Table Of ContentBETTI NUMBERS OF PARABOLIC U(2,1)-HIGGS BUNDLES
MODULI SPACES.
MARINA LOGARES
6
0
0
2
Abstract. LetX be a compactRiemannsurfacetogether witha finite setofmarked
n points. We use Morse theoretic techniques to compute the Betti numbers of the para-
a bolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked
J
point showing that the Poincar´e polynomials depend on the system of weights of the
8
parabolic bundle.
1
]
G 1. Introduction
A
The moduli spaces of stable parabolic Higgs bundles have been studied in [BY, T, Y]
.
h
andhavearichstructure, partiallyduetoitsrelationwiththerepresentationspaceofthe
t
a
fundamental group of a punctured Riemann surface. This relationship was established
m
by Simpson in [S].
[
The topology of the moduli U of stable U(p,q)-parabolic Higgs bundles with fixed
2
v generic weights and degrees has already been considered in [GLM]. This moduli space
2 is a submanifold of the moduli space M of stable parabolic Higgs bundles of fixed
4
degree, which was analised in the rank 2 case by Boden and Yokogawa in [BY] and in
3
1 the rank 3 case by Garc´ıa-Prada, Mun˜oz and Gothen in [GGM]. In these two papers,
0
the authors obtained the Betti numbers of the moduli spaces M. Our purpose here
6
0 is to calculate Betti numbers of U when p + q = 3. Note that in this case, U is a
/ submanifold of the moduli M studied in [GGM]. It is known that for fixed rank, the
h
t moduli spaces M of stable parabolic Higgs bundles, corresponding to different choices
a
m of degrees and generic weights, are diffeomorphic [GGM, T]. Our computations for
: p+q = 3 produce counterexamples to this type of phenomena for the submanifolds U,
v
that is, they provide an example of the dependence of these moduli spaces on the generic
i
X
weights of the parabolic structure.
r
a We will use Morse theory one step forward than in [GLM] thanks to fixing the rank
equal to 3. Higher ranks need to develop another tool called parabolic chains and will be
done inthe future. We start in Section 2, explaining the necessary definitions and results
for the Morse theory involved and defining also the Morse function that we are going
to use. In Section 3 we study certain critical subvarieties of this Morse function before
and in Section 5 we introduce parabolic triples for another type of critical subvarieties.
2000 Mathematics Subject Classification. 14D20, 14H60.
Key words and phrases. Parabolic bundles, Higgs bundles, moduli spaces, Betti numbers.
Partially supported by Ministerio de Educaci´on y Tecnolog´ıa through Acci´on Integrada Hispano-
Lusa HP-2000-0015 and by The European Contract Human Potential Programme, Research Training
Network HPRN-CT-2000-00101.
1
2 MARINALOGARES
Sections 4 and 6 give explicit computations for the case of one puncture and Section 7
summarizes the results and give some low genus examples.
Acknowledgements. We wish to thank Vicente Mun˜oz for very useful comments and
corrections, and to Luis A´lvarez-Consul for his help.
2. Definitions and Morse theory
Let X be a compact Riemann surface of genus g ≥ 0 together with a finite set of
marked distinct points x ,...,x . We denote D = x +···+x the divisor on X defined
1 s 1 s
by the punctures.
A parabolic bundle E over X consists of a holomorphic bundle with a parabolic
structure, that is, weighted flags, one for each puncture in X,
E = E ⊃ ··· ⊃ E ⊃ 0,
x x,1 x,r(x)
0 ≤ α (x) < ··· < α < 1.
1 r(x)
The set of all weights for all x ∈ D, α = {α (x);i = 1,...,r(x)}, is called parabolic
i
system of weights of E.
A holomorphic map f : E → E′ between parabolic bundles is called parabolic if
α (x) > α′(x) implies f(E ) ⊂ E′ for all x ∈ D, and f strongly parabolic if α (x) ≥
i j x,i x,j+1 i
α′(x) implies f(E ) ⊂ E′ for all x ∈ D, where we denote by α′(x) the weights on
j x,i x,j+1 j
E′. Also ParHom(E,E′) and SParHom(E,E′) will denote respectively the bundles of
parabolic and strongly parabolic morphisms from E to E′. Finally, a parabolic subbundle
of a parabolic bundle is a subbundle which inherits its parabolic structure from the
parabolic bundle.
We write m (x) = dim(E /E ) for the multiplicity of the weight α (x) at x. The
αi x,i x,i+1 i
parabolic degree and parabolic slope of E are defined as
r(x)
pardeg(E) = deg(E)+ m (x)α (x),
αi i
x∈D i=1
XX
pardeg(E)
parµ(E) = .
rk(E)
A parabolic bundle is called (semi)-stable if for every parabolic subbundle F of E, the
parabolic slope satisfies parµ(F) ≤ parµ(E) (resp. parµ(F) < parµ(E)).
For parabolic bundles E there is a well-defined notion of parabolic dual E∗. It consists
of the bundle Hom(E,O(−D)) and at each x ∈ D a weighted filtration
∗ ∗ ∗
E = E ⊃ ···E ⊃ 0,
x x,1 x,r(x)
0 < 1−α (x) < ··· < 1−α < 1.
r(x) 1
In the case α = 0 we choose the following weights for the filtration,
1
0 ≤ α < 1−α (x) < ··· < 1−α < 1.
1 r(x) 2
With this definition E∗∗ = E and pardeg(E∗) = −pardeg(E).
BETTI NUMBERS OF PARABOLIC U(2,1)-HIGGS BUNDLES MODULI SPACES. 3
A GL(n,C)-parabolic Higgs bundle is a pair (E,Φ) consisting of a parabolic bundle E
and Φ ∈ H0(SParEnd(E)⊗K(D)), i.e. Φ is a meromorphic endomorphism valued one-
form with simple poles along D whose residue at p ∈ D is nilpotent with resect to the
flag. A parabolic Higgs bundle is called (semi)-stable if for every Φ-invariant subbundle
F of E, its parabolic slope satisfies parµ(F) ≤ parµ(E) (resp. parµ(F) < parµ(E)).
We shall say that the weights are generic when every semistable Higgs bundle is stable,
that is, there are no properly semistable parabolic Higgs bundles.
A U(p,q)-parabolic Higgs bundle on X is a parabolic Higgs bundle (E,Φ) such that
E = V ⊕W, where V and W are parabolic vector bundles of rank p and q respectively,
and
0 β
Φ = : (V ⊕W) → (V ⊕W)⊗K(D),
γ 0
(cid:18) (cid:19)
where the non-zero components β : W → V ⊗ K(D) and γ : V → W ⊗ K(D) are
strongly parabolic morphisms. Hence a U(p,q)-parabolic Higgs bundle is (semi)-stable
if the slope (semi)-stability condition is satisfied for all Φ-invariant subbundles of the
form F = V′ ⊕W′, i.e. for all subbundles V′ ⊂ V and W′ ⊂ W such that
(1) β : W′ → V′ ⊗K(D)
(2) γ : V′ → W′ ⊗K(D).
Let us fix generic weights and topological invariants rk(E) and deg(E). The mod-
uli space MGL(n,C) of stable GL(n,C)-parabolic Higgs bundles was constructed using
Geometric Invariant Theory by Yokogawa [Y], who also showed that it is a smooth
irreducible complex variety.
By definition there is an injection from the moduli U of stable U(p,q)-parabolic
(p,q)
Higgs bundles to the moduli MGL(p+q,C) of stable GL(p+q,C)-parabolic Higgs bundles.
Moreover, such an injection is an embedding, as shown in [GLM], so U is in fact a
(p,q)
submanifold of M . When it does not induce confusion, we will denote U
GL(p+q,C) (2,1)
and MGL(3,C) by U and M.
The Toledo invariant for the moduli of U(p,q) parabolic Higgs bundles is studied in
[GLM]anddefinedasτ = 2(qpardeg(V)−ppardeg(W))/(p+q). Thus, given(E,Φ) ∈ U
we have
2
(3) τ = (∆−3b+ α (x)+α (x)−2η(x)),
1 2
3
x∈D
X
where we denote a = deg(V), b = deg(W), α (x) and α (x) the parabolic weights on V
1 2
and η(x) the parabolic weights on W over the punctures x ∈ D, and ∆ = a+b. We will
use this notation in the following.
Proposition 1. The map V ⊕W → (V ⊕W)⊗L, where L is a parabolic line bundle,
induces an isomorphism from the moduli space U (a,b) of parabolic U(p,q)-Higgs bun-
(p,q)
dles with fixed degrees (a,b) to the moduli space U (a′,b′) of parabolic U(p,q)-Higgs
(p,q)
bundles with fixed degrees (a′,b′), where a′ = a+pl and b′ = b+ql.
The map V ⊕W → V∗⊕W∗ induces an isomorphism of moduli spaces, from U (a,b)
(p,q)
to U′ (a′,b′), where a′ = −a and b′ = −b. (cid:3)
(p,q)
4 MARINALOGARES
Let us to assume that ∆ = a′ +b′ ≡ 0(3) and that the parabolic Toledo invariant τ
satisfies τ ≥ 0.
The moduli space U of stable U(p,q)-parabolic Higgs bundles has been studied in
[GLM], where the number of connected components is calculated using Bott-Morse the-
ory. Here we shall fix later p = 2 and q = 1 to go one step further and give topological
information about this moduli space.
Consider the action of C∗ on U given in [GLM] as
(4) ψ : C∗ ×U → U
(5) (λ,(E,Φ)) 7→ (E,λΦ).
This restricts to a Hamiltonian action of S1 ⊂ C∗ on U and the moment map associated
to this Hamiltonian action is defined by
1 1
(6) f([E,Φ]) = kΦk2 = kβk2 + kγk2,
π π
where we are using a suitable Sobolev metric for the norm given by the Hermite-Einstein
equations for the parabolic Higgs bundle (E,Φ) (see [S]).
Observe that f : U → R is the restriction of the moment map f : M → R used in
[GGM]. That map was proper. Hence, f is also proper since U is a closed submanifold of
M. This fact together with a result of Frankel [F], proving that a proper moment map
for a Hamiltonian circle action on a K¨ahler manifold is a perfect Bott-Morse function,
give us that f is a perfect Bott-Morse function.
Hence, we have the following formula for the Poincar´e polynomial of the manifold U,
(7) P (U) = tλNP (N),
t t
N
X
where the sum runs over all critical submanifolds N of U for f and λ is the Morse
N
index of f on N.
The critical points of f are exactly the fixed points of the circle action. Moreover, the
Morse index of f at a critical point equals the dimension of the negative weight space
of the circle action on the tangent space [F].
Simpson’s theorem gives us a criterion for (E,Φ) to be a critical point for the Morse
function.
Theorem 2 ([S], Thm.8). The equivalence class of a stable parabolic Higgs bundle (E,Φ)
is fixed under the action of S1 if and only if it is a parabolic complex variation of Hodge
structure. This means that E has a direct sum decomposition
E = E ⊕E ⊕···⊕E
0 1 m
as parabolic bundles, such that Φ is strongly parabolic and of degree one with respect to
this decomposition, in other words, the restriction Φ = Φ| ∈ H0(SParHom(E ,E )⊗
l El l l+1
K(D)). Also Φ 6= 0 and the weight of ψ on E is one plus the weight of ψ on E .
l l+1 l
Finally, the Morse index of f is calculated using the following result.
BETTI NUMBERS OF PARABOLIC U(2,1)-HIGGS BUNDLES MODULI SPACES. 5
Proposition 3 ([GLM]). The dimension of the eigenspace of the action of ψ on the
tangent space for the eigenvalue −k equals the first hypercohomology group of a complex
C• : U → U¯ ⊗K(D)
k k k
where
(8) U = ⊕ ParHom(E ,E ) U¯ = ⊕ SParHom(E ,E )
k j−i=2k i j k j−i=2k+1 i j
Thus in the U(2,1) case we have the following possibilities. If (E,Φ) is a critical point
then it can be of one of these three following forms:
E = E ⊕E , rk(E ) = 1, rk(E ) = 2
0 1 0 1
E = E ⊕E , rk(E ) = 2, rk(E ) = 1
0 1 0 1
E = E ⊕E ⊕E rk(E ) = 1, i = 1,2,3.
0 1 2 i
These form, critical subvarieties of types (rk(E ),rk(E ) or (rk(E ),rk(E ),rk(E ))
0 1 0 1 2
particularly in this case (1,2), (2,1), and (1,1,1) respectively. And this critical subva-
rieties can be identified with triples of type (1,2,d ,d ;α ,α ,η), (2,1,d ,d ;η,α ,α )
1 0 1 2 1 0 1 2
and chains of type (1,1,1,d ,d ,d ;α ,η,α ). Where by type of a triple or a chain
2 1 0 ̟(1) ̟(2)
we mean, a system of numbers that give some topological invariants of this objects,
they are the ranks, degrees and parabolic systems of weights of each parabolic bundle
conforming the triples or the chain respectively.
Observe that the critical varieties of type (1,2) and type (2,1) consist of parabolic
Higgs bundles for which either γ = 0 or β = 0 respectively. From (6) and using the
definition of τ we get that they are minima for the Morse function f and, as proved in
[GLM], they are the only ones. Hence its Morse index is zero.
In the cases (2,1) and (1,2) where E = E ⊕ E the critical submanifold will be
0 1
identified with certain moduli spaces of parabolic triples. However in the third case,
where E = E ⊕E ⊕E , we will be dealing with parabolic chains. This is the reason for
0 1 2
restricting attention to p = 2 and q = 1. If we would like to compute the Betti numbers
for higher values of p and q we will have to deal with more general parabolic chains that
the ones appearing here, and this tool has not been developed yet. This is left to future
work.
In the following sections we will calculate the Poincar´e polynomials which take part
in the formula in (7), that is for the moduli space U of parabolic U(2,1)-parabolic Higgs
bundles
P N +P N for τ > 0
(9) P (U) = t (2,1) t (1,1,1)
t P N +P N for τ < 0
t (1,2) t (1,1,1)
(cid:26)
where we denote P N the contribution on the Poincar´e polynomial of U of the sub-
t (1,2)
variety of type (1,2), P N is the contribution of the subvariety of type (2,1) and
t (2,1)
P N is the contributions from all critical subvarieties of type (1,1,1). Through
t (1,1,1)
these sections our computations will depend on some variables that we have mentioned
above: the Toledo invariant τ of the moduli space, and the degree ∆ = a+b of E. Recall
that by Proposition 1 we can suppose τ < 0 and ∆ ≡ 0(3).
It is known that for fixed rank, and for different choices of degrees and generic weights
the modulispaces of parabolicHiggs bundles M have the same Poincar´e polynomial (see
6 MARINALOGARES
[GGM]), so it is possible to choose the weights conveniently for such calculation for M.
We have seen thatU ⊂ M is a subvariety, and our calculation of its Poincar´e polynomial
will show that the same phenomenon does not happen for U. The Poincar´e polynomial
of U depends on the generic weights. We shall see this very explicitly in our calculations
for one marked point.
3. Contribution to Poincar´e polinomial from critical subvarieties of
type (1,1,1).
We start with the case where E = V ⊕W splits in three line bundles E = E ⊕E ⊕E
0 1 2
where E and E are contained in V, together with strongly parabolic homomorphisms
0 2
Φ = γ| : E → E ⊗K(D) and Φ = β| : E → E ⊗K(D).
0 E0 0 1 1 E1 1 2
We denote along this section d = deg(E ) so ∆ = d + d + d = d + b + d i.e.
i i 0 1 2 0 2
a = d +d and b = d .
0 2 1
The distributions of the weights for E and E are given by a set of injective maps
0 2
̟ = {̟ : {1,2} → {1,2}; x ∈ D} such that the weight of E at x ∈ D is α (x)
x 0 ̟(1)x
and the weight of E at x ∈ D is α (x).
2 ̟(2)x
Proposition 4. The Morse index for the critical submanifolds of type (1,1,1) depends
on d and ̟, and it is given by
0
(10) λ (d ,̟) = 2g −2+2(2d −∆+b)+2(s−v))
N(1,1,1) 0 0
where v = ♯{x ∈ D; α (x) > α (x)}, and s is the number of marked points.
̟x(1) ̟x(2)
Proof. By proposition 3 the Morse index equals the dimension of H1(C•) where C• is
1 1
the complex
ParHom(E ,E ) → 0.
0 2
UsingthelongexactsequenceforthiscomplexwegetH0(C•) = 0sinceitisisomorphic
1
to H0(ParHom(E ,E )) and, the last is equal to zero since its degree is less than zero.
0 2
Hence,
1
λ = dimT U = dimH1(C•)
2 N(1,1,1) E <0 1
= dimH1(ParHom(E ,E )) = −χ(ParHom(E ,E ))
0 2 0 2
= −deg(ParHom(E ,E ))−rk(ParHom(E ,E ))(1−g)
0 2 0 2
= d −d +s− dimParHom(E ,E ) +g −1.
0 2 0 2 x
x∈D
X
Hence, λ = 2g−2+2(2d +b−∆)+2(s−v), where v = ♯{x ∈ D;α (x) ≤
N(1,1,1) 0 ̟x(1)
α (x)}. (cid:3)
̟x(2)
Remark 5. The Proposition above proves also that λ depend only on d and ̟,
N(1,1,1) 0
the data that give us how splits V into E and E . So we may decompose N(1,1,1) =
0 2
N(d ,̟).
d0,̟ 0
S
BETTI NUMBERS OF PARABOLIC U(2,1)-HIGGS BUNDLES MODULI SPACES. 7
From now on we denote
v = ♯{x ∈ D; α (x) < η(x)}
1 ̟(1)
v = ♯{x ∈ D; η(x) < α (x)}.
2 ̟(2)
Proposition 6. Assume τ < 0, let N be the union of critical submanifolds of type
(1,1,1)
(1,1,1) parametrized by d and ̟, i.e. N = N(d ,̟). The map
0 (1,1,1) d0,̟ 0
N(d ,̟) → JacdS0 X ×Sm1X ×Sm2X
0
(E ⊕E ⊕E ,Φ ,Φ ) 7→ (E ,div(Φ ),div(Φ ))
0 1 2 0 1 0 0 1
where
m = deg(SParHom(E ,E )⊗K(D)) = b−d +2g −2+v
1 0 1 0 1
m = deg(SParHom(E ,E )⊗K(D)) = ∆−d −2b+2g −2+v
2 1 2 0 2
is an isomorphism, in particular there is only one component for fixed d and ̟. Fur-
0
¯
thermore, d the degree of E is lower bounded by d , that is,
0 0 0
1
¯
(11) d ≥ d = ∆+ (η(x)+α (x)−2α (x)) +1
0 0 3 ̟x(2) ̟x(1)
" ! #
x∈D
X
where [k] denote the entire part of k.
Proof. The isomorphism is obvious (see [GGM]). The stability condition on E applied
on the subbundles E and E ⊕E , together with the formula d = ∆−b−d gives the
2 1 2 2 0
following two bounds for d :
0
(12) 2∆−3b− (α (x)+η(x)−2α (x)) < 3d
̟x(1) ̟x(2) 0
x∈D
X
(13) ∆− (2α (x)−η(x)−α (x)) < 3d .
̟x(1) ̟x(2) 0
x∈D
X
To determine which is the appropriate bound we subtract these two inequalities. This
¯
subtraction gives a multiple of τ , hence d depends on wether τ is negative or positive.
0
(cid:3)
Remark 7. The condition on the weights being generic implies that τ can not be zero.
This is because τ = 0 implies that 2η(x)−α (x)−α (x) = ∆−3b, and if that happens
1 2
then there is a U(2,1)-parabolic Higgs subbundle (V,Φ = 0) non-stable but semistable.
¯
Remark 8. Note that the values m and m depend on ̟ and d .
1 2 x 0
Remark 9. We chose τ < 0 for computability reasons.
Denote ̟ = {̟ } .
x x∈D
Theorem 10. The Poincar´e polynomial of the critical submanifold N(d ,̟) is
0
(1+xt)2g (1+yt)2g
P (N(d ,̟)) = (1+t)2gCoeff ·
t 0 x0y0 (1−x)(1−xt2)xm1 (1−y)(1−yt2)ym2
(cid:18) (cid:19)
where m and m are the same as in Proposition 6.
1 2
8 MARINALOGARES
Proof. Use Macdonald’s formula for the Poincar´e polynomial of the symetric product
(cid:3)
(see [M]).
Now, in order to get the contribution of all the subvarieties of type (1,1,1) in P (U)
t
¯
we have to sum over all d ≥ d and all possibilities of ̟.
0 0
Pt(N(1,1,1)) = tλN(d0,̟)Pt(N(d0,̟))
dX0,̟
(1+xt)2g (1+yt)2g
= t2g−2+2(b−∆)+4d0+2(s−v)Coeff ·
x0y0 (1−x)(1−xt2)xm1 (1−y)(1−yt2)ym2
dX0,̟(cid:18) (cid:18) (cid:19)(cid:19)
t2g−2+2b−2∆+2s(1+xt)2g(1+yt)2g t4d¯0xd¯0yd¯0
= Coeff ·
x0y0 (1−x)(1−xt2)xb+2g−2(1−y)(1−yt2)y∆−2b+2g−2 t2vxv1yv2
!
̟
X
t2g−2+2b−2∆+2s(1+xt)2g(1+yt)2g t4d¯0xd¯0yd¯0
= Coeff ·
x0y0 (1−x)(1−xt2)xb+2g−2(1−y)(1−yt2)y∆−2b+2g−2 t2vxv1yv2
!
̟
X
Thus, we have to compute the following sum
t4d¯0xd¯0yd¯0
(14) .
t2vxv1yv2
X̟x
The variables depend also on the weights α (x), α (x) and η(x), and the distribution
1 2
functions ̟ .
x
4. Computations for one puncture for N .
(1,1,1)
From now on we consider the case of one puncture to get more explicit formulas, so
we denote α = α (x) for i = 1,2 and η = η(x). We abbreviate ̟ to ̟.
i i x
We have to consider the following cases for the possible distributions of the weights,
Table 1. Weight distributions.
S η < α < α S α < η < α
1 1 2 2 1 2
S (a) α −α > α −η S η = α < α
1 2 1 1 4 1 2
S (b) α −α < α −η S α < α = η
1 2 1 1 5 1 2
S α < α < η S α = α = η
3 1 2 6 1 2
S (a) α −η > α −α S η < α = α
3 2 2 1 7 1 2
S (b) α −η < α −α S α = α < η
3 2 2 1 8 1 2
Theorem 11. The contributions to the Poincar´e polynomial of the union of the subva-
rieties of type (1,1,1) when τ < 0 and one marked point are classified by the possibilities
for the distribution of the weights of E shown in Table 1. They are the following,
BETTI NUMBERS OF PARABOLIC U(2,1)-HIGGS BUNDLES MODULI SPACES. 9
(i) For S (a):
1
t2b−2∆+2gx2−b+∆−2g(1+tx)2gy1+2b−2∆−2g(1+ty)2g (1+t2xy)
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(ii) For S (b) :
1
t−2+2b−2∆+2g (1+t2) x2−b+∆−2g(1+tx)2gy1+2b−2∆−2g(1+ty)2g
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(iii) For S :
2
t2b−2∆+2g (1+t2) x2−b+∆−2g(1+tx)2gy2+2b−2∆−2g(1+ty)2g
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(iv) For S (a) :
3
t2+2b−2∆+2g (1+t2) x2−b+∆−2g(1+tx)2gy3+2b−2∆−2g(1+ty)2g
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(v) For S (b):
3
t2b−2∆+2gx1−b+∆−2g(1+tx)2gy2+2b−2∆−2g(1+ty)2g (1+t2xy)
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(vi) For S :
4
t2b−2∆+2gx2−b+∆−2g(1+tx)2g (1+t2x) y2+2b−2∆−2g(1+ty)2g
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(vii) For S :
5
t2b−2∆+2gx2−b+∆−2g(1+tx)2gy2+2b−2∆−2g(1+ty)2g (1+t2y)
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(viii) For S :
6
2t2+2b−2∆+2gx3−b+∆−2g(1+tx)2gy3+2b−2∆−2g(1+ty)2g
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(ix) For S :
7
2t−2+2b−2∆+2gx2−b+∆−2g(1+tx)2gy1+2b−2∆−2g(1+ty)2g
3 3 3
Coeff .
x0y0
(−1+x) (−1+t2x) (−1+y) (−1+t2y)
!
(x) For S :
8
2t−2+2b+4(1+∆3)−2∆+2gx2−b+∆3−2g(1+tx)2gy3+2b−23∆−2g(1+ty)2g
Coeff .
x0y0
(1−x) (1−t2x) (1−y) (1−t2y)
!
10 MARINALOGARES
¯
Proof. Compute the values of d , v . v and v for each possible distribution of the
0 1 2
weights, then we obtain the values for the sum in (14) for each case S .
i
¯
The value for d depends on the distribution of the weights and is, in the case ̟ = Id,
0
d¯ = ∆ +1 for all S except for S (b) and S where d¯ = ∆. When ̟ 6= Id, d¯ = ∆ for
0 3 i 1 7 0 3 0 3
all S except for S (a), S and S where it is d¯ = ∆ +1.
i 3 6 8 0 3
(cid:3)
5. Poincar´e Polynomial for critical subvarieties of type (1,2).
Following our previous discussion the critical subvarieties of type (1,2) and (2,1) can
be identified with the moduli of (2g −2)-stable parabolic triples of type (2,1,a+4g −
4,b;α ,α ,η) and (1,2,b + 2g − 2,a;α ,α ,η) respectively. So we recall the basics of
1 2 1 2
parabolic triples from [GGM].
¿From Proposition 1 we restrict to the case when τ > 0, note that by definition the
Morse function f forces γ = 0 when τ > 0. Hence, for our analysis we only have to
consider the critical subvarieties of type (1,2), that is (2g − 2)-stable parabolic triples
of type (2,1,a+4g −4,b;α ,α ,η).
1 2
A parabolic triple is a holomorphic triple T = (T ,T ,φ) where T and T are parabolic
1 2 1 2
bundles over X, and φ : T → T (D) is a strongly parabolic homomorphism, i.e. an
2 1
element φ ∈ H0(SParHom(T ,T (D))). We callparabolic system of weights for thetriple
2 1
(T,φ)tothevectorα = (α1,α2)whereαi isthesystem ofweights ofT withi = 1,2. The
i
type ofaparabolictripleisan-tuple(r1,r2,d1,d2;α1(x),...,αr(x)(x),η1(x),...,ηr′(x)(x),
where r = rk(T ), d = deg(T ), α is the parabolic system of weights of T and η is the
i i i i 1
parabolic system of weights of T .
2
A parabolic triple T′ = (T′,T′,φ′) is a parabolic subtriple of T = (T ,T ,φ) if T′ ⊂ T
1 2 1 2 i i
are parabolic subbundles for i = 1,2 and φ′(T′) ⊂ T′(D) where φ′ is the restriction of φ
2 1
to T′.
2
For any σ ∈ R the σ-parabolic degree of T is defined to be
pardeg (T) = pardeg(T )+pardeg(T )+σrk(T ).
σ 1 2 2
In the following we denote r = rk(T ) and r = rk(T ). Thus we have a notion of
1 1 2 2
stability for a fixed parameter. Let σ be a real number. We define the σ-slope of a triple
(T ,T ,φ) as
1 2
pardeg T +pardeg T r
1 2 2
(15) parµ (T) = +σ .
σ r +r r +r
1 2 1 2
T is called σ-stable (resp. σ-semistable) if for any non-zero proper subtriple T′ we have
parµ (T′) < parµ (T) (resp.≤).
σ σ
Proposition 12. Subvarieties of type (1,2) and type (2,1) correspond with σ-stable
triples for σ = 2g −2.
