A CONJECTURE OF HAUSEL ON THE MODULI SPACE OF
HIGGS BUNDLES ON A CURVE
JOCHEN HEINLOTH
To Gérard Laumon
Abstract. In this survey, we review some conjectures on the cohomology of
the moduli space of Higgs bundles on a curve and explain joint work with
O. Garcia-Prada and A. Schmitt [9], [10] resulting in a recursive algorithm
to determine the cohomology of moduli spaces of semi-stable Higgs bundles
on a curve (in the coprime situation). This method allows us to confirm a
conjecture of T. Hausel who predicted a formula for the y–genus of the moduli
space.
1. Introduction
The moduli space of Higgs bundles on a projective curve C was introduced by
Hitchin [17], who discovered several remarkable geometric properties of this space.
Roughly the space can be thought of as the cotangent space to the moduli space
of vector bundles on a given curve and Hitchin showed that this space carries the
structure of an integrable system. If C is defined over the complex numbers, the
moduli space admits a natural family of complex structures, e.g., it turns out to be
diffeomorphic to the so called character variety, i.e., the space of representations of
the fundamental group of the curve.
Much later, Ngô [22] exhibited another use of this moduli space by observing, that
for curves defined over finite fields, the adelic description of the stack of Higgs
bundles on the curve is closely related to spaces occurring in the study of the trace
formula.
Despite this wealth of structure, the cohomology of the moduli space has not yet
been determined. For bundles of small rank (n = 2, 3) the answer has been known
for a long time by work of Hitchin and Gothen, but from these results it did not seem
possible to come up with a general prediction. It was therefore a surprise, when T.
Hausel and F. Rodriguez-Villegas managed to formulate a conjecture, predicting
the additive structure of the cohomology.
In this note we want to survey joint work with O. Garcia-Prada and A. Schmitt
([9],[10]), in which we found an algorithm for the computation of the cohomology of
the moduli space of Higgs bundles on a curve (if rank and degree are coprime) and
applied this to prove a specialization of the conjecture by Hausel and RodriguezVillegas.
Let me briefly summarize the structure of the article. In Section 2 we recall the
basic definitions and results on moduli spaces of Higgs bundles. Since Hausel’s
conjectures originated from insights on mirror symmetry, we take the opportunity to
relate one of these conjectures to Ngô’s support theorem. In Section 3 we recall the
conjecture of Hausel and Rodriguez-Villegas giving a formula for the cohomology
of the moduli space of Higgs bundles. Here we use the reformulation introduced
by S. Mozgovoy in terms of the Grothendieck ring of varieties. We end the section
by explaining how Hausel used this conjecture to deduce an explicit conjecture for
the Hirzebruch y–genus of the moduli space. In Section 4 we then review how
1
2
J. HEINLOTH
we obtained the aforementioned algorithm and deduce Hausel’s conjecture on the
y–genus.
2. Some properties of the moduli space of Higgs bundles and its
cohomology
In this section we will recall the definition and some of the basic properties of the
moduli space of Higgs bundles on a curve. These definitions and most of the results
are due to Hitchin [17]. We then deduce some basic results on the cohomology of
the moduli space. Almost all of these results seem to be well known, but we use the
opportunity to relate one of Hausel’s conjectures to Ngô’s support theorem [23].
We will denote by C a fixed smooth projective, geometrically connected, algebraic
curve of genus g defined over some field k. For our application we will only need to
consider k = C, but sometimes the possibility to choose k to be a finite field was
useful for us. We will often abbreviate cohomology of a variety X by H ∗ (X). If
k = C we want to understand that H ∗ (X) = H ∗ (X, Q) is singular cohomology with
rational coefficients, equipped with its natural mixed Hodge structure on H ∗ (X)⊗Q
∗
C. If k is a general field, we abbreviate H ∗ (X) = Hét
(Xk , Q` ) for some prime
number ` prime to the characteristic of k. If F is a coherent sheaf on X we will
denote by H ∗ (X, F) the usual cohomology groups of coherent sheaves.
A Higgs bundle on C is a pair (E, θ : E → E ⊗ ΩC ), where E is a vector bundle on
C, θ is an OC –linear map and ΩC is the sheaf of differentials on C. To motivate
this definition, recall that deformations of a vector bundle E on C are parametrized
by H 1 (C, End(E)) and by Serre duality, this space is dual to H 0 (C, End(E) ⊗ ΩC ),
so one can think of a Higgs bundle as a point in the cotangent space to the moduli
stack of vector bundles. However, as we will see below, one has to be careful when
using this interpretation.
We denote by Mn,d the moduli stack of Higgs bundles of rank n and degree d on
C, i.e., the stack classifying pairs (E, θ : E → E ⊗ ΩC ) as above such that deg(E) =
d, rk(E) = n.
As an immediate warning let us mention that this stack is very big. Already the
stack of vector bundles on a curve is only locally of finite type, but for any fixed
degree and any N ≥ 0 the stack of vector bundles has an open substack of finite
type, such that the complement has codimension > N (see e.g.[4]). This property
usually fails for the stack of Higgs bundles (even for C = P1 and n = 2).
To see where this problem comes from, let us recall that (by a result of Biswas and
Ramanan [6]) infinitesimal deformations of a Higgs bundle (E, θ) are described by
the cohomology of the complex
C• (E, θ) := (End(E) → End(E) ⊗ ΩC ).
The tangent space of the stack at (E, θ) is the quotient stack
[H 1 (C, C• (E, θ))/H 0 (C, C• (E, θ)].
Now Serre duality implies that the dimension of H 0 of the complex is equal to the
dimension of H 2 , so whenever the Higgs bundle admits non-trivial automorphisms
the dimension of this tangent space will increase. In particular, the stack of Higgs
bundles will be singular in general.
Moreover, recall that in the deformation theory of vector bundles whenever the
dimension of H 1 (C, End(E)) increases, the dimension of the automorphism group
of the bundle also increases to cancel this variation. For Higgs bundles, we see
that automorphisms of Higgs bundles only compensate half of the variation of the
dimension of H 1 , which causes the problem that Mn,d is much bigger than expected.
Since stability of Higgs bundles allows us to forget about these problems, we will
need to recall this notion. For a vector bundle E on C the slope of E is defined
ON A CONJECTURE OF HAUSEL
3
as µ(E) := deg(E)
rk(E) . A Higgs bundle (E, θ) is called semistable if for all subsheaves
F ⊂ E with θ|F : F → F ⊗ ΩC we have µ(F) ≤ µ(E). A Higgs-bundle is called
stable if this last inequality is a strict inequality for all proper (F, θ|F ) ( (E, θ).
Note that as in the case of vector bundles the notions of semistability and stability
coincide, whenever rk(E) and deg(E) are coprime.
(Semi-)stability is an open condition on families of Higgs bundles, so we can consider
the substack of semistable Higgs bundles Mss
n,d ⊂ Mn,d . As in the case of vector
bundles, stable Higgs bundles only admit scalar endomorphisms, so that for a stable
Higgs bundle we have dim H 2 (C, End(E) → End(E) ⊗ ΩC ) = 1. Since the Euler
characteristic of the cohomology H ∗ (C, End(E) → End(E) ⊗ ΩC ) is
n2 (1 − g) − (n2 (2g − 2) + n2 (1 − g)) = −2n2 (g − 1).
We see that the stack of stable Higgs bundles Mstable
is a smooth stack of dimension
n,d
2n2 (g − 1) + 1 and it is a Gm -gerbe over its coarse moduli space Mn,d , which is
therefore smooth of dimension 2n2 (g − 1) + 2.
We will also need to introduce a the stack MdPGLn and Md,ss
PGLn of (semistable)
PGLn -Higgs bundles. Recall that by Tsen’s theorem any PGLn bundle E on Ck is
induced from a GLn bundle and the corresponding vector bundle E is determined
by E up to tensoring E with a line bundle. Therefore the degree of E induces a well
defined invariant deg(E) ∈ Z/nZ.
A PGLn –Higgs bundle of degree d ∈ Z/nZ is a pair (E, θ), where E is a PGLn
bundle of degree d on C together with θ ∈ H 0 (C, Ad(E) ⊗ Ω). It turns out that
d,ss
Md,ss
by the action of M01 ([13]).
PGLn is the quotient of Mn
The Hitchin map h : Mn,d → A = An := ⊕ni=1 H 0 (C, Ω⊗i
C ) mapping (E, θ) to
(tr ∧i (θ)) turns out to be a proper map of pure relative dimension df = 12 dim Mn,d =
n2 (g − 1) + 1 and it is Lagrangian with respect to the natural symplectic form on
Mn,d (these results are due to Hitchin, Faltings and Laumon).
Finally we will also use the natural Gm –action on Mn,d given by scalar multiplication on θ. The Hitchin map h is equivariant if we let Gm act on H 0 (C, Ω⊗i
C ) with
weight i.
For n, d coprime, we will see below that the above properties imply that, although
Mss
n,d is not proper its cohomology is pure. This was first proven by Markman and
Hausel (see [12]) by different methods.
To see why this follows from the above, let us recall a localization result, which appeared in several variants (e.g. in [7, Lemma 6], the underlying geometric argument
also appears in Brylinski’s article [8]):
Lemma 1. Let ρ : Gm × An → An be a linear action of Gm on an affine space,
such that all weights of the action are positive. Denote by s0 : Spec k → An the
inclusion of the origin in An and p : An → Spec(k) the projection.
Let K be a Gm –equivariant complex of `–adic sheaves on An . Then
Rp∗ K = s∗0 K and Rp! K = s!0 K.
From this one deduces:
Proposition 2. Let X be a smooth variety with an action of the multiplicative
group Gm . Assume that f : X → AN is a proper map, equivariant with respect to a
linear action of Gm on the affine space An , such that all weights of this action are
positive. Denote by X0 := f −1 (0) the fiber of f over 0 = Spec(k) ⊂ An . Then
H ∗ (X, Q` ) ∼
= H ∗ (X0 , Q` ).
Since X is smooth and X0 is proper, this implies that H ∗ (X, Q` ) is pure and furthermore that H ∗ (X, Q` ) vanishes for ∗ > 2 dim X0 .
4
J. HEINLOTH
Proof. Since f is Gm –equivariant, we can apply the localization Lemma 1 to K =
Rf∗ Q` and by the base change theorem for proper maps we find:
H ∗ (X, Q` ) = H ∗ (An , Rf∗ Q` ) = H ∗ (Spec k, s∗0 (Rf∗ Q` )) = H ∗ (X0 , Q` ).
Applying this to Mn,d we find:
Theorem 1 (Markman, Hausel). If n and d are coprime, the cohomology groups
∗
H ∗ (Mn,d ) and H ∗ (Mss
n,d ) are pure and H (Mn,d ) = 0 for ∗ > dim Mn,d .
Proof. Since the Hitchin map is proper, Gm –equivariant and Mn,d is smooth Propositon 2 implies that H ∗ (Mn,d ) ∼
= H ∗ (h−1 (0)) is pure. Because h is of relative dimen1
sion 2 dim(Mn,d ) the cohomology groups H ∗ (h−1 (0)) vanishes for ∗ > dim(Mn,d ).
Finally, Mst
n,d is a Gm –gerbe over Mn,d , so its rational cohomology is isomorphic
to H ∗ (Mn,d ) ⊗ H ∗ (BGm ), which is pure.
In a similar way one can reformulate many of the conjectures on the cohomology
of Mn,d in terms of the cohomology of the Hitchin fibration. Let us illustrate this
for another conjecture of Hausel:
Conjecture 3. Hausel [14, Conjecture 5.13] If (n, d) = 1 and g > 1 then the
canonical map Hc∗ (Mn,d ) → H ∗ (Mn,d ) is 0.
Note that we have seen in Theorem 1 that this conjecture is only about ∗ =
dim(Mn,d ) = 2 dim h−1 (0).
Let us rephrase this conjecture in terms related to Ngô’s support theorem:
Proposition 4. Let n, d be coprime, then Conjecture 3 holds if and only if for
all i, the perverse cohomology sheaf p Hi (Rh∗ Q` [dim(Mn,d )]) is the intermediate
extension of its restriction to A − 0.
Moreover, this statement holds true for all i 6= 0.
Proof. Let us abbreviate dh := 21 dim(Mn,d ) = rel. dim(h) = dim A. Recall that
by the decomposition theorem [5, 5.4.5] we have
Rh∗ Q` [dim(Mn,d )] =
dh
M
p
Hi (Rh∗ Q` [dim(Mn,d )])[−i]
i=−dh
is isomorphic (non-canonically) to a direct sum of shifted semisimple perverse
sheaves, all of which are Gm –equivariant. Thus, by Lemma 1, the conjecture says
that for every irreducible summand K of these perverse sheaves the canonical map
from Rp! K = s!0 K to Rp∗ K = s∗0 K vanishes.
By definition, for an irreducible perverse sheaf K on A the complex s∗0 (K) is concentrated in cohomological degrees [−n, 0] and by duality s!0 K is contained in cohomological degrees [0, n]. Moreover, cohomology in degree 0 appears if and only if
K is concentrated at 0. Thus K is isomorphic to the intermediate extension of its
restriction to A − 0 if and only if the canonical map s!0 K → s∗0 K is vanishes, i.e. if
the canonical map Hci (A, K) → H i (A, K) is 0. This proves the first claim.
Now, H ∗ (Mn,d , Q` [dim(Mn,d )]) = 0 for ∗ > 0, so that for i > 0 the complex
s∗0 ( p Hi (Rh∗ Q` [dim(Mn,d )])) must be concentrated in cohomological degrees [−n, −i].
In particular p Hi (Rh∗ Q` [dim(Mn,d )]) cannot contain summands concentrated in 0
for i > 0. For i < 0 the same must hold, because by the relative hard Lefschetz
theorem [5, 5.4.10] we have
p −i
H (Rh∗ Q` [dim(Mn,d )]) ∼
= p Hi (Rh∗ Q` [dim(Mn,d )])(i).
This proves the proposition.
ON A CONJECTURE OF HAUSEL
5
3. The conjecture of Hausel and Rodriguez-Villegas
Next we need to recall the conjecture of Hausel and Rodriguez-Villegas briefly in the
motivic version formulated by S. Mozgovoy [21]. The Grothendieck ring of varieties
K0 (Vark ) is defined to be the free abelian group generated by isomorphism classes
[X] of quasi projective varieties over k subject to the relation [X] = [X − Z] + [Z]
whenever Z ⊂ X is a closed subvariety of X. The product [X] · [Y ] := [X × Y ]
defines a ring structure on this group. One usually writes L := [A1 ]. For us the
main use of this ring is that it allows us to express the following two invariants of
Mn,d in the same terms: If k = Fq is a finite field, the map [X] 7→ #X(Fq ) defines
a morphism K0 (V ark ) → Z.
If k = C, the cohomology of any variety over C carries a natural mixed Hodge
structure, which was defined by Deligne. This defines an invariant:
E(X, u, v) :=
2 dim
XX
(−1)k
X
dim Hck;p,q (X)up v q .
p,q∈N
k=0
The long exact sequence for cohomology with compact supports for a pair Z ⊂ X
implies that this extends to a map
E : K0 (VarC ) → Z[u, v].
The following examples will appear in our formulas: E(L, u, v) = uv and if C is a
curve of genus g we have E(C, u, v) = 1 − gu − gv + uv. The Picard variety Pic of
C will have E(Pic, u, v) = (1 − u)g (1 − v)g , because its cohomology is the exterior
algebra on H 1 (C).
b 0 (Vark )[[t]] of our curve C which is
Also we need the zeta function Z(C, t) ∈ K
defined as
∞
X
[Symi (C)]ti .
ZC (t) :=
i=0
It turns out [18],[16] that ZC is a rational function
ZC (t) =
P (t)
(1 − t)(1 − Lt)
P2g
where P (t) = i=0 [Symi (C − P1 )]ti is a polynomial satisfying P (1) = [Pic] and
E(P (t)) = (1 − tu)g (1 − tv)g .
For our computations it is essential to define similar invariants also for certain algebraic stacks, e.g., the stack of vector bundles of fixed rank and degree Bundn on
a curve, which are only locally of finite type. To do this one introduces the dib 0 (Vark ) of K0 (Vark ), i.e., the completion of K0 (Vark )[L−1 ]
mensional completion K
according to the descending filtration defined by the subgroups generated
Qn−1by classes
L−n [X] such that dim[X] − n ≤ N . In this ring the class of [GLn ] = i=0 (Ln − Li )
becomes invertible and this allows to define classes [X ] for quotient stacks X =
[X]
X/GLn by [X/GLn ] := [GL
(see [4]).
n]
Example 1. Behrend and Dhillon computed the class of the moduli stack Bundn
of vector bundles of rank n and degree d on C and found:
n
2
[Pic0 ] Y
[Bundn ] = L(n −1)(g−1)
ZC (L−k ).
L−1
k=2
Using the functional equation [16, Section 3] for ZC this is equivalent to:
[Bundn ] =
n−1
[Pic0 ] Y
ZC (Lk ).
L−1
k=1
6
J. HEINLOTH
The E–polynomial extends to a morphism
b 0 (Vark ) → Z[u, v][[(uv)−1 ]].
E: K
To state the conjecture we finally need some combinatorial ingredients.
For any partition λ = (λ1 , λ2 , . . . ) with λ1 ≥ λ2 ≥ · · · ≥ 0 we denote by d(λ) the
Young tableau of λ, i.e. the diagram having λi boxes in the i-th line. For any
box x = (i, j) ∈ d(λ) the arm length a(x) := λi − j is the number of boxes to the
right of x and the leg length l(x) is the number of boxes
Pbelow x. One denotes by
h(x) = a(x) + l(x) + 1 the hook length of x and |λ| := λi .
For any λ Mozgovoy defines (following [15]):
Y
Hλ (t) :=
t(1−g)(2l(x)+1) ZC (th(x) La ).
x∈d(λ)
Finally we will need the plethystic logarithm, defined in any complete λ–ring (which
b 0 (Vark ) in our examples) as
will be K
Log(1 + a) = −
∞
X
1X
µ(d)ψd (−a)n/d ,
n
n=1
d|n
where µ denotes the Möbius µ–function and ψd is the d-th Adams operation. Instead of defining the Adams operations, let us only note that the effect of ψd on
the E–polynomial is simply to replace all variables of the polynomial by their d-th
power.
b 0 (Vark )[[t]] by the
Conjecture 5 ([21, Conjecture 2]). Define elements Hn (t) ∈ K
formula
!
X t(1−g)n2 Hn (t)
X
n
|λ|
T = Log
Hλ (t)T
(1 − t)(1 − tL)
n≥1
λ
then Hn (1) = [Mnd ] for any d such that (n, d) = 1.
Remark 6. This conjecture was deduced in a remarkable way. In [15] the number of
points over a finite field is computed for the character variety which is diffeomorphic,
over the complex numbers to Mn,d . E.g., for n = 1 this variety would be G2g
m and
(C∗ )2g is indeed diffeomorphic to T ∗ Pic over C. In this case point counting would
give (q − 1)2g .
Even in this simple example, we see that the cohomology of the character variety
will never be pure, as purity would imply that terms of the form q n would appear
in even degree 2n, but the cohomology ring of H ∗ (G2g
m ) is generated by elements of
degree 1.
Nevertheless, Hausel and Rodriguez-Villegas managed to predict the E polynomial
for the variety Mn,d , which has pure cohomology, from the non-pure point counting
formula.
The above formula looks quite complicated, but Hausel observed that it simplifies
dramatically if one applies the y–genus. The (compactly supported) Hirzebruch
y–genus of a variety is a specialization of the E–polynomial, it is defined as:
Hy ([X], y) := E([X], 1, y).
For example we have
(1) Hy (ZC (t), y) = (1 − t)g−1 (1 − ty)g−1 ,
(2) Hy (C, y) = (1 − g)(1 + y)
(3) Hy (Pic, y) = 0
ON A CONJECTURE OF HAUSEL
7
Actually both sides of the formula will then vanish since all terms are divisible
by [Pic]. However,
for the moduli space of PGLn –Higgs bundles one should have
Hn (t) d
for any d coprime to n. Let us explain why for these expres[MPGLn ] = H1 (t) t=1
sions the y–genus will turn out to be non-trivial:
We have already seen that Hy (ZC (th La )) = (1 − th y a )g−1 (1 − th y a+1 )(g−1) is a
polynomial, which vanishes at t = 1 if a = 0. In particular
ψm (Hλ )ψn (Hµ ) Hy
= 0 for all λ, µ, m, n
H1
t=1
λ) can only be non-zero if λ = k is a partition with a single line.
and Hy ( ψmH(H
)
1
t=1
In that case we find:
!g−1 Qk−1 −m
m(a+1) ma
m(a+2) m(a+1)
(1 − t
y )(1 − t
y
)
ψm (Hk )(t) a=0 t
Hy
=
H1 (t)
(1 − t)(1 − ty)
t=1
t=1
= m(g−1)
mk (g−1) k−1
Y
(1 − y )
(1 − y)(g−1)
(1 − y ma )2
(g−1)
a=1
Therefore:


P
P
∞
|λ| n/d
X
X
Log( λ Hλ (t)T |λ| )
1
ψ
(−
H
(t)T
)
d
λ
 |t=1
Hy
|t=1 = Hy −
µ(d)
H1 (t)
n
H
(t)
1
n=1
d|n
∞
X
µ(n) X
ψn (Hk (t))
=−
−Hy
T kn |t=1
n
H
(t)
1
n=1
k

!
∞
X X µ(m)
ψk (H N (t))
m

 T N |t=1
Hy
=
m
H1 (t)
N =1
m|N
Putting these two calculations together T. Hausel finds:
Conjecture 7 (Hausel’s conjecture for the y–genus). Let n ∈ N and d ∈ Z be
coprime. Then the y–genus of the moduli space of semistable PGLn –Higgs bundles
of degree d on a curve of genus g is given by:
n
g−1 X
−1
g−1
1 − yn
µ(m) mY
d,ss
N
m
(1 − y jm )2
Hy (M (PGLn ), y) =y
,
1−y
m
j=1
m|n
2
where N = (n − 1)(g − 1) =
µ–function.
1
2
d,ss
dim(M
(PGLn )) and µ denotes the Möbius
In the following sections we would like to explain a proof of this conjecture.
4. A variant of Hitchin’s approach to the cohomology
Hitchin [17] suggested to compute the cohomology of Mn,d by localization with
respect to the Gm –action as follows: For a smooth variety X with a Gm –action
ρ : Gm × X → X, for which all limit points limt→0`ρ(t).x exist, Bialynicki-Birula
constructed a decomposition into subvarieties X = Xi+ such that the fixed point
locus of the Gm –action is the disjoint union of strata Xi and the locally closed
subvarieties Xi+ of X are affine bundles Xi+ → Xi over the fixed point strata.
ss
For X = Mn,d
this allows us to write
X
ss
Lni [Xi ],
[Mn,d
]=
i
8
J. HEINLOTH
ss
where the Xi are the components of the fixed points of the Gm –action on Mn,d
.
Hitchin and Simpson observed that the fixed point loci admit a simple modular
description (see [12]). Namely, a family of isomorphisms (E, θ) ∼
= (E, λθ) induces an
action of Gm on E, which one can use to decompose E into weight spaces E = ⊕Ei
such that θ will also decompose into a sum of maps θi : Ei → Ei−1 ⊗ ΩC . This data
is equivalent to the data Ẽi := Ei ⊗ Ω−i
C and maps φi : Ẽi → Ẽi−1 .
One therefore defines the stack of chains of rank n ∈ Nr+1 and degree d ∈ Zr+1 :
Ei vector bundle on C
Chaindn := (Er → Er−1 → · · · → E0 ) |
deg(Ei ) = di , rk(Ei ) = ni
Any chain defines a Higgs bundle and we will call a chain (semi-)stable if the underlying Higgs bundle is (semi-)stable. This defines open substacks Chaind,ss
⊂
n
Chaindn . Laumon proved ([20]) that the flow given by the Gm –action is Lagrangian,
which implies that the exponents ni occurring in the Bialynick-Birula decomposition are all equal to 12 dim Mn,d . Thus we have:
[Mnd,ss ] = Ln
2
(g−1)
X
[Chainnd,ss ].
n,d
It will be important for our argument, that since the Picard group of the stack
Chainnd contains Zr+1 (since one can pull back line bundles via the forgetful maps
to Bunni ), one can vary the stability condition on chains of vector bundles. Given
α ∈ Rr+1 and a chain E• := (Ei , φi ) the α–slope of E• is defined as:
Pr
deg(Ei ) + αi rk Ei
.
µα (E• ) := i=0 P
rk(Ei )
Since α–slope
Pr only depends on the numerical invariants of E• we will also write
d +α n
µα (n, d) = i=0P ini i i .
Accordingly a chain E• is called α–(semi-)stable if for all proper subchains E•0 ⊂ E•
we have
µα (E• ) < µα (E• ), resp. µα (E• ) ≤ µα (E• ).
The corresponding substacks of α–semistable chains are denoted by Chainnd,α−ss .
Note that the stability condition we defined using Higgs bundles corresponds to
α = (0, (2g − 2), . . . , r(2g − 2)).
The same argument used for vector bundles shows that for α–semistable chains
E•0 , E•00 with µα (E 0 ) > µα (E 0 ) one has
HomChains (E•0 , E•00 ) = 0.
This allows one to prove that that unstable chains admit a canonical Harder–
Narasimhan filtration
E•1 ( · · · ( E•h = E•
defined by the condition that the subquotients E•i /E•i−1 are α–semistable with decreasing α–slopes. The numerical invariants rk(E•i /E•i−1 ), deg(E•i /E•i−1 ) are called
the type of the Harder–Narasimhan filtration. The Harder–Narasimhan stratum
d
Chaind,t
n ⊂ Chainn is the substack parametrizing chains that admit a Harder–
Narasimhan filtration of a given type t.
This suggests to try to compute [Chainnd,α−ss ] in a way analogous to the computations of the cohomology of the moduli spaces of semistable vector bundles [11],[3],
namely: first describe Chaindn and then use the geometric decomposition
Chainnd,α−ss = Chaind,α−ss
− ∪t Chainnd,t .
n
ON A CONJECTURE OF HAUSEL
9
The Harder–Narasimhan strata should be easier to compute, because they parametrize extensions of semistable chains of lower rank. To make this precise we recall
from [2] that R HomChains (E•00 , E•0 ) is computed by the cohomology of the complex:
M
M
0
0 → Hom(E•00 , E•0 ) →
Hom(Ei00 , Ei0 ) →
Hom(Ei00 , Ei−1
)
i
1
→ Ext
(E•00 , E•0 )
→
M
i
1
Ext
(Ei00 , Ei0 )
→
M
i
0
Ext1 (Ei00 , Ei−1
)
i
→ Ext2 (E•00 , E•0 ) → 0.
A key observation for our approach is that applying Serre duality to the above
complex one obtains:
00
Lemma 8. Let E•00 ,E•0 be chains and denote by E•−1
the chain shifted such that the
00
i-th bundle of the shifted chain is Ei+1 . Then we have
00
Ext2 (E•00 , E•0 )∨ ∼
⊗ ΩC ).
= Hom(E•0 , E•−1
From this we can immediately deduce:
Corollary 9. Let E•00 ,E•0 be chains and assume that one of the following conditions
is satisfied:
(1) α is a stability parameter satisfying αi − αi−1 ≥ 2g − 2 for all i and E•0 , E•00
are α-semistable chains with µα (E 0 ) > µα (E 0 )
(2) the maps φ00i of the chain E•00 are injective for all i = 1, . . . , r
(3) the maps φ0i of the chain E•0 are injective for all i = 1, . . . , r
then
Ext2 (E•00 , E•0 ) = 0.
This shows that for any type t = (nj , dj ) of a Harder–Narasimhan filtration the
natural map
Y
dj ,α−ss
Chainnj
Chaind,t
n →
mapping a chain to the subquotients of the Harder–Narasimhan filtration is a
smooth map of algebraic stacks. More precisely one finds:
Lemma 10 ([9, Proposition 4.8]). Let t = (nj , dj ) be a type of a Harder–Narasimhan
filtration of chains of rank n, d with respect to a stability parameter α ∈ Rr+1 satisfying αi − αi−1 ≥ 2g − 2 for all i. Then we have
[Chainnd,t ] = Lχ(t)
r
Y
dj ,α−ss
[Chainnj
]
i=0
where
χ(t) =
X
r
X
0≤i<j≤r k=0
njk nik (g −1)−njk dik +nik djk −
r
X
njk nik−1 (g −1)−njk dik−1 +nik−1 djk .
k=1
However, as for the stack of all Higgs bundles, the stacks of Chaindn are often very
b 0 (Vark ). Therefore, in order to apply our
big, so that they do not define classes in K
strategy we will need to truncate the stack of all chains in a way that is compatible
with the Harder–Narasimhan stratification.
Before we explain the truncation procedure, let us give some examples of well
behaved substacks. For fixed n, d us denote by Chaind,inj
the stack of chains E• of
n
rank n and degree d such that all maps φi : Ei → Ei−1 are injective. E.g., for r = 1
and n = (n, n) this is the stack of modifications E1 ⊂ E0 of vector bundles. For
these stacks we find:
10
J. HEINLOTH
Proposition 11.
(1) Let n = (n, n) and d ∈ Z2 then Chaind,inj
= ∅ unless d1 ≤ d0 and in that
n
case we have
b 0 (Vark )
[Chainnd,inj ] = [Bundn0 ][Symd0 −d1 (C × Pn−1 )] ∈ K
0
(2) For n = (n0 , n1 ) with n1 < n0 and d ∈ Z2 we have
b 0 (Vark ).
[Chainnd,inj ] = Ln1 d0 −n0 d1 −n0 n1 (1−g) [Bundn11 ][Bundn00 ] ∈ K
(3) If n ∈ Nr+1 satisfies nr ≤ · · · ≤ n0 we have
Y
Y
Lχi [Bundnii ]
[Symdi−1 −di (C × Pni −1 )],
[Chainnd,inj ] = [Bundn00 ]
i=1...r
ni <ni−1
i=1...r
ni =ni−1
where χi = ni di−1 − ni−1 di − ni−1 ni (1 − g).
Proof. For (1) we will only prove the corresponding statement for the cohomology
of the corresponding stacks:
H ∗ (Chainnd,inj ) = H ∗ (Bundn ) ⊗ H ∗ (Sym(C × Pn−1 )),
since we will need to apply the result for the corresponding E–polynomials in this
note. A different geometric argument for the full statement can be found in [9].
Note that for d0 − d1 = 1 the forgetful map
Chainnd,inj → Bundn00 ×C
which maps (E1 → E0 ) 7→ E0 , supp(E0 /E1 ) identifies Chainnd,inj with the projectivization of the universal bundle on Bundn00 ×C. This already proves (1) for d0 − d1 ≤ 1.
The general case can be deduced from this using an argument of Laumon [19] using
a variant of the Springer resolution:
Let us abbreviate d := d0 − d1 . Following [19] let us denote by Cohd0 the stack of
coherent sheaves of rank 0 and degree d on C and by
d
g := hT1 ( T2 ( . . . Td |Ti ∈ Cohi i.
Coh
0
0
Similarly let us denote the stack classifying iterated modifications of vector bundles
by:
d,inj
^ n := h(E1 ( E11 ( · · · ( E1d0 −d1 = E0 )i.
Chain
d,inj
^n
Again Chain
admits forgetful maps:
d,inj
^n
Chain
d
g
f
r
g
/ Bundn1 ×Coh
0
1
/ Cd
gr
/ Bundn1 × Cohd0
1
/ Symd C
.
p
Chaind,inj
n
By definition the left hand square is cartesian and the map gr identifies Chainnd,inj
with an open substack of the stack parametrizing extensions (E1 → F → T ) of a
torsion sheaf T and a vector bundle E1 . Since the stack of extensions is smooth over
Bundn11 × Cohd0 , this shows that the map gr is smooth as well. By [19] the forgetful
d
g → Cohd is small and it is a Sd –bundle over the inverse image of the
map Coh
0
0
complement of the diagonals in Symd C. In particular Rp∗ Q` is the intermediate
extension of its restriction to any open subset and therefore it carries a natural
Sd –action, for which we have
d,inj
^n
H ∗ (Chainnd,inj ) = H ∗ (Chaind,inj
, (Rp∗ Q` )Sd ) = H ∗ (Chain
n
)Sd .
ON A CONJECTURE OF HAUSEL
11
Moreover, from the first case we know that the composition
d,inj
^n
Chain
d
g → Bund1 ×
→ Bundn11 ×Coh
0
n1
d
Y
Coh10
i=1
induces a surjective map on cohomology. We have H (Coh10 ) = H ∗ (C)⊗H ∗ (BGm ) =
d,inj
^n )
H ∗ (C) ⊗ Q` [x] and the induced map H ∗ (Bundn ) ⊗ (H ∗ (C)[x])⊗d → H ∗ (Chain
is compatible with the natural Sd action on both sides and on each of the H ∗ (C)[x]
the map factorizes through H ∗ (C)[x]/(xn ) = H ∗ (C × Pn−1 ). This proves the cohomological version of (1). To deduce (2) we define a stratification of Chainnd,inj
by fixing the degree of the saturation of the image of the map d0 := deg(φ1 (E1 )sat )
of a chain E• . The strata parametrize pairs (E1 ⊂ E10 , E10 → E0 → E000 ), where
E10 = φ1 (E1 )sat is a vector bundle of rank n1 and degree d0 ≥ d1 and E10 → E0 → E000
is an extension of vector bundles, where rank(E 00 ) = n0 − n1 and deg(E 00 ) = d0 − d0 .
0
0
The stack of extensions (E10 → E0 → E000 ) is a vector bundle stack over Bundn1 × Bunnd00−d
−n1
with fibers of dimension
∗
dim Ext1 (E000 , E10 ) − dim Hom(E000 , E10 ) = n1 d0 − n0 d0 + n1 (n0 − n1 )(g − 1).
Thus the stack of such extensions has class
0
0
0
b
Ln1 d0 −n0 d +n1 (n0 −n1 )(g−1) [Bundn1 ][Bunnd00−d
−n1 ] ∈ K0 (Vark )
Combining this with (1) one finds:
X
1 −d
[Symd (C × Pn−1 )]Ln1 d0 −n0 (d1 +d)+n1 (n0 −n1 )(g−1) [Bundn11 ][Bunnd00−d
[Chainnd,inj ] =
−n1 ]
d≥0
n1 d0 −n0 d1 +n1 (n0 −n1 )(g−1)
1
Z(C × Pn1 −1 , L−n0 )
= [Bundn11 ][Bundn00−d
−n1 ]L
=
2
[Bundn11 ]L((n0 −n1 ) +n1 (n0 −n1 ))(g−1)+n1 d0 −n0 d1
nY
n −n
1 −1
[Pic] 0Y 1
−k
ZC (Li−n0 )
ZC (L )
L−1
i=0
k=2
= Ln1 d0 −n0 d1 −n0 n1 (1−g) [Bundn11 ][Bundn00 ].
(3) follows by induction on r using (1) and (2).
Remark 12.
(1) Part (2) of the preceding proposition shows that [Chainnd,inj ] equals the class
of a vector bundle of rank χ(H om(E1 , E0 )) over Bundn11 × Bundn00 , although
the natural forgetful map is far from being a vector bundle.
b 0 (Vark ) the Proposition implies that
(2) Since [Bundn ] is divisible by [Bund1 ] in K
d
[Chaind,inj
]
is
divisible
by
[Bun
]
as
well.
n
1
For any n, d one can stratify the stack Chainnd according to the ranks and degrees
of the images of the maps φi and their compositions and obtain a stratification for
which the preceding proposition allows one to compute the class of each stratum
b 0 (Vark ). However, the sum of all these strata will not converge in K
b 0 (Vark ).
in K
To circumvent this issue we will vary the stability parameters. Let us recall that
α ∈ Rr+1 is called critical value for n, d if there exist n0 ∈ Nr+1
and d0 ∈ Zr+1 such
0
that
µα (n0 , d0 ) = µα (n, d), and µγ (n0 , d0 ) 6= µγ (n, d) for some γ ∈ Rr+1 .
The first observation is that it is possible to compare the classes [Chaind,α−ss
] for
n
different choices of α. On the level of coarse moduli spaces this phenomenon is
usually called wall crossing. For us it is more convenient to phrase this in terms of
stacks, because we can then avoid using Jordan–Hölder filtrations:
12
J. HEINLOTH
Proposition 13 ([10, Proposition 2]). Fix n, d as before, let α ∈ Rr+1 be a critical
value and δ ∈ Rr+1 arbitrary. Then there exists > 0 such that Chainnd,α+tδ−ss is
⊆ Chaind,α−ss
. Moreover, the
independent of t for 0 < t < and Chaind,α+tδ−ss
n
n
complement is a union of α + tδ HN-strata:
Chaind,α−ss
− Chainnd,α+tδ−ss = ∪t∈I (α + tδ) − HN-Strata of type t,
n
where


I = (ni , di ) 

P
ni = n, di = d

.
µα (ni , di ) = µα (n, d) ∀i

µα+tδ (ni , di ) > µα+tδ (ni+1 , di+1 ) ∀i
P
Since the cohomology of HN–strata can be computed inductively by Lemma 10, it
will be sufficient to find for any n, d a stability parameter α for which the computation of [Chaind,α−ss
] can be done.
n
As in the preceding proposition we use different methods depending whether n
happens to be constant n = (n, . . . , n) or not.
If n is such that ni 6= ni−1 for some i, we can find an find parameters such that
there are no α–semistable chains. This follows from the following proposition:
Proposition 14 ([10, Proposition 4]). For α ∈ Rr+1 with αi − αi−1 ≥ 0 for all i
an α–semistable chain E• of rank n and degree d can only exist if
(1) For all j ∈ {0, . . . , r − 1} we have
Pj
i=0 di + αi ni
≤ µα (n, d)
Pj
i=0 ni
(2) For all j such that nj = nj−1 we have
dj
dj−1
≤
.
nj
nj−1
(3) For all 0 ≤ k < j ≤ r such that nj < min{nk , . . . , nj−1 } we have:
P
Pj
i6∈[k,j] di + αi ni + (j − k + 1)dj +
i=k αi nj
P
≤ µα (n, d).
i6∈[k,j] ni + (j − k + 1)nj
(4) For all 0 ≤ k < j ≤ r such that nj > max{nk , . . . , nj−1 } we have:
Pj
i=k+1 di − dk + αi (ni − nk )
≤ µα (n, d).
Pj
i=k+1 (ni − nk )
The proof of this proposition only exhibits natural destabilizing subchains of the
given slopes. These are easy to find if all maps of the considered chain happen
to be maximal. The condition on α is then used to show that in case some of
the maps are not of maximal rank a kernel or cokernel of the map would also
produce a destabilizing subchain or quotient chain. For small values of r and n
similar conditions already appear implicitly in [2]. It would be interesting to know
whether the above necessary conditions could be sufficient to have Chainnd−ss 6= ∅
for curves of genus > 1.
From this proposition, it is easy to deduce that if there exists i such that ni 6= ni−1
one can find δ = (0, . . . , 0, 1 . . . , 1) and t > 0 such that Chaind,α+tδ−ss
= ∅. By
n
d,ss
Proposition 13 and Lemma 10 this allow to express [Chainn ] in terms of classes
of moduli spaces of chains of smaller rank.
If n = (n, . . . , n) is constant, then we can avoid the convergence issue by showing
that for δ = (0, 1, . . . , r) there exists t > 0 such that Chaind,α+tδ−ss
⊂ Chainnd,inj
n
di ,inj
and moreover, all subquotients of HN-filtrations are also contained in Chainni
.
ON A CONJECTURE OF HAUSEL
13
In particular in this situation, all subquotients of Harder–Narasimhan filtrations
will also be of constant rank ni = (ni , . . . , ni ) ([9, Proposition 6.9]). Thus in
that case [Chainnd,α+tδ−ss ] can be computed inductively from the formulas given in
Proposition 11 (see [9, Corollary 6.10]).
Summing up we found the following algorithm that allows us to compute [Mss
n,d ] ∈
b 0 (Vark ) for (n, d) = 1:
K
(1) For any partition n of n use Proposition 14 to determine a finite set of d
P
P
with
di = d − ini (2g − 2) such that Chainnd,ss can be nonempty.
(2) For any n, d occurring in (1) determine either δ = (0, . . . , 0, 1 . . . , 1) and
t 0 such that [Chainnd,α+tδ−ss ] or determine t 0 and δ = (0, 1, . . . , r)
] is computed by [9, Corollary 6.10].
such that [Chaind,α+tδ−ss
n
(3) For n, d, δ, t as in (3) determine for any partition (ni , di ) of n, d the finite set
of 0 ≤ ti ≤ t such that α+ti δ is a critical value. For these ti use Proposition
]
13 and Lemma 10 to express [Chainnd,α+ti δ−ss ] in terms of [Chaind,α+tδ−ss
n
and classes of stacks of semistable chains of smaller rank. For these the
classes can be computed by induction using (2) and (3).
Luckily, this algorithm simplifies significantly if we only want to compute the
d
: We have seen already that all terms occurring in this algoy–genus of MPGL
n
d
) we can therefore use the
rithm are divisible by [Bund1 ]. To compute Hy (MPGL
n
d,ss
d
above algorithm, divide the classes [Chainn ] by [Bun1 ] and then apply Hy . Since
d
).
Hy ([Bund1 ], y) = 0, terms divisible by [Bund1 ]2 will not contribute to Hy (MPGL
n
By Lemma 10 this applies to all Harder–Narasimhan strata, so step (3) can be
d
skipped and moreover non-constant partitions n will not contribute to Hy (MPGL
)
n
in step (2). For constant partitions m = (m, . . . , m) the same argument shows that
Hy
[Chaind,ss
m ]
!
[Chaind,inj
m ]
= Hy
[Bund1 ]
!
[Bund1 ]
! r
[Bundm ] Y
Hy ([Symdi−1 −di (C × Pm−1 )]).
[Bund1 ] k=1
= Hy
n n
Thus, putting N = (n2 − 1)(g − 1) and c(g, n, m) = m m
( m − 1)(g − 1) we find:
d,ss
Hy (M
(PGLn ), y) = y
=y
N
N
X
X
m|n
n
d∈Z m
P
di =d−c(g,n,m)
X
Hy
m|n
[Bund ]
We know Hy ( [Bunmd ] ) =
1
Qm−1
i=2
[Bundm ]
[Bund1 ]
Hy
d,ss
[Chainm
]
!
[Bund1 ]
!
X
n −1
k∈N m
P
iki ≡d mod n
m
r
Y
Hy ([Symki (C × Pm−1 )]).
k=1
((1 − y i )(1 − y i+1 ))(g−1) and
m−1
Y
Hy (Z(C × Pm−1 , t)) = Hy (
i=0
Z(C, Li t)) =
m−1
Y
(1 − ty i )g−1 (1 − ty i+1 )g−1 .
i=0
14
J. HEINLOTH
Let us fix l =
n
m
l−1
Y
X
k ,...,kl−1 ≥0
P1
iki ≡d mod l
and ζ := e
2πi
l
a primitive l-th root of unity. Then
l
ki
m−1
Hy (Sym (C × P
i=1
1 X −jd
), y) =
ζ
l j=1
X
l−1
Y
ζ jiki Hy (Symki (C × Pm−1 ), y)
k1 ,...,kl−1 ≥0 i=1
=
l
l−1
1 X −jd Y
ζ
Hy (Z(C × Pm−1 , ζ ji ), y)
l j=1
i=1
=
l
m−1 l−1
1 X −jd Y Y
ζ
(1 − y r ζ ji )g−1 (1 − y r+1 ζ ji )g−1
l j=1
r=0 i=1
=
m−1
Y (1 − y rl )2 g−1 1 − y n g−1
1
.
µ(l)lg−1
l
(1 − y r )2
1 − ym
r=1
Summing up we find:
Theorem (Hausel’s conjecture for the y–genus). Let n ∈ N and d ∈ Z be coprime.
Then the y–genus of the moduli space of semistable PGLn –Higgs bundles of degree
d on a curve of genus g is given by:
d,ss
Hy (M
(PGLn ), y) =y
N
where N = (n2 − 1)(g − 1) =
µ–function.
1 − yn
1−y
1
2
g−1 X
m|n
n
−1
g−1
µ(m) mY
m
(1 − y jm )2
,
m
j=1
dim(Md,ss (PGLn )) and µ denotes the Möbius
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Universität Duisburg–Essen, Fachbereich Mathematik, Universitätsstrasse 2, 45117 Essen, Germany
E-mail address: [email protected]