Variations of Hodge Structures
of Rank Three k-Higgs Bundles
and Moduli Spaces of Holomorphic Triples
August 28, 2020
Ronald A. Zúñiga-Rojas1
Centro de Investigaciones Matemáticas y Metamatemáticas CIMM
Escuela de Matemática, Universidad de Costa Rica,
San José 11501, Costa Rica
e-mail: ronald.zunigarojas@ucr.ac.cr
ORCID: 0000-0003-3402-2526
Abstract
There is an isomorphism between the moduli spaces of σ-stable holomorphic triples
and some of the critical submanifolds of the moduli space of k-Higgs bundles of
rank three, whose elements (E,ϕk) correspond to variations of Hodge structure,
VHS. There are special embeddings on the moduli spaces of k-Higgs bundles of
rank three. The main objective here is to study the cohomology of the critical
submanifolds of such moduli spaces, extending those embeddings to moduli spaces
of holomorphic triples.
Keywords: Higgs bundles, holomorphic triples, moduli spaces, variations of Hodge structure.
MSC classes: Primary 14F45; Secondaries 14D07, 14H60.
Introduction
Consider a compact connected Riemann surface X of genus g > 2. Algebraically, X
is a complete irreducible non-singular curve over C. Let N = N (r, d) be the moduli
space of polystable vector bundles of rank r and degree d over X. In this paper, we
consider the co-prime condition GCD(r, d) = 1, which ensures that polystable implies
stable. This space has been widely worked by Atiyah & Bott [2], Desale & Ramanan [8],
Earl & Kirwan [9], among other authors. Here, we consider it as the corresponding
minimal critical submanifold of M(r, d), the moduli space of polystable Higgs bundles.
A Higgs bundle over X is a pair (E,ϕ) where E → X is a holomorphic vector bundle
and ϕ : E → E ⊗ K is an endomorphism twisted by the cotangent bundle K = T ∗X.
Fixing rank r and degree d of the underlying vector bundle E, the isomorphism classes
of polystable Higgs bundles are parametrized by a quasiprojective variety: the moduli
space of polystable Higgs bundles Mps(r, d). Again, since GCD(r, d) = 1, polystability
implies stability and then, the spaceMps(r, d) =Mss(r, d) =Ms(r, d) becomes a smooth
projective variety. These spaces were first worked by Hitchin [20] and Simpson [27]. Since
then, they have been around for more than thirty years, and have been studied extensively
by a number of authors: e.g. [7, 15, 16, 17, 18, 19, 28].
1 Supported by Universidad de Costa Rica through Escuela de Matemática, specifically through CIMM (Centro de
Investigaciones Matemáticas y Metamatemáticas), Project 820-B8-224. This work is partly based on the Ph.D. Project [29]
called “Homotopy Groups of the Moduli Space of Higgs Bundles”, supported by FEDER through Programa Operacional
Factores de Competitividade-COMPETE, and also supported by FCT (Fundação para a Ciência e a Tecnologia) through
the projects PTDC/MAT-GEO/0675/2012 and PEst-C/MAT/UI0144/2013 with grant reference SFRH/BD/51174/2010.
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Higgs bundles are an interesting topic of research because they have links with many
other areas of mahtematics such as integrable systems, mirror symmetry, Langlands pro-
gramme, Hodge theory, among others. We are interested on their link to Hodge theory.
The work of Simpson [26, 27], Hausel [14], and Hausel & Thaddeus [18, 19] shall be
particularly useful for our purposes.
There is a Morse function f : Mk(3, d)→ R defined∫by
f(E,ϕ) = 12 ‖ϕ‖
2 = i tr(ϕϕ∗
π L2 2 )π X
applied to the moduli spaces of stable k-Higgs bundlesMk(r, d). We study the stabiliza-
tion of the cohomology groups of the critical submanifolds from this Morse function f , for
the case of rank r = 3. The co-prime condition (3, d) = 1 implies that the moduli space
Mk(3, d) is smooth. A k-Higgs bundle or Higgs bundle with poles of order k, (E,ϕk),
is a Higgs bundle where the morphism ϕk is twisted by Lp k-times, where p ∈ X is an
arbitrary fixed point and Lp = OX(p) is its associated line bundle (local structure sheaf):
ϕk : E → E ⊗K ⊗ L⊗kp = E ⊗K(k · p).
According to Simpson [26] the critical points of f , are variations of the Hodge structure
(VHS), a decomposition of the form:
⊕n
E = Ej such that ϕ : Ej → Ej+1 ⊗K for 1 6 j 6 n− 1. (0.1)
j=1
for general rank r. In our case, for Mk(3, d), there are three kind of variations of Hodge
structure:
i. (1, 2)-VHS: ( ( )0 0 )
E ⊕ E , k k1 2 φ 0 ∈ Fd1 ⊆M (3, d).
ii. (2, 1)-VHS: ( ( )0 0 )
E k k2 ⊕ E1, 0 ∈ Fφ d2 ⊆M (3, d).
iii. (1, 1, 1)-VHS:
(
⊕ ⊕ 
 
 0 0 0 )L1 L2 L3, φ21 0 0  ∈ F k km1m2 ⊆M (3, d),
0 φ32 0
Here, F k k kd1 , Fd2 and Fm1m2 denote the respective critical submanifolds of the moduli space
Mk(3, d). The first two, F kd1 and F
k
d2 , are close related to the space Nσ(r1, r2, d1, d2), the
moduli space of σ-stable holomorphic triples of type (r,d) = (r1, r2, d1, d2).
A holomorphic triple T = (E1, E2, φ) onX consists of a pair of holomorphic vector bundles
E1 → X and E2 → X, of ranks r1, r2 and degrees d1, d2 respectively, and a holomorphic
map φ : E2 → E1. The stability for triples depends on a parameter σ ∈ R, which gives
a collection of moduli spaces Nσ(r1, r2, d1, d2) widely worked by several authors: e.g.
[4, 5, 6, 11, 22, 23]. The range of σ is an interval [σm, σM ] ⊆ R split by a finite number
2
Ronald A. Zúñiga-Rojas
of critical values σc. The reader may see Bradlow, Garćıa-Prada, Gothen [5], Muñoz,
Oliveira, Sánchez [22], or Muñoz, Ortega, Vásquez-Gallo [23] for the interval details.
This paper works with a very particular framework. We study holomorphic triples on
X of the form T = (Ẽ1, Ẽ2, φ) with type (2, 1, d̃1, d̃2), where ranks r1 = 2, r2 = 1 and
degrees d̃1, d̃2 are in terms of (1, 2)-VHS described before:
( ( )
⊕ 0 0
)
E E k k1 2, ∈ F ⊆M (3, d),φ 0 d1
where Ẽ1 = E2 ⊗ K(kp), Ẽ2 = E1, φ : E1 → E2 ⊗ K(kp), and so the degrees become
d̃1 = deg(Ẽ1) = d2 + 2(2g − 2 + k), d̃2 = d1.
We study as well the holomorphic triples T = (Ẽ1, Ẽ2, φ) with type (1, 2, d̃1, d̃2), related
to (2, 1)-VHS of the form
( ( )0 0 )
E ⊕ E , ∈ F k k2 1 0 d2 ⊆M (3, d),φ
where, in this case Ẽ1 = E1 ⊗ K(kp), Ẽ2 = E2, φ : E2 → E1 ⊗ K(kp), and the degrees
become d̃1 = deg(Ẽ1) = d1 + 2g − 2 + k, d̃2 = d2.
Finally, we also study (1, 1, 1)-VHS
(  0 0 0 )
L1 ⊕ L2 ⊕ L , φ 0 0  ∈ F k k3 21 m1m2 ⊆M (3, d),
0 φ32 0
and those are related to symmetric products of the form
Symm1(X)× Symm2(X)× J d3(X)
where Jd3(X) is the Jacobian of X, the moduli space of stable line bundles of degree d3,
and m1,m2 will be described below as the corresponding degrees of auxiliar bundles.
Our estimates are based on embeddings Mk(3, d) ↪→Mk+1(3, d) defined by
[ ] [ ]
i : (E,ϕkk ) −7 → (E,ϕk ⊗ sp)
where 0 6= sp ∈ H0(X,Lp) is a nonzero fixed section of Lp = OX(p).
The paper is organized as follows: in section 1 we recall some basic facts about holomor-
phic triples, Higgs bundles and k-Higgs bundles; in section 2, we present the effect of the
embeddings on σ-stable triples; in subsection 2.1, we show that the embeddings preserve
σ-stability, in subsection 2.2, we discuss the effect of the embeddings considering the flip
loci, and present an original result, the so-called “Roof Theorem” :
Theorem 0.1 (Theorem 2.6). There exists an embedding
ĩk : Ñσc(k) ↪→ Ñσc(k+1)
3
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
such that the following diagram commutes:
Ñσc LL(k+1)
 
Nσ−(k+1) Nσ+c LL c LL (k+1)
∃ĩk
ik
Ñ ikσc(k)
 
Nσ− N +c (k) σc (k)
where Ñσc(k) is the blow-up of Nσ−(k) = Nσ−(k)(2, 1, d̃1, d̃2) along the flip locus Sσ−(k) and,c c c
at the same time, represents the blow-up of Nσ+(k) = Nσ+(k)(2, 1, d̃1, d̃2) along the flipc c
locus Sσ+c (k).
Here, σc(k) ∈]σm, σM [ is a σ-critical value depending on the parameter k, that lies in the
interval mentioned above, where σm = µ1 − µ2 = d̃1/2− d̃2 and σM = 4σm.
In section 3 we present the cohomology main results for triples: in subsection 3.1 ap-
pear some useful results about the cohomology of the symmetric product Symk(X). In
subsection 3.2 we present the stabilization of the cohomology (Theorem 3.9) for certain
indices:
Theorem 0.2 (Theorem 3.9). There is an isomorphism
ĩ∗
∼
k : Hj(Ñ
= j
σc(k+1),Z) −−{−−→ H (Ñσc(k),Z) ∀(j 6 n(k) ) }
at the blow-up level, where n(k) = min d̃1 − dM − d̃2 − 1, 2 d̃1 − 2d̃2 − (2g− 2) + 1 .
And hence, the cohomology stabilization of the moduli spaces of triples:
Corollary 0.3 (Corollary 3.11). There is an isomorphism
∗ j ∼ik : H (Nσc(k+1),Z) −−−
=−→ Hj(Nσc(k),Z) ∀j 6 n(k)
where n(k) as above.
In subsection 3.3 we show the stabilization of the (1, 2)-VHS cohomology using the iso-
morphisms between them and the moduli spaces of triples:
Corollary 0.4 (Corollary 3.16). There is an isomorphism
Hj(F k+1
∼
d1 ,Z) −−−
=−→ Hj(F kd1 ,Z)
for all j 6 σH(k)− 2(µ1 − µ)− 1.
4
Ronald A. Zúñiga-Rojas
Here, σH(k) ∈]σm, σM [ is a particular σ(-critical )value depending on the parameter k:
σH(k) = deg K(k · p) = 2g − 2 + k.
In subsection 3.4 we show the analogous dual result for (2, 1)-VHS:
Corollary 0.5 (Corollary 3.19). For k large enough, there is an isomorphism
Hj(F k+1
∼
d2 ,Z) −−−
=−→ Hj(F kd2 ,Z)
for all j 6 σH(k)− 4(µ2 − µ)− 1.
Finally, in subsection 3.5 we described the cohomology for (1, 1, 1)-VHS and its relation-
ship with the spaces Symm1(X)× Symm2(X)× J d3(X):
Corollary 0.6 (Corollary 3.21). There is an isomorphism
Hj(F∞
∼
m1m2 ,Z) −−−
=−→ Hj(F k
( ) m1m2
,Z)
for all j 6 min m̄1 + k, m̄2 + k − 1.
1 Preliminary definitions
Let X be a compact connected Riemann surface of genus g > 2 and let K = T ∗X be
the canonical line bundle of X. Note that, algebraically, X is also a nonsingular complex
projective algebraic curve.
The k-th symmetric product Symk(X) is a smooth projective variety of dimension k ∈ N,
that could be interpretated as the moduli space of degree k effective divisors. In other
words, Symk(X) = Xk/Sk, the symmetric product with quotient topology, is the quotient
of Xk the k-times cartesian product by the action of Sk the k-symmetric group. Obviously
Sym1(X) = X.
Definition 1.1. For a (smooth or holomorphic) vector bundle E → X, we denote the
rank of E by rk(E) = r and the degree of E by deg(E) = d. Its slope is defined to be
( ) = deg(E) dµ E rk( ) = . (1.1)E r
A vector bundle E → X is called semistable if µ(F ) 6 µ(E) for any nonzero F ⊆ E.
Similarly, a vector bundle E → X is called stable if µ(F ) < µ(E) for any nonzero F ( E.
Finally, E is called polystable if it is the direct sum of stable subbundles, all of the same
slope.
1.1 Holomorphic Triples
Definition 1.2. A holo(morphic triple) on X is a triple T = (E1, E2, φ) consisting of twoholomorphic vector bundles E1 → X and E2 → X and a homomorphism φ : E2 → E1,
i.e. an element φ ∈ H0 Hom(E2, E1) .
5
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Definition 1.3. A homomorphism from a triple T ′ = (E ′ ′1, E2, φ′) to another triple T =
(E1, E2, φ) is a commutative diagram of the form:
′
E ′
φ // ′
2 E1
 φ 
E2 // E1
where the vertical arrows represent holomorphic maps.
Definition 1.4. A triple T ′ = (E ′1, E ′ , φ′2 ) is a subtriple of T = (E1, E2, φ) if
i. E ′j ⊆ Ej is a coherent subsheaf for j = 1, 2
ii. φ′ = φ| ′
E′
, i.e. φ is the restriction of φ.
2
In other words, we get the commutative diagram
′
E ′
φ //
2 E
′
1
 φ 
E2 // E1
where the vertical arrows are injective inclusions this time. In such a case, we denote
T ′ ⊆ T .
If E ′ = E ′1 2 = 0 we call the subtriple T ′ ⊆ T as the trivial subtriple.
T ′ is a non-trivial proper subtriple if 0 6= T ′ ( T .
Remark 1.5. For stability criteria, it will be enough to consider saturated subsheaves. In
our case, since X is a Riemann surface, saturated subsheaves are precisely subbundles.
Definition⊕1.6. A triple⊕T = (E1, E2, φ) ⊕is reducible if there are direct sum decomposi-n n n
tions E1 = E1i, E2 = E2i, and φ = φi such that φi ∈ Hom(E2i, E1i). In such a
i=1 i=1 i=1
case, T has a direct sum decomposition
⊕n
T = Ti of subtriples Ti = (E1i, E2i, φi).
i=1
If T = (E1, E2, φ) is not reducible, we say that T is irreducible.
Remark 1.7. We adopt Bradlow and Garćıa-Prada [4] convention that if E2i = 0 or
E1i = 0 for some i, then φi is the zero map.
Definition 1.8. σ-Stability, σ-Semistability and σ-Polystability:
i. For any σ ∈ R, the σ-degree and the σ-slope of T = (E1, E2, φ) are defined as:
degσ(T ) = deg(E1) + deg(E2) + σ · rk(E2),
and
µ ( degT ) = σ(T ) rk(E2)σ rk( ) + rk( ) = µ(E1 ⊕ E2) + σE1 E2 rk(E1) + rk(E2)
respectively.
6
Ronald A. Zúñiga-Rojas
ii. T is then called σ-stable [respectively, σ-semistable] if µ (T ′σ ) < µσ(T ) [respectively,
µσ(T ′) 6 µσ(T )] for any proper subtriple 0 6= T ′ ( T .
iii. A triple is called σ-polystable if it is the direct sum of σ-stable triples of the same
σ-slope.
Now we may use the following notation for moduli spaces of triples:
i. Denote r = (r1, r2) and d = (d1, d2), and then regard
Nσ = Nσ(r,d) = Nσ(r1, r2, d1, d2)
as the moduli space of σ-polystable triples T = (E1, E2, φ) such that rk(Ej) = rj and
deg(Ej) = dj.
ii. Denote by N s sσ = Nσ(r,d) the open subspace of σ-stable triples.
iii. Call (r,d) = (r1, r2, d1, d2) the type of the triple T = (E1, E2, φ).
The moduli space of σ-stable triples N sσ = N sσ(r,d) = N sσ(r1, r2, d1, d2) is formally con-
structed by Bradlow and Garćıa-Prada [4] using dimensional reduction. There is also a
direct construction by Schmitt [25] using Geometric Invariance Theory (GIT). The reader
also may consult the work of Bradlow, Garćıa-Prada and Gothen [5]; Muñoz, Oliveira
and Sánchez [22]; or Muñoz, Ortega and Vázquez-Gallo [23] for the following details.
There are certain necessary conditions on σ for σ-stable triples to exist. For triples of
type (r,d) = (r1, r2, d1, d2), consider the slopes µj = dj for j = 1, 2 and definerj
σm = µ1 − µ2, (1.2)
and ( )
σM = 1 +
r1 + r2 (µ1 − µ2), for r1 =6 r| − | 2, (1.3)r1 r2
Theorem 1.9 ([4, Th. 6.1.]). The moduli space of σ-stable triples N sσ(r1, r2, d1, d2) is
a complex analytic variety, which is projective when σ ∈ Q. A necessary condition for
N sσ(r1, r2, d1, d2) to be non-empty is
0 6 σm 6 σ 6 σM , if r1 6= r2,
0 6 σm 6 σ, if r1 = r2.
Remark 1.10. If µ1 = µ2 and r1 6= r2 then σm = σM = 0 and so, N sσ(r1, r2, d1, d2) is
empty unless σ = 0.
Proposition 1.11 ([5, Prop. 2.4.]). The σ-stability of T = (E1, E2, φ) is equivalent to
the σ-stability of the dual triple T ∗ = (E∗, E∗2 1 , φ∗), where φ∗ represents the conjugate
transpose of φ. The map T 7→ T ∗ defines an isomorphism
N s(r , r , d , d ) ∼= N sσ 1 2 1 2 σ(r2, r1,−d2,−d1).
The last result is frequently used to restrict the study of triples to r1 > r2 and appeal
to duality when r1 < r2. We shall use this duality result later to study and compare the
cohomology of (1, 2)-VHS and (2, 1)-VHS.
7
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Definition 1.12. For triples of type (r1, r2, d1, d2), the number σ ∈ [σm,∞[ is a critical
value if there exist integers r′ , r′ , d′1 2 1 and d′2 such that
(r ′ ′ ′ ′= 1 + r2)(d1 + d2)− (r1 + r2)(d1 + d2)σ
r′1r
′
2 − r1r2
or equivalently
d1 + d2 + σ · r
′ ′ ′
2 d1 + d2 σ · r2
r + r r + =r r′ ′ + ′ ′1 2 1 2 1 + r2 r1 + r2
with 0 6 r′ 6 r , (r′ , r′ , d′ , d′ ) =6 (r , r , d , d ), (r , r ) =6 (0, 0) and r′ r =6 r r′j j 1 2 1 2 1 2 1 2 1 2 1 2 1 2.
We denote σ = σc if it is critical.
The number σ ∈ [σm,∞[ is called generic if it is not critical.
Proposition 1.13 ([5, Prop. 2.6.]). Fix the type (r1, r2, d1, d2).
i. The critical values σc form a discrete subset of the interval [σm,∞[.
ii. If r1 6= r2 the critical values σc are finite and lie in the interval [σm, σM ].
iii. The stability criteria for two values of σ lying between two consecutive critical values
are equivalent; thus, the moduli spaces are isomorphic.
iv. If σ is generic and GCD(r2, r1 + r2, d1 + d2) = 1, then σ-semistability is equivalent
to σ-stability.
1.2 Higgs Bundles
Definition 1.14. A Higgs bundle over X is a pair (E,ϕ) where E → X is a holomorphic
vector bundle and ϕ : E → E⊗K is an endomorphism of E twisted by K, which is called
a Higgs field. Note that ϕ ∈ H0(X; End(E)⊗K).
Definition 1.15. A subbundle F ⊆ E is said to be ϕ-invariant if ϕ(F ) ⊆ F ⊗ K. A
Higgs bundle is said to be semistable [respectively, stable] if µ(F ) 6 µ(E) [respectively,
µ(F ) < µ(E)] for any nonzero ϕ-invariant subbundle F ⊆ E [respectively, F ( E].
Finally, (E,ϕ) is called polystable if it is the direct sum of stable ϕ-invariant subbundles,
all of the same slope.
Fixing the rank rk(E) = r and the degree deg(E) = d of a Higgs bundle (E,ϕ), the
isomorphism classes of polystable bundles are parametrized by a quasi-projective variety:
the moduli space M(r, d). Constructions of this space can be found in the work of
Hitchin [20], using gauge theory, or in the work of Nitsure [24], using algebraic geometry
methods.
Hitchin [20] works with the Yang-Mills self-duality equations (SDE) FA + [ϕ, ϕ∗] = 0 (1.4)
( ) ∂̄Aϕ = 0,
where ϕ ∈ Ω1,0 X,End(E) is a complex auxiliary field and FA is the curvature of a
connection dA which is compatible with ∂̄A, the holomorphic structure of the bundle E =
(E , ∂̄A), where E is a smooth complex bundle of rank rk(E) = 2 and degree deg(E) = 1.
8
Ronald A. Zúñiga-Rojas
Hitchin calls ϕ Higgs field, because it shares a lot of the physical and gauge properties of
those of the Higgs boson. Here, ϕ∗ denotes the adjoint of ϕ with respect to the hermitian
metric on E,2 and [·, ·] denotes the extension of the Lie bracket to Lie algebra-valued
forms.
The set of solutions ( )
β(E) = {(∂̄A, ϕ)| solution of (1.4)} ⊆ A0,1(E)× Ω1,0 X(,End(E) )
where A0,1(E) denotes the space of holomorphic structures on E , Ω1,0 X,End(E) denotes
(1, 0)-forms of X with values on End(E), and the collection
βps(2, 1) = {β(E)|E polystable, rk(E) = 2, deg(E) = 1},
allow Hitchin to construct the Moduli space of solutions to SDE (1.4)
MYM(2, 1) = β (2, 1)/GCps ,
and
MYMs (2, 1) = βs(2, 1)/GC ⊆MYM(2, 1),
the moduli space of stable solutions to SDE (1.4), where GC represents the complex gauge
group, which acts by conjugation on βps(2, 1) and βs(2, 1).
Remark 1.16. Since GCD(2, 1) = 1, then βps(2, 1) = β(s2, 1) and so
MYM(r, d) =MYMs (r, d).
Using definition 1.14, and the notion of stability 1.15, Hitchin [20] presents an alternative
algebro-geometric construction of the moduli space of polystable Higgs bundles:
MH(2, 1) = {(E,ϕ)| E polystable }/GC
and the subspace
MHs (2, 1) = {(E,ϕ)| E stable }/GC ⊆MH(2, 1),
of stable Higgs bundles.
Remark 1.17. Again, GCD(2, 1) = 1 implies MH(2, 1) =MHs (2, 1).
Finally, Hitchin [20] concludes:
Theorem 1.18. [20] There is a homeomorphism of topological spaces
MH(2, 1) ∼=MYM(2, 1). 
Because of the last homeomorphism, from now on it will be enough to denoteM(2, 1) =
MH(2, 1) ∼=MYM(2, 1) for brief. Hitchin [20] computes the real dimension of the moduli
space of stable rank two pairs (E,ϕ):
Theorem 1.19 ([20, Th. 5.8.]). Let X = Σg be a compact Riemann surface of genus
g > 1. The moduli space M(2, 1) of all stable pairs (E,ϕ), where E → X is a rank two
holomorphic vector bundle of degree one, and ϕ is a trace free holomorphic section of
End(E)⊗K, is a smooth real mani(fold of di)mension
dimR M(2, 1) = 12(g − 1).
2By Hitchin [20], there is a hermitian metric on E.
9
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Corollary 1.20. The space M(2, 1() is a qua)si–projective variety of complex dimension
dimC M(2, 1) = 3(2g − 2).
Nitsure [24] constructs the moduli space of Higgs bundles of general rank r and degree d
using Geometric Invariant Theory (GIT), and computes its dimension:
Theorem 1.21 ([24]). The space(M(r, d))is a quasi–projective variety of complex dimen-sion
dimC M(r, d) = (r2 − 1)(2g − 2).
Remark 1.22. Note that the result of Nitsure [24] coincides with the result of Hitchin [20]
for rank two Higgs bundles.
Simpson [27] calls the pair (E,ϕ) as Higgs bundle. His work contributes generalizing Higgs
bundles to higher dimensions and proving an analogous proposition to Theorem 1.19 for
general rank, with the same notion of stability in mind, considering the moduli space of
Higgs bundles as the quotient
MH(r, d) = {(E,ϕ)| E polystable }/GC
and the subspace
MHs (r, d) = {(E,ϕ)| E stable }/GC ⊆MH(r, d),
of stable Higgs bundles.
Remark 1.23. Once again, GCD(r, d) = 1 impliesMH(r, d) =MHs (r, d). See Simpson [27]
for details.
Theorem 1.24 ([27, Prop. 1.5]). There is a homeomorphism of topological spaces
MH(r, d) ∼=MYM(r, d). 
Also for general rank, we denote M(r, d) = MH(r, d) ∼= MYM(r, d) for brief, or even
M =M(r, d) when the rank r and the degree d are clear. Recall that we are considering
the co-prime case GCD(r, d) = 1, in order for M =M(r, d) to be a smooth variety. An
important feature of M(r, d) is that it carries an action of C∗: z · (E,ϕ) = (E, z · ϕ).
According to Hitchin [20], (M, I, Ω) is a Kähler manifold, where I is its complex structure
and Ω its corresponding Kähler form. Furthermore, C∗ acts onM biholomorphically with
respect to the complex structure I by the aforementioned action, where the Kähler form
Ω is invariant under the induced action eiθ · (E,ϕ) = (E, eiθ · ϕ) of the circle S1 ⊆ C∗.
Besides, this circle action is Hamiltonian, with proper moment map f : M→ R defined
by:
( 1 i
∫
f E, ϕ) = 2 ‖ϕ‖
2
L2 = 2 tr(ϕϕ
∗) (1.5)
π π X
where ϕ∗ is the adjoint of ϕ with respect to the hermitian metric on E, and f has finitely
many critical values.
There is another important fact mentioned by Hitchin [20] (see the original version in the
work of Frankel [10], and its application to Higgs bundles in [20]): the critical points of
f are exactly the fixed points of the circle action on M.
If (E,ϕ) = (E, eiθϕ) and ϕ = 0, then the critical value is c0 = 0. The corresponding
critical submanifold is F0 = f−1(c0) = f−1(0) = N , the moduli space of stable bundles
10
Ronald A. Zúñiga-Rojas
(see Hitchin [20], Simpson [27], or Bradlow, Garćıa-Prada, Gothen [6] for details). On
the other hand, when ϕ 6= 0, there is a type of algebraic structure for Higgs bundles
introduced by Simpson [26, 27]: a variation of Hodge structure, or simply a VHS, for a
Higgs bundle (E,ϕ) is a decomposition:
⊕n
E = Ej such that ϕ : Ej → Ej+1 ⊗K for 1 6 j 6 n− 1. (1.6)
j=1
It has been proved by Simpson [26] that the fixed points of the circle action on M(r, d),
and so, the critical points of f , are these variations of the Hodge structure VHS, where
the critical values cλ = f(E,ϕ) will depend on the degrees dj of the components Ej ⊆ E,
and λ denotes the index of the critical point for the Morse-Bott function f . By Morse
theory, we can stratifyM in such a way that there is a non-minimal critical submanifold
Fλ = f−1(cλ) for each nonzero critical value 0 6= cλ = f(E,ϕ) where (E,ϕ) represents(a fixed point of the)circle action, or equivalently, a VHS. We then say that (E,ϕ) is a
rk(E1), . . . , rk(En) -VHS.
The Morse indexes of the critical submanifolds of the moduli space of stable Higgs bundles
M(r, d) for general rank r were calculated by Bradlow et al. [6]:
Proposition 1.25 ([6, Prop. 3.10.]). Let (E,ϕ) be a stable Higgs bundle which cor-
responds to a critical point of f . Then the Morse index of the corresponding critical
submanifold (E,ϕ) ∈ Fλ is ∑ ( ( ))
index(Fλ) = 2 dim H1 C•k(E,ϕ)
k>0
where ( ( )) ( )
dim H1 C•k(E,ϕ) = −χ C•k(E,ϕ)
and C•k(E,ϕ) is the deformation complex of the pair (E,ϕ). 
Proposition 1.26 ([6, Prop. 3.12.(2)]). For M(r, d)
index(Fλ) > (r − 1)(2g − 2)
for every non-minimal critical submanifold Fλ ⊆M(r, d). 
Proposition 1.27 ([6, Prop. 3.14.]). Let F0 ⊆M(r, d) be the set of local minima. Then
F0 = {(E,ϕ) ∈M(r, d)| ϕ = 0} .
Hence, F0 coincides with N (r, d), the moduli space of semistable bundles of rank r and
degree d, which equals the subvariety N s(r, d) ⊆ N (r, d) corresponding to stable bundles
if GCD(r, d) = 1. 
1.3 k-Higgs Bundles
Definition 1.28. Fix a point p ∈ X, and let Lp = OX(p) be the associated line bundle to
the divisor p ∈ Sym1(X) = X. A k-Higgs bundle (or Higgs bundle with poles of order k)
is a pair (E,ϕk) where:
k
E −−ϕ→ E ⊗K ⊗ L⊗kp = E ⊗K(k · p)
11
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
( )
and where the morphism ϕk ∈ H0 X,End(E)⊗K(k ·p) is what we call a Higgs field with
poles of order k. The moduli space of k-Higgs bundles of rank r and degree d is denoted
by Mk(r, d). For simplicity, we will suppose that GCD(r, d) = 1, and so, Mk(r, d) will
be smooth.
Remark 1.29. So far, everything we have said for Higgs bundles and the moduli space
M(r, d) also hold for k-Higgs bundles and the moduli spaces Mk(r, d).
There is a new tool for k-Higgs bundles: an embedding of the form
i : Mkk (r, d)→Mk+1(r, d) : [(E,ϕk)] 7−→ [(E,ϕk ⊗ sp)] (1.7)
where 0 6= s 0p ∈ H (X,Lp) is a nonzero fixed section of Lp.
When the rank is r = 2 or r = 3, the map ik induces embeddings of the form
i
F kλ −−−
k−→ F k+1λ ∀λ,
for non-minimal3 critical submanifolds F kλ , where λ is the Morse index of the submanifold.
ForMk(2, 1), when r = 2, the Morse index is λ = 2(g+2d1−2)+k, which depends just on
the parameter d ∈]1 , g− 11 2 2 +
k
2 [∩Z since d = deg(E) = 1 (co-prime case GCD(r, d) = 1),
g > 2 is fixed, and k is the order of the pole.
Hence, we may index the (1, 1)-critical submanifolds as F kd1 , and the embeddings are well
defined:
F k ∼= Symd̄1+k(X) // Symd̄1+k+1d1 (X) ∼= F
k+1
d1
ik : D  // D + p
where D ∈ Symd̄1+k(X) is a divisor and d̄1 = 2g− 2d1− 1 for simplicity. The reader may
see Bento [3], Hausel [14], Hausel and Thaddeus [18, 19] or Hitchin [20] for details.
ForMk(3, d), when r = 3, we have three types of critical submanifolds. For (1, 2)-critical
submanifolds F kλ , the Morse index is given by λ = 2(3d1 − d + 2g − 2 + k) where once
again d1 = deg(E1) is the degree of the maximal destabilizing line bundle E1 ⊆ E, and
so, we are in a very similar situation than before. Without lost of generality, we may pick
the index d1 for the (1, 2)-critical submanifolds, and the embeddings become
( i : F k //k d1 ( )) F k+1d1 ( )0 0 ( 0 0 )(E,ϕk) = E1 ⊕ E2,  //k 0 (E,ϕk ⊗ sp) = E1 ⊕ Eφ 2, φk ⊗ sp 0
where φk : E1 → E2 ⊗ K(kp) and d < d < d3 1 3 + g − 1 +
k
2 as we shall see below (see
Bento [3], Gothen [12] or Z-R [29] for interval details). Moreover
(φk ⊗ sp)(E1) ⊆ φk(E1)⊗ Lp ⊆ E2 ⊗K ⊗ L⊗kp ⊗ Lp = E2 ⊗K ⊗ L⊗k+1p
and therefore
i (F k ) ⊆ F k+1k d1 d1 .
3For stable pairs in F k0 = Nk, the embeddings are trivial. Cf. Hausel [14, Ch. 3. Sec. 3.4.].
12
Ronald A. Zúñiga-Rojas
Lemma 1.30 ([3, Lema 2.3.1.]). Let (E,ϕk) ∈ F k(d1 be a k)-Higgs bundle of the form( )
(E,ϕk) = E1 ⊕
0 0
E2, φk 0 .
Hence, (E,ϕk) is stable if and only if (the holom)orphic triple T = (E ⊗K(k · p), E , φk2 1 )
is σH-stable where σH = σH(k) = deg K(k · p) = 2g − 2 + k.
Proof. The pair (E,ϕk) is stable if and only if the holomorphic chain
( ) C : E1 = E1 → E2 = E2 ⊗K(k · p)
is α = σH(k), 0 stable, which means that any proper subchain C ′ : E ′ → E ′1 2 has α-slope
µ (C ′α ) < µα(C); considering a subbundle E ′ ′ ′1 ⊆ E1 = E1 (with degr)ee deg(E∗ 1) = d1 and a
subbundle E ′2 ⊆ (E with degree )deg(E ′ ) = d′2 2 2, then E ′2 ⊗ K(k · p) ⊆ E2 is a subbundle
with degree deg E ′ ⊗K(k · p)∗2 = d′2 − r′2(2g − 2 + k), and we have
′ ′ ′
(E,ϕk) stable ⇐⇒ d1 + d2 − r2(2g − 2 + k) d1 + d2
r′1 +
<
r′2 3
where r′j = rk(E ′j). ( ( ))
On the othe(r hand, suppose ( k) = ⊕
0 0
E,ϕ) E1 E2, k 0 is stable. The holomorphicφ
(triple T = E2 ⊗ K(k ·)p), E , φk1 is σ-stable if and only if any proper subtriple T ′ =
E ′ ⊗K(k · p), E ′ , (φk)′2 1 satisfies µσ(T ′) < µσ(T )⇔( )
deg(E ′1) + deg E ′2 ⊗K(k · p) + rk(E ′1) · σ
r(k(E
′
1) + rk(E ′2)
<
)
deg(E1) + deg E2 ⊗K(k · p) + rk(E1) · σ
rk(E1) + rk(
⇔
E2)
d′1 + d′2 + r′ (2g − 2 + k) + r′2 1σ d1 + d2 + 2(2g − 2 + k) + σ
r′1 +
<
r′2 1 + 2
⇔ d
′ ′
1 + d2 + r′2σH(k) + r′1σ d1 + d2 + 2σH(k) + σ< .
r′ ′1 + r2 3
Since (E,ϕk) is stable, it is enough to take
r′2 · σ ′H(k) + r1 · σ 2 · σH(k) + σ ′ ′ ′ ′
r′1 + ′
=
r2 3
⇔ r2σH(k)− 2r1σH(k) = r2σ − 2r1σ
⇔ (r′2 − 2r′ ′ ′1)σH(k) = (r2 − 2r1)σ ⇔ σH(k) = σ
and so, the triple T = (E2 ⊗K(k · p), E1, φk) is σH(k)-stable.
Remark 1.31. Note that, at the last part of the proof above, the equality r′2 = 2r′1 does
not hold, because of the stability of (E,ϕk).
13
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
] [
Proposition 1.32 ([3, Proposição 2.3.2.]). For each d d d σH(k)1 ∈ 3 , 3 + 2 ∩ Z there is a
(1, 2) critical subman(ifold of M
k
((3, d) of{ )
t)he form }
F kd1 = (
0 0
E,ϕk) = E1 ⊕ E2, k 0 : dφ 1 = deg(E1), rk(E1) = 1, rk(E2) = 2 .
Furthermore, there is an isomorphism
F k ∼d1 = NσH(k)(2, 1, d− d1 + 2σH(k), d1)
with the moduli space of σH(k)-stable triples of this type.
Proof. The isomorphism is given by:
F kd ( ) // Nσ (k)(2, 1, d− d + 2σ (k), d )1 H 1 H 1
(E,ϕk) = ( 0 0E  k1 ⊕ E2, ) // (E ⊗K(k · p), E , φ )φk 0 2 1
where σH(k) = 2g − 2 + k as above.
In general, for the critical va(lues σc, we kno)w that the interval is σm 6 σc 6 σM where
deg E2 ⊗K(k · p)
= − = − deg(E1) = d− d1 + 2σH(k)σm µ2 µ1 − d1
r2 r1 2
and ( )
= 1 + r2 + r1
( )
σM (µ2 − µ1) = 4σm = 2 d− 3d1 + 2σH(k)|r2 − r1|
(see [5]). So, in particular we have
( ) = 2 d− d1 + 2σH(k) dσH k g − 2 + k > σm = 2 − d1 ⇐⇒ d1 > 3 .
On the other hand, we have
( )
( ) = 2 − 3 + 2 ( ) ⇐⇒ d + σH(k)σH k < σM d d1 σH k d1 < 3 2 .
Therefore, ] [
∈ d d + σH(k)d1 3 , 3 2 ∩ Z.
Remark 1.33. In general, for the critical values σc, the interval [σm, σM ] is closed. Nev-
ertheless, for our particular case of interest σm < σH(k) < σM , so the interval will be
open.
For (2, 1)-critical submanifolds F kλ = F kd2 , the Morse index is given by λ = 2(3d2 − 2d +
2g− 2 + k); here d2 = deg(E2) is the degree of the maximal destabilizing bundle E2 ⊆ E
of rank two this time, and so, we are in a very similar situation than before:
( i : F k ( )) // k+1k d2 Fd2 ( ( ))
(E,ϕk) = E2 ⊕
0 0 0 0
E1,
 // (E,ϕk ⊗ s ) = E ⊕ E ,
φk 0 p 2 1 φk ⊗ sp 0
14
Ronald A. Zúñiga-Rojas
with φk : E2 → E1⊗K(kp) and 2d 2d k3 < d2 < 3 +g−1+ 2 instead (see Bento [3], Gothen [12]
or Z-R [29]). Furthermore,
(φk ⊗ sp)(E2) ⊆ φk(E2)⊗ L ⊆ E ⊗K ⊗ L⊗k ⊗ L = E ⊗K ⊗ L⊗k+1p 1 p p 1 p
and hence
i (F k ) ⊆ F k+1k d2 d2 .
Lemma 1.34 ([3, Lema 2.3.5.]). Let (E,ϕk) ∈ F k
( (d2
be a k)-Higgs bundle of the form)
( k) = 0 0E,ϕ E2 ⊕ E1, .φk 0
Hence, (E,ϕk) is stable if and only if (the holom)orphic triple T = (E1 ⊗K(k · p), E , φk2 )
is σH-stable where σH = σH(k) = deg K(k · p) = 2g − 2 + k.
Proof. The proof is very similar to that presented for the (]1, 2)-case in Le[mma 1.30.
Proposition 1.35 ([3, Proposição 2.3.6.]). For each d ∈ 2d , 2d σH(k)2 3 3 + 2 ∩ Z there is a
(2, 1) critical subman(ifold of M
k
((3, d) of)t)he form{ }
F k k
0 0
d2 = (E,ϕ ) = E2 ⊕ E1, k 0 : d2 = deg(Eφ 2), rk(E2) = 2, rk(E1) = 1 .
Furthermore, there is an isomorphism
F k ∼d2 = Nσ (k)(1, 2, d− d2 + σH(k), dH 2)
with the moduli space of σH(k)-stable triples.
Proof. In this case, the isomorphism is given by:
F kd ( ) // Nσ (k)(1, 2, d− d2 + σH(k), d )2 H 2
(E,ϕk) = (E2 ⊕
0 0
E , 1 //k 0 ) (E1 ⊗K(k · p), E2, φ
k)
φ
The rest of the proof is very similar to the (1, 2)-case presented in Proposition 1.32.
Finally, we consider the (1,1, 1)-critical submanifolds of the form( 0 0 0 )
F kλ = F kd1d2d3 =  L1 ⊕ L2 ⊕  L3, φk k21 0 0   ⊆M (3, d),
0 φk32 0
where Lj ⊆ E is a line bundle for j = {1, 2, 3}, we denote dj = deg(Lj) and so, the
degree of E → X could be write as deg(E) = d = d1 + d2 + d3. Using the fact that
d3 = d − d1 − d2 and considering auxiliar bundles Mj = L∗j ⊗ Lj+1 ⊗ K(k · p) → X of
degree mj = deg(Mj) = dj+1 − dj + σH(k), we may write, for simplicity ϕ ∈ H0j (Mj)
where ϕk k k k1 = φ21 and ϕ2 = φ32, and hence Mj = O(Dj) where Dj = div(ϕj).
Note that ϕj =6 0⇒ mj > 0. Furthermore
d3 = d− d1 −
d+ 2m2 +m1 − 3σH(k)
d2 ⇐⇒ d3 = 3
15
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
and hence d+m1 + 2m2 = 0 mod 3.
Using all the above notation, we can re-writethe (1, 1, 1)-critical submanifolds as(  0 0 0 )F kλ = F km m =  L1 ⊕ L2 ⊕ L3, ϕk 0 01 2 1   ⊆Mk(3, d),
0 ϕk2 0
and conclude that
Proposition 1.36 ([3, Proposição 2.3.9.]). For each pair (m1,m2) ∈ Ω, there is a
(1, 1, 1)-critical submanifold F k km1m2 ⊆M (3, d), where
 d+ x+ 2y = 0 mod 3 
Ω = (x, y) ∈ N∗ × N∗ : 2x+ y < 3σH(k) .
x+ 2y < 3σH(k)
Proof. The stability conditions in this case are
d2 + d3 d d
µ(L2 ⊕ L3) < µ(E)⇐⇒ 2 < 3 and µ(L3) < µ(E)⇐⇒ d3 < 3 .
In terms of mj > 0 we get
d
d3 < 3 ⇐⇒ 2m2 +m1 < 3 ( ) and
d2 + d3 d
σH k 2 < 3 ⇐⇒ 2m1 +m2 < 3σH(k)
Remark 1.37. For the particular case of (1, 1, 1)-critical submanifolds, note that
(ϕkj ⊗ sp)(Lj) ⊆ (ϕkj )(Lj)⊗ Lp = Lj+1 ⊗K ⊗ L⊗k ⊗k+1p ⊗ Lp = Lj+1 ⊗K ⊗ Lp
and hence
i (F k ) ⊆ F k+1k m1m2 m1m2 .
Theorem 1.38. There is an isomorphism
F k ∼= Symm̄1+km1m2 (X)× Sym
m̄2+k(X)× J d3(X)
for each pair (m1,m2) ∈ Ω, where Jd3(X) is the Jacobian of X, the moduli space of stable
line bundles of degree d3, and m̄j = mj − k = dj+1 − dj + 2g − 2.
Proof. It is enough to take
F k // Symm̄1+km1m2   (X)× Sym
m̄2+k(X)× J d3(X)
0 0 0
(E,ϕk) = (L1 ⊕ L k2 ⊕ L3, ϕ1 0 0 )  // (div(ϕk1), div(ϕk2), L3)
0 ϕk2 0
In thi(s case, the Morse index for)(1, 1, 1)-critical submanifolds F k = F kλ m1m2 is given by
λ = 2 4(2g − 2)−m1 −m2 + 3k . The reader may consult Gothen [12] or Bento [3] for
details.
16
Ronald A. Zúñiga-Rojas
Remark 1.39. Lemma 1.30, Proposition 1.32, Lemma 1.34, Proposition 1.35, and Propo-
sition 1.36 are presented by Bento [3] for the general case of rank three Hitchin pairs.
Here, we presented them for the particular case of rank three k-Higgs bundles.
From the embeddings
k −−−iF k−→ F k+1λ λ ∀λ,
above mentioned, we get induced isomorphisms in cohomology:
Hj
∼
(F k+1λ ,Z) −−−
=−→ Hj(F kλ ,Z)
for all λ, for certain values of j in terms of k. Our goal is to find the range of j for which
these isomorphisms hold.
The embeddings restricted to (1, 1)-critical submanifolds in the rank two case, were stud-
ied by Hausel [14] and presented by Hausel and Thaddeus [19]. Here, we focus on rank
three.
If we restrict the embeddings to critical manifolds of type (1, 2):
F k −−−ik−→ F k+1
( ( )d)1 (d1 ( )) (1.8)
0 0
E1 ⊕ E2, k 0 −7 → ⊕
0 0
E1 E ,ϕ 2 ϕk21 21 ⊗ sp 0
then, the isomorphisms
∼
F kd1 −−−
=−→ N
( ( )) σH(k)
(2, 1, d̃1, d̃2)
0 0
E1 ⊕ E2, k 0 7−→ (V1, V2, ϕ)ϕ21
between (1, 2) critical submanifolds and moduli spaces of triples, where we denote by
V1 = E2⊗K(kp), by V2 = E1, by ϕ = ϕk21 and σH(k) = deg(K(kp)) = 2g− 2 + k, induce
another embeddings:
ik : Nσ (k)(2, 1, d̃1, d̃2)→ NH σH(k+1)(2, 1, d̃1 + 2, d̃2)
(V1, V2, ϕ) 7→ (V1 ⊗ Lp, V2, ϕ⊗ sp)
where d̃1 = deg(V1) = d2 + 2σH(k) and d̃2 = deg(V2) = d1, and so, induce embeddings on
the flips:
ik : Nσ−(k)(2, 1, d̃1, d̃2) ↪→ Nσ−(k+1)(2, 1, d̃1 + 2, d̃2)H H
and
ik : Nσ+ (k)(2, 1, d̃1, d̃2) ↪→ Nσ+ (k+1)(2, 1, d̃1 + 2, d̃2).H H
The situation with critical manifolds of type (2, 1)
F k −−−ik−→ F k+1
( ( )d)2 (d2 ( )) (1.9)
⊕ 0 0 0 0E2 E1, ϕk 0 7−→ E2 ⊕ E1, ϕk21 21 ⊗ sp 0
17
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
is very similar to the (1, 2) situation, using now isomorphisms
∼
F k =
( ( )d2
−−−−→ Nσ (k)(1, 2, d̃) H 1
, d̃2)
E2 ⊕
0 0
E1, k 0 7−→ (V1, V2, ϕ)ϕ21 ( )
where by V1 = E1 ⊗ K(k · p), by V2 = E2, by ϕ = ϕk21 and σH(k) = deg K(k · p) =
2g − 2 + k, and the induced embeddings become:
ik : Nσ (k)(1, 2, d̃1, d̃H 2)→ Nσ (k+1)(1, 2, d̃1 + 1, d̃H 2)
(V1, V2, ϕ) 7→ (V1 ⊗ Lp, V2, ϕ⊗ sp)
where d̃1 = deg(V1) = d1 + σH(k) and d̃2 = deg(V2) = d2. Hence, for the flips on σH(k),
the induced embeddings become:
ik : Nσ−(k)(1, 2, d̃1, d̃2) ↪→ NH σ−H(k+1)(1, 2, d̃1 + 1, d̃2)
and
ik : Nσ+ (k)(1, 2, d̃1, d̃2) ↪→ Nσ+ (k+1)(1, 2, d̃1 + 1, d̃2).H H
Critical submanifolds of type (1, 1, 1) are different from the other two. The embeddings
F k
ik
m1m2 −−−−→ F
k+1
  m1m2   0 0 0   0 0 0(L1 ⊕ L2 ⊕ L3, ϕ1 0 0 ) 7−→ ( L1 ⊕ L2 ⊕ L3, ϕ1 ⊗ sp 0 0 )
0 ϕ2 0 0 ϕ2 ⊗ sp 0
(1.10)
together with the isomorphisms
F k ∼ m̄1+km1m2 = Sym (X)× Sym
m̄2+k(X)× J d3(X)
induce embeddings of the form:
Symm̄1+k(X)× Sy(mm̄2+k(X)× J d3(X)) → (Symm̄1+k+1(X)× Symm̄2+k+)1(X)× J d3(X)
div(ϕk1), div(ϕk2), L3 →7 div(ϕk1 + p), div(ϕk2 + p), L3 .
2 Stable Holomorphic Triples and Roof Theorem
2.1 σ-Stability
For (2, 1, d̃2, d̃1)-triples and (1, 2, d̃1, d̃2)-triples, the embeddings ik preserve σ-stability:
Lemma 2.1. A triple T of type (2, 1, d̃2, d̃1) or type (1, 2, d̃1, d̃2) is σ-stable ⇔ ik(T ) is
(σ + 1)-stable.
18
Ronald A. Zúñiga-Rojas
Proof. We will show the result holds for (2, 1, d̃2, d̃1)-triples, the proof of (1, 2, d̃1, d̃2)-
triples is analogous.
Recall that T = (V1, V2, ϕ) is σ-stable if and only if µσ(T ′) < µσ(T ) for any T ′ proper
subtriple of T .
Denote by S = ik(T ) = (V1 ⊗ Lp, V2, ϕ⊗ sp). Is easy to check that µσ+1(S) = µσ(T ) + 1:
( ) = degσ+1(S)µσ+1 S rk( =V1 ⊗ Lp)⊕ rk(V2)
deg(V1 ⊗ Lp) + deg(V2) + (σ + 1) rk(V2)
1 + 2 =
deg(V1) + deg(Lp) + deg(V2) + σ rk(V2) + rk(V2)
3 =
deg(V1) + deg(V2) + σ rk(V2) + deg(Lp) + rk(V2)3 3 = µσ(T ) + 1
since deg(Lp) = 1 and rk(V2) = 2.
Any S ′ proper subtriple of S is of the form S ′ = i (T ′k ) for some T ′ subtriple of T , or
equivalently:
S ′ = (V ′1 ⊗ L ′p, V2 , ϕ⊗ sp)
and there are injective sheaf homomorphisms V ′1 → V1 and V ′2 → V2. This statement is
justified since the following diagram commutes:
)
S // // S ′ //
  (
T ′ // T
) =
V  //  // 
(
2 B B // V2
ϕ⊗sp ϕ⊗s (ϕ⊗s )⊗s
−1 ϕ
p p p
 )  ⊗ (

V1 L
 //  // A⊗ L∗ p A //p V1
where the first floor of the diagram contains the first entries of the triples, second floor
contains the second entries, the diagonal arrows are the coresponding morphisms, and
we consider the subbundles A = V ′1 ⊗Lp, B = V ′2 and T ′ = (V ′1 , V ′2 , ϕ) ⊆ (V1, V2, ϕ) = T .
So, there is a one–to–one correspondence between the proper subtriples S ′ ⊆ S and the
proper subtriples T ′ ⊆ T . We can easily see that µσ+1(S ′) = µ ′σ(T ) + 1 and hence:
µ ′ ′σ+1(S ) < µσ+1(S)⇔ µσ(T ) + 1 < µσ(T ) + 1⇔ µσ(T ′) < µσ(T ).
Therefore, T is σ-stable ⇔ S = ik(T ) is (σ + 1)-stable.
Corollary 2.2. The embedding
ik : Nσ(k)(2, 1, d̃1, d̃2)→ Nσ(k+1)(2, 1, d̃1 + 2, d̃2)
is well defined for any σ(k) such that σm < σ(k) < σM . In particular, the embedding ik
restricted to F kd1 (see (1.8)) is well defined and we have a commutative diagram of the
19
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
form:
(E1 ⊕ E2, ϕk)  // (E1 ⊕ E2, ϕk ⊗ sp)
F k
ik // k+1
dOO 1
Fd1OO
=∼ =∼
 
N //σ NH(k) σ (k+1)i Hk
( Ẽ1, Ẽ2, ϕk21) // (Ẽ1, Ẽ k2, ϕ21 ⊗ sp),
where Ẽ1 = E2 ⊗K(k · p), Ẽ2 = E1, and ϕk21 : E1 → E2 ⊗K(k · p). 
Corollary 2.3. The embedding
ik : Nσ(k)(1, 2, d̃2, d̃1)→ Nσ(k+1)(1, 2, d̃2 + 1, d̃1)
is well defined for any σ(k) such that σm < σ(k) < σM . In particular, the embedding ik
restricted to F kd2 (see (1.9)) is well defined and we have a commutative diagram of the
form:
(E2 ⊕ E1, ϕk)  // (E2 ⊕ E1, ϕk ⊗ sp)
F k
ik // F k+1dOO 2 d2OO
∼= ∼=
 
Nσ (k) // NH σH(k+1)ik
(Ẽ1, Ẽ , ϕk2 21)
 // (Ẽ1, Ẽ2, ϕk21 ⊗ sp),
where Ẽ1 = E1 ⊗K(k · p), Ẽ2 = E2, and ϕk21 : E2 → E1 ⊗K(k · p). 
These results allow us to conclude that there is an interesting and important correspon-
dence between the σ-stability values of moduli spaces of holomorphic triples:
σm(k) σH(k) σM(k)
| ∗ | · · | //
ik ik ik ik
    
· | · ∗ | · · | · · | //
σm(k + 1) σ ′H(k + 1) σ σM(k + 1)
20
Ronald A. Zúñiga-Rojas
where σm(k) = µ̃1 − µ̃2, σM(k) = 4(µ̃1 − µ̃2), σH(k) = deg(K(kp)) = 2g − 2 + k, and the
correspondence gives us σm(k+ 1) = σm(k) + 1, σ′ = σM(k) + 1, σM(k+ 1) = σM(k) + 3,
and σH(k + 1) = σH(k) + 1. First and second floor are representations of the real line,
where the second floor of the diagram corresponds to the interval [σm, σM ] for poles of
order k, and the first floor for poles of order (k + 1) after the embedding ik.
Remark 2.4. An interesting fact from the correspondence represented by last diagram is
that
ik : σM(k) 7→ σ′ < σM(k + 1).
2.2 Blow-up and The Roof Theorem
At this point, a brief description of the flip loci of the moduli spaces of holomorphic
triples will be useful to understand the coming results and notation. The reader may see
Muñoz et al. [23] for details.
Fixing the type (r1, r2, d1, d2) for the moduli spaces of holomorphic triples, we shall de-
scribe the differences between Nσ1(r1, r2, d1, d2) and Nσ2(r1, r2, d1, d2) where σ1 and σ2
are separated by a critical value σc ∈ [σm, σM ]. Here, we suppose r1 6= r2, since for our
purposes, the case r1 = 2 and r2 = 1 will be particularly useful.
Let
σ+c = σc + ε and σ−c = σc − ε
where ε > 0 is small enough so that σ ∈ ]σ−c c , σ+c [ is the only critical value in that
subinterval.
Definition 2.5. Define the flip loci as the sets
{ }
Sσ+ = T ∈ Nσ+| T is σ−c − unstable ⊆ Nσ+(r1, r2, dc c c 1, d2)
and { }
S = T ∈ N | T is σ+σ− σ− c − unstable ⊆ Nσ−(rc c c 1, r2, d1, d2),
and denote Ss± = Sσ± ∩ N s±(r1, r2, d1, d2) as the stable part of the flip loci, where σ±σc c σ cc
means any of both σ+c or σ−c .
Denote Ñσ−(k) as the blow-up of Nσ−(k) = Nσ−(k)(2, 1, d̃1, d̃2) along the flip locus Sc c c σ− ,c (k)
which is isomorphic to Ñσ+(k), the blow-up of Nc σ+(k) = Nσ+(k)(2, 1, d̃1, d̃2) along the flipc c
locus Sσ+(k). From now on, we will denote just Ñσc(k) whenever no confusion is likely toc
arise.
Theorem 2.6. For each k, there exists an embedding at the blow-up level
ĩk : Ñσc(k) ↪→ Ñσc(k+1)
21
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
such that the following diagram commutes:
Ñσc LL(k+1)
 
Nσ−c LL (k+1)
Nσ+c LL (k+1)
∃ĩk
ik
Ñ ikσc(k)
 
Nσ−c (k) Nσ+c (k)
where Ñσc(k) is the blow-up of Nσ−(k) = Nσ−(k)(2, 1, d̃1, d̃2) along the flip locus Sσ−(k) and,c c c
at the same time, represents the blow-up of Nσ+(k) = Nσ+(k)(2, 1, d̃1, d̃2) along the flipc c
locus Sσ+c (k).
Proof. Recall that T is σ-stable if and only if ik(T ) is (σ+1)-stable. Furthermore, by [23],
note that any triple
T = (V1, V2, ϕ) ∈ Sσ+c (k) ⊆ Nσ+c (k)(2, 1, d̃1, d̃2)
is a non-trivial extension of a subtriple T ′ ⊆ T of the form T ′ = (V ′1 , V ′2 , ϕ′) = (M, 0, ϕ′)
by a quotient triple of the form T ′′ = (V ′′1 , V ′′2 , ϕ′′) = (L, V2, ϕ′′), where M is a line bundle
of degree deg(M) = dM and L is a line bundle of degree deg(L) = dL = d̃1−dM . Besides,
also by [23], the non-trivial critical values σc 6= σm for σm < σc < σM are of the form
σc = 3dM − d̃1 − d̃2. Then, we can visualize the embedding ik : T ↪→ ik(T ) as follows:
0 // T ′ // T // T ′′ // 0
0 // 0 // =V2 // V2 // 0
ϕ′
ϕ
ϕ′′
  
0 //M // V // //_1 L 0
ik

0 // 0 // =V2 // V2 // 0
ϕ′⊗sp
ϕ⊗sp
ϕ′′⊗sp
  
0 //M ⊗ L // V1 ⊗ L //p p L⊗ L //p 0
where deg(V1 ⊗ Lp) = d̃1 + 2 and deg(M ⊗ Lp) = dM + 1, and so L⊗ Lp verifies that
deg(L⊗ Lp) = deg(V1 ⊗ Lp)− deg(M ⊗ Lp):
deg(L⊗ Lp) = dL + 1 = d̃1 − dM + 1 =
22
Ronald A. Zúñiga-Rojas
(d̃1 + 2)− (dM + 1) = deg(V1 ⊗ Lp)− deg(M ⊗ Lp).
Hence, σc(k + 1) verifies that σc(k + 1) = σc(k) + 1:
σc(k + 1) = 3 deg(M ⊗ Lp)− deg(V1 ⊗ Lp)− deg(V2) =
3dM + 3− d̃1 − 2− d̃2 = (3dM − d̃1 − d̃2) + 1 = σc(k) + 1
and where ik(T ′) = (M ⊗ L ′p, 0, ϕ ⊗ sp) is the maximal σ+c (k + 1)-destabilizing subtriple
of ik(T ), verifying exactness at the image level of the embedding.
Similarly, also by [23], any triple T ∈ Sσ−(k) ⊆ Nσ−(k)(2, 1, d̃1, d̃2) is a non-trivial extensionc c
of a subtriple T ′ ⊆ T of the form T ′ = (V ′, V ′1 2 , ϕ′) = (L, V2, ϕ′) by a quotient triple of the
form T ′′ = (V ′′, V ′′1 2 , ϕ′′) = (M, 0, ϕ′′), where M is a line bundle of degree deg(M) = dM
and L is a line bundle of degree deg(L) = dL = d̃1 − dM . Then, the embedding
ik : T ↪→ ik(T )
looks like:
0 // T ′ // T // T ′′ // 0
0 // =V2 // V2 // 0 // 0
ϕ′
ϕ
ϕ′′
  
0 // L // V1 //M //_ 0
ik

0 // =V2 // V2 // 0 // 0
ϕ′⊗sp
ϕ⊗sp
ϕ′′⊗sp
  
0 // L⊗ L // V1 ⊗ L //p p M ⊗ L //p 0
where i (T ′k ) = (L, V , ϕ′) is the maximal σ+2 c (k + 1)-destabilizing subtriple of ik(T ).
Hence, ik restricts to the flip loci Sσ+(k) and Sσ−(k). Recall that, by definition, the blow-upc c
of Nσ+(k) along the flip locus Sσ+(k), is the space Ñc c σc(k) together with the projection
π : Ñσc(k) → Nσ+c (k)
where π restricted to Nσ+(k) − Sσ+(k) is an isomorphism and the exceptional divisorc c
E+ = π−1(Sσ+(k)) ⊆ Ñσc(k) is a fiber bundle over Sσ+(k) with fiber Pn−k−1, wherec c
n = dim(Nσ+(k)) and k = dim(Sσ+(k)). So, the embedding can be extended to E+ inc c
a natural way. Same argument remains valid when we consider Ñσc(k) as the blow-up
of Nσ−(k) along the flip locus Sσ−(k) with exceptional divisor E− = π−1(Sσ−(k)) ⊆ Ñc c c σc(k).
Therefore, the embedding can be extended to the whole Ñσc(k).
Recall that there is an isomorphism
Nσ(1, 2, d1, d2) ∼= Nσ(2, 1,−d2,−d1)
for all σ by Proposition 1.11. Hence, the following corollary represents the analogous
dual Roof-Theorem for the (1, 2)-case, and also holds:
23
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Corollary 2.7. For each k, there exists an embedding at the blow-up level
ĩk : Ñσc(k) ↪→ Ñσc(k+1)
such that the following diagram commutes:
Ñσc LL(k+1)
 
Nσ− N +c LL (k+1) σc LL (k+1)
∃ĩk
ik
Ñ ikσc(k)
 
Nσ−c (k) Nσ+c (k)
where Ñσc(k) is the blow-up of Nσ−(k) = Nσ−(k)(1, 2, d̃2, d̃1) along the flip locus Sc c σ−c (k) and,
at the same time, represents the blow-up of Nσ+(k) = Nσ+(k)(1, 2, d̃2, d̃1) along the flipc c
locus Sσ+c (k).
Proof. Follows from Theorem 2.6 and Proposition 1.11.
Remark 2.8. The construction of the blow-up may be found in the book of Griffiths and
Harris [13].
3 Cohomology
We want to show that the embeddings i k k+1k : Fλ ↪→ Fλ induce covariant isomorphisms in
cohomology:
∼
Hj(F k+1,Z) −−−=−→ Hj(F kλ λ ,Z)
for all λ and certain j. To do that, we need to study F kd1 , F
k k
d2 and Fm1m2 separately.
Because of Proposition 1.11, the cohomology of F k kd1 and Fd2 are similar, so it will be
enough to analyze F kd1 . The cohomology of F
k
m1m2 will be completely different.
We shall start by describing the cohomology of Symk(X) = Xk/Sk, the k-th symmetric
product in subsection 3.1, which is related to the cohomology of the rank three VHS.
For (1, 2)-VHS, we will prove that the embeddings ik : F kd1 ↪→ F
k+1
d1 induce isomorphisms
in cohomology:
Hj(F k+1
∼
d1 ,Z) −−−
=−→ Hj(F kd1 ,Z)
for certain j, or equivalently:
Hj(N k+1
∼
σ ,Z) −−−
=−→ Hj(N kσ ,Z),H H
24
Ronald A. Zúñiga-Rojas
where we denote N kσ = Nσ (k)(2, 1, d̃1, d̃2). We do that in two steps. First, in subsectionH H
3.2, we get that
Hj
∼
(N k+1,Z) −−−=−→ Hj(N kσ σ ,Z)c c
for all critical σc = σc(k) such that σm(k) < σc(k) < σM(k), and for all j 6 n(k), where
the bound n(k) is known. We first analize the embedding restricted to the flip loci,
ik : Sσ−(k) ↪→ Sσ−c c (k+1) and ik : Sσ+c (k) ↪→ Sσ+ . For simplicity, we will denote fromc (k+1)
now on Sk− = Sσ− and Sk(k) + = Sσ+(k) whenever no confusion is likely to arise about thec c
critical value.
In subsection 3.3, we stabilize the cohomology of the (1, 2)-VHS, using useful results from
the work of Bradlow, Garćıa-Prada, Gothen [5]. In subsection 3.4, we present the dual
results for (2, 1)-VHS.
Finally, in subsection 3.5, we study the case of the (1, 1, 1)-VHS.
3.1 Cohomology of Symmetric Products
We begin by recalling some cohomology features of Symk(X) = Xk/Sk, the symmetric
product with quotient topology, where Xk is the k-times cartesian product and Sk is the
order k symmetric group. Obviously Sym1(X) = X.
As mentioned before, the k-th symmetric product Symk(X) is a smooth projective variety
of dimension k ∈ N, that could be interpretated as the moduli space of degree k effective
divisors.
It is well known that
H0(X,Z) = Z, H1(X,Z) = Z2g, H2(X,Z) = Z.
There is a generator β ∈ H2(X,Z) induced by the orientation of X. Moreover, there are
2g generators α1, α2, . . . , α 12g ∈ H (X,Z) such that
αi ∪ αj = −αj ∪ αi = 0 if i− j 6= ±g for i, j ∈ {1, . . . , 2g}
and
αi ∪ αi+g = −αi+g ∪ αi = β for i ∈ {1, . . . , g}
with the usual cup product ∪. Hence
αi ∪ β = β ∪ αi = 0 and β2 = β ∪ β = 0.
For the usual cartesian product Xk, we get that the ring H∗(Xk,Z) ∼= H∗(X,Z)⊗k is
generated by {α }2gir i=1 and βr with 1 6 r 6 k, which are elements of the form
αir = 1⊗ . . .⊗ 1⊗ αi ⊗ 1⊗ . . .⊗ 1 ∈ H1(Xk,Z)
and
βr = 1⊗ . . .⊗ 1⊗ β ⊗ 1⊗ . . .⊗ 1 ∈ H2(Xk,Z)
where αi and β fill the r-th entry of αir and βr respectively, and they are subject to the
relations
αir ∪ αjr = −αjr ∪ αir = 0 if i− j 6= ±g for i, j ∈ {1, . . . , 2g}
and
αir ∪ αi+g r = −αi+g r ∪ αir = βr for i ∈ {1, . . . , g}.
25
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Hence
αir ∪ βr = βr ∪ αir = 0 and β2r = βr ∪ βr = 0.
Besides, each βr commutes with every element of H∗(Xk,Z).
Finally, the symmetric product Symk(X) has a cohomology ring H∗(Symk(X),Z) gener-
ated by elements of the form
∑k
ζ = α + · · ·+ α = α ∈ H1 ki i1 ik ir (Sym (X),Z) for 1 6 i 6 2g
r=1
and ∑k
η = β1 + · · ·+ βk = βr ∈ H2(Symk(X),Z)
r=1
where
ζi ∪ ζj = −ζj ∪ ζi and ζi ∪ η = η ∪ ζi
for any i and j. The reader may consult Macdonald [21] or Arbarello-Cornalba-Griffiths-
Harris [1] for details.
According to Arbarello∣ et al. [1], there is 4k ∈ Sym
k+1(X) a universal divisor such that
∣
4 ∣ kk∣ = D for every divisor D ∈ Sym (X).
{D}×X
Therefore, the first Chern class c (4 ) ∈ H2(Symk+11 k (X),Z) of this universal divisor is
given by
∑g
c (4 ) = γ ⊗ k + (ζ ⊗ α − ζ ⊗ α 2 k+11 k i i+g i+g i) + η ⊗ 1 ∈ H (Sym (X),Z) (3.1)
i=1
where ∑2
H2(Symk+1(X),Z) = Hj(Symk(X),Z)⊗H2−j(X,Z)
j=0
and ∑g
γ = ζi ∪ ζ 2 ki+g ∈ H (Sym (X),Z).
i=1
Macdonald [21] compute the Poincaré polynomi(al of H
∗(Symk(X
( ) )
),Z):
Symk( ) = Coeff (1 + xt)
2g
Pt X xk (1− x)(1− xt2) . (3.2)
For k > 2g − 2 there is the Abel–Jacobi map Symk(X) → J k, which is a locally trivial
fibration with fibre Pk−g, and gives the(Poincaré polynomial:( ) 2g 2(k−g+1) )
Pt Symk( ) =
(1 + t) (1 + t )
X (1 2) . (3.3)− t
The reader may see Macdonald [21], Arbarello et al. [1], or Hausel [14] for details.
Our embedding ik : F k k+1λ → Fλ is in fact related to the embedding
Symk(X)→ Symk+1(X)
26
Ronald A. Zúñiga-Rojas
D 7→ D + p
for a fixed point p ∈ X. We will abuse notation and call this last embedding also ik. We
get a sequence
X = Sym1(X) ⊆ Sym2(X) ⊆ . . . ⊆ Symk(X) ⊆ . . .
and so, we may consider its direct limit
Sym∞(X) = lim Symk(X),
k→∞
which is a P∞-bundle over the Jacobian J , and(hence its)Poincaré polynomial is:( )
Sym∞( ) = (1 + t)
2g
Pt X (1 . (3.4)− t2)
The reader may consult Hausel [14] for all the details.
Theorem 3.1. The pull-back
i∗k : H∗(Symk+1(X),Z)→ H∗(Symk(X),Z)
induced by the embedding ik : Symk(X)→ Symk+1(X), is surjective.
Proof. It is enough to see that the cohomology ring H∗(Symk(X),Z) is generated by the
universal classes {ζ gi}i=1 and η mentioned above, and that the universal divisor 4k has
first Chern class of the form 3.1. See Hausel [14] for details.
Corollary 3.2. The cohomology ring of the direct limit Sym∞(X) is the covariant limit
H∗(Sym∞(X),Z) = lim H∗(Symk(X),Z)
∞←k
which is a graded commutative free algebra generated by the classes {ζ }gi i=1 and η.
Proof. This is a consequence of Theorem 3.1 and the Poincaré polynomial (3.4) found by
Hausel [14].
Theorem 3.3 ([21, (12.2)]). There is a cohomology isomorphism
Hj(Symk+1(X),Z)→ Hj(Symk(X),Z)
for all j 6 k − 1. 
Corollary 3.4. There is an isomorphism
Hj(Sym∞(X),Z)→ Hj(Symk(X),Z)
for all j 6 k − 1.
Proof. It follows directly from Theorem 3.1, Corollary 3.2 and Theorem 3.3.
27
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
3.2 Cohomology of Triples
A few words about notation. Recall that we are using d̃j = deg(Vj) because of the
correspondence
V1 = E2 ⊗K(k · p) and V2 = E1
through the isomorphism F k ∼( d1 = Nσ) (k)(2, 1, d̃1, d̃2) whereH
d̃1 = deg(V1) = deg E2 ⊗K(k · p) = d2 + 2σH(k) and d̃2 = deg(V2) = deg(E1) = d1.
Similarly, the notation becomes
V1 = E1 ⊗K(k · p) and V2 = E2
through the isomorphism F k ∼d2 = Nσ (k)(1, 2, d̃1, d̃2) for the dual cases, and soH
d̃1 = deg(V1) = deg(E1 ⊗K(k · p)) = d1 + σH(k) and d̃2 = deg(V2) = deg(E2) = d2.
Theorem 3.5. There is an isomorphism
∼
i∗ j k+1 = j kk : H (S− ,Z) −−−−→ H (S−,Z)
for all j 6 d̃1− dM − d̃2− 1 = d2− d1 + 2σH(k)− dM , where dj = deg(Ej), d̃j = deg(Vj),
M → X is a line bundle of degree dM = deg(M), and σH(k) = deg(K(kp)) = 2g− 2 + k.
Proof. Recall that, according to [23, Theorem 4.8.], Sk− = P(V) is the projectivization of
a bundle V → N ′ ×N ′′ of rank rk(V) = −χ(T ′′σ σ , T ′), wherec c
N ′σ = Nσc(1, 1, d̃ − d , d̃ ) ∼= J d̃2 × Symd̃1−dM−d̃21 M 2 (X)c
and
N ′′σ = Nσc(1, 0, d , 0) ∼M = J dM (X)c
where any triple T = (V , V , ϕ) ∈ Sk1 2 − ⊆ Nσ−(k)(2, 1, d̃1, d̃2) is a non-trivial extension ofc
a subtriple T ′ ⊆ T of the form T ′ = (V ′1 , V ′2 , ϕ′) = (L, V2, ϕ′) by a quotient triple of the
form T ′′ = (V ′′1 , V ′′2 , ϕ′′) = (M, 0, ϕ′′), where M is a line bundle of degree deg(M) = dM
and L is a line bundle of degree deg(L) = dL = d̃1 − dM .
Then, the embedding ik : T → ik(T ) restricts to:
(V ′, V ′ ′  // ′ ′ ′1 2 , ϕ ) (V1 ⊗ Lp, V2 , ϕ ⊗ sp)
N ′ ik // N ′
OOσc σcOO +1
=∼ =∼
 
J d̃2 × Symd̃1−dM−d̃2(X) // J d̃2 × Symd̃1−dM−d̃2+1( ) ik ( )(X)
[V ′], div(ϕ′)  // [V ′], div(ϕ′2 2 ⊗ sp)
because σc(k + 1) = σc(k) + 1, and dM(k + 1) = dM(k) + 1, and because, by the proof of
the Roof Theorem 2.6, ik restricts to the flip locus Sk−.
28
Ronald A. Zúñiga-Rojas
Recall that in our case σ ′ ′ ′ ′c = σc(k) > σm. Then, for subtriples o(f the form T) = (V1 , V2 , ϕ )
we get that ϕ′ =6 0 and so, they are entirely parametrized by [V ′2 ], div(ϕ) . That is why
the map from N ′ to J d̃2 × Symd̃1−dM−d̃2σ (X) is an isomorphism at the Jacobian.c
Similarly, ik restricts to:
(V ′′1 , 0, 0)
 // (V ′′1 ⊗ Lp, 0, 0)
N ′′ ik //σ N ′′OO c σcOO +1
∼= =∼
 
J dM // dMi Jk
[V ′′]  //1 [V ′′1 ⊗ Lp]
Here, the quotient triples of the form T ′′ = (V ′′1 , 0, 0) are trivially parametrized by [V ′′1 ]
and so, the map from N ′′ dMσ to J is also an isomorphism at the Jacobian.c
Hence, by Corollary 3.3,
i∗
∼
k : Hj(N ′σ +1,Z) −−−
=−→ Hj(N ′σ ,Z) ∀j 6 d̃1 − dM − d̃2 − 1,c c
and hence
∼
i∗ : Hjk (Sk+1− ,Z) −−−
=−→ Hj(Sk−,Z) ∀j 6 d̃1 − dM − d̃2 − 1.
Similarly, for the flip locus Sk+ = Sσ+(k) we have:c
Theorem 3.6. There is an isomorphism
i∗
∼
k : Hj(Sk+1+ ,Z) −−−
=−→ Hj(Sk+,Z)
for all j 6 d̃1− dM − d̃2− 1 = d2− d1 + 2σH(k)− dM , where dj = deg(Ej), d̃j = deg(Ẽj),
M → X is a line bundle of degree dM = deg(M), and σH(k) = deg(K(kp)) = 2g− 2 + k.
Proof. Quite similar argument to the one presented above, except for the detail that this
time is the other way around: according also to [23, Theorem 4.8.], Sk+ = P(V) is the
projectivization of a bundle V → N ′c × N ′′c of rank rk(V) = −χ(T ′′, T ′), but this time
N ′c = Nc(1, 0, dM , 0) ∼= J dM (X), andN ′′c = Nc(1, 1, d̃ −d , d̃ ) ∼= J d̃2×Symd̃1−dM−d̃21 M 2 (X)
where any triple T = (V1, V2, ϕ) ∈ Sk+ ⊆ Nσ+(k)(2, 1, d̃1, d̃2) is a non-trivial extension ofc
a subtriple T ′ ⊆ T of the form T ′ = (V ′1 , V ′2 , ϕ′) = (M, 0, ϕ′) by a quotient triple of the
form T ′′ = (V ′′, V ′′ ′′1 2 , ϕ ) = (L, V2, ϕ′′), where M is a line bundle of degree deg(M) = dM
and L is a line bundle of degree deg(L) = dL = d̃1 − dM .
Theorem 3.7. There is an isomorphism
∼ ( )
i∗ : Hj(N ,Z) −−−=−→ Hjk σ−(k+1) (Nσ−(k),Z) ∀j 6 2 d̃1 − 2d̃2 − (2g − 2) + 1.c c
Since the behavior of Nσ− , where σ−c = σc − ε, is the same that the one of Nσ+ , where
σ+
c m
m = σm + ε, is enough to prove the following lemma:
29
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Lemma 3.8. The relative cohomology groups
( Hj(Nσ+m(k+1),)Nσ+m(k);Z) = 0
are trivial for all j 6 2 d̃1 − 2d̃2 − (2g − 2) .
Proof. Note that N − = ∅, hence N + = Skσ (k) σ (k) +, and according to [23, Theorem 4.10.],m m
any triple T = (V1, V2, ϕ) ∈ Sk+ = Nσ+ (k)(2, 1, d̃1, d̃2) is a non-trivial extension of am
subtriple T ′ ⊆ T of the form T ′ = (V ′1 , V ′2 , ϕ′) = (V1, 0, 0) by a quotient triple of the form
T ′′ = (V ′′, V ′′1 2 , ϕ′′) = (0, V2, 0). Hence, there is a map
π : Nσ+ → N (2, d̃ )× J d̃2(X)m 1
( (V1, V2, ϕ) 7→ ([V1)], [V2])
where the inverse image π−1 N (2, d̃ ) × J d̃2(X) = PN1 has rank N = −χ(T ′′, T ′) =
d̃1 − 2d̃2 − (2g − 2), and the proof follows.
Theorem 3.9. There is an isomorphism
ĩ∗
∼
: Hj(Ñ ,Z) −−−=−→ Hjk σc(k+1) (Ñσc(k),Z) ∀(j 6 n(k) )
at the blow-up level, where n(k) = min(d̃1 − dM − d̃2 − 1, 2 d̃1 − 2d̃2 − (2g − 2) + 1).
Proof. By the Roof Theorem 2.6, ik lifts to the blow-up level. We will denote N k− =
N k kσ−(k)(2, 1, d̃1, d̃2) and Ñ = Ñc σc(k) its blow-up along the flip locus S− = Sσ− . Recallc (k)
that, from the constru(ction of t)he blow-up, there is a map (π : Ñ k− )→ N k− such that
0→ π∗ Hj− (N k−) → Hj(Ñ k)→ Hj(Ek)/π∗ j k− H (S−) → 0
splits where Ek = π−1(Sk− −) is the so-called exceptional divisor. Hence, the following
diagram ( ) ( )
0 // π∗− Hj(N k−) // Hj(Ñ k) // Hj(Ek)/π∗ Hj(Sk ) //− − 0 (3.5)
OO OO OO
=∼ ĩ∗ ∼( ) k =( )
0 // π∗ Hj(N k+1) // Hj− − (Ñ k+1) // Hj(Ek+1)/π∗− Hj(Sk+1) //− 0
commutes for all j 6 n(k), and the theorem follows.
Corollary 3.10. There is an isomorphism
i∗
∼
k : Hj(Nσ+(k+1),Z) −−−
=(−→ Hj(Nσ+(k),Z) )∀j 6 n(k)c c
where n(k) = min(d̃1 − dM − d̃2 − 1, 2 d̃1 − 2d̃2 − (2g − 2) + 1) as before.
Proof. Recall that Ñ k = Ñσc(k) is also the blow-up of N k+ = Nσ+(k)(2, 1, d̃1, d̃2) along thec
flip locus Sk+ = S k kσ+(k),(so there )is a map π+ : Ñ → Nc + suc(h that )
0→ π∗+ Hj(N k+) → Hj(Ñ k)→ Hj(Ek)/π∗ Hj+ (Sk+) → 0
splits: ( ) ( )
Hj(Ñ k) = π∗ Hj(N k j k ∗ j k+ +) ⊕H (E )/π+ H (S+) ,
and by Theorem 3.6 and Theorem 3.9, the result follows.
Corollary 3.11. There is an isomorphism
i∗
∼
k : Hj(N
=
σc(k+1),Z) −−−−→ Hj(Nσc(k),Z) ∀j 6 n(k). 
30
Ronald A. Zúñiga-Rojas
3.3 Cohomology of the (1, 2)-VHS
So far, we stabilize the cohomology of Nσc(k) for any critical σc(k). Here and after, σL
respresents the largest critical value in the open interval ]σm, σM [, and Nσ+ (respectivelyL
N s+) denotes the moduli space of σL-polystable (respectively σL-stable) triples for valuesσL
σL < σ < σM . The space Nσ+ is so-called the ‘large σ’ moduli space (see [5]). TheL
following results will allow us to generalize the stabilization for all σ ∈ ]σm(k), σM(k)[:
d d
Theorem 3.12 ([5, Th. 7.7.]). Assume that r1 > r2 and 1 2> . Then the moduli space
r1 r2
N s+ = N s+(r1, r2, d1, d2) is smooth of dimensionσL σL
(g − 1)(r21 + r22 − r1r2)− r1d2 + r2d1 + 1,
and is birationally equivalent to a Pñ-fibration over N s(r1−r2, d1−d2)×N s(r2, d2), where
N s(r, d) is the moduli space of stable bundles of degree r and degree d, and
ñ = r2d1 − r1d2 + r1(r1 − r2)(g − 1)− 1.
In particular, N s+(r1, r2, d1, d2) is non-empty and irreducible.σL
If GCD(r1 − r2, d1 − d2) = 1 and GCD(r2, d2) = 1, the birational equivalence is an
isomorphism.
Moreover, in all cases, Nσ+ = Nσ+(r1, r2, d1, d2) is irreducible and hence, birationallyL L
equivalent to N s+. σL
Theorem 3.13 ([5, Th. 7.9.]). Let σ be any value in the range σm < 2g − 2 6 σ < σM ,
then N sσ is birationally equivalent to N sσ+. In particular it is non-empty and irreducible.L

Corollary 3.14 ([5, Cor. 7.10.]). Let (r,d) = (r1, r2, d1, d2) be such that
GCD(r2, r1 + r2, d1 + d2) = 1.
If σ is a generic value satisfying σm < 2g−2 6 σ < σM , then Nσ is birationally equivalent
to Nσ+, and in particular it is irreducible.L
Proof. Nσ = N sσ if GCD(r2, r1 + r2, d1 + d2) = 1 and σ is generic. In particular, we have
Nσ+ = N s+ , and the result follows from the last theorem. The reader may see the fullL σL
details in [5].
Theorem 3.15. There is an isomorphism
i∗
∼
k : Hj(N k+1,Z) −−−
=−→ Hjσ (N kσ ,Z) ∀j 6 σH(k)− 2(µ1 − µ)− 1H H
where N kσ = Nσ (k)(2, 1, d̃1, d̃2), σH = σH(k) = 2g−2+k, and µ1 = µ(E1) > µ(E) = µ.H H
Proof. In this case GCD(1, 3, d̃1 + d̃2) = 1 trivially, and σH = σH(k) is a σ-critical value
that satisfies
σm < 2g − 2 6 σH(k) < σM .
31
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Therefore, by the description of Muñoz et al. [23] of the critical values ([23] Lemma 5.2.
and Lemma 5.3.), the line bundle M → X satisfies in this case, the following:
σm < σH(k) = 3dM − d̃1 − d̃2
equivalently
dM = σH(k) + µ
and hence
d̃1 − dM − d̃2 − 1 = σH(k)− 2(µ1 − µ)− 1.
In such a case
d̃1 − 2d̃2 − (2g − 2) = σH(k)− 2(µ1 − µ) + k > σH(k)− 2(µ1 − µ)− 1
and then
2(d̃1 − 2d̃2 − (2g − 2)) > d̃1 − dM − d̃2 − 1.
Therefore, in this case
n(k) = d̃1 − dM − d̃2 − 1 = σH(k)− 2(µ1 − µ)− 1.
Finally, by Theorem 3.13 and by Corollary 3.14, the space N kσ = NH σ (k)(2, 1, d̃H 1, d̃2) is
birationally equivalent to Nσ+(k) = Nσ+(k)(2, 1, d̃1, d̃2), which is equal to the moduli spaceL L
N s = N s+( ) +( )(2, 1, d̃1, d̃2) of holomorphic stable triples also by Theorem 3.13, whereσL k σL k
σ+L (k) is the maximal critical value, depending on k in this case. The isomorphism then
follows by Corollary 3.11.
Corollary 3.16. There is an isomorphism
Hj(F k+1
∼
d1 ,Z) −−−
=−→ Hj(F kd1 ,Z)
for all j 6 σH(k)− 2(µ1 − µ)− 1 induced by the embedding 1.8. 
3.4 Cohomology of the (2, 1)-VHS
Because of the duality
Nσ(1, 2, d̃1, d̃2) ∼= Nσ(2, 1,−d̃2,−d̃1)
from Proposition 1.11, we get
Theorem 3.17. There is an isomorphism
i∗ : Hj
∼
(N = jk σc(k+1),Z) −−−−→( H (Nσc(k),Z) ∀j 6 )m(k)
where m(k) = min(−d̃1 − dM + d̃2 − 1, 2 − d̃1 + 2d̃2 − (2g − 2) + 1).
Proof. The result follows as the analogous to Corollary 3.11 varying d̃1 and d̃2 according
to the duality from Proposition 1.11.
Theorem 3.18. For k large enough, there is an isomorphism
i∗
∼
k : Hj(N k+1
= j k
σ ,Z) −−−−→ H (Nσ ,Z) ∀j 6 σH(k)− 4(µ2 − µ)− 1H H
where N kσ = Nσ (k)(1, 2, d̃1, d̃2), σH = σH(k) = 2g−2+k, and µ2 = µ(EH H 2) > µ(E) = µ.
32
Ronald A. Zúñiga-Rojas
Proof. In this case, by the duality from Proposition 1.11, and by the description of the σc
critical values (Muñoz et al. [23] Lemma 5.2. and Lemma 5.3.), the line bundle M → X
satisfies in this case, the following:
σm < σH(k) = 3dM + d̃1 + d̃2
equivalently
dM = −µ > −µ2
and hence
−d̃1 − dM + d̃2 − 1 = σH(k)− 4(µ2 − µ)− 1,
where, once again, σH = σH(k) is a σ-critical value satisfying
σm < 2g − 2 6 σH(k) < σM .
In such a case
−d̃1 + 2d̃2 − (2g − 2) = σH(k)− 6(µ2 − µ) + k >
σH(k)− 4(µ2 − µ) > σH(k)− 4(µ2 − µ)− 1
if k > 2(µ2 − µ) > 0 is large enough. Then
2(−d̃1 + 2d̃2 − (2g − 2)) > −d̃1 − dM + d̃2 − 1.
Therefore, in this case
m(k) = −d̃1 − dM + d̃2 − 1 = σH(k)− 4(µ2 − µ)− 1.
Hence, the result follows as the dual analogous to Theorem 3.15.
Corollary 3.19. For k large enough, there is an isomorphism
Hj
∼
(F k+1d2 ,Z) −−−
=−→ Hj(F kd2 ,Z)
for all j 6 σH(k)− 4(µ2 − µ)− 1 induced by the embedding 1.9. 
3.5 Cohomology of the (1, 1, 1)-VHS
Theorem 3.20. The pull-back
i∗ : H∗(F k+1 ∗ kk m1m2 ,Z)→ H (Fm1m2 ,Z)
induced by the embedding ik : F km1m2 → F
k+1
m1m2, is surjective.
Proof. This is a direct consequence of Theorem 1.38, Theorem 3.1 and Corollary 3.3.
Corollary 3.21. There is an isomorphism
Hj
∼
( (F
∞
)m1m2 ,Z) −−−
=−→ Hj(F km1m2 ,Z)
for all j 6 min m̄1 + k, m̄2 + k − 1.
Proof. It follows directly from Theorem 1.38, Corollary 3.3 and Corollary 3.4.
33
VHS of rank 3 k-Higgs bundles and M.S. of holomorphic triples
Acknowledgement
I would like to thank Peter B. Gothen for introducing me to the beautiful subject of
Higgs bundles. I thank Vicente Muñoz and André Gamma Oliveira for enlightening
discussions about the moduli space of triples; I thank Steven Bradlow too, for the time
and discussions about stable pairs and triples. I am grateful to Joseph C. Várilly for
helpful discussions.
Financial support from Fundação para a Ciência e a Tecnologia (FCT), and from Vicer-
rectoŕıa de Investigación de la Universidad de Costa Rica, is acknowledged.
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35