Φ i ( O p ) , π ∗ ( O p )) ∼ = Ext ∗ ( π ∗ ( O p ) , Φ i ( O p ) ⊗ Λ V ) ∨ ∼ = Ext ∗ ( O p , π ∗ (Φ i ( O p ) ⊗ Λ V )) ∨ = 0 , where the last equality follows because Φ i ( O p ) ⊗ Λ V has no invariants (see Corollary2 for p = 0 , and Proposition 4 for p = 0 ).The argument is slightly more involved for the second statement. We only illustratethe case i = 2 and j = 3 , the others are analogous. A resolution of Φ ( O ) is the Koszulcomplex C [ x, y ] ⊗ χ χ χ y − x x y −−−−−−−−→ C [ x, y ] ⊕ C [ x, y ] ⊗ χ χ χ · (cid:16) x y x − y (cid:17) −−−−−−−−−−−−−→ C [ x, y ] where χ χ χ is the weight of x − y . Applying Hom( − , Φ ( O )) yields(3) Φ ( O ) y − x −−−−−−−−→ Φ ( O ) ⊕ Φ ( O ) ⊗ χ χ χ · (cid:16) x − y (cid:17) −−−−−−−−−−−→ Φ ( O ) ⊗ χ χ χ . Recall from Proposition 4 that Φ ( O ) ∼ = C [ x, y ] / ( xy, ( x + y ) ) . The kernel of multipli-cation by y − x is the submodule (cid:0) x + y (cid:1) ∼ = C (cid:8) x + y , x , y , x (cid:9) ∼ = χ χ ⊕ V ⊕ χ . OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 11
The cokernel of multiplication by x − y is C [ x, y ] / ( xy, x − y ) ⊗ χ χ χ ∼ = χ χ χ ⊕ V ⊕ χ . Observe that the middle cohomology is the sum of the above kernel and cokernel. Then,none of the cohomologies of (3) have invariant summands. (cid:3)
Next, we focus on the zero dimensional loci and compare them to the positive dimen-sional ones. We start from this computation:
Lemma 2.
For each i = 1 , , , , we have isomorphisms in D ( BG ) : RHom( O , Φ ( O )) ∼ = χ χ [0] ⊕ ( χ χ ) ⊕ [ − ⊕ χ χ [ − O , Φ ( O )) ∼ = χ [0] ⊕ ( χ ⊕ χ χ ) [ − ⊕ χ χ [ − O , Φ ( O )) ∼ = χ [0] ⊕ ( χ ⊕ χ χ )[ − ⊕ χ χ [ − O , Φ ( O )) ∼ = χ χ [0] ⊕ ( χ χ ⊕ χ χ )[ − ⊕ χ χ [ − . Proof.
We again only compute the case i = 3 . It is technically easier to use duality first: Ext ∗ [ X/G ] ( O , Φ i ( O )) ∼ = Ext −∗ [ X/G ] (Φ i ( O ) , O ⊗ Λ V ) . We resolve Φ ( O ) (computed in Proposition 4) by the Koszul complex noting that xy has weight χ : Ext ∗ [ X/G ] (Φ i ( O ) , O ⊗ Λ V ) ∼ = H ∗ ( O χ χ → O χ χ ⊕ O χ → O χ ) and hence the statement. (cid:3) Finally, we study Hom spaces between zero dimensional loci:
Lemma 3.
Let ρ, σ be irreducible representations of G . Then, the position ( ρ, σ ) is markedby a X in the following table if and only if Ext ∗ [ X/G ] ( O ⊗ ρ, O ⊗ σ ) = 0 .ρ σ χ χ χ χ χ χ χ χ χ χ χ χ V V ∨ X X X X X X X χ X X X X X X X χ X X X X X X X χ X X X X X X X χ χ X X X X X X X χ χ X X X X X X X χ χ X X X X X X X χ χ χ X X X X X X X V X X X X V ∨ X X X X
Proof.
Twisting the (2) we obtain
Ext ∗ [ X/G ] ( O ⊗ ρ, O ⊗ σ ) = (cid:2) ( [0] ⊕ V [ − ⊕ Λ V [ − ⊗ ( ρ ∨ ⊗ σ )[0] (cid:3) G . and one checks which choices of ( ρ, σ ) produce invariant summands, using Proposition2. (cid:3) It follows from Lemma 2 that O , O ⊗ χ , and O ⊗ V ∨ are in the left orthogonal ofthe images of the positive-dimensional loci. In particular, the sheaf M := C [ x, y ] / ( x + y , x − y ) is also in the left orthogonal, since as a representation(4) M ∼ = ⊕ V ∨ ⊕ χ . Proposition 6.
The collection (5) ( O ⊗ χ , M, O ⊗ χ χ , O ⊗ χ χ χ , O ⊗ V, O ) is an exceptional collection in D [ X/G ] .Proof. First observe that all objects are exceptional: the twists of O by characters areimmediate, the twist by V follows from Proposition 2, we check M now. Its endomor-phism algebra is computed by resolving M with a Koszul complex and computing thecohomology of M −→ M χ χ ⊕ M χ χ χ −→ M χ ( χ χ and χ χ χ are the weights of x + y , x − y respectively), so by (4) it only hasone invariant summand in degree 0, i.e. M is exceptional.We now check the orthogonalities. All the conditions not involving M follow fromLemma 3, and all the ones involving M are similar: we only compute Ext ∗ ( M, O ⊗ χ ) G = 0 here. Once again, resolve M with a Koszul complex and consider the cohomol-ogy of O χ → O χ ⊕ O χ χ → O χ χ , which has no invariant summands. (cid:3) For i = 1 , ..., , let C i denote the image of Φ i : D ( ¯ X i ) → D [ X/G ] . For i ≥ , define C i to be the image of the twisted functors Φ ′ = Φ ( − ) ⊗ χ Φ ′ = Φ ( − ) ⊗ χ χ χ Φ ′ = Φ ( − ) ⊗ M Φ ′ = Φ ( − ) ⊗ V Φ ′ = Φ ( − ) ⊗ χ χ Φ ′ = Φ ( − ) . Theorem 4.
There is a motivic semiorthogonal decomposition (6) D [ X/G ] = h C , C , C , C , C , C , C , C , C , C i . Proof.
For all i = 1 , ..., , the component C i is equivalent to D ( ¯ X i ) . This is the contentof Proposition 5 (the generators of C , ..., C are still exceptional objects by Proposition6). Proposition 6 also shows that C , ..., C are semiorthogonal. Moreover, it immedi-ately follows from Lemma 2 that Ext ∗ [ X/G ] ( C j , C i ) = 0 for i = 1 , , , and j = 5 , ..., .The first statement of Lemma 1 implies Ext ∗ [ X/G ] ( C i , C ) = 0 for i = 2 , , . Now let i, j be distinct elements of { , , } . Then, the Fourier-Mukai kernels Z i and Z j onlyintersect at the origin, therefore the second statement of Lemma 1 implies the vanishing Ext ∗ [ X/G ] ( C i , C j ) = 0 .It remains to show that the category T := h C , ..., C i coincides with D [ X/G ] . Itsuffices to show that T contains the spanning class Ω := {O p ⊗ ρ | p ∈ X, ρ ∈ Irrep( G p ) } . In this case, in fact, every object E ∈ T ⊥ satisfies Hom( ω, E ) = 0 for all ω ∈ Ω , whichimplies E = 0 by definition of a spanning class, and therefore T ⊥ = { } . OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 13 If p = 0 , then G p ∼ = G , and we recover all irreducible representations as follows: fromthe exceptional collection (5) we get χ , χ χ , χ χ χ , V, , and V ∨ (the last one from the composition series of M , whose factors are ( χ , V ∨ , ) ). Withthese, and using Corollary 2, we obtain χ , χ , χ χ , and χ χ .If p = 0 , then G p is either trivial or µ , in which case p ∈ X i for some i = 2 , , . If G p = 0 , then Φ ( O π ( p ) ) contains O p as a direct summand (Proposition 4), hence O p ∈ T .If p has µ stabilizer, we apply Proposition 4: on the one hand, the pull-back along X i → ¯ X i contains O p as a direct summand. On the other hand, the pull-back π ∗ ( O π ( p ) ) is non-reduced of length 2. Then, the non-trivial irreducible representation is obtained asthe kernel of the map π ∗ ( O π ( p ) ) → O p . (cid:3) Motivic Semiorthogonal Decomposition for Type (C)
Setup.
We recall here the construction of Type (C) quotients, following [ALQ18](see Theorem 3). Let
Λ = Z [ i ] be the Gaussian lattice and E ∼ = C / Λ the correspondingelliptic curve with µ automorphism group. The action of G := G (4 , , on A ≃ C described in Section 3 preserves the lattice Λ ⊕ and descends to an action on B := E . Asit turns out, the quotient B/G is not smooth, but one considers another Abelian surface A := E and an isogeny π : B → A defined by the matrix ν = (cid:18) − i − (cid:19) . Let t denote the only non-trivial ξ -invariant element of E , then the kernel of π is ker( π ) = ∆ = { ( t , t ) } , and we have B/ ∆ ∼ = A . Moreover, G acts trivially on ∆ , so we have an action of ∆ × G on B . This action descends to an action of G on A and induces a natural iso-morphism B/ ∆ ⋊ G ≡ A/G . Explicitly, every element g of G acts on A via the matrix νλν − where λ is the matrix expression of g acting on A . In particular, the generators ( − , , , ( − i, i, , (1 , , σ ) act on A via the matrices: α = (cid:18) − i (cid:19) , β = (cid:18) − i i − i (cid:19) , γ = (cid:18) − i − (cid:19) . Fixed loci and stabilizers.
We use the numbering of Prop. 3 for the conjugacyclasses and the centralizers of G , and we list representatives of conjugacy classes andfixed loci. By E [ n ] we denote the set { x ∈ E | nx = 0 } of n -torsion points. Theconjugacy classes are: D = [Id] D = [ γ ] D = [ α ] D = [ βγ ] D = [ − Id] D = [ αβ ] D = [ − αβ ] D = [ αβγ ] D = [ αγ ] D = [ β ] . The corresponding fixed loci are: A = AA = { ( x, y ) ∈ A | x = ix } ∼ = { e, t } × EA = { ( x, y ) ∈ A | x = (1 + i ) y } ∼ = E [2] × EA = { ( x, y ) ∈ A | x = iy } ∼ = EA = E [2] A = ... = A = { e, t } . We point out that the only points of A with non-trivial stabilizers are:Points Stabilizer { e, t } GE [2] \ { e, t } µ × µ ( A ∪ A ∪ A ) \ E [2] µ .Finally, we compute the quotients ¯ A i . Observe that the group G acts on E [2] with 7orbits. Four orbits are given by elements of { e, t } × { e, t } , which are fixed by G . Theremaining three orbits (let a := and b := i ) are the sets: { e, t } × { a, b }{ a, b } × { e, t }{ a, b } . This shows immediately that ¯ A = E [2] /G consists of the 7 orbits above, and thatthe quotients ¯ A = ... = ¯ A = { e, t } . We have ¯ A = P by [ALQ18, Theorem 1.1]. It is left to compute the quotients of theone dimensional loci. The computations are similar for i = 2 , , . For example, we have ¯ A = { ( x, y ) ∈ A | x = t y + E [2] } / h α, β i , and one sees that, for any t ∈ E [2] , we have β (cid:18) t y + ty (cid:19) = (cid:18) t ( iy ) − itiy (cid:19) In other words, β acts by identifying two copies of E (and by negation on each of them)and by the order 4 automorphism on the other two copies: ¯ A ≃ P ∪ P ∪ P . Similarly, one sees ¯ A = P ∪ P and ¯ A = P .4.3. The motivic semiorthogonal decomposition.
We construct Fourier-Mukai func-tors as in Section 3.2, by taking the structure sheaves of (equivariantized) graphs of theinclusions of the fixed loci in A i × A . We denote by Φ i : D ( ¯ A i ) → D [ A/G ] the cor-responding Fourier-Mukai functors. The fixed loci A i with i ≥ only contain pointsthat are fixed by G . The action of G locally at each of these points coincides with thatdescribed in Section 3, so we can modify the Φ i by twisting Φ ′ = Φ ( − ) ⊗ M Φ ′ = Φ ( − ) ⊗ V Φ ′ = Φ ( − ) ⊗ χ χ Φ ′ = Φ ( − ) , Φ ′ = Φ ( − ) ⊗ χ χ χ OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 15 where M is the torsion sheaf supported at { e, t } whose fiber at every point is locallyisomorphic to the module C [ x, y ] / ( x + y , x − y ) (see Section 3.3).We then modify Φ by only twisting by a character on points with stabilizer G , anddefining for X ∈ D ( ¯ A ) : Φ ′ ( X ) := ( Φ ( X ) ⊗ χ if Supp( X ) ⊂ { e, t } Φ ( X ) if Supp( X ) ⊂ ¯ A \ { e, t } and extending additively to D ( ¯ A ) . Then we have: Theorem 5.
In the case of a smooth quotient
A/G of type (C), the functors Φ , ..., Φ , Φ ′ , ..., Φ ′ give rise to a motivic semiorthogonal decomposition (7) D [ A/G ] = (cid:10) D ( ¯ A ) , ..., D ( ¯ A ) (cid:11) . Proof of Theorem 5.
Everything can be checked locally around each point p ∈ A . If thestabilizer of p is not µ × µ , (7) restricts to the semiorthogonal decomposition (6), andthe statement follows from Theorem 4.On the other hand, suppose p has a µ × µ stabilizer: the local model around p is A where each copy of µ acts by negating a coordinate, and it has a motivic semiorthogonaldecomposition(8) D [ A /µ × µ ] = hD ( ¯ A ) , D ( ¯ A ) , D ( ¯ A ) , D (pt) i . When restricted to p , the categories in (7) yield the semiorthogonal decomposition (8).For example, let p = ( e, a ) . We have p ∈ A ∩ A ∩ α ∗ A ∩ A . Then we have local isomorphisms Φ ( O p ) ≃ C [ u, v ] / ( u , v )Φ ( O p ) ≃ C [ u, v ] / ( u , uv, v )Φ ( O p ) ≃ C [ u, v ] / ( u, v ) , which satisfy the correct orthogonalities by (8) and generate D [ A/G ] at p . (cid:3) Corollary 3.
The category D [ A/G ] admits a full exceptional collection.Proof. In fact, each of the pieces D ( ¯ A i ) appearing in (7) admits a full exceptional col-lection: the ¯ A i are computed in Section 4.2 and are unions of projective spaces andpoints. (cid:3) Motivic semiorthogonal decompositions for D [ A/G ] We will break the proof up into three parts. In §5.1, we prove the case of zero-dimensionalfixed loci where G acts by group automorphism. In §5.2, we prove the case of positivedimensional fixed loci where G acts by group automorphism. In §5.3, we use the resultsof §5.1 and §5.2 to prove the case where G is an arbitrary finite group of automorphisms,i.e. not necessarily fixing the identity. Zero dimensional fixed loci.
Suppose G acts on A by group automorphisms. Weassume G acts irreducibly on T e ( A ) and hence dim( A G ) = 0 . Theorem 6.
Suppose G is a finite group of automorphisms of an Abelian variety A thatfix the identiy. If G acts irreducibly on T e ( A ) , then D [ A/G ] has a motivic semiorthogonaldecomposition.Proof. The case G ∼ = C n ⋊ S n with C = 1 has already been done in [PVdB19].We now consider the case G ∼ = S n +1 and A ⊂ X = E n +1 is the divisor given by x + · · · + x n +1 = 0 . We apply [LP17, Thm 1.2.1]. Let W ( λ ) = C ( λ ) / ( λ ) . Then W ( λ ) acts on X λ and we set X frλ to be the free locus. Then we need to show that A ∩ X frλ isdense in A ∩ X λ .To that end, write λ = (1 r )(2 r ) · · · ( k r k ) . Then we can identify W λ ∼ = S r × S r ×· · · × S r k . We can then identify X λ ∼ = E r × E r × · · · E r k then we can identify A ∩ X λ as the set of points e ∈ X such that e + · · · + e r + 2 e r +1 + · · · + 2 e r + r + · · · + ke r + ··· + r k − +1 + · · · + ke r + ··· + r k = 0 . Consequently A ∩ X frλ is the subset of A ∩ X λ such that e = · · · 6 = e r , e r +1 = · · · 6 = e r + r , . . . , e r + ··· + r k − +1 = · · · 6 = e r + ··· + r k . Solving for e r + ··· + r k we see that this is the open set { e ∈ X λ | e r + ··· + r k ∈ − e −· · ·− e r − e r +1 −· · ·− e r + r −· · ·− ke r + ··· + r k − + E [ k ] } where E [ k ] is the k -torsion of E . Since any non-free point can be moved while stayingin this set, it is open on each connected component of A ∩ X frλ . Thus it is dense. Weconclude the MSOD conjecture holds for [ A/S n +1 ] .Lastly, if A is a surface of Type (C), this is Theorem 5. (cid:3) Positive dimensional fixed loci.
Suppose now that dim( A G ) > . Let A ⊂ A G be the connected component containing the identity and P G a complementary Prymsubvariety . Set ∆ = P G ∩ A , then there is a short exact sequence of Abelian varieties → ∆ → P G × A → A → and an isomorphism P G × A / ∆ ∼ = A where ∆ acts freely on A . Set ˜ A = P G × A for convenience. Note, the ∆ -equivariantderived category of ˜ A is equivalent to the derived category of A : D ( ˜ A ) ∆ ∼ = D ( A ) via the pushforward mapMoreover, since ∆ ⊂ A G , we see that ∆ and G commute with eachother inside Aut( P G × A ) . So we can consider [ ˜ A/G × ∆] and the corresponding derived equivalence: D [ ˜ A/G ] ∆ ∼ = D [ ˜ A/G × ∆] ∼ = D [ A/G ] . OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 17
Note, the action is diagonal as ∆ acts on both P G and A . The quotient stack [ ˜ A/G ] possesses a motivic semiorthogonal decomposition. Consider the functors defining it: Φ λ : D ( P λG /C G ( λ ) × A ) ֒ → D [ ˜ A/G ] . The functor Φ λ corresponds to a G -equivariant kernel K λ on ˜ A λ × ˜ A , where G acts onthe right. Theorem 7.
For each λ ∈ G/ ∼ , the functors Φ λ are fully-faithful and there exists atotal order on G/ ∼ such that the functors give a motivic semiorthogonal decomposition of D [ A/G ] Proof.
We show the functors defining motivic semiorthogonal decompositions for D [ ˜ A/G ] descend to a semiorthogonal decomposition of ∆ -equivariant derived categories by show-ing that they are invariant under the action of ∆ . To that end, notice that there are iso-morphisms ˜ A λ /C G ( λ ) × ∆ ∼ = A/C G ( λ ) which give derived equivalences D ( ˜ A λ /C G ( λ )) ∆ ∼ = D ( A/C G ( λ )) and D [ ˜ A/G ] ∆ ∼ = D [ A/G ] . Since
A/G is smooth, so is P G /G . If G does not act irreducible on T e ( P G ) , then asbefore the motivic decompositions will decompose as well. So we can assume that [ P G /G ] is of types A, B, or C.In each of the irreducible cases, the kernels defining Fourier-Mukai functors are sup-ported on: Z λ = [ g ∈ G g · Γ with its reduced structure. In particular, the image of the Fourier-Mukai functor Φ λ ( D ( ¯ A λ )) is supported on ∪ µ ∼ λ A µ . Since the action of ∆ preserves the support, i.e. µ ( δ ( x )) = δ ( x ) for all x ∈ X µ and δ ∈ ∆ , each of the images of Φ λ ( D ( ¯ A λ )) is ∆ -invariant. Thus the motivic semiorthog-onal decompositions we have constructed for D [ ˜ A/G ] descend to a semiorthogonal de-composition for D [ A/G ] , [AE21, Corollary 3.18]. Finally, the ∆ -equivariant derived cate-gory of D ( P λG /C G ( λ ) × A ) is equivalent to D [ A λ /C G ( λ )] as [( P λG /C G ( λ ) × A ) / ∆] ∼ = A λ /C G ( λ ) . (cid:3) Translations.
Suppose now that G does not fix the origin. That is, G possesses anormal subgroup of translations. Let T ⊂ G be the subgroup generated by translationsand H be the subgroup that fixes the origin. Notice that T acts freely and G admits asemi-direct product decomposition: G = T ⋊ H. Let A T = A/T , then the action of H on A descends to an action on A T . Given by h (¯ a ) = h ( a ) where a is any preimage of ¯ a under the quotient map π : A → A T . This is well-defined as T is a normal subgroup. Our goal will be to construct a motivic semiorthogonaldecomposition of D [ A/G ] by using the one we know exists for D [ A T /H ] . In fact, we can prove a more general statement. Suppose a finite group G = K ⋊ H actseffectively on a smooth quasi-projective variety X with K acting freely so that X K = X/K is also a smooth quasi-projective variety. Then H acts on X K . Lemma 4.
Pick conjugacy class representatives ( k , h ) , . . . , ( k r , h ) so that ( k, h ) is conju-gate to one of ( k i , h ) . Then there is an isomorphism of varieties r a i =1 X ( k i ,h ) /C G (( k i , h )) → X hK /C H ( h ) . Proof.
Since all varieties involved are quasi-projective and normal, it is enough to checkthat it is a bijection on closed points. For surjectivity, we take ¯ x ∈ X hK and show thereexists k ∈ K, h ∈ H and x ∈ X so that ( k, h )( x ) = x and ¯ x is conjugate to the imageof x under the quotient X → X K . Pick any lift x of ¯ x . Since h (¯ x ) = ¯ x , h ( x ) is in thesame orbit as x under the K action. That is, there exists k ∈ K so that k ( h ( x )) = x .There exists ( k ′ , h ′ ) so that conjugating ( k, h ) by ( k ′ , h ′ ) is one of the ( k i , h ) classes. Itfollows that x is conjugate to the image of k i ( h ( x )) under the action of H . This provessurjectivity.Since the action of K is free, if k = k ′ , then X ( k,h ) ∩ X ( k ′ ,h ) = ∅ . Indeed, any x inthe intersection would force k − k ′ h ( x ) = h ( x ) and hence k − k ′ stabilizes a point. Butonly the identity stabilizes a point. Hence, we have X ( k i ,h ) ∩ X ( k j ,h ) = ∅ for i = j .Now if x i ∈ X ( k i ,h ) and x j ∈ X ( k j ,h ) , then the images of x i and x j are not identifiedby an element in the centralizer of C H ( h ) . Indeed, suppose they were, i.e. that thereexists h ′ ∈ H with h ′ h = h ′ h and h ′ (¯ x i ) = ¯ x j . Then h ′ ( x i ) ∈ X ( k j ,h ) . So x i = (1 , h ′ ) − ( k j , h )( h ′ ( x i )) = ( k h ′ j , h )( x i ) . Hence, ( k j , h ) is conjugate to ( k i , h ) but this cannot happen if i = j . It follows thatdisjoint components have disjoint images.Now if x, y ∈ X ( k i ,h ) are such that ¯ x = ¯ y . Then, since K acts freely, there existsa unique k ∈ K such that k ( x ) = y . We need to see that ( k, centralizes ( k i , h ) .Conjugation of ( k i , h ) by ( k, fixes x : ( k i , h ) ( k, ( x ) = ( k, − ( k i , h )( k, x ) = ( k, − ( h i , k )( y ) = ( k, − ( y ) = x. We conclude that the fixed locus of ( k i , h ) ( k, and ( k i , h ) intersect. But this only happensif ( k i , h ) ( k, = ( k i , h ) and hence ( k, centralizes ( k i , h ) . (cid:3) The following is immediate.
Corollary 4.
Let ( k , h ) , . . . , ( k r , h ) be as in Lemma 4. Then D ( X hK /C H ( h )) ∼ = r M i =1 D ( X ( k i ,h ) /C G ( k i , h )) . Lemma 5.
There is an equivalence of categories: coh ( X ) G ∼ = coh ( X K ) H induced by the quotient map π : X → X K . This functor can also be described by taking T -invariants. In particular, this equivalence of Abelian categories induces a derived equiva-lence: D [ X/G ] ∼ = D [ X K /H ] . OTIVIC SEMIORTHOGONAL DECOMPOSITIONS FOR ABELIAN VARIETIES 19
Proof.
This is straightforward. If F is a sheaf on X , then a G -equivariant structure is thedata of a K -equivariant structure with a compatible H -equivariant structure. The dataof a K -equivariant structure is the data of a pullback of a sheaf ¯ F on X K . The data of acompatible H -equivariant structure is the data of an H -equivariant structure on ¯ F . Thedetails are left to the reader. (cid:3) Theorem 8.
Suppose G = K ⋊ H acts on a smooth quasi-projective variety X so that K acts freely. Consider the induces action of H on the quotient X K = X/H and suppose D [ X K /H ] admits a motivic semiorthogonal decomposition of the form D [ X K /H ] = hD ( ¯ X h K ) , . . . , D ( ¯ X h t K ) i . Then D [ X/G ] admits a motivic semiorthogonal decomposition.Proof. The equivalence of Lemma 5 gives a derived equivalence D [ X/G ] ∼ = D [ X K /H ] . The motivic semiorthogonal decomposition of D [ X K /H ] directly induces one of D [ X/G ] .The pieces D ( X h /C H ( h )) of the motivic semiorthogonal decomposition of D [ X K /H ] decompose into D ( X ( k i ,h ) /C G ( k i , h )) . Since these are pairwise completely orthogonal,any total order suffices. Thus we get a semiorthogonal decomposition D [ X/G ] = h r M i =1 D ( X ( k i ,h ) /C G (( k i , h ))) , . . . , r t M i =1 D ( X ( k i ,h t ) /C G (( k i , h t )) i . Finally D ( X/G ) linearity follows from D ( X K /H ) -linearity using the canonical isomor-phism X/G ∼ = X K /H . (cid:3) Combining Theorem 8 with our earlier work gives motivic semiorthogonal decompo-sitions for all Abelian varieties with smooth quotients.
Theorem 9.
Let G = T ⋊ H be a finite group of automorphisms of an Abelian variety A such that A/G is smooth. Then there exists a motivic semiorthogonal decomposition for D [ A/G ] . Examples
Curves.
Quotients of an elliptic curve E by a finite group of group automorphismsare weighted projective lines in the sense of Geigle and Lenzing [GL87]. The group G is µ n for n = 2 , , , or . All quotients have P as a coarse space. There are 4 µ stabilizersfor the generic case n = 2 and 3 stabilizers for the other cases, of orders (3 , , , (4 , , ,and (6 , , respectively. From the point of view of representation theory, the categories coh ([ E/G ]) are the module categories over a canonical algebra in the sense of Ringel[Rin90]. In this context, the motivic semiorthogonal decompositions of Theorem 1 recoverthe exceptional collections constructed in [Mel95].6.2. Surfaces.
In this section, we explicitly compute the motivic semiorthogonal decom-positions constructed here and in [PVdB19] when A is a surface and the action of G fixes e and acts irreducibly on T e ( A ) . Such an action is one of three types by Theorem 3. Wehave already written out Type (C) in Section 4. We begin with Type (A): Example 2.
Let A ∼ = E and G = µ n ⋊ S for n = 1 , , , , . The generic case, n = 1 ,has coarse quotient Sym ( E ) and the non-trivial element fixes the diagonal. Hence, D [ A/G ] = hD (Sym ( E )) , D ( E ) i . In all other cases, [ A/G ] admits a full exceptional collection. The coarse quotient is Sym ( P ) ≃ P .If n = 2 , there are 5 conjugacy classes, whose fixed loci and centralizers are as follows:Representative g A g C ( g ) ¯ A g (1 , , A G P ( − , − , E [2] × E [2] G pt( − , , E [2] × E h ( − , , , ( − , − , i E [2] × P (1 , , σ ) E h (1 , , σ ) , ( − , − , i P (1 , − , σ ) E [2] h (1 , − , σ ) i E [2] Thus we have D [ A/G ] = hD ( P ) , D ( P ) , ..., D ( P ) | {z } copies , D (pt) , ..., D (pt) | {z } copies i . A similar computation (omitted) computes the MSOD in the other cases. For n = 3 there are 9 conjugacy classes. The corresponding MSOD has one two-dimensional com-ponent, 3 one-dimensional components, and 5 zero-dimensional components: D [ A/G ] = hD ( P ) , D ( P ) , ..., D ( P ) | {z } copies , D (pt) , ..., D (pt) | {z } copies i . For n = 4 there are 14 conjugacy classes. The corresponding MSOD has one two-dimensional component, 4 one-dimensional components, and 9 zero-dimensional com-ponents: D [ A/G ] = hD ( P ) , D ( P ) , ..., D ( P ) | {z } copies , D (pt) , ..., D (pt) | {z } copies i . For n = 6 there are 27 conjugacy classes. The corresponding MSOD has one two-dimensional component, 6 one-dimensional components, and 20 zero-dimensional com-ponents: D [ A/G ] = hD ( P ) , D ( P ) , ..., D ( P ) | {z } , D (pt) , ..., D (pt) | {z }
26 copies i . For type (B):
Example 3.
Consider [ E /S ] . The coarse quotient is P . The group S acts on T e ( E ) by (1 ,
2) = (cid:18) (cid:19) , (1 , ,
3) = (cid:18) − − (cid:19) . The conjugacy class corresponding to (1 , fixes the diagonal ∆ ⊂ E × E . The conju-gacy class corresponding to (1 , , fixes the diagonal copy of the three torsion subgroup E [3] . The centralizers act trivially on the fixed loci. By Theorem 6, there is a motivicsemiorthogonal decomposition of the form D [ E /S ] = hD ( P ) , D ( E ) , D ( E [3]) i . References [AE21] Benjamin Antieau and Elden Elmanto,
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Email address : [email protected] FR: Department of Mathematics, Rutgers, Piscataway, NJ 08854, USA
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