arXiv:1108.6165v2 [math.AG] 21 Mar 2015 Moduli spaces of principal bundles on singular varieties Adrian Langer 05.01.2012 ADDRESS : Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland To the memory of Professor Masaki Maruyama. Abstract Let k be an algebraically closed field of characteristic zero. Let f : X → S be a flat, projective morphism of k-schemes of finite type with integral geometric fibers. We prove existence of a projective relative moduli space for semistable singular principal bundles on the fibres of f . This generalizes the result of A. Schmitt who studied the case when X is a nodal curve. 1 Introduction Let X be a smooth projective variety defined over an algebraically closed field k of characteristic 0. In [14] and [15] M. Maruyama, generalizing Gieseker’s result from the surface case, constructed coarse moduli spaces of semistable sheaves on X (in fact the construction worked in some other cases). Later these moduli 2000 Mathematics Subject Classification. Primary: 14D20, 14D22; Secondary: 14H60, 14J60. 1
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Moduli spaces of principal bundles on singularvarieties
Adrian Langer
05.01.2012
ADDRESS:Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa,Poland
To the memory of Professor Masaki Maruyama.
Abstract
Let k be an algebraically closed field of characteristic zero. Letf : X →S be a flat, projective morphism ofk-schemes of finite type with integralgeometric fibers. We prove existence of a projective relative moduli spacefor semistable singular principal bundles on the fibres off .
This generalizes the result of A. Schmitt who studied the case whenX isa nodal curve.
1 Introduction
Let X be a smooth projective variety defined over an algebraicallyclosed fieldkof characteristic 0. In [14] and [15] M. Maruyama, generalizing Gieseker’s resultfrom the surface case, constructed coarse moduli spaces of semistable sheaveson X (in fact the construction worked in some other cases). Laterthese moduli
spaces were also constructed for arbitrary varieties (see C. Simpson’s paper [21])and in an arbitrary characteristic (see [11] and [12]). Since the moduli space ofsemistable sheaves compactifies the moduli space of (semistable) vector bundles,it is an obvious problem to try to construct similar compactifications in case ofprincipal bundles. This problem was considered by many authors (see [20] andthe references within) and it was solved in case of smooth varieties. However, incase of singular varieties the problem is still open in spiteof some partial results(see, e.g., [3] and [18]). The aim of this paper is to solve this problem in thecharacteristic zero case.
Let ρ :G→ GL(V) be a faithfulk-representation of the reductive groupG. Inthe following we assume that image of the representationρ is contained in SL(V).
A pseudo G-bundleis a pair(A ,τ), whereA is a torsion freeOX-moduleof rank r = dimV andτ:Sym∗(A ⊗V)G → OX is a nontrivial homomorphismof OX-algebras. In [3] U. Bhosle, following earlier work of A. Schmitt [16] inthe smooth case, constructed the moduli space of pseudoG-bundles in caseXsatisfies some technical condition, which she showed to holdfor seminormal orS2-varieties. However, it is easy to see that this condition isalways satisfied (seeLemma 2.3).
Giving the homomorphismτ is equivalent to giving a section
σ : X →Hom(A ,V∨⊗OX)//G= Spec(Sym∗(A ⊗V)G).
Let UA denotes the maximum open subset ofX whereA is locally free. We saythat the pseudo-G-bundle(A ,τ) is asingular principal G-bundleif there exists anon-empty open subsetU ⊂UA such thatσ(U)⊂ Isom(V ⊗OU ,A
∨ |U)/G.In case whenX is smooth, A. Schmitt showed in [17] that the moduli space of
δ -semistable pseudoG-bundles parametrizes only singular principalG-bundles(for large values of the parameter polynomialδ ). In a subsequent paper [18],he also showed that in case whenX is a curve with only nodes as singularities,the moduli space constructed by Bhosle parameterizes only singular principalG-bundles. Moreover, under some mild assumptions on the representationρ , heproved thatσ(UA )⊂ Isom(V ⊗OU ,A
∨ |UA)/G (in this case we say that(A ,τ)
is anhonest singular principal G-bundle).In this paper we prove that the same result holds for all the varieties: the mod-
uli space constructed by Bhosle (for large values of the parameter polynomialδ )parameterizes singular principalG-bundles for all varietiesX and all representa-tionsρ . More precisely, we prove the following theorem:
THEOREM 1.1. Let f : X → S be a flat, projective morphism of k-schemes of finitetype with integral geometric fibers. Assume that k has characteristic zero. Let us
Moduli spaces of principal bundles on singular varieties 3
fix a polynomial P and a faithful representationρ :G → SL(V) ⊂ GL(V) of thereductive algebraic group G.
1. There exists a projective moduli space MρX/S,P → S for S-flat families of
semistable singular principal G-bundles on X→ S such that for all s∈ Sthe restrictionA |Xs has Hilbert polynomial P.
2. Let P correspond to sheaves of degree0. If the fibres of f are Gorensteinand there exists a G-invariant non-degenerate quadratic form ϕ on V thenMρ
X/S,P → S parameterizes only honest singular principal G-bundles.
Since the fibre ofMρX/S,P → Sovers∈ S is equal toMρ
Xs,P this theorem showsthat moduli spaces of singular principal bundles are compatible with degeneration.
Our approach is similar to the one used in [5], [6] as explained in [20]: weprove a global boundedness result for swamps (this part of our paper works in anycharacteristic). Then we use this fact to prove the semistable reduction theoremin the same way as in the case of smooth varieties. The above mentioned bound-edness result is the main novelty of the paper. It is obtainedby proving that thetensor product of semistable sheaves on a variety is not far from being semistable.
The second part of the theorem follows from careful computation of Hilbertpolynomials of dual sheaves on Gorenstein varieties.
Unfortunately, the above approach does not work in positivecharacteristicbecause we still do not know how to construct moduli spaces ofswamps for rep-resentations of typeρa,b,c:GL(V)→ GL((V⊗a)⊕b⊗ (detV)−c) for c 6= 0. In caseof characteristic zero, to construct the moduli space of pseudoG-bundles it wassufficient to use moduli spaces ofρa,b,c-swamps forc= 0. But the constructionused the Reynolds operator which is not available in positive characteristic.
Moreover, in positive characteristic there appears a serious problem with defin-ing the pull-back operation for families of pseudoG-bundles on non-normal vari-eties (see [20, Remark 2.9.2.23]).
The structure of paper is as follows. In Section 2 we recall some definitionsand results, and we show that Bhosle’s condition is satisfiedfor all varieties. InSection 3 we study Picard schemes in the relative setting andwe state some ex-istence results for moduli spaces of swamps. Section 4 is a technical heart of thepaper: we prove that the tensor product of semistable sheaves on non-normal va-rieties is close to being semistable. Then in Section 5 we show that in many casessingular principal bundles of degree 0 are honest. In Section 6 we use all these
4 A. Langer
results to prove semistable reduction theorem and to show existence of projectiverelative moduli spaces for (honest) singular principal bundles.
Notation.All the schemes in the paper are locally noetherian. Avarietyis an irreducible
and reduced separated scheme of finite type over an algebraically closed field.
2 Preliminaries
2.1 Basic definitions
Let X be ad-dimensional projective variety over an algebraically closed fieldk.Let OX(1) be an ample line bundle onX.
We say that a coherent sheafE on X is torsion freeif it is pure of dimensiond. For a torsion free sheafE we can write its Hilbert polynomial as
P(E)(m) := χ(X,E⊗OX(m)) =d
∑i=0
αi(E)mi
i!.
The rank of E is defined as the dimension ofE⊗K(X), whereK(X) is the fieldof rational functions. It is denoted by rkE and it is equal toαd(E)/αd(OX). Wealso define thedegreeof E as
degE = αd−1(E)− rkE ·αd−1(OX)
(see [9, Definition 1.2.11]). Theslopeµ(E) is, as usually, defined as the quotientof the degree ofE by the rank ofE.
For two coherent sheavesE,F on X we set
E⊗F = E⊗F/Torsion.
LEMMA 2.1. If X is a normal variety and E and F are torsion free sheaves on Xthen
µ(E⊗F) = µ(E)+µ(F).
Proof. If E is a torsion free sheaf then for a general choice of hyperplanesH1, ...,Hd ∈|OX(1)| we have
P(E)(m) =d
∑i=0
χ(E|⋂j≤i H j )
(m+ i −1
i
)
Moduli spaces of principal bundles on singular varieties 5
(see [9, Lemma 1.2.1]). It follows that the rank and degree ofE depend only onχ(E|⋂
j≤i H j ) for i = d andi = d−1.If X is a normal variety then by assumptionE is locally free outside of a closed
subset of codimension≥ 2. For a general choice of hyperplanesH1, ...,Hd ∈|OX(1)| the intersection
⋂j≤d H j is a union of points and
⋂j≤d−1H j is a smooth
curve. Therefore the sheavesE|⋂j≤i H j for i = d and i = d− 1 are locally free.
Similarly, the sheavesF|⋂j≤i H j for i = d and i = d−1 are locally free. Since in
case of points and smooth curves our assertion is clear, we get the lemma.
If X is normal then we can define the determinant of a torsion free sheafE asthe reflexivization of
∧rkE E. In this case the degree degE is equal to the degreeof the determinant. This fact follows immediately from the proof of the abovelemma.
2.2 Serre’s conditionsSk
We say that a coherent sheafE on a schemeX satisfiescondition Sk if for all pointsx∈ X we have depthx(Ex)≥ min(dimEx,k).
The following lemma is quite standard but we need a more general versionthan usual. In case of smooth projective varieties it is essentially equivalent to [9,Proposition 1.1.6].
LEMMA 2.2. Let X be a Cohen–Macaulay scheme of finite type over a field. Then
1. E xtqX(E,ωX) is supported on the support of E and for all points x∈ X wehaveE xtqX(E,ωX)x= 0 if q< codimxE. Moreover,codimxE xtqX(E,ωX)≥qfor q≥ codimxE.
2. E satisfies condition Sk if and only if for all points x∈X we havecodimxE xtqX(E,ωX)≥q+k for all q> codimxE.
Proof. By assumptionX is Cohen–Macaulay and every local ringOX,x is a quo-tient of a regular local ring, so we can apply the local duality theorem (see [8,Theorem 6.7]) to prove thatE xtqX(E,ωX)x 6= 0 if and only if H dimx X−q
x (E) 6= 0.
But the local cohomologyH dimxX−qx (E) vanishes if dimxX−q> dimxE, which
proves the first part of 1. Ifq= codimxE then codimx(E xtqX(E,ωX))≥ q is equiv-alent to the obvious inequality dimx(E xtqX(E,ωX))≤ dimxE. Hence, since everysheaf satisfiesS0, the second part of 1 follows from 2.
6 A. Langer
To prove 2 note that by [8, Theorem 3.8] depthx(Ex) ≥ min(dimEx,k) if andonly if H i
x (E) = 0 for all i <min(dimEx,k). By the local duality theorem this lastcondition is equivalent toE xtqX(E,ωX)x = 0 for q> max(codimxE,dimOX,x−k).This is equivalent to saying that forq> codimxE a non-vanishing ofE xtqX(E,ωX)x
implies dimOX,x ≥ q+k.
Let k be an algebraically closed field. LetX be ad-dimensional pure (i.e.,OX satisfiesS1) scheme of finite type overk. Let C be a smooth curve definedover k and let us fix a closed point 0∈ C. By pX : Z = X ×C → X we denotethe projection. LetY be a non-empty proper closed subscheme ofX ×{0} (inparticular, we assume thatX has dimension≥ 1), and leti : Y → Z denote thecorresponding closed embedding. Let us also setU = Z−Y and let j : U → Zdenote the corresponding open embedding.
LEMMA 2.3. If E is a pure sheaf of dimension d on X then we have a canonicalisomorphism p∗XE ≃ j∗ j∗(p∗XE). In particular, OZ ≃ j∗OU and for any locallyfree sheaf F on Z we have F≃ j∗ j∗F.
Proof. Let us setF = p∗XE. Since we have a canonical mapF → j∗ j∗F, the asser-tion is local and hence we can assume thatX andY are affine. By [8, Proposition2.2] we have an exact sequence
0→ i∗H0
Y (F)→ F → j∗ j∗F → i∗H1
Y (F)→ 0.
To prove thati∗H iY(F) = 0 for i = 0,1, it is sufficient to prove that for every point
y∈Y, the depth ofFy is at least 2 (see [8, Theorem 3.8]). Now, let us take a localparameters∈OC,0. ThenFy/sFy ≃Ey has depth at least 1 (because by assumptionE satisfiesS1), so the required assertion is clear.
Remark2.4. The above lemma shows in particular that every variety satisfies con-dition (2.19) in the sense of Bhosle (see [3, Definition 2.8]).
2.3 Moduli spaces of pseudoG-bundles
Let us fix a faithful representationρ :G → SL(V) ⊂ GL(V), r = dimV, of a re-ductive algebraic groupG.
A pseudo G-bundleis a pair(A ,τ), whereA is a torsion freeOX-moduleof rank r and τ:Sym∗(A ⊗V)G → OX is a nontrivial homomorphism ofOX-algebras. Givingτ is equivalent to giving a section
σ : X →Hom(A ,V∨⊗OX)//G= Spec(Sym∗(A ⊗V)G).
Moduli spaces of principal bundles on singular varieties 7
A weighted filtration(A•,α•) of A is a pair consisting of a filtration
A• = (0⊂ A1 ⊂ . . .⊂ As ⊂ A )
by saturated subsheaves (i.e., such that the quotientsA /Ai are torsion free) ofincreasing ranks and ans-tuple
α• = (α1, . . . ,αs)
of positive rational numbers. To every weighted filtration(A•,α•) one can asso-ciate the polynomial
M(A•,α•) :=s
∑i=1
αi(P(A ) · rk(Ai)−P(Ai) · rk(A )).
If (A•,α•) is a weighted filtration of a pseudoG-bundle(A ,τ) then one canalso define the numberµ(A•,α•,τ) describing stability of the SL(A ⊗K(X))-group action on Hom(A ⊗K(X),V∨⊗K(X))//G (see, e.g., [19, 3.3.2]).
Let us fix a positive polynomialδ with rational coefficients and of degree≤ dimX−1. Then we say that a pseudoG-bundle(A ,τ) is δ -(semi)stableif A
is torsion free and for any weighted filtration(A•,α•) of A we have inequality
M(A•,α•)+δ ·µ(A•,α•,τ)(≥)0.
To define the slope version of (semi)stability instead ofM(A•,α•) one usesthe rational number
L(A•,α•) :=s
∑i=1
αi(degA · rk(Ai)−degAi · rk(A )).
The following theorem follows from the results of Schmitt [16] (in the smoothcase) and from the results of Bhosle [3] and Lemma 2.3 in general:
THEOREM 2.5. Let (X,OX(1)) be a polarized projective variety defined over analgebraically closed field of characteristic zero. Then there exists a projectivemoduli space Mρ,δ
X,P for δ -semistable pseudo G-bundles(A ,τ) on X, such thatAhas Hilbert polynomial P (with respect toOX(1)).
8 A. Langer
2.4 Semistability of singular principal G-bundles
Let (A ,τ) be a pseudoG-bundle. Let us recall that givingτ is equivalent to givinga section
σ : X →Hom(A ,V∨⊗OX)//G= Spec(Sym∗(A ⊗V)G).
Let UA denotes the maximum open subset ofX whereA is locally free. Thepseudo-G-bundle(A ,τ) is a singular principal G-bundleif there exists a non-empty open subsetU ⊂UA such that
σ(U)⊂ Isom(V ⊗OU ,A∨ |U)/G.
If A has degree 0 andσ(UA ) ⊂ Isom(V ⊗OUA,A ∨ |UA
)/G then we say that(A ,τ) is anhonest singular principal G-bundle.
Let us recall that a singular principalG-bundle(A ,τ), via the following pull-back diagram, defines a principalG-bundleP(A ,τ) over the open subsetU :
P(A ,τ) //
��
Isom(V ⊗OU ,A∨ |U)
��
Uσ|U
// Isom(V ⊗OU ,A∨ |U)/G.
If X is smooth then every singular principalG-bundle is honest (see [19,Lemma 3.4.2]). Note that our definitions are slightly different to those appearingin previous literature (which changed in time to the one close to our definitions).
Let (A ,τ) be a singular principalG-bundle and letλ : Gm → G be a one-parameter subgroup ofG. Let
QG(λ ) := {g∈ G : limt→∞
λ (t)gλ (t)−1 exists inG}.
A reductionof (A ,τ) to λ is a sectionβ : U ′ → P(A ,τ)/QG(λ ) defined oversome non-empty open subsetU ′ ⊂U . Such reduction defines a reduction of struc-ture group of a principal GL(V)-bundle associated toA |U ′ to the parabolic sub-groupQGL(V)(λ ), so we get a weighted filtration(A ′
• ,α•) of A |U ′.Let j : U ′ → X denote the open embedding. Then fori = 1, ...,swe defineAi
as saturation ofA ∩ j∗(A ′i ). In particular, we get a weighted filtration(A•,α•)
of A .We say that a singular principalG-bundle(A ,τ) is (semi)stableif A is torsion
free and for any reduction of(A ,τ) to a one-parameter subgroupλ : Gm → G wehave inequality
M(A•,α•)(≥)0.
Moduli spaces of principal bundles on singular varieties 9
3 Moduli spaces of swamps revisited
In this section we recall and reprove some basic results concerning existence ofthe relative Picard scheme and its compactifications. Then we apply these resultsto existence of moduli spaces of swamps.
We interpret the compactified Picard scheme as the coarse moduli space ofstable rank 1 sheaves and we use Simpson’s construction of these moduli spaces toprove existence of the universal family (i.e., the Poincaresheaf) under appropriateassumptions. This approach, although very natural, seems to be hard to find inexisting literature, especially in the relative case.
The notation in this section is as follows.R denotes a universally Japanesering. We also fix a projective morphismf : X → Sof R-schemes of finite type withgeometrically connected fibers. We assume thatf is of pure relative dimensiond.By OX(1) we denote anf -very ample line bundle onX. We also fix a polynomialP.
3.1 Universal families on relative moduli spaces
Let us define the moduli functorMX/S,P : (Sch/S)−→ (Sets) by sendingT → Sto
MX/S,P(T) =
isomorphism classes ofT-flat families of Giesekersemistable sheaves with Hilbert polynomialPon the geometric fibres ofp : T ×SX → T
/∼,
where∼ is the equivalence relation∼ defined byF ∼ F ′ if and only if there existsan invertible sheafK on T such thatF ≃ F ′⊗ p∗K.
THEOREM 3.1. (see [14], [15], [21], [11] and [12])There exists a projectiveS-scheme MX/S,P, which uniformly corepresents the functorMX/S,P. Moreover,there is an open subscheme Ms
X/S,P ⊂ MX/S,P that universally corepresents thesubfunctorM s
X/S,P of families of geometrically Gieseker stable sheaves.
We are interested when the moduli schemeMsX/S,P represents the functorM s
X/S,P.This is equivalent to existence of a universal family onMs
X/S,P×SX.Let us recall that the moduli schemeMs
X/S,P is constructed as a quotient ofan appropriate subschemeRs of the Quot-scheme Quot(H ;P) by PGL(V). Letq∗H → F denote the universal quotient onRs×SX.
the functorM sX/S,P if and only if there exists aGL(V)-linearized line bundle A
on Rs on which elements t of the centre Z(GL(V)) ≃ Gm act via multiplicationby t. If such A exists thenH om(p∗A, F) descends to a universal family and anyuniversal family is obtained in such a way.
3.2 Existence of compactified Picard schemes in the relativecase
For simplicity we assume that all geometric fibers off are irreducible and reduced(hence they are varieties) and thatS is connected.
Let us fix a polynomialP. For all locally noetherianS-schemesT → S let usset
P ic′X/S,P(T)=
{isomorphism classes of invertible sheavesL on XT = T ×SXsuch thatχ(Xt ,Lt(n)) = P(n) for every geometrict ∈ T
}.
Note that ifP ic′X/S,P(T) is non-empty then the highest coefficient ofP is thesame as the highest coefficient of the Hilbert polynomial ofOXs for anys∈ S.
As before we introduce an equivalence relation∼ on P ic′X/S,P(T) by L ∼ L′
if and only if there exists an invertible sheafK onT such thatL ≃ L′⊗ p∗K. Thenwe can definethe Picard functor
P icX/S,P : (Sch/S)−→ (Sets)
by sending anS-schemeT to P icX/S,P(T) = P ic′X/S,P(T)/∼
Let us also define the compactified relative Picard functors.There are twodifferent methods of compactification of the Picard scheme.We can compactifythe Picard scheme by adding all the rank 1 torsion free sheaves on the fibres ofX or only those rank 1 torsion free sheaves that are locally free on the smoothlocus of the fibres. The second method has the advantage of producing a smallerscheme.
Let us set
P ic′X/S,P(T) =
isomorphism classes ofT-flat sheavesL on XT = T ×SXsuch thatLt is a torsion free, rank 1 sheaf onXt
andχ(Xt ,Lt(n)) = P(n) for every geometrict ∈ T
.
Moduli spaces of principal bundles on singular varieties 11
As before we definethe compactified Picard functor
P icX/S,P : (Sch/S)−→ (Sets)
by sending anS-schemeT to P icX/S,P(T) = P ic′X/S,P(T)/∼.We also definethe small compactified Picard functor
P icsmX/S,P : (Sch/S)−→ (Sets)
by sending anS-schemeT to
P icsmX/S,P(T) =
{L ∈ P ic′X/S,P(T) such thatL is locally freeon the smooth locus ofXT/T
}/∼ .
THEOREM 3.3. Assume that f: X → S has a section g: S→ X.
1. There exists a quasi-projective S-schemePicX/S,P that represents the PicardfunctorP icX/S,P.
2. If g(S) is contained in the smooth locus of X/S then there exists a pro-jective S-schemePicX/S,P that represents the compactified Picard functor
P icX/S,P. Moreover,PicX/S,P contains a closed S-subschemePicsmX/S,P that
represents the small compactified Picard functorP icsmX/S,P.
Proof. First let us remark that all the Picard functorsP icX/S,P, P icX/S,P and
P icsmX/S,P are subfunctors of the moduli functorMX/S,P. In fact, from our as-
sumptions it follows thatP icX/S,P = M sX/S,P = MX/S,P. Now we can construct
PicX/S,P, PicX/S,P andPicsmX/S,P as Geometric Invariant Theory quotients of appro-
priate subschemesRPic⊂RPicsm⊂RPic=Rs=Rssof the Quot-scheme used to con-
struct the moduli spaceMsX/S,P by GL(V). In fact all these quotients are PGL(V)-
principal bundles. To prove thatPicsmX/S,P is a closed subscheme ofPicX/S,P it is
sufficient to see thatRPicsm is a closed subscheme ofRPic. This follows from [2,
Lemma on p. 37] applied to the universal quotient restrictedto the smooth locusof RPic×SX → RPic.
To prove 1 by (a slight generalization of) Proposition 3.2 itis sufficient to showexistence of a GL(V)-linearized line bundleAPic on RPic on which the centre ofGL(V) acts with weight 1.
12 A. Langer
Let us setAPic= detp∗(F ⊗q∗Og(S)), whereF comes from the universal quo-tient onRPic×SX. The definition makes sense sinceF is a line bundle onRPic×SXandp∗(F ⊗q∗Og(S)) = (idRPic×Sg)∗F is also a line bundle. The centre of GL(V)acts on the fibre ofAPic at ([ρ ],x) ∈ RPic×SX with weightχ(OXf (x)
|x) = 1, whichimplies the first assertion of the theorem.
Now assume thatg(S) is contained in the smooth locus ofX/S. Then the sameargument as above gives existence of the Poincare sheaf onPic
smX/S,P. Existence
of the Poincare sheaf onPicX/S,P is slightly more difficult. First let us show thatthere exists a resolution
0→ En → . . .→ E0 → Og(S) → 0,
whereEi are locally free sheaves onX. Since there are sufficiently many locallyfree sheaves onX we can construct the resolution up to stepEn−1, wheren is therelative dimension ofX/S. Then the kernel ofEn−1 → En−2 is also locally free.Indeed, it is sufficient to check it on the geometric fiberXs overs∈ S, where onecan use the fact that the homological dimension ofOg(s) is equal ton (this followsfrom the smoothness assumption).
Tensoring with a high tensor powerOX(m) we can assume that all the higherdirect images ofF ⊗q∗(Ei(m)) under the projectionp vanish. In particular, allsheavesp∗(F ⊗q∗(Ei(m))) are locally free. Then we can set
APic= detp!(F ⊗q∗(Og(S)(m))) =⊗
i
(detp∗(F ⊗q∗(Ei(m))))(−1)i .
Obviously, the centre of GL(V) still acts on the fibres ofAPic with weight 1. Hencethe theorem follows from Proposition 3.2.
Remark3.4. Note that the second part of Theorem 3.3 does not immediatelyfol-low from [1] and [2]. Representability of (compactified) Picard functors is proventhere only in etale topology or after rigidification (see, e.g., [2, Theorems 3.2 and3.4]). Rigidification of the compactified Picard functor amounts in our case torestricting to the open subset ofRPic, where the restriction ofF to g(S) is invert-ible. Then by the same argument as in the proof of 1 of Theorem 3.3 we canconstruct the scheme representing the corresponding rigidified Picard functor ob-taining [2, Theorem 3,4]. However, we prefer to make a stronger assumption asin 2 to construct the projective Picard scheme.
Moduli spaces of principal bundles on singular varieties 13
3.3 Moduli spaces of swamps
Let us fix non-negative integersa andb and consider a GL(V)-module(V⊗a)⊕b.Let ρa,b:GL(V) → GL(V⊗a)⊕b) be the corresponding representation. IfA is asheaf of rankr = dimV then we can associate to it a sheafAρa,b = (A ⊗a)⊕b. Onthe open set whereA is locally free,Aρa,b is a locally free sheaf associated to theprincipal bundle obtained by extension from the frame bundle ofA .
Let us recall that aρa,b-swampis a triple(A ,L,ϕ) consisting of a torsion freesheafA on X, a rank 1 torsion free sheafL on X and a non-zero homomorphismϕ : Aρa,b → L.
Let us fix a positive polynomialδ of degree≤ d−1 with rational coefficients.Let us writeδ (m) = δ md−1
(d−1)! +O(md−2).
For a weighted filtration(A•,α•) of A we setr i = rkAi and we consider avectorγ ∈Qr defined by
γ = ∑αi(r i − r, ..., r i − r︸ ︷︷ ︸
r i×
, r i, ..., r i︸ ︷︷ ︸(r−r i)×
).
Let γ j denote thejth component ofγ. We set
µ(A•,α•;ϕ
)=−min
{γi1 + · · ·+ γia
∣∣(i1, ..., ia) ∈ I : ϕ|(Ai1⊗···⊗Aia)⊕b 6≡ 0
},
whereI = {1, ...,s+1}×a is the set of all multi-indices.Let us recall that aρa,b-swamp(A ,L,ϕ) is δ -(semi)stableif for all weighted
filtrations(A•,α•) we have
M(A•,α•)+µ(A•,α•;ϕ
)δ (≥)0.
A ρa,b-swamp(A ,L,ϕ) is slopeδ -(semi)stableif for all weighted filtrations(A•,α•) we have
L(A•,α•)+µ(A•,α•;ϕ
)δ (≥)0.
Now we can state the most general existence result for modulispaces ofswamps. We keep the notation from the beginning of this section.
THEOREM 3.5. Let us fix an S-flat familyL of pure sheaves of dimension d onthe fibres of f: X → S. Assume that either d= 1 or f has only irreducible andreduced geometric fibres. Then there exists a coarse S-projective moduli space forδ -semistable S-flat families ofρa,b-swamps(A ,L ,ϕ) such that for every s∈ Sthe restrictionA |Xs has Hilbert polynomial P.
14 A. Langer
In case whenX is a smooth complex projective variety this theorem wasproved by Gomez and Sols in [7], and later generalized by Bhosle to singularcomplex varieties satisfying Bhosle’s condition in [3]. Note that in [7] and [3]the authors considered only the case whenL is locally free. However, this is notnecessary due to Lemma 2.3 and it is sufficient to assume thatL is torsion free.Generalization to the relative case in arbitrary characteristic follows from [11] and[12]. We need only to comment why one does need to require thatthe fibres off are irreducible or reduced in the curve case. This fact follows from [9, Remark4.4.9]: torsion submodules for sheaves on curves are detected by any twist of itsglobal sections. This allows to omit using [3, Proposition 2.12] in the curve case.In particular, this shows that all the results of Sorger [22]are now a part of themore general theory.
We also have another variant of the above theorem (cf. [20, Theorem 2.3.2.5]):
THEOREM 3.6. Let us fix a Hilbert polynomial Q. Assume that all geometricfibers of f are irreducible and reduced and assume that f: X → S has a sectiong : S→ X such that g(S) is contained in the smooth locus of X/S. Then there existsa coarse moduli space forδ -semistable S-flat families ofρa,b-swamps(A ,L ,ϕ)such that for every s∈ S the restrictionA |Xs has Hilbert polynomial P and therestrictionL |Xs has Hilbert polynomial Q. This moduli space is projective overPicX/S,Q.
4 Tensor product of semistable sheaves on non-normalvarieties
Let (X,OX(1)) be ad-dimensional polarized projective variety defined over analgebraically closed fieldk.
Let ν : X → X denote the normalization ofX and letE be a coherentOX-module. Sinceν is a finite morphism, there exists a well defined coherentOX-moduleν !E corresponding to theν∗OX-moduleH om(ν∗OX,E). If E is torsionfree then we haveH omOX(ν∗OX/OX,E) = 0. Hence
ν∗(ν !E) = H omOX(ν∗OX,E)⊂ H omOX(OX,E) = E
andν !E is also torsion free.
Moduli spaces of principal bundles on singular varieties 15
LEMMA 4.1. There exists a constantα (depending only on the variety X) suchthat for any rank r torsion free sheaf E on X we have
0≤ µ(E)−µ(H om(ν∗OX,E))≤ α.
Proof. We have an exact sequence
0→ H omOX(ν∗OX,E)→ E → E xt1OX(ν∗OX/OX,E).
For largem we have
P(H omOX(ν∗OX,E))(m)≤ P(E)(m)
and, sinceH omOX(ν∗OX,E) andE have the same rank, we have
Note thatE xt1OX(ν∗OX/OX,E) is supported on the support ofν∗OX/OX. Let
Y1, . . . ,Yk denote codimension 1 irreducible components of the supportof ν∗OX/OX.Thenαd−1(E xt1OX
(ν∗OX/OX,E)) can be bounded from the above using the ranksof E xt1OX
(ν∗OX/OX,E) atY1, . . . ,Yk. Hence by the above inequality, to prove thelemma it is sufficient to bound these ranks.
There exists a subsheafG⊂E such thatG is locally free (we need only locallyfree in codimension 1) andE/G is torsion (i.e., equal to zero at the generic pointof X). This can be constructed by takingr general sections ofE(m) for largemand twisting the image ofO r
X ⊂ H0(E(m))⊗OX → E(m) by OX(−m).Then we have an exact sequence
0= H om(ν∗OX/OX,E)→ H om(ν∗OX/OX,E/G)→ E xt1(ν∗OX/OX,G)
Note that the sheaves in this sequence are supported on⋃
Yi and the rank ofE xt1(ν∗OX/OX,G) on Yi is the same as the rank ofE xt1(ν∗OX/OX,O
rX) on Yi .
In particular, it depends only on the rankr and it is independent ofE. Hencethe dimensions ofH om(ν∗OX/OX,E/G) at the generic points ofY1, . . . ,Yk arebounded from the above by a linear function ofr. But this implies that the ranksof E/G, and hence also ofE xt1(ν∗OX/OX,E/G), onY1, . . . ,Yk are bounded inde-pendently ofE. Now we can use the sequence
E xt1(ν∗OX/OX,G)→ E xt1(ν∗OX/OX,E)→ E xt1(ν∗OX/OX,E/G)
to bound the ranks ofE xt1OX(ν∗OX/OX,E) onY1, . . . ,Yk.
16 A. Langer
COROLLARY 4.2. Let us setβ = αd−1(OX)−αd−1(OX). Then for any rank rtorsion free sheaf E on X we have
β ≤ µ(E)−µ(ν !E)≤ α +β ,
where the slopes are computed with respect toOX(1) on X andν∗OX(1) onX.
COROLLARY 4.3. For any rank r torsion free sheaf E on X we have
β ≤ µmax(E)−µmax(ν !E)≤ α +β .
Proof. If G⊂ E is a subsheaf ofE thenν !G⊂ ν !E and hence
µ(G) ≤ µ(ν !G)+α +β ≤ µmax(ν !E)+α +β .
This proves thatµmax(E)≤ µmax(ν !E)+α +β .
Now if F ⊂ ν !E thenν∗F ⊂ ν∗(ν !E)⊂ E. Therefore
µ(F) = µ(ν∗F)−β ≤ µmax(E)−β ,
which implies thatµmax(ν !E)≤ µmax(E)−β .
Moduli spaces of principal bundles on singular varieties 17
For a torsion free sheafE on X we setν♯E = ν∗E/Torsion. Thenν∗ν♯E =(ν∗ν∗E)/Torsion.
Note thatν ! is an equivalence of categories of sheaves onX andX whereasν♯ has much worse properties. Butν♯ has the following important property: sinceν∗(E1⊗E2) = ν∗E1⊗ν∗E2 we haveν♯(E1⊗E2) = ν♯E1⊗ν♯E2.
Let C = Ann(ν∗OX/OX) ⊂ OX and CX = C ·OX ⊂ OX denote conductorideals of the normalisation.
LEMMA 4.4. For any torsion free sheaf E on X we have
µ(ν♯E)≤ µ(ν !E)−µ(CX).
Proof. Note thatC =H omOX(ν∗OX,OX). Therefore for any coherentOX-moduleE we have a canonical map
C ⊗E = H omOX(ν∗OX,OX)⊗H om(OX,E)→ H omOX(ν∗OX,E) = ν∗(ν !E)
given by composition of homomorphisms. Sinceν∗ andν∗ are adjoint functorsthis map induces
ν∗C ⊗ν∗E → ν !E.
SinceE is torsion free andCX = ν♯C we get
CX⊗ν♯E ≃ CX ·ν♯E → ν !E.
Since this inclusion is an isomorphism at the generic point of X we have the fol-lowing inequality
µ(CX⊗ν♯E)≤ µ(ν !E).
Now Lemma 2.1 gives
µ(CX⊗ν♯E) = µ(ν♯E)+µ(CX),
which implies the required inequality.
COROLLARY 4.5. For any rank r torsion free sheaf E on X we have
−β ≤ µ(ν♯E)−µ(E)≤−β −µ(CX),
where the slopes are computed with respect toOX(1) on X andν∗OX(1) on X.
18 A. Langer
Proof. The canonical mapE → ν∗(ν∗E) leads to the inclusion
E → ν∗(ν♯E).
This givesµ(E) ≤ µ(ν∗(ν♯E)) = µ(ν♯E)+β ,
where the last equality follows from proof of Lemma 4.2. Thisbounds the differ-enceµ(ν♯E)−µ(E) from below. To get the bound from the above it is sufficientto use Lemma 4.4 and Corollary 4.2.
Remark4.6. By Lemma 4.4 and the above corollary we have
µ(ν !E)≥ µ(ν♯E)+µ(CX)≥ µ(E)−β +µ(CX).
This allows to take in Lemma 4.1α = −µ(CX). The proof of Lemma 4.1 alsogives a related and explicit bound onα.
The above corollary can be used to prove the following corollary:
COROLLARY 4.7. For any rank r torsion free sheaf E on X we have
−β ≤ µmax(ν♯E)−µmax(E)≤−β −µ(CX).
Proof. If G⊂ E is a subsheaf ofE thenν♯G⊂ ν♯E and hence
µ(G) ≤ µ(ν♯G)+β ≤ µmax(ν♯E)+β .
This proves thatµmax(E)≤ µmax(ν♯E)+β .
Now if F ⊂ ν♯E then by the proof of Lemma 4.4 we have
CX⊗F ⊂ CX⊗ν♯E → ν !E.
Together with Lemma 2.1 and Corollary 4.3, this gives
µ(F)≤ µmax(ν !E)−µ(CX)≤ µmax(E)−β −µ(CX),
which implies that
µmax(ν♯E)≤ µmax(E)−β −µ(CX).
Moduli spaces of principal bundles on singular varieties 19
Sinceν∗(E1⊗E2) = ν∗E1⊗ν∗E2 we haveν♯(E1⊗E2) = ν♯E1⊗ν♯E2. There-fore [13, Introduction] or [6, Lemma 3.2.1] imply the following proposition.
PROPOSITION 4.8. There exists an explicit constantγ (depending only on thepolarized variety(X,OX(1))) such that for any two torsion free sheaves E1 andE2 on X of ranks r1, r2, respectively, we have
µmax(E1⊗E2)≤ µmax(E1)+µmax(E2)+(r1+ r2)γ.
5 Honest singular principal bundles
In this sectionX is ad-dimensional projective variety defined over an algebraicallyclosed fieldk with a fixed ample line bundleOX(1).
The main aim of this section is proof of the following generalization of [18,Proposition 3.4]:
PROPOSITION5.1. Assume that X is Gorenstein (i.e., a Cohen–Macaulay schemewith invertible dualizing sheafωX) and there exists a G-invariant non-degeneratequadratic formϕ on V. Then every degree0 singular principal bundle is an honestsingular principal bundle.
Proof. Let (A ,τ) be a degree 0 singular principal bundle. As in the proof of [18,Proposition 3.4] one can easily show that there exists an injective mapA → A ∨
induced by the formϕ. By Lemma 5.3 we see that the Hilbert polynomials ofA andA ∨ are the same up to the terms of orderO(md−2). HenceA → A ∨
is an isomorphism in codimension 1. Now let us recall that foreachx ∈ X twofinitely generated modules over a local ringOX,x satisfyingS2 that coincide incodimension 1 are equal. In particular, at each pointx whereA is locally free themapA → A ∨ is an isomorphism. As in the proof of [18, Proposition 3.4] thisimplies that
σ(UA )⊂ Isom(V ⊗OUA,A ∨ |UA
)/G.
The following lemma generalizes a well known equality from smooth varietiesto singular ones.
LEMMA 5.2. For any rank r coherent sheaf E and a line bundle L we have
deg(E⊗L) = degE+ r (L ·OX(1)d−1).
20 A. Langer
Proof. We use the notation from Kollar’s book [10, Chapter VI.2]. In particu-lar, Ki(X) stands for the subgroup of the Grothendieck group ofX generated bysubsheaves supported in dimension at mosti. We have
L⊗E(m) =d
∑i=0
c1(L)i ·E(m)
(see, e.g., [10, Chapter VI.2, Lemma 2.12]). On the other hand, by [10, ChapterVI.2, Corollary 2.3] we have
E ≡ r OX modKd−1(X).
Note that
L⊗E(m) = E(m)+ r c1(L) ·OX(m)+c1(L) · (E− r OX)(m)+∑i≥2
c1(L)i ·E(m)
andc1(L) · (E− r OX)+∑i≥2c1(L)i ·E ∈ Kd−2(X) by [10, Chapter VI.2, Proposi-tion 2.5]. Therefore by [10, Chapter VI.2, Corollary 2.13] we have
By Lemma 2.2 we have dimE xtqX(E,ωX) ≤ d−2 for q > 0, so by [10, ChapterVI, Corollary 2.14]
χ(X,E xtqX(E,ωX)⊗OX(−m)) = O(md−2)
for q> 0. SinceωX is invertibleH om(E,ωX) = E∨⊗ωX and we get
χ(X,E(m)) = (−1)dχ(X,E∨⊗ωX(−m))+O(md−2).
In particular, we have
αd−1(E∨⊗ωX) =−αd−1(E).
Therefore by Lemma 5.2
degE∨ = deg(E∨⊗ωX)− r c1(ωX) ·c1(OX(1))d−1
= αd−1(E∨⊗ωX)− rαd−1(OX)− r c1(ωX) ·c1(OX(1))d−1
= −degE−2rαd−1(OX)− r c1(ωX) ·c1(OX(1))d−1.
Applying this equality forE = OX we see that
−2αd−1(OX)− c1(ωX) ·c1(OX(1))d−1 = 0,
so degE∨ =−degE.
6 Semistable reduction for singular principalG-bundles
The following global boundedness of swamps on singular varieties can be provenin the same way as in the case of smooth varieties (see [5, Theorem 4.2.1], [6,Theorem 3.2.2] or [20, Theorem 2.3.4.3]). The only difference is that we needProposition 4.8 (instead of, e.g., [6, Lemma 3.2.1]).
22 A. Langer
THEOREM 6.1. Let us fix a polynomial P, integers a, b and a class l in the Neron–Severi group of X. Then the set of isomorphism classes of torsion free sheavesA on X with Hilbert polynomial P and such that there exists a positive rationalnumberδ and a slopeδ -semistableρa,b-swamp(A ,L,ϕ) with L of class l isbounded.
This boundedness result implies the following semistable reduction theorem(see [5, Theorem 5.4.4], [6, Theorem 4.4.1] or [20, Theorem 2.4.4.1]). We skipthe proof as it is the same as in the smooth case.
THEOREM 6.2. Assume that k has characteristic zero. Then there exists a polyno-mial δ∞ such that for every positive polynomialδ > δ∞ everyδ -semistable pseudoG-bundle(A ,τ) is a singular principal G-bundle.
Let us recall that a singular principalG-bundle is semistable if and only if theassociated pseudoG-bundle isδ -semistable forδ > δ∞ (see [5, Theorem 5.4.1]).Therefore the above semistable reduction theorem and Theorem 2.5 imply thefollowing corollary.
COROLLARY 6.3. Assume that k has characteristic zero and let us fix a polyno-mial P. Then there exists a projective moduli space Mρ
X,P for semistable principalG-bundles(A ,τ) on X such thatA has Hilbert polynomial P.
Now let us consider the relative case. Letf : X → S be a flat, projectivemorphism ofk-schemes of finite type with integral geometric fibers. Assume thatk has characteristic zero and fix a polynomialP.
THEOREM 6.4. Let us fix a faithful representationρ :G→ GL(V) of the reductivealgebraic group G.
1. There exists a projective moduli space MρX/S,P → S for S-flat families of
semistable singular principal G-bundles on X→ S such that for all s∈ Sthe restrictionA |Xs has Hilbert polynomial P.
2. Let P correspond to sheaves of degree0. If the fibres of f are Gorensteinand there exists a G-invariant non-degenerate quadratic form ϕ on V thenMρ
X/S,P → S parameterizes only honest singular principal G-bundles.
The first part of this theorem follows directly from the abovecorollary (rewrit-ten in the relative setting). The second part is a direct consequence of Proposition5.1. Since proof in the relative setting is essentially the same as usual (cf. [9,Theorem 4.3.7]) we skip the details.
Moduli spaces of principal bundles on singular varieties 23
Acknowledgements
The author would like to thank Alexander Schmitt for useful conversations. Hewould also like to thank the Alexander von Humboldt Foundation for supporting,via the Bessel Research Award, his visit to the University ofDuisburg-Essen,where most of this paper was written. The author was partially supported by aPolish MNiSW grant (contract number N N201 420639).
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