Stratifications on the Nilpotent Cone
of the moduli space of Hitchin pairs
15/06/2020
Peter B. Gothen1
Centro de Matemática da Universidade do Porto and
Departamento de Matemática
Faculdade de Ciências da Universidade do Porto
Rua do Campo Alegre, s/n
4197-007 Porto, Portugal
e-mail: pbgothen@fc.up.pt
Ronald A. Zúñiga-Rojas2
Centro de Investigaciones Matemáticas y Metamatemáticas CIMM
Escuela de Matemática, Universidad de Costa Rica UCR
San José 11501, Costa Rica
e-mail: ronald.zunigarojas@ucr.ac.cr
Abstract. We consider the problem of finding the limit at infinity (corresponding to
the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural
C∗-action on the moduli space. For general rank we provide an answer for Higgs bundles
with regular nilpotent Higgs field, while in rank three we give the complete answer. Our
results show that the limit can be described in terms of data defined by the Higgs field,
via a filtration of the underlying vector bundle.
Keywords: Higgs Bundles, Hitchin Pairs, Hodge Bundles, Moduli Spaces, Nilpotent
Cone, Vector Bundles.
MSC 2010 classification: Primary 14H60; Secondary 14D07.
Introduction
Over thirty three years ago, Hitchin [10] introduced Higgs bundles on Riemann surfaces
through dimensional reduction of the self-duality equations from R4 to R2, and they
appeared in the work of Simpson [16] motivated by uniformisation problems for higher
dimensional varieties. Since then, the moduli space of Higgs bundles has become an
important topic of research in many areas of geometry and mathematical physics and
there are even ramifications to number theory via the Langlands programme. Much
more detailed information and many references to relevant work can be found in the
following selection of (mainly) expository papers: [3], [4], [5], [8], [14], [15].
A Higgs bundle on a Riemann surface is a pair consisting of a holomorphic vector
bundle together with an endomorphism valued holomorphic one-form, called the Higgs
field.
1Partially supported by Centro de Matemática da Universidade do Porto (CMUP), financed by
national funds through FCT—Fundação para a Ciência e a Tecnologia, I.P., under the project
UIDB/00144/2020.
2Supported by Universidad de Costa Rica through Escuela de Matemática, specifically through
CIMM, through Projects 820-B5-202 and 820-B8-224. Partially supported by FCT (Portugal) through
grant SFRH/BD/51174/2010.
2 Stratifications on the Nilpotent Cone
Taking the characteristic polynomial of the Higgs field defines the Hitchin map, which
is a proper map from the moduli space of Higgs bundles to a vector space. It makes
the moduli space of Higgs bundles into an algebraic completely integrable Hamiltonian
system, and thus the generic fibre of the Hitchin map is an abelian variety. On the other
hand, the fibre over zero, named the nilpotent cone by Laumon XX, is highly singular
and it encodes many important properties of the moduli space: for example, the moduli
space deformation retracts onto it.
Another important attribute of the moduli space of Higgs bundles is that it carries
an action of the non-zero complex numbers C∗ via multiplication on the Higgs field. The
limit of the action on a Higgs bundle of z ∈ C∗ as z → 0 always exists, and thus the
moduli space has an associated Bial ynicki-Birula stratification. On the other hand, the
limit as z →∞ exists if and only if the Higgs bundle belongs to the nilpotent cone. These
limits are fixed points of C∗-action. Such fixed points are known as Hodge bundles and
are all contained in the nilpotent cone.
In our earlier work [6, 20] (see also [19, 21]) we investigated the limit as z → 0 of
any Higgs bundle and its relation to the Harder–Narasimhan filtration of the underlying
vector bundle, in order to better understand the relation between the Bia lynicki-Birula
and Shatz stratifications of the moduli space (the latter being defined by the Harder–
Narasimhan type). The case of rank two had already considered by Hitchin [10], who
observed that in this case the two stratifications coincide. This is no longer the case in
higher rank and, indeed, the general problem is quite intricate; a complete solution is
given in [6] for rank 3. The companion problem of finding the limit of Higgs bundle in
the nilpotent cone as t→∞ was also considered in the second author’s PhD thesis [21]
and the result of the present article are essentially contained there. We have decided to
write them up here in view of recent interest in the fine structure of the Bia lynicki-Birula
stratification of the nilpotent cone.
Our main results are as follows. In the case when the Higgs field of a Higgs bundle
in the nilpotent cone is a regular nilpotent, there is an associated graded Higgs bundle
induced from the filtration obtained by taking the kernels of iterates of Φ. This Higgs
bundle is in fact a Hodge bundle and we show that it is exactly the limit of the action of
z ∈ C∗ on the original Higgs bundle as z →∞. The precise statement is in Theorem 2.1
below. On the other hand, when the Higgs field is not a regular nilpotent, the situation
is again more intricate. We analyse the situation completely in the case of rank 3 and
show that there is a refinement of the aforementioned filtration obtained using also the
image of the Higgs field, which allows to identify the limit as a function of topological
invariants of the filtration. It is notable that the answer depends only on properties of
the Higgs field and not on the stability properties of the underlying vector bundle (as
opposed to situation for z → 0). The precise statement is in Theorem 3.1 below.
We mention that in this paper we work with the moduli space of Hitchin pairs, since
our results and methods are in this generality: this means that we allow the Higgs field
to be twisted by any holomorphic line bundle of degree greater than or equal to that of
the canonical bundle of the Riemann surface, rather than just the canonical bundle.
This paper is organised as follows. In Section 1 we give some necessary preliminaries
about Hitchin pairs, Higgs bundles and their moduli spaces, and we introduce the Hitchin
map, the Nilpotent Cone and the C∗-action. Then, in Section 2, we present the result in
general rank for Hitchin pairs with regular nilpotent Higgs field. Finally, in Section 3, we
give the complete result for Hitchin pairs of rank 3 with nilpotent Higgs field.
2
P.B. Gothen and R.A. Zúñiga-Rojas 3
1 Preliminaries on Hitchin pairs and their moduli
In this section we review some standard facts about Hitchin pairs and their moduli.
Details can be found in, for example, Hitchin [10, 11], Nitsure [13] and Simpson [17].
Let X be a compact, connected and oriented Riemann surface of genus g > 2 and let
L→ X be a holomorphic line bundle.
Definition 1.1. A Hitchin pair over X is a pair (E,Φ) where the underlying vector
bundle E → X is a holomorphic vector bundle and the Higgs field Φ : E → E ⊗ L is
holomorphic.
If we need to specify the line bundle L, we say that the Hitchin pair (E,Φ) is twisted
by L.
Definition 1.2. A Higgs bundle over X is a Hitchin pair (E,Φ) twisted by the canonical
line bundle K = K = T ∗X X.
The slope of a vector bundle E is the quotient between its degree and its rank:
µ(E) = deg(E)/ rk(E).
Recall that a vector bundle E is semistable if µ(F ) 6 µ(E) for all non-zero holomorphic
subbundles F ⊆ E, stable if it is semistable and strict inequality holds for all non-zero
proper F , and polystable if it is the direct sum of stable bundles, all of the same slope.
The slope of a Hitchin pair is the slope of its underlying vector bundle and the stability
condition is defined analogously to the vector bundle situation, except that the slope
condition is applied only to Φ-invariant subbundles, i.e., holomorphic subbundles F ⊆ E
such that Φ(F ) ⊆ F ⊗ L.
The moduli space ML(r, d) of S-equivalence classes of semistable rank r and degree
d Higgs bundles was first constructed by Nitsure [13]. The points ofML(r, d) correspond
to isomorphism classes of polystable Hitchin pairs. When r and d are co-prime any
semistable Hitchin pair is automatically stable. Henceforth we shall assume that we are
in this situation and that deg(L) > 2g−2. ThenML(r, d) is a smooth complex manifold
of complex dimension
r2 deg(L) + 1 + dimH1(X,L).
The moduli space is non-compact but there is a proper map, the so-called Hitchin map,
defined by:
χ :M[ L(r, d]) −→ H( 0(X,L)⊕ . . .⊕H) 0(X,Lr)
(E,Φ) −7 → tr(Φ) det(Φ) (1.1), . . .,
whose components are holomorphic sections obtained as the coefficients of the (fibrewise)
characteristic polynomial of Φ. When L = K, the moduli space is a holomorphic sym-
plectic manifold and the Hitchin map endows is an algebraically completely integrable
Hamiltonian system whose generic fibre is an abelian variety. (For general L, this has
been generalised to the Poisson setting by Bottacin [2] and Markman [12].) On the other
hand, the fibre of the Hitchin m{a[p over z]ero, }
χ−1(0) := (E,Φ) ∈ML(r, d) | χ(Φ) = 0
is known as the Nilpotent Cone in the moduli space, and has a complicated structure
with several irreducible components.
3
4 Stratifications on the Nilpotent Cone
Next we review some standard facts about the holomorphic action of the multiplicative
group C∗ on ML(r, d). The action is defined by the multiplication:
z · (E,Φ) 7→ (E, z · Φ).
The limit (E0, ϕ0) = lim(E, z · Φ) exists for all (E,Φ) ∈ M(r, d). On the other hand, it
z→0
follows from the properties of the Hitchin map that the limit (E∞,Φ∞) = lim (E, z · Φ)
z→∞
exists if and only if (E,Φ) belongs to the nilpotent cone χ−1(0). When the limit of
(E, z · Φ) as z → 0 or z → ∞ exists it is fixed by the C∗-action. Moreover, a Hitchin
pair (E,Φ) is a fixed poin⊕t of the C∗-action if and only if it is a Hodge bundle, i.e., thereis a decomposition E = pj=1 Ej with respect to which the(Higgs field has weig)ht one:
Φ: Ej → Ej+1 ⊗ L. The type of the Hodge bundle (E,Φ) is rk(E1), . . . , rk(Ep) .
We shall consider the moduli space from the complex analytic point of view. For this,
fix a C∞ complex vector bundle E → X of rank r and degree d. A holomorphic structure
on E is given by a ∂̄-operator
∂̄E : A0(E)→ A0,1(E)
and we thus obtain a holomorphic vector bundle E = (E , ∂̄E). A Hitch(in pair (E,Φ)) arises
from a pair (∂̄E,Φ) consisting of a ∂̄-operator and a Higgs field Φ ∈ A0 End(E)⊗L which
is holomorphic, i.e., ∂̄E,LΦ = 0, where ∂̄E,L denotes the ∂̄-operator on the underlying
smooth bundle of End(E)⊗L defining the holomorphic structure. The natural symmetry
group is the complex g{auge group }
GC = g : E → E | g is a C∞-bundle isomorphism ,
which acts on pairs (∂̄E,Φ) in the standard way:
g · (∂̄E,Φ) = (g ◦ ∂̄ −1E ◦ g , g ◦ Φ ◦ g−1).
The moduli space{can then be viewed as the quotient
3
}
ML(r, d) = (∂̄E,Φ) | Φ is holomorphic and (E,Φ) is polystable /GC.
2 Limit at infinity for regular nilpotent Higgs field
Let (E,Φ) be a stable Hitchin pair of rank r and degree d which represents a point in
the nilpotent cone χ−1(0) ⊆ ML(r, d). Let p ∈ N be the least positive integer such that
Φp = 0 and Φp−1 6= 0. Then p 6 r and Φ is regular if p = r. Since we are working over
a Riemann surface, taking the saturation of the kernel sheaf of Φp−j+1 : E → E ⊗Lp−j+1
defines a subbundle Ej ⊂ E. We obtain in this way a filtration of E,
E = E1 ⊃ E2 ⊃ · · · ⊃ Er ⊃ Er+1 = 0 (2.1)
and, clearly,
Φ(Ej) ⊆ Ej+1 ⊗ L. (2.2)
3See Atiyah & Bott [1, Section 14] for general holomorphic bundles, and Hausel & Thaddeus [9,
Section 8] for the particular cases of Hitchin pairs and Higgs bundles.
4
P.B. Gothen and R.A. Zúñiga-Rojas 5
Define Ēj = Ej/Ej+1. Then, in view of (2.2), Φ induces a map ϕj : Ēj → Ēj+1⊗L. Note
that if Φ is regular then the inclusions in (2.1) are all strict of co-dimension one. Thus,
when Φ is regular, we obtain a Hodge bundle of rank r and degree d of type (1, . . . , 1):
 
( ) (  0 . . . . . . . . . 0⊕ ∑−1 ⊕ 

ϕ1 0 . . . . . . 0 )r r r 
(Ē, Φ̄) = Ēj, ϕj = Ē , 0 ϕj 2 0 . . . 0  . (2.3)j=1 j=1 j=1 ... . . . . . . . . . ...
0 . . . 0 ϕr−1 0
Theorem 2.1. Let (E,Φ) be a stable Hitchin pair of rank r and degree d which represents
a point in the nilpotent cone χ−1(0) ⊆ML(r, d) and assume that Φ is a regular nilpotent,
i.e., Φr−1 =6 0. Then lim (E, z ·Φ) = (Ē, Φ̄), where (Ē, Φ̄) is given by (2.3). In particular
z→∞
the limit is a Hodge bundle of type (1, . . . , 1).
Proof. Using the notation introduced above we may consider a smooth splitting
⊕r
E ∼= Ē
∞ j
.
C
j=1
Then the Higgs field takes the triangular form:
  0 . . . . . . . . . 0

ϕ21 0 . . . . . . 0 
Φ =  ϕ31 ϕ32 0 . . . 0  ... . . . . . . . . . ... 
ϕr,1 . . . ϕr,r−2 ϕr,r−1 0
where ϕij : Ēj → Ēi ⊗ L and we note that ϕj,j−1 = ϕj in the notation introduced above.
The ∂̄-operator defining the holomorphic structure on E is of the form:
  ∂̄1 0 . . . 0 . . =  β . . ∂̄  21
∂̄2 . . 
E
... . .

. . . . 0 
βr,1 . . . βr,r−1 ∂̄r
( )
where ∂̄j is the corresponding holomorphic structure of Ēj, and β ∈ Ω0,1ij X,Hom(Ēj, Ēi) .
We now define a family of complex C∞-gauge transformations g(z) ∈ GC by:
 
 1 0 . . . 0
( ) = 

0 .z . . ... 
g z  ... . .  .. . . . 0 
0 . . . 0 zr−1
5
6 Stratifications on the Nilpotent Cone
Then
g−1(z)(z · Φ)g(z)    1 0 . . . 0  0

. . . . . . 0 1 0 . . . 0
 0 z−1
. . ..
= . . .. . . . . 
 
 zϕ 0 . . . 0  . . ..  21  0 z . .. .. . . . . . . . 0 . . ...  ... . . . . . . 0 
 0 . . . 0 z1−r zϕr,1 . . . zϕr,r−1 0 0 . . . 0 zr− 
1
 0 . . . . . . . . . 0   0 . . . . . . . . . 0  ϕ21 0 . . . . . . 0   ϕ21 0 . . . . . . 0 =  z
−1ϕ31 ϕ32 0 . . . 0
 .. . . . . . . .. 

 −−−→  0 ϕ32 0 . . . 0  =: Φ∞,z→∞. . . . .   ... . . . . . . . . . ... 
z1−pϕr,1 . . . z
−1ϕr,r−2 ϕr,r−1 0 0 . . . 0 ϕr,r−1 0
and also
g−1(z) ∂̄E g(z)    1 0 . . . 0  ∂̄1 0 . . . 0 
1 0 . . . 0 
=  0
.
z−1 . . ...  . .β ∂̄ . . .21 2 . . . ... . . . . . . 0  ... . . . . . . 0 
 0 z . . ..  ... . . . . . . 0 
0 . . . 0 z1−r βr,1 . . . βr,r−1 ∂̄r  0 . . . 0 zr− 
1
 ∂̄1 0 . . . 0   ∂̄1 0 . . . 0
=  . .  z−1β . .21 ∂̄2 . .  −−−→  0
.
∂̄ . . ... 
 2  ∞... . . . . . . 0  z→∞  ... . . . . . . 0  =: ∂̄E ,
z−rβr,1 . . . z
−1βr,r−1 ∂̄r 0 . . . 0 ∂̄r
where the limits are taken in the configuration space of all pairs (∂̄E,Φ), up to gauge
equivalence. Moreover, the fact that ∂̄EΦ = 0 immediately implies that ∂̄∞ ∞E Φ = 0 and,
clearly, the Hitchin pair defined by (∂̄∞ ∞E ,Φ ) is (Ē, Φ̄). Hence, in order to prove that the
stated limit is valid in the moduli space, it only remains to prove that this Hitchin pair
is stable. For this we observe that the only Φ̄-invariant subbundles of Ē are those of the
form
Ēl ⊕ Ēl+1 ⊕ · · · ⊕ Ēr ⊆ Ē
and note that the slope of such a subbundle equals that of El ⊆ E because they are
isomorphic as C∞-bundles. Thus, since the subbundle El ⊆ E is Φ-invariant, the stability
of (Ē, Φ̄) follows from that of (E,Φ).
Remark 2.2. Since in rank two a nilpotent Higgs field is either zero or regular, the pre-
ceding theorem, together with the results of our previous paper [6], gives a complete
description of the closure of the C∗-orbit of a rank 2 Hitchin pair in the nilpotent cone.
Indeed, as we have just seen, the type of the limiting VHS as z → ∞ is determined
by the Higgs field and, from [6, Corollary 3.2], the type of the limiting VHS as z → 0
is determined by the Harder–Narasimhan type of the underlying vector bundle. These
observations were already made by Hausel [7].
6
P.B. Gothen and R.A. Zúñiga-Rojas 7
3 Rank Three Hitchin Pairs in the Nilpotent Cone
In this section we determine the limit lim (E, z ·Φ) for any rank 3 Hitchin pair (E,Φ) in
z→∞
the nilpotent cone χ−1(0) ⊂ (E,Φ) ∈ ML(3, d). Since the case Φ = 0 is trivial and the
case when Φ is a regular nilpotent has already been covered, it only remains to consider
the case when Φ 6= 0 and Φ2 ≡ 0. For completeness we state the full result.
Theorem 3.1. Let (E,Φ) be a stable Hitchin pair of rank 3 and degree d which represents
a point in the nilpotent cone χ−1(0) ⊆ ML(3, d). Then one of the following alternatives
holds:
(a) The Higgs field Φ vanishes identically and lim (E, z · Φ) = (E,Φ) = (E, 0).
z→∞
(b) The Higgs field Φ is a regular nilpotent (i.e., Φ2 =6 0) and there is a filtration
E = E1 ⊃ E2 ⊃ E3 ⊃ E4 = 0
with each step of co-dimension one and such that Φ(Ej) ⊂ Ej+1 ⊗ L for j = 1, 2, 3.
In this case,
(  0 0 0 )
(E∞,Φ∞) = lim (E, z · Φ) = Ē1 ⊕ Ē2 ⊕ Ē3, ϕ 0 0 1  (3.1)
z→∞
0 ϕ2 0
is a Hodge bundle of type (1, 1, 1) where
Ēj = Ej/Ej+1 and ϕj : Ēj−1 → Ēj ⊗ L
is induced by Φ.
(c) The Higgs field Φ satisfies Φ2 = 0 but does not vanish identically, and there is a
filtration
E = E1 ⊃ E2 ⊃ E3 ⊃ E4 = 0
with each step of co-dimension one and satisfying Φ(Ej) ⊂ Ej+2 ⊗ L for j = 1, 2.
The topological invariants of E2 and E3 are constrained by the inequalities
µ(E)− deg(L)/2 < µ(E/E2 ⊕ E3) < µ(E) + deg(L)/2. (3.2)
Moreover,
(c.1.) if µ(E1/E2 ⊕ E3) < µ(E) then ( ( ))
(E∞,Φ∞) = lim (E, z · Φ) = E1/E2 ⊕
0 0
E2, 0 (3.3)z→∞ ϕ
is a Hodge bundle of type (1, 2) where ϕ : E1/E2 → E2⊗L is induced by Φ and,
(c.2.) if µ(E1/E2 ⊕ E3) > µ(E) then ( ( ))
(E∞,Φ∞) = lim ( 0 0E, z · Φ) = E1/E3 ⊕ E3, 0 (3.4)z→∞ ϕ
is a Hodge bundle of type (2, 1) where ϕ : E1/E3 → E3 ⊗ L is induced by Φ.
7
8 Stratifications on the Nilpotent Cone
Proof. If Φ vanishes identically it is clear that the statement of case (a) holds and, when
Φ is a regular nilpotent, the statement of case (b) follows from Theorem 2.1 with r = 3.
It remains to consider the case when Φ 6= 0 and Φ2 ≡ 0. Then, we may consider:
E2 = k̃er(Φ) ⊂ E1 = E and E −13 = ĩm(Φ)⊗ L ⊂ E2,
where the tildes indicate taking the saturation of a subsheaf. We note that, necessarily
from our assumptions on Φ, that rk(E2) = 2, rk(E3) = 1, and that we obtain a filtration
with the properties stated in case (c).
We proceed to prove the constraints (3.2). From stability of (E,Φ) we have the
inequalities
µ(E3) < µ(E) ⇐⇒ 3 deg(E3) < d, (3.5)
µ(E2) < µ(E) ⇐⇒ 3 deg(E2) < 2d, (3.6)
since E2 and E3 are Φ-invariant subbundles of E. Moreover, Φ induces a non-zero map
of line bundles E/E2 → E3 ⊗ L and hence
deg(E3) + deg(L) > d− deg(E2). (3.7)
Now, using (3.7) and (3.6) we obtain
2µ(E/E2 ⊕ E3) = d− deg(E2) + deg(E3)
> 2d− 2 deg(E2)− deg(L)
2
> 3d− deg(L)
which is the first of the inequalities (3.2). Similarly, from using (3.7) and (3.5) we obtain
2µ(E/E2 ⊕ E3) = d− deg(E2) + deg(E3)
6 2 deg(E3) + deg(L)
2
< 3d+ deg(L)
which is the second of the inequalities (3.2).
It remains to identify the limit of (E, z · Φ) as z → ∞. For this we take, as usual, a
smooth splitting
E ∼= E
∞ 1/E2 ⊕ E2/E3 ⊕ E3.C
With respect to this splitting we have, from the definitions of E2 and E3, that
Φ = 

 0 0 00 0 0  .
ϕ 0 0
With respect to each of the smooth splittings E ∼= E1/E2 ⊕E2 and E ∼= E1/E3 ⊕E3 we
can take a family of smooth complex gauge(transfo)rmations g(z) ∈ GC defined by
( ) = 1 0g z 0 z
(interpreting each entry as a block of the appropriate size). Exactly the same argument
as in the proof of Theorem 2.1 shows that we have the convergence in the configuration
space, up to gauge equivalence, stated in each of the sub-cases (c.1.) and (c.2.). It remains
to prove that the convergence also holds in the moduli space, i.e., that the Hitchin pairs
in (3.3) and (3.4) are stable under the respective hypotheses on µ(E/E2 ⊕ E3).
8
P.B. Gothen and R.A. Zúñiga-Rojas 9
Case (c.1.) The proper non-trivial Φ∞-invariant subbundles F ⊂ E∞ are of two kinds:
(1) F ⊆ E2 ⊆ E∞ any non-zero subbundle (which may equal E2). In this case F defines
a Φ-invariant subbundle of the stable Hitchin pair (E,Φ) and hence µ(F ) < µ(E) =
µ(E∞) as desired.
(2) F = E1/E3 ⊕ E3 ⊆ E∞. In this case µ(F ) = µ(E1/E3 ⊕ E3) < µ(E) = µ(E∞) by
hypothesis.
Case (c.2.) Again, the proper non-trivial Φ∞-invariant subbundles F ⊂ E∞ are of two
kinds:
(1) F = L⊕ E3 ⊆ E∞ for a proper subbundle L ⊆ E1/E3 (which may be zero). In this
case we can lift L to a subbundle L̃ ⊂ E and we note that E3 ⊆ L̃. Hence V ⊆ E is
Φ-invariant and µ(L⊕ E3) = µ(V ) < µ(E) = µ(E∞) as we wanted.
(2) F = E2/E3 ⊆ E1/E ∞3 ⊆ E . In this case µ(F ) = µ(E2/E3) and we have
( ⊕ ) = 1
( )
µ E1/E2 E3 2(3µ(E1)− 2µ(E2) +)µ(E3)
= 12 3µ(E)− µ(E2/E3) .
Hence the hypothesis µ(E1/E2 ⊕ E3) > µ(E) is equivalent to µ(E2/E3) < µ(E), as
desired.
Acknowledgement
The authors are members of the Vector Bundles and Algebraic Curves (VBAC) research
group.
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