The question is whether the other Nakajima operators also have analogues in the form of Pn−1-functors
H`,n:DbS`(X×X
`)∼=Db(X×X[`])→Db(X[n+`])∼=Db
Sn+`(X n+`).
A rst natural guess for a generalisation ofH0,n would be the functor
H`,n0 :DbS`(X×X`)−−→triv DbSn×S`(X×X `) δ[n]∗
−−−→DbSn×S`(X
n×X`) Inf
−→DbSn+`(Xn+`); (7.2) see Section7.2.1 for details on the ination functorInf and its adjointRes. Here, for a subset
J ⊂ [n+`] = {1, . . . , n+`} of cardinality |J| = n the morphism δJ denotes the closed
embedding of the partial diagonal
X×X` ∼= ∆J =
(y1, . . . , yn+`)|ya=yb for alla, b∈J ⊂Xn+`.
In fact, for every bijectionµ: [`]→J¯:= [n+`]\J there is the embeddingδJ,µ:X×X` →Xn+`
onto∆J given by
δJ,µ(x, x1, . . . , x`) = (y1, . . . , y`+n) , yj =x∀j∈J , yµ(i)=xi∀i∈[`]
and we set δJ = δJ,e where e: [`]→ J¯denotes the unique strictly increasing bijection. The
functorH`,n0 is given on objects by H`,n0 (E) = M
J⊂[n+`],|J|=n
δJ∗E for E ∈DbS`(X×X
The rst part of the composition (7.2), namely δ[n]∗◦triv, can be rewritten as H0,n idX`.
SinceH0,n is a Pn−1-functor so isδ[n]∗◦trivand (δ[n]∗◦triv)R(δ[n]∗◦triv)∼= ¯S −[0,n−1] X :=id⊕S¯ −1 X ⊕ · · · ⊕S¯ −(n−1) X where we write S¯X := ( )⊗(ωX O
X`)[2] even in the case that X is not projective. But
for the computation ofH`,n0R◦H`,n also the other summands of (7.3) besidesδ[n]∗E have to be taken into account which yields
H`,n0RH`,n0 (E)∼= ¯SX−[0,n−1](E)⊕ terms supported on partial diagonals of X×X`
. (7.4)
The approach is to adaptH`,n0 slightly in order to erase the error term in (7.4). We succeed
in doing so by replacingH`,n0 by a complex of functors
H`,n := 0→H`,n0 →H`,n1 → · · · →H`,n` →0).
More concretely, this means the following. First, we have to setH`,n0 :=Inf◦δ[n]∗◦Man◦triv.
That means that the denition ofH`,n0 diers from (7.2) by
Man := ( )⊗Can:D b Sn×S`(X×X `)→Db Sn×S`(X×X `)
where the alternating representation an is the non-trivial character of Sn. The FM kernel of
H`,n0 is given by P`,n0 = M J⊂[n+`],|J|=n µ: [`]→J¯bijection OΓ δJ,µ ⊗aJ ∈D b S`×Sn+`(X×X `×Xn+`).
In Section7.2.4we construct a complexP`,n = (0→ P0
`,n→ · · · → P`,n` →0)whose termsP`,ni
for 1≤i≤`are direct sums of structure sheaves of certain subvarieties of the graphs ΓδJ,µ.
Then we setH`,n:=FMP`,n. The denition of the complexP`,n∈D b
S`×Sn+`(X×X
`×Xn+`)
makes sense forX a variety of arbitrary dimension and our rst result is
Proposition A. LetX =C be a smooth curve andn >max{`,1}. 1. We have HR
`,n◦H`,n ∼=id which means that H`,n is fully faithful.
2. Let`0, n0 be positive integers with n0+`0 =n+` and`0 > `. Then HR
`0,n0◦H`,n= 0.
Corollary B. Letm≥2. Form even we setr = m2 −1 and form odd we setr = m2−1. For every smooth curve C there is a semi-orthogonal decomposition
DbSm(Cm) =hA0,m,A1,m−1, . . . ,Ar,m−r,Bi
where A`,m−` :=H`,m−`(DbS`(C×C
`))∼=Db
S`(C×C `).
We will see in Section7.5.8that the fully faithful functors H`,m−` also induce autoequiva-
lences ofDbSm(Cm). Note that forC=Ean elliptic curve the canonical bundle of the product Em is trivial. But as a Sn-bundle it is given by OEn ⊗an which means that the canonical
bundle of the quotient stack [En/Sn] is non-trivial. Otherwise, Db([Em/Sm]) ∼= DbS m(E
m)
could not allow a semi-orthogonal decomposition.
To us, the case of most interest is that of a smooth quasi-projective surface due to the BKRH equivalence Db(X[m])∼=DbS
n(X m).
Theorem C. Let X be a smooth surface and n > max{`,1}. Then H`,n is a Pn−1-functor
with P-cotwist S¯X. In particular, HR
`,n◦H`,n ∼= ¯S
−[0,n−1]
X .
In the case that X = A is an abelian variety, there is also the following variant of the
above construction. For m≥2 we consider theSm-invariant subvariety
Am−1 ∼=Nm−1A:=
(a1, . . . , am)|a1+· · ·+am = 0 ⊂Am.
IfAis an abelian surface there is a variant of the BridgelandKingReidHaiman equivalence
as an equivalence Db(Km−1A) ∼= DbSm(Nm−1A) where Km−1A ⊂ A[m] is the generalised
Kummer variety; see [Nam02] or [Mea15]. We also consider the S`-invariant subvariety
M`,n :=
(a, a1, . . . , a`)|n·a+a1+· · ·+a`= 0 ⊂A×A`.
Then for all n ≥ 2 the functor H`,n: DbS`(A ×A
`) → Db
Sn+`(A
n+`) restricts to a functor
ˆ
H`,n:DbS`(M`,nA)→DbSn+`(Nn+`−1A); see Section7.4.12for details.
For ` = 0 we have M0,nA = An ⊂ A where An denotes the set of n-torsion points.
The functor Hˆ0,n is given by sending the skyscraper sheaf C(a) of a ∈ An to the object C(a, . . . , a) ⊗an ∈ DbSn(Nn−1A). As shown in [6], for A an abelian surface the objects
C(a, . . . , a)⊗an are Pn−1-objects in the sense of [HT06]. Similarly, for A = E an elliptic
curve they are exceptional.
In contrast, for`≥1we have a (notS`-equivariant) isomorphismM`,nA∼=A` so that the
domain category of the functorHˆ`,n is indecomposable.
Proposition A'. Let A=E be an elliptic curve and n >max{`,1}. 1. We have HˆR
`,n◦Hˆ`,n ∼=id which means that Hˆ`,n is fully faithful.
2. Let`0, n0 be positive integers with n0+`0 =n+` and`0 > `. Then HˆR
`0,n0◦Hˆ`,n= 0.
Corollary B'. For every elliptic curveE there is a semi-orthogonal decomposition
DbSm(Nm−1E) =hC(a, . . . , a)⊗an|a∈Am,Aˆ1,m−1, . . . ,Aˆr,m−r,Biˆ whereAˆ`,m−` := ˆH`,m−`(Db S`(M`,m−`A)) ∼ =DbS `(M`,m−`A).
Theorem C'. LetA be an abelian surface and n >max{`,1}. Then Hˆ`,n is a Pn−1-functor
with P-cotwist[−2]. In particular, H`,nR ◦H`,n ∼=id⊕[−2]⊕ · · · ⊕[−2(n−1)].