SHEAVES ON ABELIAN SURFACES AND STRANGE DUALITY 1. Introduction Let

30 Pages·2012·0.4 MB·English

SHEAVES ON ABELIAN SURFACES AND STRANGE DUALITY
ALINA MARIAN AND DRAGOS OPREA
Abstract. Weformulatethreeversionsofastrange dualityconjectureforsectionsof
theThetabundlesonthemodulispacesofsheavesonabeliansurfaces. Assupporting
evidence, we check the equality of dimensions on dual moduli spaces, answering a
question raised by Go¨ttsche-Nakajima-Yoshioka [GNY].
1. Introduction
Let(A,H)beapolarizedabeliansurface. Inthispaper,weconsiderthemodulispaces
of Gieseker H-semistable sheaves on A, and sections of the Theta line bundles defined
over them.
It will be convenient to bookkeep coherent sheaves E on A by their Mukai vectors,
setting
v(E) = r+c (E)+χ(E)ω ∈ H2(cid:63)(A),
1
whereω standsfortheclassofapoint. Ascustomary, wewillequiptheevencohomology
H2(cid:63)(A) with the Mukai pairing. For any two vectors x = (x ,x ,x ) ∈ H2(cid:63)(A) and
0 2 4
y = (y ,y ,y ) ∈ H2(cid:63)(A), we set
0 2 4
(cid:90) (cid:90)
(cid:104)x,y(cid:105) = − x∨∪y = (x y −x y −x y ).
2 2 0 4 4 0
A A
It follows that for any two sheaves E and F, we have
2
(cid:88)
χ(E,F) = (−1)i Exti(E,F) = −(cid:104)v(E),v(F)(cid:105).
i=0
For an arbitrary v ∈ H2(cid:63)(A), let us write M for the moduli space of Gieseker H-
v
semistable sheaves E on A, with Mukai vector v. To keep things simple, we will make
the following assumption throughout:
Assumption 1. (i) The polarization H is generic i.e., it belongs to the complement
of a locally finite union of hyperplane walls in the ample cone of A;
(ii) The vector v is a primitive element of the lattice H2(cid:63)(A,Z);
(iii) The vector v is positive i.e., one of the following is true:
– rank (v) > 0;
1
2 ALINAMARIANANDDRAGOSOPREA
– rank (v) = 0 and c (v) is effective, χ(v) (cid:54)= 0, and (cid:104)v,v(cid:105) =(cid:54) 0,4.
1
The choice of generic polarization will play only a minor role in what follows, and as
such, we will supress it from the notation. Note that (i) and (ii) together imply that
M consists of stable sheaves only. Therefore, by [Muk3], M is a smooth manifold of
v v
dimension 2d +2, with
v
1
(1) d = (cid:104)v,v(cid:105).
v
2
The main characters of our story will be a collection of naturally defined Theta line
bundles on M . Consider the subgroup v⊥ inside the holomorphic K-theory of A gener-
v
ated by the sheaves F whose Mukai vectors w are orthogonal to v:
(2) χ(v⊗w) = −(cid:104)v∨,w(cid:105) = 0.
There is a morphism
Θ : v⊥ → Pic(M ), [F] → Θ ,
v F
constructed and studied by Li and Le Potier [Li] [LP] in the case of surfaces, and also
by Dr´ezet-Narasimhan in the case of curves [DN]. The construction is easiest to explain
assuming that M is a fine moduli space, such that E is the universal sheaf on M ×A.
v v
In this case, for a sheaf F with Mukai vector w, we set
(3) Θ = detRp(E ⊗q(cid:63)F)−1,
F !
where p and q are the two projections from M ×A. The orthogonality condition (2) is
v
used to obtain a well defined line bundle Θ , even in the absence of universal structures,
F
by descent from the Quot scheme.
Note that even though the line bundle Θ depends on the K-theory class of F, the
F
Chernclassc (Θ )dependsonlyontheMukaivectorw. Forsimplicityofnotation,inall
1 F
cohomological computations below, we will write Θ for any one of the line bundles Θ
w F
as above. The finer dependence of the line bundles Θ on the sheaf F will be discussed
F
in more detail in Section 2.
Our goal in this paper is to compute the Euler characteristics of the line bundles Θ ,
w
which can be interpreted as the K-theoretic Donaldson invariants of the abelian surface
A. We will provide a simple expression for these Euler characteristics, valid in any rank.
We will thus answer a question raised in [GNY], as part of a general study of the rank
two K-theoretic Donaldson invariants of surfaces.
SHEAVES ON ABELIAN SURFACES AND STRANGE DUALITY 3
Toexplaintheresults,letusfirstfixareferencelinebundleΛonAwithc (Λ) = c (v).
1 1
Then, we have a well-defined determinant morphism
(4) αΛ+ : Mv → A(cid:98)= Pic0(A), E → detE ⊗Λ−1.
Its fiber over the origin is the moduli space M+(Λ) of sheaves with fixed determinant Λ.
v
The choice of the determinant is unimportant for our arguments, and therefore we will
omit it from our notation when no confusion is likely to arise. We will show:
Theorem 1. For any vectors v and w satisfying Assumption 1, and such that χ(v⊗w) =
0, we have
1c (v⊗w)2 (cid:18)d +d (cid:19)
(5) χ(M+,Θ ) = χ(M+,Θ ) = 1 v w .
v w w v 2 d +d d
v w v
There is yet another moduli space of interest to us, which is Fourier-Mukai ‘dual’ to
the one considered above. Letting P denote the normalized Poincar´e bundle on A×A(cid:98),
the Fourier-Mukai transform is defined by
(6) RS(E) = Rp!(P ⊗q(cid:63)E) ∈ D(A(cid:98)),
where p,q are the two projections. With this understood, let us set
α− : M → A, E → detRS(E)⊗detRS(Λ)−1.
Λ v
Thefiberofthemorphismα− overtheoriginisdenotedbyM−,andparametrizessheaves
Λ v
E with fixed determinant of the Fourier-Mukai transform. We will prove:
Theorem 2. For any vectors v and w as in Theorem 1, we have
1c (vˆ⊗wˆ)2(cid:18)d +d (cid:19)
(7) χ(M−,Θ ) = χ(M−,Θ ) = 1 v w .
v w w v 2 d +d d
v w v
Here vˆ and wˆ denote the Fourier-Mukai transforms of the two vectors v and w.
Finally, we may consider the morphism
av = (αΛ+,αΛ−) : Mv → A(cid:98)×A.
This is the Albanese map of the moduli space M , cf. [Y1]. Its fiber over the origin will
v
henceforth be denoted by K . We will show:
v
Theorem 3. Assume that the N´eron-Severi group of A has rank 1. With the same
hypotheses as in Theorem 1, we have
d2 (cid:18)d +d (cid:19)
(8) χ(K ,Θ ) = χ(M ,Θ ) = v v w .
v w w v
d +d d
v w v
4 ALINAMARIANANDDRAGOSOPREA
The manifest symmetry of the formulas in Theorems 1, 2 and 3 matches first of all
that of their counterpart for the case of sheaves on a K3 surface. Indeed, the theta Euler
characteristics for the moduli space of sheaves on a K3 were shown to be [GNY][OG]
(cid:18) (cid:19)
d +d +2
v w
(9) χ(M ,Θ ) = χ(M ,Θ ) = .
v w w v
d +1
v
This symmetry further suggests a general strange duality for surfaces, reminiscent of
the case of moduli spaces of bundles on curves. There, the analogous invariance of the
Verlinde formula reflects a geometric isomorphism between generalized theta functions
with dual ranks and levels [B][MO1]. In the case of sheaves on an abelian or K3 surface,
it is tempting to assert, similarly, that whenever defined and nonzero, the morphisms
SD± : H0(M±,Θ )∨ → H0(M±,Θ )
v w w v
are isomorphisms. The same considerations should apply to the companion morphism
SD : H0(K ,Θ )∨ → H0(M ,Θ ).
v w w v
In the above, for each of the three pairs of moduli spaces i.e., (M±,M±) and (K ,M ),
v w v w
the line bundle Θ stands for any one of the Θ ’s, for E in the corresponding moduli
v E
space of sheaves with Mukai vector v; similarly, Θ is any one of the line bundles Θ ,
w F
for F in the dual moduli space of sheaves with Mukai vector w. We will review the
definition of the three strange duality morphisms in Section 2 below, assuming that
Assumption 2. For any two (semi)-stable sheaves E and F with Mukai vectors v and
w, we have
H2(E ⊗F) = 0.
This is automatic if c (v⊗w)·H > 0, by Serre duality and stability.
1
In order to use the numerics provided by Theorems 1, 2 and 3, one has to assume in
addition that the line bundle Θ has no higher cohomology on the various moduli spaces
w
considered. The vanishing of higher cohomology is a delicate question, which can be
answeredsatisfactorilyonlyinfewcases. Forsmoothmodulispaces,orformodulispaces
with mild singularities - e.g. rational - one may invoke standard vanishing theorems.
These require the understanding of the positivity properties of Θ i.e., determining
w
whether Θ is big and nef. In the case under study, smoothness is assumed, bigness
w
is easy to detect, and nefness is hoped for. This last point of nefness is a subtle issue,
even though the presence of the holomorphic symplectic structure on the moduli spaces
considered here makes the question more tractable. A study for the Hilbert scheme
of two points on K3 surfaces and their deformations can be found for instance in [HT].
SHEAVES ON ABELIAN SURFACES AND STRANGE DUALITY 5
Nevertheless, LePotier[LP]andLi[Li]provedthefollowingresults, whichcanbeviewed
as a higher dimensional generalization of the ampleness of the determinant line bundle
on the moduli space of bundles over a curve.
Fact 1. (i) If w has positive rank, and c (w) is a high multiple of the polarization
1
H, then Θ is relatively ample on the fibers of the determinant map α+.
w
(ii) If w has rank 0, and c (w) is a positive multiple of the polarization, then Θ is
1 w
big and nef on the fibers of α+.
Similar results should hold for the morphism α−. When the Picard rank of A is 1,
this is obtained for free in many cases, using the remarks following Conjecture 2(ii).
One may then speculate
Conjecture 1. When Assumptions 1 and 2 are satisfied, the three morphisms SD+,SD−
and SD are either isomorphisms or zero.
This has the immediate
Corollary 1. As E varies in K , the Theta sections Θ on the dual moduli space M
v E w
span the linear series |Θ |. Same statements apply to the moduli spaces M+ and M−,
v w w
letting E vary in M+ and M− respectively.
v v
The conjecture was demonstrated in a number of cases, in this and other geometric
setups. An overview of the already existing arguments, as well as proofs of new cases,
can be found in [MO2].
1.1. Acknowledgements. The calculations presented in this paper have as starting
point Kota Yoshioka’s extensive previous work on the subject.
We would like to thank Jun Li for many conversations related to moduli spaces of
sheaves on surfaces, and the NSF for financial support. A.M. is grateful to Jun Li
and the Stanford Mathematics Department for making possible a very pleasant stay at
Stanford in the spring of 2007.
2. Preliminaries on Theta divisors
We review here the definition of the strange duality morphisms. We will fix Λ an
arbitrary determinant, and we let Λ(cid:98) = detRS(Λ). We recall the notation of the Intro-
duction, letting M+ and M− denote the moduli spaces of sheaves with determinant Λ
v v
and determinant of the Fourier-Mukai transform equal to Λ(cid:98) respectively; Kv consists of
sheaves satisfying both requirements.
6 ALINAMARIANANDDRAGOSOPREA
We explained in the Introduction that the line bundle Θ only depends on the K-
F
theory classof the referencesheafF. We establish herethe following more preciseresult:
Lemma 1. Consider a sheaf F of Mukai vector w, and consider the line bundle Θ on
F
the moduli space M . Then,
v
(i) for F ∈ M+, the restriction of Θ to M+ is independent of the choice of F;
w F v
(ii) for F ∈ M−, the restriction of Θ to M− is independent of the choice of F;
w F v
(iii) for F ∈ M , the restriction of Θ to K is independent of the choice of F.
w F v
Proof. To prove (i), pick two sheaves F and F with Mukai vector w and the same
1 2
determinant. Considering the virtual element in K-theory
f = F −F ,
1 2
we need to show that
Θ ⊗Θ−1 = Θ
F1 F2 f
is trivial. We will show that in K-theory,
(10) f = O −O
Z W
for two zero-dimensional schemes Z and W, which necessarily have to be of the same
length. This follows by induction on the rank of the F’s. The rank 0 case is obvious.
When the rank is 1, then
F = L⊗I ,F = L⊗I
1 Z 2 W
with L = detF = detF , and the result is immediate. For the inductive step, note that
1 2
it suffices to replace F and F by the twists F (D) and F (D), for some ample divisor
1 2 1 2
D. In this case, we reduce the rank by constructing exact sequences
0 → O → F (D) → F(cid:48) → 0
A i i
with F(cid:48), and F(cid:48) of the same lower rank and the same determinant. The claim then
1 2
follows by the induction hypothesis applied to F(cid:48) − F(cid:48). Once (10) is understood, it
1 2
suffices to assume that Z and W are supported on single points i.e., that f is a formal
sum
(cid:88)
f = (O −O ).
z w
z,w
In this case, we will check Θ is trivial by testing against any S-family E → S ×A of
f
sheaves with fixed determinant Λ. In particular, the latter requirement implies that
detE ∼= M (cid:2)Λ
SHEAVES ON ABELIAN SURFACES AND STRANGE DUALITY 7
for some line bundle M on S. Then, the pullback of Θ under the classifying morphism
f
S → M+ is
v
detp!(E ⊗q(cid:63)f)−1 = (cid:16)detE(cid:12)(cid:12)S×{z}(cid:17)−1⊗detE(cid:12)(cid:12)S×{w} ∼= M−1⊗M ∼= OS,
completing the proof of (i).
For (ii), the same reasoning applies to
RS(f) = RS(F )−RS(F ),
1 2
which can be assumed to be the difference of the structure sheaves of z and w on the
dual abelian variety A(cid:98). Therefore,
f = P −P ,
z w
where P and P are the line bundles on A represented by z and w. We need to check
z w
that
detp(E ⊗q(cid:63)P ) ∼= detp(E ⊗q(cid:63)P ).
! z ! w
Consider the relative Fourier-Mukai sheaf on S ×A(cid:98)
detp13!(p(cid:63)12E ⊗p(cid:63)23P) ∼= M (cid:2)Λ(cid:98),
for some line bundle M on S. Here the pushforward and pullbacks are taken along
the projections from S ×A×A(cid:98). The conclusion follows by looking at the isomorphic
pullbacks of this sheaf over S ×{z} and S ×{w}.
Finally, the third statement is obvious since K is simply connected [Y1]. Indeed, it
v
sufficestocheckthatc (Θ )isindependentofF. ThisisaGrothendieck-Riemann-Roch
1 F
computation, using the defining formula (3), with E replaced by a quasi universal sheaf,
if needed.
Remark 1. The Lemma above is sufficient for our purposes. It would be useful to have
a more detailed understanding of how the Theta line bundles Θ vary with F. For the
F
case of curves, the requisite formulas were established by Dr´ezet-Narasimhan [DN]. We
speculate that the following holds:
Conjecture 2. Consider two sheaves F and F with the same Mukai vector orthogonal
1 2
to v.
(i) On M+, we have
v
Θ = Θ ⊗(cid:0)α−(cid:1)(cid:63)(cid:0)detF ⊗detF−1(cid:1).
F1 F2 1 2
(ii) On M−, we have
v
Θ = Θ ⊗(cid:0)(−1)◦α+(cid:1)(cid:63)(cid:0)detRS(F )⊗detRS(F )−1(cid:1).
F1 F2 1 2
8 ALINAMARIANANDDRAGOSOPREA
(iii) If c (v) = 0, then on M we have
1 v
Θ = Θ ⊗(cid:0)(−1)◦α+(cid:1)(cid:63)(cid:0)detRS(F )⊗detRS(F )−1(cid:1)⊗(cid:0)α−(cid:1)(cid:63)(cid:0)detF ⊗detF−1(cid:1).
F1 F2 1 2 1 2
Formula (i) is easily checked on the Hilbert scheme of points using that
Θ = (detF) ⊗E rankF,
F (n)
where E is the exceptional divisor and (detF) is the pullback of the symmetrization
(n)
of detF on the symmetric product [EGL]. Assuming (i), evidence for (ii) is provided by
the change of the Theta line bundles under Fourier-Mukai transform. Indeed, when the
Picard number of A is 1, Yoshioka [Y1] exhibited very general examples of birational iso-
morphisms between the moduli spaces M± and M∓ on A and A(cid:98), interchanging the maps
v v
(cid:98)
α+ and α−; arguments of Maciocia [Ma] can be used to show that under this isomor-
phism the line bundle Θ corresponds to Θ , at least for generic F satisfying WIT.
F (−1)(cid:63)F(cid:98)
Finally item (iii) is consistent with (i) and (ii), and with Grothendieck-Riemann-Roch.
It may be possible to prove all three formulas using suitable degeneration arguments.
Assuming Lemma 1, it is now standard to define the three strange duality morphisms.
The construction is contained in [D] and [OG], but we will review it briefly here for the
sake of completeness.
Recall that for any pair of sheaves (E,F) ∈ M ×M we have
v w
χ(E ⊗F) = 0.
We will assume furthermore that
Assumption 2. (modified version)
(a) either H2(E ⊗F) = 0; by stability this happens if c (E ⊗F).H > 0;
1
(b) or H0(E ⊗F) = 0; by stability this happens if c (E ⊗F).H < 0.
1
To treat all cases at once, let us denote by M and M any one of the three pairs
v w
(M+,M+), (M−,M−) and (K ,M ). We set
v w v w v w
(cid:40)
Θ , for F ∈ M , if Assumption 2 (a) holds,
F w
(11) L =
w Θ−1, for F ∈ M , if Assumption 2 (b) holds.
F w
By Lemma 1, this is a well-defined line bundle on M . We similarly define the line
v
bundle L on M .
v w
Descent arguments, presented in detail in Dˇanilˇa’s paper [D], show the existence of a
natural divisor
∆ (cid:44)→ M ×M
v,w v w
SHEAVES ON ABELIAN SURFACES AND STRANGE DUALITY 9
which is supported set-theoretically on the locus
∆ = (cid:8)(E,F) ∈ M ×M ,such that h1(E ⊗F) (cid:54)= 0(cid:9).
v,w v w
This divisor is obtained as the vanishing locus of a section of a naturally defined line
bundle Θ on M ×M . The splitting
v,w v w
Θ = L (cid:2)L
v,w w v
follows from Lemma 1 by the see-saw theorem. Therefore, ∆ becomes an element of
v,w
H0(M ,L )⊗H0(M ,L )
v w w v
inducing the duality morphism
H0(M ,L )∨ → H0(M ,L ).
v w w v
Note that when Assumption 2 (a) holds, the construction gives rise to the three duality
morphisms of the Introduction:
(12) SD± : H0(M±,Θ )∨ → H0(M±,Θ ), and
v w w v
(13) SD : H0(K ,Θ )∨ → H0(M ,Θ ).
v w w v
3. Euler characteristics on Albanese fibers
3.1. The Albanese map. In this section, we will compute the Euler characteristics of
line bundles on K . We begin by reviewing a few facts about the Albanese map of M .
v v
Recall from the Introduction the modified determinant morphism
αΛ+ : Mv → A(cid:98),E → detE ⊗Λ−1,
and its Fourier-Mukai ’dual’
αΛ− : Mv → A, E → detRS(E)⊗Λ(cid:98)−1.
Putting these two morphisms together, we obtain the map
(14) av = (α+,α−) : Mv → A(cid:98)×A.
Yoshioka proved that a is the Albanese map of the moduli space M [Y1].
v v
The morphism (14) is easiest to understand for the vector v = (1,0,n). Then, the
moduli space Mv is isomorphic to the product A(cid:98)×A[n] of the dual abelian variety and
the Hilbert scheme A[n] of points on A. The morphism a can be identified with
v
1×s : A(cid:98)×A[n] → A(cid:98)×A
10 ALINAMARIANANDDRAGOSOPREA
where the first map is the identity, while the second is induced by summation on the
abelian surface. That is, for a zero-cycle Z supported on points z with length n , we let
i i
(cid:88)
s : A[n] → A, s([Z]) = n z .
i i
i
The fiber of s over (0,0) is the generalized Kummer variety K of dimension 2(n−1).
n−1
In general, Yoshioka studied the fiber of the Albanese map a over the origin
v
K = a−1(0,0),
v v
under the assumption that the vector v is primitive and positive, in the sense discussed
in the Introduction. In this situation, and when (cid:104)v,v(cid:105) ≥ 6, Yoshioka proved in [Y1] that
• K is an irreducible holomorphic symplectic manifold, deformation equivalent to
v
the generalized Kummer surface K , with n = (cid:104)v,v(cid:105)/2.
n−1
• There is an isomorphism
H2(K ,Z) ∼= v⊥.
v
Here v⊥ is computed in the cohomology of the surface A. Under the above
isomorphism, Mukai vectors w ∈ v⊥ of sheaves orthogonal to v correspond to the
Chern class c (Θ ) ∈ H2(K ,Z).
1 w v
• Moreover, if one endows H2(K ,Z) with the Beauville-Bogomolov form, and v⊥
v
with the intersection form, the above isomorphism is an isometry.
3.2. GeneralizedKummervarieties. WestartthecalculationofEulercharacteristics
by considering the case of line bundles on the generalized Kummer varieties. In other
words, we assume that v = (1,0,n). It may be possible to derive the answers from
the calculations of [BN], but for completeness we include a direct argument, which also
expresses the numerics in a form more convenient to us.
ForanydivisorDontheabeliansurfaceA,weletD bethedivisoronA[n] consisting
(n)
of zero-cycles which intersect D. This divisor is a pull-back under the support morphism
f : A[n] → A(n),
from the symmetric power A(n) of A,
D = f(cid:63)(D(cid:2)D(cid:2)...(cid:2)D)Sn.
(n)
Further,letE betheexceptionaldivisorofA[n] consistingofschemeswithtwocoincident
points in their support. Any line bundle on the Hilbert scheme is of the form D ⊗Er.
(n)
We will denote by the same symbol the restriction of these bundles to the generalized