Quasi-universal families

2

P

i=0

(−1)ⁱch(q∗Hom_p^∗_A(p^∗Mi, N)).

By assumptions X×Y is projective, so the projectionq is proper. We can therefore apply the Grothendieck-Riemann-Roch theorem forq. Since the higher direct images ofHom_p^∗_A(p^∗M_i, N) underq∗ vanish by the choice of the resolution, we get:

ch(q∗Hom_p^∗_A(p^∗M_i, N)) =q∗(ch(Hom_p^∗_A(p^∗M_i, N))p^∗td(X)).

As p^∗A is an Azumaya algebra, this class can be simplied using the isomorphism given by (1.53): and as the p^∗Mi are locally projective, we get by [HL10, Lemma II.6.1.3] :

ch(

wherech(−)^∨ is the dual class. Putting everything together, we see that the class is given by q∗(ch(p^∗M)^∨ch(N)ch(p^∗A)⁻¹p^∗td(X)).

Using ch(p^∗M) =p^∗ch(M)shows, that it does depend only on ch(M) and ch(N). 1.7 Quasi-universal families

Assume X is a smooth projective surface with a distinguished line bundle O_X(1), so that X is a polarized projective scheme. As the moduli space M_A/X,P is a coarse moduli space, there is no universal family on it. In this section we want to prove that at least a quasi-universal family exists. We will use the notation from [HL10, Chapter 4.6] and adapt the proof given there to our situation. If we have a family ofA-modules parametrized by a scheme S, then we have the the projections p:X×S →S and q:X×S →X.

Denition 1.86:

A at family E of torsion-free A-modules of rank one on X parametrized by MA/X,P is called quasi-universal, if the following holds: ifF is a family of torsion-freeA-modules of rank one with Hilbert polynomial P over S and if φF :S →M_A/X,P, s7→ [F_s]is the induced morphism, then there is a locally free O_S-module W such that F ⊗p^∗W ∼=φ^∗_F,XE, where φ^∗_F,X := (id_X ×φF)^∗.

Remark 1.87:

By denition for every point y ∈ M_A/X,P the A-module E_y on X is isomorphic to M^⊕n for a torsion-freeA-moduleM of rank one with Hilbert polynomialP, which denes the isomorphism class given byy∈M_A/X,P. Herenis the rank of the stalk ofW aty. IfM_A/X;c1,c2 is connected then this numberndoes not depend onyand is called the similitude ofE, see [Muk87, Appendix, Denition A.5].

LetR be the locally closed subscheme in Quot(A(−m)^N, P) used in the construction of MA,P

and letFe be the universal quotient on X×R, that is the restriction toX×R of the universal family on X ×Quot(A(−m)^N, P). Fe is a GL(N)-linearized sheaf on X ×R and since the centerZ ofGL(N) acts trivially onR, see [HS05, Proposition 2.2 (v)], it has the structure of a Z-representaion and decomposes into weight spaces.

Theorem 1.88:

Quasi-universal families exist.

Proof:

We rst prove that there are GL(N)-linearized locally free sheaves of Z-weight one on R: if n is suciently largeA=p∗(Fe⊗q^∗O_X(n))is a locally free sheaf on Rof rank P(n) and carries a naturalGL(N)-linearization ofZ-weight one, the one induced fromFe.

So letAbe any GL(N)-linearized locally free sheaf ofZ weight one onR. NowZ acts trivially onHom(p^∗A,Fe), which therefore carries aP GL(N)-linearization and descends to a family E on X×M_A/X,P by [HL10, Theorem 4.2.14], because π₂ :R −→ M_A/X,P is a principal P GL(N) -bundle, see [HS05, Theorem 2.4] where it is stated that this morphism is locally trivial in the fppf-topology, but asP GL(N) is smooth it is also locally trivial in the étale topology.

It remains to show thatE is quasi-universal.

SupposeF is a family of torsion-free A-modules of rank one onX parametrized by a scheme S with Hilbert polynomialP. Thenp∗F(m) is a locally free O_S-module of rank P(m) =N. Let

R(F) =Isom(O^N_S, p∗F(m)) −−−−→^π1 S

be the frame bundle. Then, by [HL10, Example 4.2.6], there is a universal GL(N)-equivariant trivialization

O^N_R(F) −−−−→^∼= π^∗₁p∗F(m).

By the relative version of Serre's theorem we have a surjection

q^∗O_X(−m)⊗p^∗p∗F(m) −−−−→ F −−−−→ 0.

Now applyπ^∗_1,X to this surjection, withπ1,X =idX ×π1, and use the universal trivialization to get a quotient

r^∗O_X(−m)^N ⊗s^∗OR(F) −−−−→ π_1,X^∗ F −−−−→ 0

onX×R(F), wherer=q◦π_1,X :X×R(F)→Xands:X×R(F)→R(F)are the projections, and we havep◦π1,X =π1◦s. More exactly we have:

π_1,X^∗ p^∗p∗F(m) = (p◦π_1,X)^∗p∗F(m) = (π₁◦s)^∗p∗F(m) =s^∗(π^∗₁p∗F(m))

40 1.7 Quasi-universal families

and the last module can be replaced using the universal quotient.

We can tensor the quotient withπ_1,X^∗ A_S to get a quotient:

r^∗O_X(−m)^N ⊗s^∗OR(F)⊗π_1,X^∗ A_S −−−−→ π^∗_1,XF ⊗π_1,X^∗ A_S −−−−→ 0.

Since tensor products and pullback commute and π_1,X^∗ A_S =π_1,X^∗ q^∗A=r^∗A, we get in fact the following quotient:

r^∗A(−m)^N ⊗s^∗OR(F) −−−−→ π_1,X^∗ (F ⊗ A_S) −−−−→ 0. (7) ButF is by denition an A_S-module, so we get a surjection

F ⊗ A_S −−−−→ F −−−−→ 0

using theA_S-module structure. Applyingπ_1,X^∗ to the last equation and combining this with (7), we get a quotient:

r^∗A(−m)^N⊗s^∗OR(F) −−−−→ π_1,X^∗ F −−−−→ 0 of A(−m)^N on X×R(F).

AsQuot(A(−m)^N, P)represents the Quot-functor, this quotient gives rise to a map φeF :R(F)→Quot(A(−m)^N, P).

NowGL(N) acts onR(F) from the right by composition, so that the frame bundle is in fact a principal GL(N)-bundle, see [HL10, Example 4.2.3]. The morphism φeF is GL(N)-equivariant and by construction we have φeF(R(F))⊂R.

Thus we can consider the commutative diagram

R(F) −−−−→^φeF R

π1



 y



 y^π2 S −−−−→^φF MA/X,P

Since π^∗_2,XE ∼=Hom(p^∗A,Fe) we have:

π_1,X^∗ φ^∗_F,XE ∼=φe^∗_F,Xπ^∗_2,XE ∼=φe^∗_F,XHom(p^∗A,Fe)∼=Hom(φe^∗_F,Xp^∗A,φe^∗_F,XFe). Now by denition: φe^∗_F,XFe∼=π_1,X^∗ F andφe^∗_F,Xp^∗A∼=s^∗φe^∗_FA.

Since φe^∗_FA is GL(N)-linearized in a natural way and π1 : R(F) → S is a GL(N)-principal bundle, there is a locally free sheafB on S such that there is an isomorphism φe^∗_FA∼=π^∗₁B. Furthermore we have s^∗φe^∗_FA∼=π^∗_1,Xp^∗B. So we conclude:

π_1,X^∗ φ^∗_F,XE ∼=Hom(π_1,X^∗ p^∗B, π^∗_1,XF) =π^∗_1,XHom(p^∗B,F)

which is equivariant. Using [HL10, 4.2.14] we see that this map descends to an isomorphism:

φ^∗_F,XE ∼=Hom(p^∗B,F)∼=p^∗B^∨⊗ F. So E is in fact a quasi-universal family.

2 Moduli spaces over K3 and abelian surfaces

If X is a smooth K3 or abelian surface and E a coherent O_X-module, then Mukai dened in [Muk87] a class v(E)∈H^ev(X,Q) =

Herech(E) is the Chern Character of E and td(X) is the Todd class of X, both classes belong to H^ev(X,Q). For every u = (a, b, c) and u⁰ = (a⁰, b⁰, c⁰) in H^ev(X,Q) he dened a symmteric bilinear form(−,−), using the cup product, by:

(u, u⁰) =b∪b⁰−a∪c⁰−a⁰∪c

means taking theH⁴(X,Q)componenet of the product. For two coherentO_X-modules E and F the product (v(E), v(F)) equals the H⁴ component of −ch(E)^∨.ch(F).td(X). Using the Hirzebruch-Riemann-Roch theorem he proved:

Proposition 2.1:

AssumeE and F are coherent O_X-modules and put χ(E, F) =

Denote byM(v) the moduli space of stable sheaves onX with Mukai vectorv, then it is known thatM(v) is smooth and that the there is canonical isomorphism

T_[E]M(v)∼=Ext¹_O

SoM(v) has dimension(v, v) + 2. Using these results Mukai proved the following two theorems:

Theorem 2.2:

Assumev is a Mukai vector with (v, v) =−2, then M(v) is either empty or a reduced point.

Theorem 2.3:

Assumevis an isotropic Mukai vector, that is(v, v) = 0. IfM(v)contains a connected component M which is compact, then we have:

42 2.1 Euler characteristic and Mukai vectors for modules over orders

1. M(v) is irreducible;

2. every semistable sheaf K with v(K) =v is stable.

In this section we will adapt these denitions to the situation of torsion-freeA-modules on aK3 or an abelian surface and obtain similiar results if Ais unramied, that is an Azumaya algebra on X.

2.1 Euler characteristic and Mukai vectors for modules over orders Denition 2.4:

Assume A is a terminal order on a smooth projective surface and let M and N be coherent A-modules, then we dene the A-Euler characteristic of the pair (M, N) by:

χA(M, N) :=

2

P

i=0

(−1)ⁱdimk(Extⁱ_A(M, N)) Lemma 2.5:

AssumeA is a terminal order on a smooth projective surfaceX. If0→M⁰→M →M⁰⁰→0 is an exact sequence of coherent A-modules, then we have:

χA(M, N) =χA(M⁰, N) +χA(M⁰⁰, N) Proof:

Since the categoryR−M odof leftR-module has enough injectives for any ringR, we see that we can adapt the proof of [Har77, Proposition III.2.2] to see that the category M od(A) has enough injectives. So we choose an injective resolution0→N →I^•.

Since theI^j are injective, the functorsHomA(−, I^j)are exact, so we get a short exact sequence of complexes

0→HomA(M⁰⁰, I^•)→HomA(M, I^•)→HomA(M⁰, I^•)→0.

Taking the long exact sequence of cohomology groups gives the long exact ExtA-sequence. By (1.52) we have Extⁱ_A(M, N) = 0 for i≥3. Using the fact that Euler characteristic in an exact sequence is zero and grouping the Extaccording toM,M⁰ and M⁰⁰ shows

χA(M⁰, N)−χA(M, N) +χA(M⁰⁰, N) = 0.

Lemma 2.6:

Assume A is a terminal order on a smooth projective surface X. Let M be a coherent locally projective A-module and letN be a coherent A-module. Then we have:

χA(M, N) =χ(X,HomA(M, N)) =R

Xch(HomA(M, N)).td(X). Proof:

Since M is locally projective the local-to-global spectral sequence shows that we have Extⁱ_A(M, N) =Hⁱ(X,Hom_A(M, N)).

This implies that theA-Euler characteristic for M and N agrees with the usual Euler charac-teristic forHomA(M, N), meaning χA(M, N) =χ(X,HomA(M, N)). Now the last assertion is just the Hirzebruch-Riemann-Roch theorem forHom_A(M, N).

Denition 2.7:

Let M be a coherent A-module, then we dene theA-Chern character by:

chA(M) :=ch(M).p

ch(A^∗)⁻¹. Remark 2.8:

By √

− we mean the positive square root, that is the degree zero component is positive. Since rk(A) =r² withr≥1the classp

ch(A)⁻¹ is well dened and can be found with a power series expansion.

Lemma 2.9:

Assume A is a terminal order on a smooth projective surface X. If M is a coherent locally projectiveA-module and N is a coherent A-module, then we have:

ch(HomA(M, N)) =chA(M^∗).chA(N).

Proof:

By (1.53) we have an isomorphism A^∗⊗O_X HomA(M, N)∼=HomO_X(M, N). This implies ch(A^∗⊗_OX Hom_A(M, N)) =ch(Hom_OX(M, N)).

SinceA^∗ is a locally free O_X-module, the Chern character is multiplicative with respect to the tensor product:

ch(A^∗⊗_OXHom_A(M, N)) =ch(A^∗)ch(Hom_A(M, N)).

ButM is a locally projectiveA-module, so by (1.55) it is a locally freeO_X-module, which implies thatHom_OX(M, N) =M^∗⊗_OX N. Using again the multiplicativity ofch, we get

ch(Hom_OX(M, N)) =ch(M^∗)ch(N). So, nally, we see:

ch(Hom_A(M, N)) =ch(Hom_OX(M, N))ch(A^∗)⁻¹

=ch(M^∗)p

ch(A^∗)⁻¹ch(N)p

ch(A^∗)⁻¹

=chA(M^∗)chA(N).

Corollary 2.10:

Assume A is a terminal order on a smooth projective surface X. If M is a locally projective A-module andN is a coherent A-module, then we have:

χA(M, N) =R

XchA(M^∗).chA(N).td(X).

44 2.1 Euler characteristic and Mukai vectors for modules over orders Proof:

This follows from (2.6) and (2.9).

Remark 2.11:

By choosing a locally projective resolution for a coherentA-moduleM, we see that the previous corollary is in fact correct for all coherentA-modulesM and N.

Denition 2.12:

If A is a terminal order on a smooth projective surface X and M is a coherent A-module we dene its A-Mukai vector by:

vA(M) =chA(M).p td(X).

Example 2.13:

Assume X is a K3 or an abelian surface. If A is an Azumaya algebra with rk(A) = r², then A^∗ ∼=A using the trace map. Soc₁(A) =c₁(A^∗) =−c₁(A) implying c₁(A) = 0. Thus we have ch(A) =r²−c₂(A). Using (√

1−x)⁻¹ ≈1 +¹₂x shows that pch(A)⁻¹= ¹_r+_2r¹3c₂(A).

Furthermore since X is abelian or K3 we have K_X = 0 so that td(X) = 1 +χ(O_X) and with

√1 +x≈1 +¹₂x this gives

ptd(X) = 1 +¹₂χ(O_X).

(Here we have to be a little bit careful how to read this: 1∈H⁰(X,Q)but ¹₂χ(O_X)∈H⁴(X,Q)!).

Now assumeM is a coherentA-module with rk(M) =sas anO_X-module and Chern classesc₁ and c2, thench(M) =s+c1+¹₂(c²₁−2c2).

We compute theA-Mukai vector of M and get:

vA(M) = (_r^s,¹_rc1,_2r¹(c²₁−2c2+_r^s2c2(A)) + _2r^sχ(O_X)).

For example if M is a torsion-freeA-module of rank one, then rk(M) =r², so we get:

vA(M) = (r,¹_rc₁,_2r¹ (c²₁−2c₂+c₂(A)) + ^r₂χ(O_X)).

If S is an Artinian A-module of lengthlA(S) = n, then rk(S) = 0, c₁(S) = 0 and c₂(S) = −n so that

vA(S) = (0,0,ⁿ_r).

The last formula is also valid for a terminal order A on a smooth projective surface, since ch(S) =nonly lives inH⁴(X,Q).

Lemma 2.14:

Assume A is a terminal order on smooth projective surface X. If M and N are coherent A -modules, then we have

χA(M, N) =−(v_A(M), vA(N)),

where the bilinear form(−,−) on H^ev(X,Q) =L

H²ⁱ(X,Q) is given by (v, w) =−R

Xv^∨.w as described above.

Proof:

IfAis an Azumaya algebra, then, due to (2.10), the left hand side is given by theH⁴ component of

ch(M)^∨.ch(N).td(X).ch(A)⁻¹.

Usingtd(X).ch(A)⁻¹ = _r¹2 +_r¹4c₂(A) +_r¹₂χ(O_X) we see that, ifrk(M) =sand rk(N) =t, the left hand side is given by:

st

r²χ(O_X) + _r^st4c₂(A) + ^s

r²ch₂(N) +_r^t2ch₂(M)− ¹

r²c₁(N)c₁(M). Herech₂ = ¹₂(c²₁−2c₂). Now writing the Mukai vectors as

vA(M) = (^s_r,¹_rc1,_2r¹(2ch2(M) +_r^s2c2(A)) + _2r^sχ(O_X)) shows that the right hand side is given by the same term.

IfAis ramied, then c₁(A)6= 0 and the computations are tedious but show the same result.

Example 2.15:

AssumeA is an Azumaya algebra. We see that ifv= (a, b, c) then(v, v) =b²−2ac. Using the description ofvA(M)from (2.13) for a torsion-freeA-module M of rank one with Chern classes c1 and c2, we get

(vA(M), vA(M)) = _r¹2c²₁−(c²₁−2c2)−c2(A)−r²χ(O_X).

The term _r¹2c²₁−(c²₁−2c₂)can be simplied to _r¹2(2r²c₂−(r²−1)c²₁). So if∆ = 2r²c₂−(r²−1)c²₁ denotes the discriminant ofM we have

(vA(M), vA(M)) = _r¹2∆−c2(A)−r²χ(O_X)

Since a torsion-free A-module M of rank one is simple (1.49), we have HomA(M, M) = k and using Serre duality shows thatExt²_A(M, M) =HomA(M, M)⁰ =k. So we get

dim(Ext¹_A(M, M)) = 2 + (vA(M), vA(M)),

ordim(Ext¹_A(M, M)) = 2 +_r¹2∆−c₂(A)−r²χ(O_X). Now if we compare this with the dimension formula for the moduli spaceMA/X;c₁,c2 given in (3.1), we see that

dim(M_A/X;c1,c2) = 2 + (vA(M), vA(M)) which is the same type of formula as in the case of stable sheaves.

Example 2.16:

AssumeA is a terminal order and S is an ArtinianA-module of length n. Then we know that vA(S) = (0,0,ⁿ_r). So(vA(S), vA(S)) = 0which implies thatχA(S, S) = 0.

46 2.3 Two-dimensional moduli spaces

Im Dokument Moduli spaces of bundles over two-dimensional orders (Seite 38-46)