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THE BOREL SUBGROUP AND
BRANES ON THE HIGGS MODULI SPACE
EMILIO FRANCO AND ANA PE ´
ON-NIETO
Abstract. We consider two families of branes supported on the singular locus
of the moduli space of Higgs bundles over a smooth projective curve X. On the
one hand, a (BBB)-brane Car(L) constructed from the Cartan subgroup and
a topologically trivial line bundle Lon Jac0(X). On the other hand, a (BAA)-
brane Uni(L) associated to the unipotent radical of the Borel subgroup and
the previous line bundle L. We give evidence of both branes being dual under
mirror symmetry, in the sense that an ad-hoc Fourier–Mukai integral functor
relates the restriction of the hyperholomorphic bundle of the (BBB)-brane to
a generic Hitchin fibre, with the support of the (BAA)-brane. We provide
analogous constructions of (BBB)-branes and (BAA)-branes associated to a
choice of a parabolic subgroup P with Levi subgroup L, obtaining families of
branes which cover the whole singular locus of the moduli space.
Contents
1. Introduction 2
2. Preliminaries 4
2.1. Non-abelian Hodge theory 4
2.2. The Hitchin fibration 5
2.3. Rank one torsion free sheaves on connected nodal reducible curves 6
2.4. Fourier–Mukai on fine compactified Jacobians 9
3. A (BBB)-brane from the Cartan subgroup 10
3.1. A hyperholomorphic bundle on the Cartan locus 10
3.2. Spectral data for the Cartan locus 11
4. (BAA)-branes from the unipotent radical of the Borel 13
4.1. An isotropic subvariety 13
4.2. Spectral data for Uni(L) 14
5. Duality 21
6. Parabolic subgroups and branes on the singular locus 25
6.1. Levi subgroups and the singular locus 25
6.2. Parabolic subgroups and complex Lagrangian subvarieties 26
References 27
Date: September 13, 2017.
E. Franco is currently supported by the project PTDC/MAT- GEO/2823/2014 funded by
FCT (Portugal) with national funds and has been supported by the FAPESP postdoctoral grant
number 2012/16356-6 and BEPE-2015/06696-2.
A. Pe´on-Nieto was supported by a postdoctoral grant associated to the project FP7 - PEOPLE
- 2013 - CIG - GEOMODULI number: 618471, and a postdoctoral contract of the Heidelberg
Institute for Theoretical Studies.
1
arXiv:1709.03549v1 [math.AG] 11 Sep 2017
2 EMILIO FRANCO AND ANA PE ´
ON-NIETO
1. Introduction
Hitchin introduced in [H1] Higgs bundles over a smooth projective curve X
and soon it was noted that their moduli space Mn[H1, Si1, Si2, N] carries a very
interesting geometry. In particular Mncan be endowed with a hyperk¨ahler structure
(g, Γ1,Γ2,Γ3) [H1, Si1, Si2, Do, C] and fibres over a vector space Mn→H with
Lagrangian tori as generic fibres [H2]. A natural generalization is to consider Higgs
bundles for complex reductive Lie groups other than GL(n, C). After the work of
[HT, DG, DP], the moduli spaces of Higgs bundles for two Langlands dual groups
equipped with the afore mentioned fibrations become SYZ mirror partners (as
defined by [HT] based on work by [SYZ]). In this paper we focus in the case of
GL(n, C), which is Langlands self-dual.
Branes in the Higgs moduli space were introduced in [KW] and have since at-
tracted great attention. A (BBB)-brane in the the moduli space Mnis given by a
pair (N,F), where N ⊂Mnis a hyperholomorphic subvariety and Fa hyperholo-
morphic bundle on N. Additionally, a (BAA)-brane is a pair (S, W) where S⊂Mn
is a subvariety which is complex Lagrangian with respect to the holomorphic sym-
plectic form in complex structure Γ1, and W = (W, ∇W) is a flat bundle over
S. As stated in [KW], mirror symmetry is expected to interchange (BBB)-branes
with (BAA)-branes. This is explored in [H4, BS1, BG, HS, GW, BCFG, H5, Ga,
FJ, BS2], where (BBB) and (BAA)-branes are constructed and studied. The first
obstacle one encounters to prove a duality statement is the difficulty to construct
a hyperholomorphic bundle. Hitchin [H4] already provides a description of a non-
trivial hyperholomorphic bundle arising from the Dirac–Higgs bundle. This appears
also in recent work [H5, Ga, FJ], where a duality between (BBB) and (BAA)-branes
is proven in the smooth locus of the Hitchin fibration.
In this paper we construct a family of (BBB) and (BAA)-branes indexed by a
topologically trivial line bundle L → Jac0(X) and supported on the singular locus of
the Hitchin fibration. Both constructions involve the Borel subgroup B <GL(n, C)
and the Cartan subgroup C <B. We denote by Car and Bor the locus of Higgs
bundles reducing its structure group to C and B respectively. Since C is a complex
reductive subgroup of GL(n, C), we observe that Car is a hyperk¨ahler subvariety
of Mn. We construct a hyperholomorphic vector bundle on Car out of our line
bundle L → Jac0(X) and this constitutes our (BBB)-brane, that we denote by
Car(L). We also define a subvariety Uni(L)⊂Mngiven by those Higgs bundles
in Bor whose underlying vector bundle has a fixed reduction to C, defined in terms
of L. We prove that Uni(L) is complex Lagrangian, and define Uni(L) to be the
(BAA)-brane given by the trivial line bundle on Uni(L) with the trivial connection.
To study the behaviour of Car(L) and Uni(L) under mirror symmetry one
would like to transform them under a fibrewise Fourier–Mukai transform. Since
these branes are supported on the singular locus, the Hitchin fibers are not fine
compactified Jacobians, and therefore a full Fourier–Mukai transform is not known
to exist, not even after restricting ourselves to the open subset of the Cartan locus
whose associated spectral curves are nodal. We can however define an ad-hoc
Fourier–Mukai transform relating the generic loci of both branes. We expect that
the weaker form of duality proven here would be induced from the global duality if
a full Fourier–Mukai transform were to exist. In view of these results we conjecture
that Car(L) and Uni(L) are dual branes under mirror symmetry.
We finish by discussing how this construction can be generalized to a large class
of branes in the moduli space Mnof rank nHiggs bundles covering the whole
singular locus. In the (BBB)-case, the support of these branes correspond to the
image of Mr1× · · · × Mrs, or equivalently, the locus of those Higgs bundles reducing
its structure group to the Levi subgroup GL(r1,C)× · · · × GL(rs,C). We observe
THE BOREL SUBGROUP AND BRANES 3
that these subvarieties cover the singular locus of Mn. The (BAA)-brane is given
by a complex Lagrangian subvariety constructed in a similar way as before, but
substituting the Borel subgroup with the parabolic subgroup associated to the
partition n=r1+· · · +rs.
This paper is organized as follows. Section 2 gives the necessary background on
the Hitchin system, torsion-free rank one sheaves on reducible curves, and Fourier–
Mukai transforms over fine compactified Jacobians on reduced curves.
Section 3 studies the (BBB)-brane Car(L). Its construction is addressed in
Section 3.1. We consider the Cartan locus, Car, given by those Higgs bundles
whose structure group reduces to the Cartan subgroup C ∼
=(C×)n<GL(n, C). The
Cartan locus is given by the image of c: Symn(M1)→Mn, where M1is the rank
one Higgs moduli space. Since the projection M1→Jac0(X) is compatible with the
three complex structures Γ1, Γ2and Γ3, we prove that the choice of a holomorphic
bundle on Jac0(X) yields a hyperholomorphic bundle on Car (Proposition 3.1,
Lemma 3.2 and Remark 3.3). For the remainder of the paper we focus on the
case of a line bundle Lon Jac0(X), which produces the (BBB)-brane Car(L). In
Section 3.2 we analyze the spectral data for the Higgs bundles in Car (Proposition
3.7), which is crucial to study the behaviour of Car(L) under mirror symmetry.
Section 4 addresses the construction and description of the (BAA)-brane Uni(L).
In Section 4.1 we consider Bor, the locus of all the Higgs bundles reducing to
the Borel subgroup B. Taking those whose underlying vector bundle project to a
certain C-bundle determined by L, we obtain a subvariety Uni(L) which is isotropic
by gauge considerations (Proposition 4.2). In Section 4.2 we study the generic
Hitchin fibers corresponding to completely reducible spectral curves with nodal
singularities, obtaining that they are fully contained in Bor (Theorem 4.4). We
next compute the spectral data of the points of Uni(L) in Proposition 4.5, what
allows us to show that it is mid-dimensional and, therefore, Lagrangian (Theorem
4.7).
After Sections 3.2 and 4.2 we have a description of the generic restriction of
Car(L) and Uni(L) to a Hitchin fibre, which is isomorphic to the coarse com-
pactified Jacobian of a reduced but reducible curve. We study in Section 5 the
transformation under Fourier–Mukai of the restriction of Car(L), in spite of the
lack of literature on the construction of Poincar´e sheaves over coarse compactified
Jacobians. To overcome this problem, we imitate in (5.1) the construction of the
Poincar´e bundle for fine compactified Jacobians that we reviewed in Section 2.4. In
this case, instead of starting from the universal sheaf bundle for the classification
of rank one torsion free sheaves on our spectral curve, we use a universal sheaf for
the Cartan locus of the Hitchin fibre. As discussed in Remark 5.1, one expects that
a Poincar´e sheaf on the whole Hitchin fibre would restrict to this ad-hoc Poincar´e
sheaf. Hence, it is natural to study the behaviour of Car(L) under a Fourier–Mukai
integral functor constructed with it, which we do. We obtain that the generic re-
striction of Car(L) to a Hitchin fibre is sent to corresponding restriction of Uni(L)
(Corollary 5.4). This lead us to conjecture that Car(L) and Uni(L) are dual branes
under mirror symmetry.
In Section 6 we adapt the above results to arbitrary parabolic subgroups. Given
a partition n=r1+· · · +rswe consider the associated parabolic subgroup Pr<
GL(n, C) with Levi subgroup Lr<Pr. In Section 6.1 we consider the subvariety
Mrof Mn, consisting of Higgs bundles whose structure group reduces to Lr, and
describe the intersection with generic Hitchin fibers (Proposition 6.3). The variety
Mris a complex subscheme for Γ1, Γ2and Γ3, hence the support of a (BBB)-
brane. By varying the partition r, we produce families of branes covering the
4 EMILIO FRANCO AND ANA PE ´
ON-NIETO
strictly semistable locus of Mn. On the other hand, in Section 6.2 we consider Unir,
consisting of Higgs bundles with structure group reducing to Prand fixed associated
graded bundle. We prove that this is a Lagrangian submanifold (Theorem 6.5), and
so a choice of flat bundle on it produces a (BAA)-brane. A look at the spectral data
of both Mrand Unir, as well as the comparison with the case P(1,...,1), indicates
the existence of a duality.
Acknowledgements. We would like to thank P. Gothen, M. Jardim, A. Oliveira, C.
Pauly and A. Wienhard for their kind support and inspiring conversations.
2. Preliminaries
2.1. Non-abelian Hodge theory. Let Xbe a smooth projective curve over C.
A Higgs bundle over Xis a pair (E, ϕ) given by a holomorphic vector bundle
E→Xand a Higgs field ϕ∈H0(X, End(E)⊗K), a holomorphic section of the
endomorphisms bundle twisted by the canonical bundle Kof X[H1, Si, Si1, Si2].
The moduli space of rank nand degree 0 semistable Higgs bundles on X. was
constructed in [H1, Si1, Si2, N]. This is a quasi-projective variety Mnof dimension
(2.1) dim Mn= 2n2(g−1) + 2.
It can be constructed as a GIT quotient as follows: fix a topological bundle Eof
degree 0 on X. Consider Athe space of holomorphic structures on E. This is an
affine space modelled on Ω0,1(X, ad(E)) whose cotangent bundle is
T∗
A=A × Ω0(X, ad(E)⊗K)
where we have identified ad(E) and its dual by means of the Killing form (rather,
a non degenerate extension of it to the center, to which we will henceforth refer
as Killing form). Inside T∗
A=A × Ω0(X, ad(E), we consider stable pairs (T∗
A)s,
which are pairs (∂A, ϕ)∈T∗
Asatisfying that there exists a hermitian metric hon
Esuch that the Chern connection ∇hassociated with (E, ∂A) and hsatisfies:
1) ∇2
h+ [ϕ, ϕ∗h]=0
2) ∂A(ϕ)=0.
Now, the complex gauge group
Gc= Ω0(X, Aut(E))
acts on (T∗
A)s, and we may identify
Mn∼
=T∗
A//Gc= (T∗
A)s/Gc,
where the double quotient denotes the GIT quotient. This is a complex manifold
with complex structure Γ1.
Let ηbe a Hermitian metric on the topological bundle E→X. Let
G= Ω0(X, Aut(E, η)),
be the unitary gauge group of automorphisms of Epreserving the metric η. Also,
one can naturally define three complex structures e
Γ1,e
Γ2and e
Γ3on T∗
Asatisfying
the quaternionic relations, together with a hyperk¨ahler metric preserved by G. This
action defines a moment map µiassociated to each of the complex structures e
Γi,
and one can define the hyperk¨ahler quotient by the action of G. The complex
structures e
Γ1,e
Γ2and e
Γ3on T∗
Adescend naturally to complex structures Γ1, Γ2
and Γ3on the quotient. It is proven in [H1, Si1, Si2] that the moduli space of
Higgs bundles is identified with the hyperholomorphic quotient equipped with the
complex structure Γ1,
Mn∼
=µ−1
1(0) ∩µ−1
2(0) ∩µ−1
3(0).G.
THE BOREL SUBGROUP AND BRANES 5
Additionally, [Do, C] proved that the moduli space of rank nflat connections on
the C∞vector bundle Eover Xof degree 0 is isomorphic to the above hyperk¨ahler
quotient equipped with the complex structure Γ2.
The hyperk¨ahler structure defined on Mninduces a holomorphic 2-form Ω1=
ω2+ iω3on Mn, where ω2and ω3are the K¨ahler forms associated to Γ2and Γ3.
We next give the expression of Ω1by means of the gauge theoretic construction of
Mn. Let (∂A, ϕ)∈T∗
As, and consider two tangent vectors
(˙
Ai,˙ϕi)∈T(∂A,ϕ)T∗
Ai= 1,2
we have
(2.2) Ω1(˙
A1,˙ϕ1),(˙
A2,˙ϕ2)=ZX
˙
A1˙
∧˙ϕ2−˙
A2˙
∧˙ϕ1.
where to define the wedge product ˙
∧, we identy Ω(0,1)(X, ad(E)) ∼
=(Ω0(X, ad(E))⊗
Ω0,1
X) and Ω0(ad(E)⊗K)∼
=(Ω0(ad(E)⊗Ω1,0
X), and for Zi⊗ωi,i= 1,2, Zi∈
Ω0(X, ad(E)), ωi∈Ω1(X), we set
(Z1⊗ω1)˙
∧(Z2⊗ω2) = hZ1, Z2i ⊗ ω1∧ω2
with h,ithe Killing form.
2.2. The Hitchin fibration. We recall here the spectral construction given in
[H2, BNR]. Let (q1, . . . , qn) be a base of the algebra C[gl(n, C)]GL(n,C)of regular
functions on gl(n, C) invariant under the adjoint action of GL(n, C). We choose
them so that deg(qi) = i. The Hitchin map is defined by
h: Mn−→ H := Ln
i=1 H0(X, K i)
(E, ϕ)7−→ (q1(ϕ), . . . , qn(ϕ)) .
It is a surjective proper morphism [H2, N] endowing the moduli space with the
structure of an algebraically completely integrable system. In particular, its generic
fibers are abelian varieties and every fiber is a compactified Jacobian. To describe
these, consider the total space |K|of the canonical bundle and the obvious algebraic
surjection π:|K| → X. We note that the pullback bundle π∗K→ |K|admits a
tautological section λ. Given an element b∈H, with b= (b1, . . . , bn), we construct
the spectral curve Xb⊂ |K|by considering the vanishing locus of the section of
π∗Kn
λn+π∗b1λn−1+· · · +π∗bn−1λ+π∗bn.
The restriction of π:|K| → Xto Xbis a ramified degree ncover that which by
abuse of notation we also denote by
π:Xb−→ X.
Since the canonical divisor of the symplectic surface |K|is zero and Xbbelongs to
the linear system |nK|, one can compute the arithmetic genus of Xb,
(2.3) gXb= 1 + n2(g−1).
By Riemann-Roch, the rank nbundle π∗OXbis has degree
deg(π∗OXb) = −(n2−n)(g−1).
Given a torsion-free rank one sheaf Fover Xbof degree δ, where
(2.4) δ:= n(n−1)(g−1),
we have that EF:= π∗Fis a vector bundle on Xof rank nand degree 0. Since πis
an affine morphism, the natural O|K|-module structure on F, given by understand-
ing Fas a sheaf supported on |K|, corresponds to a π∗O|K|= Sym•(K∗)-module
6 EMILIO FRANCO AND ANA PE ´
ON-NIETO
structure on EF. Such structure on EFis equivalent to a Higgs field
ϕF:EF−→ EF⊗K.
This establishes a one-to-one correspondence between torsion-free rank one sheaves
on Xband Higgs bundles (EF, ϕF) such that
h((EF, ϕF)) = b.
In fact, stability is preserved under the spectral correspondence as we see in the
following theorem. We refer to the moduli space of torsion free of rank 1 and
degree δsheaves on Xbas the compactified Jacobian Jacδ(Xb). We denote by
JacδXb⊂Jac δXbthe open subset of invertible sheaves.
Theorem 2.1 ([Si2, Sch]).A torsion-free rank one sheaf Fon the spectral curve
Xbis stable (resp. semistable, polystable) if and only if the corresponding Higgs
bundle (EF, ϕF)on Xis stable (resp. semistable, polystable). Hence, the Hitchin
fibre over b∈His isomorphic to the moduli space of torsion-free rank one sheaves
of degree δ= (n2−n)(g−1) over Xb,
h−1(b)∼
=Jac δXb.
For non integral curves, [Sch, Th´eor`eme 3.1] gives an easy characterization of
semistability, modulo some corrections pointed out in [CL, Remark 4.2] and [dC,
Section 2.4]. Assuming the spectral curve is reduced, a torsion-free rank one sheaf
F → Xbis stable (resp. semi-stable) if and only if for every closed sub-scheme
Z⊂Xbpure of dimension one, and every rank one torsion free quotient sheaf
F|ZFZ, one has that
(2.5) degZFZ>(n2
Z−nZ)(g−1) (resp. ≥),
where nZ= rk(π∗OZ).
2.3. Rank one torsion free sheaves on connected nodal reducible curves.
Motivated by Section 2.2, we recall in this section some well-known facts about rank
one torsion free sheaves on connected reducible nodal curves whose singularities
always lie on two (and only two) irreducible components. Let Xbe such a curve,
and let X1, . . . , Xnbe its irreducible components. The normalization
(2.6) ν:e
X→X
satisfies e
X∼
=FXi. Let Dij =Xi∩Xjfor 1 ≤i < j ≤n, and D=Si,j Dij . By
assumption, Dij consists of simple points, which are nodal singularities of X.
Definition 2.2. Let Fbe a coherent sheaf on X. We say that Fis torsion free if
Tor (Fx)=0for all x∈X, where Tor (Fx)denotes the torsion submodule of Fx,
defined as
Tor (Fx) = nf∈ Fx:∃a∈ OX,x \Div0(OX,x),such that a·f= 0o
with Div0(OX,x )the divisors of zero of OX,x.
Since Xis nodal, then torsion free sheaves are precisely sheaves of depth one (cf.
[Se, Section 7]).
Definition 2.3. Let Fbe a torsion free sheaf. We define its degree as the integer
dappearing in the Euler characteristic
χ(X, F) = d+r(1 −g).
Definition 2.4. A torsion free sheaf F → Xis of rank rif
rk (F|Xi/Tor(F|Xi)) = r.
THE BOREL SUBGROUP AND BRANES 7
Remark 2.5.The above is the definition given in [Sch]; in other sources, such a
sheaf is called of multirank r. Note that the rank is not always well defined, but
it ensures that the Higgs bundle obtained has the right characteristic polynomial
(see discussion at the beginning of [dC, Section 2.4]).
A particular example of rank one torsion free sheaves are line bundles on X.
The variety of all such bundles is denoted by Jac(X). Line bundles admit a simple
description in terms of their pullback to the normalization, as shown in the following
lemma due to Grothendieck [Gr, Proposition 21.8.5], that we reproduce adapted to
our notation.
Lemma 2.6. Let R⊂D, and let νR:e
XR→Xbe the partial normalization at R.
Note that νD:e
XD→Xis just the normalization map (2.6). The pullback map
ˆνR: Jac(X)−→ Jac( e
XR)
L7−→ ν∗
RL
is a smooth fibration with fiber (C×)|R|−nR+1 where nRis the number of connected
components of e
XR.
Proof. Since both Jac(X) and Jac( e
XR) are torsors for the groups Jac0(X) and
Jac0(e
XR), the tangent space at any point of them is isomorphic to the tangent space
at fixed points, Land ν∗
RL, of each connected component. Then, both varieties are
clearly smooth and it is enough to prove smoothness of ˆνRat these fixed points.
Since both are stable, the tangent space of Jac(X) and Jac( e
XR) at Land ν∗
RL
are, respectively, Ext1(L, L)∼
=H1(X, OX) and Ext1(ν∗
RL, ν∗
RL)∼
=H1(e
XR,Oe
XR)
(see [HL, Corollary 4.5.2] for instance) and the differential of ˆνRis given by the
pull-back under νR. Taking the short exact sequence
(2.7) 0 −→ OX−→ νR,∗Oe
XR−→ OR−→ 0.
with associated long exact sequence
0−→ C−→ CnR−→ C|R|−→ H1(X, OX)ν∗
R
−→ H1(e
XR,Oe
XR)−→ 0,
we can easily check that ν∗
R:H1(X, OX)→H1(e
XR,Oe
XR) has maximal rank, so
it is a smooth morphism.
The rest of the statement follows naturally from the the short exact sequence
0−→ O×
X−→ νR,∗O×
e
XR−→ O×
R−→ 0,
whose associated long exact sequence reads
0−→ C×−→ (C×)nR−→ (C×)|R|−→ Jac(X)ν∗
R
−→ Jac( e
XR)−→ 0.
We define
(2.8) ˆν: Jac(X)−→ Jac( e
X)
L7−→ ν∗L,
which by Lemma 2.6 is a smooth fibration with fiber (C∗)δ−n+1.
One can give the following geometrical interpretation of Lemma 2.6. Given R=
Sj>1D1j, a line bundle L→Xconsists of a line bundle L1→X1and a line bundle
L2→XRwhere XR=Si>1Xi(namely, a line bundle on e
XR=X1tiSi>1Xi)
8 EMILIO FRANCO AND ANA PE ´
ON-NIETO
together with an identification zx: (L1)x1→(Lj)xjat each point x∈D1jwith
preimages xi∈Xi. To recover Lfrom these data, tensoring (2.7) with L, we obtain
0−→ L−→ L1⊕L2−→ L|R−→ 0
s7−→ (s|X1, s|X1)
(a, b)7−→ (a(x)−zxb(x))x∈R.
One sees that two tuples (zx)x∈Rand (z0
x)x∈Rinduce isomorphic bundles L∼
=L0if
and only if they differ by a non zero factor. Repeating the process with L|Y, one
obtains a description of line bundles on reducible curves in terms of line bundles
on each of the irreducible components, together with gluing data.
We next explain how to compute the degrees of line bundles on Xfrom the
degrees on each connected component.
Lemma 2.7. Let L→Xbe a line bundle on a connected nodal curve with ir-
reducible components Xii= 1, . . . , n such that the only singularities lie on two
components. For any L∈Jac(X),deg L=Pideg L|Xi.
Proof. We proceed by induction on the number of irreducible components of X. If
there is just one such component, the statement is trivial. Assume there are n. Let
Y=Sn−1
i=1 Xi
Let R=Sj≤n−1Djn , and consider νR:e
XR→Xthe partial normalization of
Xalong R, that is
e
XR=YtXn.
By the induction hypothesis, it is enough to prove that deg(L) = deg(L|Y) +
deg(L|Xn).
Tensoring (2.7) by Lwe get
0−→ L−→ ν0
∗Oe
XR⊗L−→ L|R−→ 0.
Now, given a coherent sheaf Fon X, let χ(X, F(m)) = Pi(−1)ihi(X , F(m))
be its Hilbert polynomial. We have that
χ(X, L(m)) −χ(X, L|R) = χ(e
XR, ν∗
RL).
Note also that
χ(X, L(m)) = m+ deg(L)−g(X)+1,
and
χ(X, L|R) = |R|.
Taking an ample divisor so that half of the points belong to Yand half to X,
χ(e
XR, ν∗
RL) = χ(Y, L(m/2)|Y) + χ(Xn, L(m/2)|Xn)
=m+ deg(L|Y) + deg(L|Xn)+1−g(Y)+1−g(Xn).
Therefore,
deg(L)−g(X)+1− |R|= deg(L|Y) + deg(L|Xn)+1−g(Y)+1−g(Xn).
From the long exact sequence induced from (2.7), we have that
g(e
XR) + |R| − 1 = h1(e
XR,Oe
XR) + |R| − 1 = h1(X, OX) = g(X)
which together with the fact that g(e
XR) = g(Y) + g(Xn) concludes the proof.
Definition 2.8. We say that the multidegree of a line bundle L→Xvis the
multidegree of ˆν(L) = ν∗L→e
Xv, that is, the degree on each of the connected
components of e
X. In the above ˆνis defined in (2.8).
THE BOREL SUBGROUP AND BRANES 9
A rank one torsion free sheaf on Xis either a line bundle or a pushforward of a
line bundle on a partial normalization of X[Se]. Geometrically, rank one torsion
free coherent sheaves on Xwhich are a pushforward from an element in Jac( e
XR)
are obtained as n-uples of line bundles Li→XR,i on each connected component of
e
XR, together with identifications at all points x∈D\R.
Lemma 2.9. With the same notation as in Lemma 2.6, for any F∈Jac( e
XR), one
has
deg(νR,∗F) = deg(F) + |R|.
Proof. We have
deg(ν∗F)+1−gX=χ(X , νR,∗F) = χ(e
XR, F ) = deg(F) + nR−ge
XR.
From (2.7),
χ(X, OX)−χ(X, νR,∗Oe
XR) + |R|= 0,
which implies ge
XR=nR−1 + gX− |R|. Substitution in the above equation proves
the statement.
2.4. Fourier–Mukai on fine compactified Jacobians. In this Section we recall
the results from [MRV1, MRV2], where a Poincar´e sheaf is built over the product
of a fine compactified Jacobian of a curve with nodal singularities and its dual,
yielding a Fourier–Mukai transform between these spaces.
Given a flat morphism f:Y→Swhose geometric fibres are curves, for any
S-flat sheaf Fon Y, we can construct the determinant of cohomology Df(E) (see
for instance (see [KM] and [Es, Section 6.1])), which is an invertible sheaf on S
constructed locally as the determinant of complexes of free sheaves locally quasi-
isomorphic to Rf∗E.
Since Jac δ(X) is a fine moduli space by hypothesis, one has a universal sheaf
U → X×Jac δ(X). Denote by U0its restriction to X×Jacδ(X). Consider the triple
product X×Jac δ(X)×Jacδ(X) and denote by fij the projection to the product of
the i-th and j-th factors. We define the Poincar´e bundle P → Jac δ(X)×Jacδ(X)
as the invertible sheaf
(2.9) P=Df23 f∗
12U ⊗ f∗
13U0−1⊗ Df23 f∗
13U0⊗ Df23 (f∗
12U).
Given J∈Jacδ(X), we have that the restriction PJ:= P|Jac δ(X)×{J}is a
line bundle over Jac δ(X). In fact, if we consider the obvious projections f1:
X×Jac δ(X)→Xand f2:X×Jac δ(X)→Jac δ(X), one has [MRV1, Lemma 5.1]
(2.10) PJ=Df2(U ⊗ f∗
1J)−1⊗ Df2(f∗
1J)⊗ Df2(U).
Using Pone can construct an associated integral functor. We set π1and π2to
be, respectively, the projection from Jac δ(X)×Jacδ(X) to the first and second
factors. Then, define
(2.11) Φ : DbJac δ(X)−→ DbJacδ(X)
E•7−→ Rπ2,∗(π∗
1E•⊗ P).
By [MRV2], one can extend the Poincar´e bundle P → Jac δ(X)×Jacδ(X) to
a Cohen-Macaulay sheaf P → Jac δ(X)×Jac δ(X). Denote by π1(resp. π2) the
projection Jac δ(X)×Jacδ(X)→Jac δ(X) to the first (resp. second) factor. Using
Pas a kernel, one can consider the integral functor
(2.12) Φ : DbJac δ(X)−→ DbJac δ(X)
E•7−→ Rπ2,∗(π∗
1E•⊗ P),
10 EMILIO FRANCO AND ANA PE ´
ON-NIETO
which is a derived equivalence by [MRV2].
3. A(BBB)-brane from the Cartan subgroup
In this section we construct a (BBB)-brane of Mn, namely, a pair (N,F) given
by:
1) A hyperholomorphic subvariety N ⊂Mn,i.e. a subvariety which is holomor-
phic with respect to the three complex structures Γ1, Γ2and Γ3.
2) A hyperholomorphic bundle Fon N, i.e. a vector bundle with a connection
whose curvature is of type (1,1) in the complex structures Γ1, Γ2and Γ3.
3.1. A hyperholomorphic bundle on the Cartan locus. The embedding of
the Cartan subgroup C ∼
=(C×)ninto GL(n, C) induces the Cartan locus of the
moduli space of Higgs bundles
Car = (E, ϕ)∈Mn∃s∈H0(X, E/C),
ϕ∈H0(X, Es(c)⊗K).,
where c= Lie(C) and Esis the principal C-bundle on Xconstructed from the
section s. Observe that Car is the image of the injective morphism
c: Symn(M1)−→ Mn.
Note also that Car is a hyperholomorphic subvariety, since in the complex structure
(Mn,Γ2), it corresponds with the locus of the moduli space of flat connections given
by those reducing its structure group to C.
Note that both, the moduli space of topologically trivial rank 1 Higgs bundles
(M1,Γ1)∼
=T∗Jac0(X) and the moduli space of rank 1 flat connections (M1,Γ2)∼
=
Loc1(X) fiber algebraically over Jac0(X). In fact, this projection
M1−→ Jac0(X)
is hyperholomorphic. As a immediate consequence, the induced map
p: Symn(M1)−→ SymnJac0(X)
is hyperholomorphic as well.
These remarks imply the following proposition.
Proposition 3.1. Suppose that Fis a vector bundle on SymnJac0(X), then
p∗Fdefines a hyperholomorphic vector bundle on the hyperholomorphic manifold
Symn(M1).
Proof. This follows from the fact that the Chern connection of a holomorphic bundle
is of type (1,1), that curvature commutes with pullbacks and that holomorphic
maps respect types under pullback.
In view of Proposition 3.1, one has that F:= c∗p∗Fis a hyperholomorphic
vector bundle on Car, the Cartan locus of the moduli space. The pair (Car,F)
constitutes a (BBB)-brane on the Higgs moduli space Mn.
Associated to a line bundle L → Jac0(X) one can define a line bundle on
Symn(Jac0(X)) as we explain in the following lemma.
Lemma 3.2. Consider
πi: (Jac0(X))×n→Jac0(X)
the projection onto the i-th factor. Let Ln:= Nn
=1 π∗
iL. Then Lndescends to a
bundle L(n)on Symn(Jac0(X)).
THE BOREL SUBGROUP AND BRANES 11
Proof. The bundle Lnis invariant by the action of Sand moreover the natural
linearization action derived from the one on the bundle ⊕n
i=1Lsatisfies that over
point p∈(Jac0(X))×nwith non trivial centraliser Zp⊂S, the centraliser Zpacts
trivially on Ln
p. It follows from Kempf’s descent lemma that Lndescends to a
bundle L(n)on Symn(Jac0(X)).
Remark 3.3.The same argument yields a hyperholomorphic bundle on Car for any
choice of a holomorphic bundle F → Jac0(X).
Given L → Jac0(X) topologically trivial, consider the associated hyperholomor-
phic line bundle L=c∗p∗L(n)on the Cartan locus Car and denote by
Car(L) := (Car,L)
the associated (BBB)-brane on Mn, which we call Cartan (BBB)-brane associated
to L.
3.2. Spectral data for the Cartan locus. In this section we compute the fibers
of the Hitchin map restricted to the hyperholomorphic subvariety Car ⊂Mn.
We call h(Car), the image of Car under the Hitchin map, the Cartan locus of
the Hitchin base. Since the polystable Higgs bundles contained in Car decompose
as direct sums of line bundles, we have that the Cartan locus of the Hitchin base
is the image of
(3.1) V := SymnH0(X, K ),
under the injection
(3.2) V→H
(α1, . . . , αn)S7−→ (q1(α1, . . . , αn), . . . , qn(α1, . . . , αn)).
Hence
(3.3) dim V = ng.
Let u∈V, and denote by Xvthe corresponding spectral curve. We define ∆ to
be the big diagonal of V,
∆ := {(α1, . . . , αn)S∈V such that αi=αjfor some i, j}.
Clearly V \∆ is dense inside V. For any two αiand αjwith i6=j, denote the
divisor Dij =αi(X)∩αj(X). Consider also the Cartan nodal locus of the Hitchin
base to be subset of V \∆
Vnod :=
(α1, . . . , αn)S∈V\∆ such that for every i < j < k
(a) there is no multiple point on Dij, and
(b) Dij ∩Dik is empty.
.
Remark 3.4.Since conditions (a) and (b) are generic, Vnod is a dense open subset
of V.
Lemma 3.5. For any v∈Vnod, with v= (α1, . . . , αn)S, the spectral curve,
(3.4) Xv=
n
[
i=1
Xi,
is a reduced curve, with nirreducible components Xi:= αi(X)isomorphic to X
and only nodal singularities at the points
I:= [
i,j
Dij
where |I|= (n2−n)(g−1) = δ.
12 EMILIO FRANCO AND ANA PE ´
ON-NIETO
Proof. The spectral curve Xis given by the equation
λn+π∗b1λn−1+· · · +π∗bn= 0,
where bi=qi(α1, . . . , αn). By the properties of the invariant polynomials qi, one
can rewrite this equation as
n
Y
i=1
(λ−π∗αi)=0,
and (3.4) follows. We only have nodal points at most by the construction of Vnod.
The two curves, Xiand Xj, intersect each other at Dij, which by definition of
Vnod is a set of 2g−2 distinct points. There are nirreducible components, so the
number of intersection points is
|I|= 2(g−1)n
2= (n2−n)(g−1) = δ,
where we have used the condition Dij ∩Dik =∅if j6=kin the definition of
Vnod.
After Theorem 2.1, we are interested in the moduli space Jac δ(Xv) of torsion-free
rank one sheaves on Xvof degree δ.
Since Xv=SXi, then the normalization ν:e
Xv→Xvis isomorphic to
e
Xv∼
=Xtn
. . . tX.
Consider the following morphisms
(3.5) e
Xv
p
ν
Xv
π
Xj
?_
ιj
oo3S
δj
ee
X.
αj
∼
=
99
Lemma 3.6. Set
(3.6) ˇν: Jac(d1,...,dn)(e
Xv)−→ Jacδ(Xv)
L7−→ ν∗L,
where (d1, . . . , dn)is the multidegree of the line bundle on e
X(cf. Definition 2.8).
Then, the map is well defined and an injection if and only if di= 0 for all
i= 1, . . . , n.
Proof. By Lemma 2.9 ii), a necessary condition is for Pn
i=1 di= 0. We need to
check which multidegrees yield semistable bundles. But since for any F∈Jac( e
X),
eπ∗F= (Ln
i=1 F|Xi,⊕iαi) (where we identify X∼
=Xi), and the only semistable
such bundles must satisfy di= 0, Theorem 2.1 allows us to conclude.
THE BOREL SUBGROUP AND BRANES 13
Proposition 3.7. For any v∈Vnod, one has
h−1(v)∩Car = ˇνJac0(e
Xv).
Moreover, under the isomorphism
(3.7) m: Jac 0(e
Xv)∼
=Jac0(X)×n
induced by the ordering (X1, X2, . . . , Xn)of the connected components of e
X.
i) the spectral datum L∈ˇνJac0(e
Xv)corresponding to Ln
i=1(Li, αi)∈Car is
taken to (L1, . . . , Ln)∈Jac0(e
X)×n. Namely: L=ν∗F=Lj(ιj)∗Ljwhere ιjis as
in (3.5) and F∈Jac( e
X)restricts to F|Xj=Lj.
ii) the restriction of L→Car to h−1(v)∩Car corresponds to Ln→Jac0(X)×n
defined in Lemma 3.2.
Proof. i) By construction, a Higgs bundle in Car decomposes as a direct sum of
line bundles,
(E, ϕ)∼
=
n
M
i=1
(Li, αi).
By the argument in the proof of Lemma 3.6, ˇν(Jac0)⊂h−1(v)∩Car. Now, let L∈
Jacδ(Xv) be the spectral datum corresponding to and element (E, ϕ)∈h−1(v)∩
Car. It is easy to see that the Higgs bundle is totally decomposable if and only if
its π∗OXv-module structure factors through a π∗ν∗Oe
X∼
=O⊕n
X-module structure.
Hence L=ν∗Ffor some F∈Jac( e
X). Lemma 3.6 finishes the proof, as the only
possible multidegree is (0,...,0).
ii) In order to prove the second statement, note that the isomorphism (3.7) is
totally determined by a choice of an ordering of the connected components of e
X, in
this case (X1, . . . , Xn). Now, the choice of such an ordering induces an embedding
j: (Jac0(X))×n→Symn(Jac0(X)) making the following diagram commute:
Jac0(X))×n
q
##
_
j
m//h−1(u)
_
i
Symn(M1)
p
c//Car
Symn(Jac0(X)),
with q=p◦jbeing the usual quotient map. We need to check that
m∗i∗L∼
=Ln.
But, since the above diagram commutes and cis an injection, the LHS is equal
to j∗c∗L=j∗c∗c∗p∗L(n)∼
=j∗p∗L(n)∼
=q∗L(n)and the statement follows by the
construction of L(n).
4. (BAA)-branes from the unipotent radical of the Borel
4.1. An isotropic subvariety. Starting from the line bundle L → Jac0(X), we
construct in this section a complex Lagrangian subvariety Uni(L) of the moduli
space of Higgs bundles, mapping to the Cartan locus V ⊂H of the Hitchin base.
Fix a Borel subgroup B <GLn(C) containing C, so that B = C nU where
U = [B,B] is the unipotent radical of B. Denote by Bor the subvariety of the
14 EMILIO FRANCO AND ANA PE ´
ON-NIETO
moduli space Mngiven by those Higgs bundles whose structure group reduces to
B,
(4.1) Bor = (E, ϕ)∈Mn∃σ∈H0(X, E/B),
ϕ∈H0(X, Eσ(b)⊗K).,
where Eσ:= σ∗Eis the principal B-bundle on Xassociated to the section σ∈
H0(X, E/B).
After fixing a point x0∈X, we have an embedding jx0:X →Jac0(X). Denote
by ˆ
Lthe restriction of Lto X⊂Jac0(X) tensored δ/n = (n−1)(g−1) times by
OX(x0),
(4.2) ˆ
L:= j∗
x0L⊗OX(x0)(n−1)(g−1) .
Now, define the closed subvariety of Bor given by those Higgs bundles (E, ϕ)
whose underlying vector bundle Ehas associated graded piece totally determined
by ˆ
L:
(4.3) Uni(L) = ((E, ϕ)∈Bor EC:= Eσ/U∼
=
n
M
i=1
(ˆ
L ⊗ K⊗i−n)).
In order to prove that Uni(L) is an isotropic submanifold of (Mn,Ω1) we first
give a description of it in gauge theoretic terms. Let Edenote the topologically
trivial rank nvector bundle; choose a reduction of the structure group to B (which
always exists), and let EBbe the corresponding principal B-bundle, so that E∼
=
EB(GL(n, C)). Define EC=EB/U. It follows from (4.3) that
(4.4)
Uni(L) =
(∂A, ϕ)∈Mn:∃g∈ Gc
1) g·∂=∂C+N,
N∈Ω0,1(X, EB(n)),
(EC, ∂C) = Ln
i=1(ˆ
L ⊗ Ki−n);
2) g·ϕ∈Ω0(X, EB(b)⊗K).
.
Remark 4.1.Both Car and Uni(L) are subvarieties of Bor, but they do not intersect,
as the elements of Car ∩Uni(L) would have underlying bundle of the form ECin
(4.3), which is unstable, and totally decomposable Higgs field, conditions which
yield unstable Higgs bundles.
Proposition 4.2. The complex subvariety Uni(L)of Mnis isotropic with respect
to the symplectic form Ω1defined in (2.2).
Proof. It is enough to prove the statement for open subset of stable points in Uni(L).
We will check this subset is non empty in Proposition 4.5.
So let (E, ϕ)∈Uni(L) be a stable point. By (4.4), a vector ( ˙
A, ˙ϕ)∈T(E ,ϕ)Mn
satisfies that, up to the adjoint action of the gauge Lie algebra,
(˙
A, ˙ϕ)∈Ω0,1(X, EB(n)) ×Ω0(X, EB(b)⊗K).
The result follows from gauge invariance of the symplectic form Ω1and the fact
that n⊂b⊥, where orthogonality is taken with respect to the Killing form.
4.2. Spectral data for Uni(L).In this section we give a description of the spectral
data of the Higgs bundles corresponding to the points of Uni(L). This will allow
us to show that this subvariety is mid-dimensional, and, after Proposition 4.2,
Lagrangian.
We begin by studying the spectral data over Vnod. We recall that this is the
subset of V whose corresponding spectral curves Xvare nodal curves. We will use
the notation from (3.5).
THE BOREL SUBGROUP AND BRANES 15
Let us fix some notation. See also Figure 4.2. Let v∈Vno d, and let R⊂D.
Consider the partial normalization along R,
(4.5) e
XR
νR//
pR!!
Xv
π
X.
Assume that R=R1t· · · tRnR+Rswith e
XR=FnR
i=1 e
XR,i being the decomposition
into connected components such that
νR,i :e
XR,i −→ XRi
is a partial normalization onto its image XRialong a non-separating divisor Ri, and
Rsis the separating divisor in R(i.e. the divisor along which connected components
are to appear after normalization). Let Dibe the ramification divisor of
pi:e
XR,i −→ X
and set
(4.6) e
XR,i
pi
νR,i
e
Xj
?_
eιj
oo
νj
R,i
∼
=
XRi
π
Xj
?_
ιj
oo
X.
αj
∼
=
88
Assume that e
XRi=Sj∈Cie
Xjhas |Ci|irreducible components. Then for v∈
Vnod
(4.7) Di=X
j,k∈Ci
Djk −Ri.
Note that
(4.8) D=X
i
(Di+Ri) + Rs.
16 EMILIO FRANCO AND ANA PE ´
ON-NIETO
Figure 1. Partial normalization along R
Choose an ordering ( e
XR,1,..., e
XR,nR) of the connected components of e
XRin-
ducing an isomorphism
Jac η(e
XR)∼
=Jacη1(e
XR,1)× · · · × JacηnR(e
XR,nR).
Consider the decomposition
(4.9) Jac η(e
XR)∼
=[
PnR
i=1 ηi=η
Jacη1(e
XR,1)× · · · × JacηnR(e
XR,nR).
Let also
Jacηi(e
XR,i) = [
Pdj
i=ηi
Jac(d1
i,...,d|Ci|
i)(e
XR,i).
Lemma 4.3. i) If Rs6=∅, the Higgs bundles whose corresponding spectral data is
in νR,∗Pic( e
XR)are strictly semistable.
ii) The pushforward map
ˇνR: Jac(d1
1,...,d|C1|
1)(e
XR,1)× · · · × Jac(d1
nR,...,d|CnR|
nR)(e
XR,nR)−→ Jac δ(e
X)
is well defined and an injection only if
ηi=
Ji
X
k=1
dk
i=|Di|.
Proof. i) Let F∈Pic( e
XR) be such that νR,∗Fis the spectral datum for a Higgs
bundle in Mn. Assume Fsatisfies eι∗
kF=Fk. Then it follows that
(4.10) πv,∗νR,∗F=pR,∗F=
nR
M
i=1
pi,∗Fi,
where the notation is as in (4.6). Note that the direct sum is invariant by the
Higgs field, since the Higgs field is equivalent to a πv,∗OXvmodule structure on
πv,∗νR,∗F, and the latter factors through a πv,∗νR,∗Oe
XR-module structure. This
proves point i), as Rs6=∅ ⇐⇒ nR≥2.
ii) Let F∈Jacη1(e
XR,1)× · · · × JacηnR(e
XR,nR)⊂Jacη(e
XR), and assume that
eι∗
kF=Fk, where the notation is as in (4.6).
Assume first that Rs=∅. We first compute the value of ηfor ˇνR(F) to have
degree δ=|D|.
THE BOREL SUBGROUP AND BRANES 17
From (2.7), we compute
g(e
XR) = gX− |R|+nR−1.
Since
χ(X, ˇνR(F)) = χ(e
XR, F ),
we find that ˇνR(F) = δif and only if deg F=δ− |R|. This proves the statement,
as in this case nR= 1 and D1=D\R, so η1=η.
Now, if Rs6=∅, from (4.10), it must happen that deg pi,∗Fi= 0 for the Higgs
bundle to be semistable.
Given that pi=πi◦νR,i, and that XRiis a totally reducible nodal spectral curve
with |Ci|irreducible components, arguing as in Lemma 3.5 (compare with (2.4))
we find that
deg πi,∗νR,i,∗Fi= 0 ⇐⇒ deg νR,i,∗Fi= (|Ci|2− |Ci|)(g−1) = X
j,k∈Ci
Djk .
Now, considering
0−→ ν∗
R,iOXRi−→ O e
XRi−→ ORi−→ 0,
we have that X
j,k∈Ci
Djk = deg νR,i,∗Fi= deg Fi+|Ri|,
which together with (4.7) implies the statement.
Theorem 4.4. One has the following: i) The image under the Hitchin map of Bor
in (4.1) is h(Bor) = V.
ii) There is an equality of subvarieties Bor ×HVnod = Mn×HVnod.
iii) For every v∈Vnod , identify Jacd(e
XR)(for suitable multidegree d) with the
corresponding subset of the Hitchin fibre h−1(v)∼
=Jac δ(Xv). Then, if Rs6=∅, the
Higgs bundles corresponding to spectral data in Jacd(e
XR)admit a reduction of their
structure group to B1×· · · ×BnR⊂Bwhere Biis the Borel subgroup of GL(|Ci|,C).
iv) Let v= (α1, . . . , αn)Sn∈Vnod and let (E, ϕ)∈h−1(v)be a Higgs bundle
whose spectral data L→Xvlies in Jacδ(Xv). For any ordering J= (j1, . . . , jn)
of {1, . . . , n}, and associated ordering αJ= (αj1, . . . , αjn)of {α1, . . . , αn}, there
exists a filtration
(EJ)•: 0 ((Ej1, ϕj1)(· · · ((Ejn, ϕjn)=(E , ϕ),
such that
(Eji, ϕji)/(Eji−1, ϕji−1)=(α∗
jiι∗
jiL⊗Ki−n, αji),
where the notation is as in (3.5).
v) Given v= (α1, . . . , αn)Sn∈Vnod, let (E , ϕ) = LnR
i=1(Ei, ϕi)∈h−1(v)∩
Jacη(e
XR)correspond to the line bundle L→e
XR. Then, for any ordering Jk=
(j1, . . . , j|Ck|)of Ck, and associated ordering αJk= (αj1, . . . , αj|Ck|)of {αj}j∈Ck,
k= 1,...nRthere exists a filtration
(EJk)•: 0 ((Ej1, ϕj1)(· · · ((E|Ck|, ϕj|Ck|)=(Ek, ϕk)
such that
(Eji, ϕji)/(Eji−1, ϕji−1) = (L|e
Xji⊗ O(−X
k≥i+1 e
Xji∩e
Xjk, αji)
where we abuse notation by identifying the subdivisors e
Xji∩e
Xjk⊂Dk(4.7) and
their images under pk, and L|e
Xjiwith its pullback under αji◦(νji
R,k)−1.
18 EMILIO FRANCO AND ANA PE ´
ON-NIETO
Proof. i) This is a consequence of the following fact: given the Jordan–Chevalley
decomposition of x=xs+xn∈gln(C) into a semisimple xsand a nilpotent piece xn,
the invariant polynomials qidefining the Hitchin fibration evaluate independently
of the nilpotent part, namely qi(x) = qi(xs).
ii) By the universal property of fibered products, we need to find a morphism
Mn×HVnod →Bor ×HVnod making the following diagram commute:
Mn×HVnod
π1
''
π2
++
Bor ×HVnod //
_
h//Vnod
_
Mnh//B.
In other words, it is enough to prove that any Higgs bundle (E, ϕ)∈Mn×HVnod
admits a full flag decomposition. This is clear from an analysis of the spectral data,
as the spectral curve is totally reducible. Indeed, let v∈Vnod , and assume first
that L∈Jac(Xv) is the spectral datum for (E, ϕ).
Define
(4.11) Yi:=
i
[
j=1
Xj, Zi:=
n
[
k=i+1
Xk.
We consider the restriction of Lto L|Ziand denote its kernel by Li,
(4.12) 0 −→ Li−→ L−→ L|Zi−→ 0.
Since Liis a subsheaf of L, it gives the Higgs subbundle (Ei, ϕi)⊂(E, ϕ) under the
spectral correspondence. Since Li−1is a subsheaf of Liwe have that (Ei−1, ϕi−1)⊂
(Ei, ϕi) and the existence of the filtration follows for Higgs bundles in h−1(v)∩
Jacδ(Xv).
Assume next that (E, ϕ)∈h−1(v)∩Jac( e
XR). Since, with the notation of (4.5),
pR=πv◦νR, the same argument as before allows us to conclude.
iii) Follows from ii) above and Lemma 4.3.
iv) Let Jbe an ordering of {1, . . . , n}, and set YJi=Si
k=1 Xjk,ZJi=Sn
k=i+1 Xjk.
Let
0−→ LJi−→ L−→ L|ZJi−→ 0.
Then, reasoning as in Point ii) above we may conclude that the filtration exists.
Note that LJi=L⊗ IX,ZJiwhere IX ,ZJidenotes the ideal defining the sub-
scheme ZJi⊂X. Now, IX,ZJi
∼
=OYJi⊗ IYJi,ZJi∩YJi, thus
LJi∼
=L|YJi⊗ IYJi,ZJi∩YJi.
Note that
0−→ LJi.LJi−1−→ L|ZJi−1−→ L|ZJi−→ 0
is exact, so that
LJi/LJi−1∼
=L|ZJi⊗ IZJi−1,ZJi
∼
=L|ZJi⊗ OXji⊗ IXji,ZJi∩Xji
∼
=L|Xji(−
n
X
r=i+1
Djijr).
THE BOREL SUBGROUP AND BRANES 19
Now, the pushforward of
0−→ LJi−1−→ LJi−→ LJi.LJi−1−→ 0
gives under the spectral correspondence
(Ei, ϕi)(Ei−1, ϕi−1)∼
= α∗
jiι∗
jiL(−
n
X
r=i+1
Djijr), αji!,
where we abuse notation by identifying the divisor Djk and its image under π.
Naturally, K∼
=OX(Djk ), which yields the result.
v) To simplify notation, take the orderings ((α1, . . . , α|C1|), . . . (α|CnR−1|, . . . , αn)).
The reasoning that follows adapts just the same way to any other choice of order-
ings. The statement is proven as iv) above, taking the following remarks into
account:
(1) The subscheme Zi⊂Xis the image of its partial normalization e
Zi⊂e
XR,
on which the filtration will be given on each of the connected components.
This restricts the proof to line bundles over connected curves e
XR.
(2) So we may assume e
XRis connected and the ordering is (α1, . . . , αn). We
obtain a full flag in the same way, the difference with iv) being that the
ideal
Ie
Zi−1,e
Zi
∼
=Oe
Xi(−e
Xi∩e
Zi)
depends on the ordering (and R) and so does
e
Xi∩e
Zi=X
k≥i+1 e
Xi∩e
Xk.
As a corollary of Theorem 4.4, we obtain the description of the Hitchin fibers
intersected with Uni(L). Before we state the result we need some extra definitions.
Given b∈V, let D⊂Xbbe the singular divisor, and let R⊂Dbe a subdivisor.
For each ordering Jof {1, . . . , n}, and each i∈ {1, . . . , n}, define the divisors
(4.13) BJ,i =X
j≥i+1
Djijk\R, BJ,i =X
j≥i+1
Djijk∩R.
Proposition 4.5. Let ˆ
Lbe defined as in (4.2). For every v∈Vnod,d∈Zn, identify
Jacd(e
XR)ss with the corresponding subset of the Hitchin fibre h−1(v)∼
=Jac δ(Xv).
Then i) For R=∅
(4.14)
Uni(L)∩Jacδ(Xv) = nL∈Jac δ(Xv) such that ν∗L=p∗ˆ
L∼
=ˆ
L,..., ˆ
Lo.
Furthermore, Higgs bundles described in (4.14) are stable.
ii) Suppose que R6=∅. Then Uni(L)∩Jacd(e
XR)is a subset of
(4.15) (L∈Jacd(e
XR)nL|e
Xi: 1 ≤i≤no=nˆ
L⊗O(BJ,i) : ji∈Jo
for some ordering Jof {1, . . . , n}).
Proof. i) The inclusion ⊂in (4.14) follows from Theorem 4.4, which implies that
the spectral datum Lof any (E, ϕ)∈Uni(L)∩Jacδ(Xv) satisfies
ˆ
L=α∗
iι∗
iL.
Now, since any line bundle on e
Xvis totally determined by its restriction to all the
connected components, it is enough to check that j∗
ip∗ˆ
L=j∗
iν∗L, which follows
from commutativity of the arrows in (3.5) and the fact that αi:X→Xiis an
isomorphism.
20 EMILIO FRANCO AND ANA PE ´
ON-NIETO
All there is left to check is that all line bundles are in the RHS of (4.14) are
stable, as by Theorem 4.4, this implies automatically that they are inside Uni(L).
Let L∈Jacδ(Xv), which by Theorem 2.1 is the spectral data of (E, ϕ), semistable
or unstable. We will check that all such line bundles satisfy the strict inequality in
(2.5). First of all, any subscheme pure of dimension 1 of Xis of the form
ZI=[
i∈I
Xi, YI=[
i∈Ic
Xi.
for some I⊂ {1, . . . , n}.
Now, Lbeing a line bundle, it follows that any rank one torsion free quotient of
L|ZImust be isomorphic to L|ZI, so it is enough to check that L|ZIsatisfies the
strict inequalities in (2.5), and therefore is stable. This is an easy computation.
ii) With the notation of (4.1), note that (E, ϕ)∈Uni(L) if and only if for some
reduction E=Eσ(Cn) of its structure group to a Borel, then the associated graded
bundle EC∼
=Ln
i=1 ˆ
L⊗ Ki−n.Now, once the image under the Hitchin map has been
fixed, the above statement is equivalent to the following: (E , ϕ)∈Uni(L)∩h−1(v),
v= (α1, . . . , αn)Sn, if and only if for some ordering Jof {1, . . . , n}, there exists a
filtration
0=(E0, ϕ0)((E1, ϕ1)(· · · ((En, ϕn) = (E , ϕ)
such that
(Ei, ϕi)(Ei−1, ϕi−1)∼
=(L ⊗ Ki−1, αji).
The statement then follows from Theorem 4.4, noting that
Ki−n=O(BJ,i)⊗ O(BJ,i )
and that
ν∗
RO(BJ,i) = X
k≥i+1
O(e
Xji∩e
Xjk).
The description of the spectral data given in Proposition 4.5 allows us to study
the dimension of Uni(L), which turns up to be one half of dim Mn.
Proposition 4.6. The complex subvariety Uni(L)of Mnhas dimension
dim Uni(L) = n2(g−1) + 1 = 1
2dim Mn.
Proof. First, we observe that Uni(L) is a fibration over V and recall that dim V =
ng. By Proposition (4.5), over the dense open subset Vnod ⊂V, the fibre of
Uni(L)|Vnod →Vnod at vhas a dense open subset
ˆν−1(ˆ
L,..., ˆ
L)⊂Jac δ(Xv)∼
=h−1(v),
where we recall (2.8). Now, by Lemma 2.6,
ˆν−1(ˆ
L,..., ˆ
L)∼
=C×δ−n+1 .
By smoothness of the point, the Hitchin fiber is transverse to the (local) Hitchin
section, so
dim Uni(L)|Vnod = dim Vnod + dim ˆν−1ˆ
L,..., ˆ
L
=ng +δ−n+ 1
=ng + (n2−n)(g−1) −n+ 1
=n2(g−1) + 1.
THE BOREL SUBGROUP AND BRANES 21
which is half of the dimension of Mn, as we recall from (2.1). This finishes the proof
since by Proposition 4.2, Uni(L) is isotropic, so its dimension can not be greater
than 1
2dim Mn.
Finally, we can state the main result of the section.
Theorem 4.7. The complex subvariety Uni(L)of Mnis complex Lagrangian with
respect to Ω1.
Proof. This is clear after Propositions 4.2 and 4.6.
By Theorem 4.7 we can define the Borel unipotent (BAA)-brane associated to L
as the triple given by
Uni(L) := Uni(L),OUni(L),∇,
where ∇is the trivial connection on OUni(L).
5. Duality
In this section we discuss about the duality under mirror symmetry of the (BBB)-
brane Car(L), and the (BAA)-brane Uni(L). Ideally, we would like to transform
them under a Fourier–Mukai transform, but such a tool is currently unavailable
for coarse compactified Jacobians of reducible curves, which is the situation that
we face in this case. We can however define an ad-hoc Fourier–Mukai transform
relating the generic loci of both branes. We expect for the weaker form of duality
proven here to be induced from the global duality if a full Fourier–Mukai transform
were to exist, and so it is a hint of the existence of the latter.
Fourier–Mukai transforms for fine compactified Jacobians of reducible curves are
studied in [MRV1, MRV2], which is the case that we review in Section 2.4. The
construction of the Poincar´e sheaf given in (2.9) does not apply to the case of coarse
compactified Jacobians, since there is no universal bundle over Xv×Jac δ(Xv) in
this case. We can consider the following bundle parametrizing the sheaves in the
Cartan locus,
UCar := (νסν)∗e
U −→ XvסνJac0(e
Xv),
where e
Uis the Universal line bundle of multidegree 0 on e
Xv. Also, one can consider
the universal line bundle of degree δover Xv,U0→Xv×Jacδ(Xv). Since Jacδ(Xv)
lies in the stable locus of Jac δ(Xv), the existence of U0follows from the existence of
a universal bundle over the stable locus [Si1, Theorem 1.21 (4)]. We already have
all the ingredients for the following definition, analogous to (2.9), of a Poincar´e
sheaf over ˇνJac0(e
Xv)×Jacδ(Xv),
(5.1) PCar := Df23 f∗
12UCar ⊗f∗
13U0−1⊗ Df23 f∗
13U0⊗ Df23 f∗
12UCar ,
where the fij are the corresponding projections from Xv׈νJac0(e
Xv)×Jacδ(Xv)
to the product of the i-th and j-th factors.
Remark 5.1.The Poincar´e bundle over fine compactified Jacobians is defined in
(2.9) using the universal sheaf U → Xv×Jac δ(Xv). In the case of coarse com-
pactified Jacobians there is no universal sheaf, but, locally, one can repeat the
construction in (2.9) using local universal sheaves U0→U×Jac δ(Xv) giving
P0→U×Jac δ(Xv). It is reasonable to expect that, if a Poincar´e bundle P →
Jac δ(Xv)×Jac δ(Xv) over coarse compactified Jacobians were to exist, the restric-
tion of it to U×Jac δ(Xv) would coincide with P0. The restriction of any local
22 EMILIO FRANCO AND ANA PE ´
ON-NIETO
universal sheaf to XvסνJac0(e
Xv)⊂Xv×Jac δ(Xv) would be isomorphic to
UCar and this justifies the construction of PCar.
Using PCar we construct a Fourier–Mukai integral functor analogous to (2.11),
(5.2) ΦCar :DbˇνJac0(e
Xv) −→ DbJacδ(Xv)
E•7−→ Rπ2,∗(π∗
1E•⊗ PCar),
where π1and π2to be, respectively, the projection from ˇνJac0(e
Xv)×Jacδ(Xv)
to the first and second factors.
Recall that our (BBB)-brane Car(L) is given by the hyperholomorphic bundle
Lsupported on Car. By Proposition 3.7, over the dense open subset Vnod of the
Cartan locus of the Hitchin base V = h(Car) ⊂H, the hyperholomorphic sheaf L
restricted to a certain Hitchin fibre Jacδ(Xv) is ˇν∗Ln, supported on ˇν(Jacδ(e
Xv)).
The main result of this section is the study of the behaviour of ˇν∗Lnunder ϕCar,
but first we need some technical results.
Fix x0and take the line bundle O(x0)(n−1)(g−1). Denote
τ: Jac0(e
X)∼
=
−→ Jacδ(e
X)
the isomorphism given, on each of the components, by tensorization by the previous
line bundle. We can define a Poincar´e bundle e
P → Jac0(e
Xv)×Jacδ(e
Xv).
Consider the projections to the first and second factors
Jac0(e
Xv)×Jacδ(e
Xv)
eπ1
vv
eπ2
((
Jac0(e
Xv) Jacδ(e
Xv),
and, using e
P, one can construct another Fourier–Mukai integral functor
e
Φ : Db(Jac0(e
Xv)) −→ Db(Jacδ(e
Xv))
E•7−→ Reπ2,∗(eπ∗
1E•⊗e
P).
Note that e
Φ is governed by the usual Fourier–Mukai transform on each of the
Jac0(Xi). We need the following lemma in order to describe the interplay between
ΦCar and e
Φ.
Lemma 5.2. One has that
(ˇν×1Jac)∗PCar ∼
=(1g
Jac ׈ν)∗e
P.
Proof. Note that (ˇν×1Jac)∗PCar is a family of line bundles over Jac0(e
X) parame-
trized by Jacδ(Xv). Since e
P → Jac0(e
Xv)×Jac0(e
Xv) is a universal family for these
objects, there exists a map
t: Jacδ(Xv)−→ Jac0(e
Xv),
such that
(ˇν×1Jac)∗PCar ∼
=(1g
Jac ×t)∗e
P.
Recall the description of PJgiven in (2.10) for each J∈Jacδ(Xv). Recall as well
the projections f1:Xv×Jac δ(Xv)→Xvand f2:Xv×Jac δ(Xv)→Jac δ(Xv),
THE BOREL SUBGROUP AND BRANES 23
and consider the following commuting diagram
e
Xv×Jac0(e
Xv)νסν//
e
f2
Xv×Jac δ(Xv)
f2
Jac0(e
Xv)ˇν//Jac δ(Xv)
We know from [Es, Proposition 44 (1)] that the determinant of cohomology com-
mutes with base change, i.e.
ˇν∗Df2∼
=De
f2(νסν)∗.
From the construction of UCar as (νסν)∗e
U, it is clear that we can choose a family
of line bundles UCar
0→XvסνJac0(e
X)such that it is a subfamily of UCar giving
the sequence
(5.3) 0 −→ UCar
0−→ UCar −→ UCar /UCar
0−→ 0,
where UCar/UCar
0→XvסνJac0(e
X)is a family of sky-scraper sheaves on Xv
supported on sing(Xv), and
(νסν)∗UCar
0∼
=e
U.
The additive property of the determinant of cohomology [Es, Proposition 44 (4)],
says that, whenever we have an exact sequence as in (5.3),
Df2(UCar)∼
=De
f2(UCar
0)⊗ D e
f2(UCar/UCar
0).
Also, trivializing Jat sing(Xv), we can see that UCar/UCar
0⊗f∗
1Jis isomorphic
to UCar/UCar
0, hence
Df2UCar/UCar
0⊗f∗
1J∼
=Df2UCar/UCar
0.
Using these properties, we can show that
e
Pt(J)∼
=ˇν∗PCar
J
∼
=ˇν∗Df2UCar ⊗f∗
1J−1⊗ Df2(f∗
1J)⊗ Df2UCar
∼
=ˇν∗Df2UCar ⊗f∗
1J−1⊗ˇν∗Df2(f∗
1J)⊗ˇν∗Df2UCar
∼
=ˇν∗Df2UCar
0⊗f∗
1J−1⊗ˇν∗Df2UCar /UCar
0⊗f∗
1J−1⊗ˇν∗Df2(f∗
1J)
⊗ˇν∗Df2UCar
0⊗ˇν∗Df2UCar /UCar
0
∼
=ˇν∗Df2UCar
0⊗f∗
1J−1⊗ˇν∗Df2UCar /UCar
0−1⊗ˇν∗Df2(f∗
1J)
⊗ˇν∗Df2UCar
0⊗ˇν∗Df2UCar /UCar
0
∼
=ˇν∗Df2UCar
0⊗f∗
1J−1⊗ˇν∗Df2(f∗
1J)⊗ˇν∗Df2UCar
0
∼
=De
f2(νסν)∗UCar
0⊗(νסν)∗f∗
1J−1⊗ D e
f2((νסν)∗f∗
1J)
⊗ D e
f2(νסν)∗UCar
0
∼
=De
f2e
U ⊗ e
f∗
1ν∗J−1⊗ D e
f2e
f∗
1ν∗J⊗ D e
f2e
U
∼
=e
P(ν∗J)
∼
=e
Pˆν(J).
Then, t= ˆνand this finish the proof.
We can now study the image of ˇν∗(Ln) under (5.2).
24 EMILIO FRANCO AND ANA PE ´
ON-NIETO
Proposition 5.3. One has the isomorphism
ΦCar(ˇν∗(Ln)) ∼
=ˆν∗e
Φ(Ln),
and furthermore, ˆν∗e
Φ(Ln)is a complex supported on degree ggiven by ˆν∗O(ˆ
Ln).
Proof. Let us also consider the following maps
Jac0(e
Xv)×Jacδ(Xv)
π0
1
vv
π0
2
((
Jac0(e
Xv) Jacδ(Xv),
and observe that
•π0
2=π2◦(ˇν×1Jac),
•π0
1=eπ1◦(1g
Jac ׈ν),
•π1◦(ˇν×1Jac) = ˇν◦π0
1, and
•eπ2◦(1g
Jac ׈ν) = ˆν◦π0
2.
Recalling Lemma 5.2, that ˇνis an injection and that ˆνis flat by Lemma 2.6, one
has the following,
ΦCar(ˇν∗(Ln)) =Rπ2,∗π∗
1ˇν∗(Ln)⊗ P Car
∼
=Rπ2,∗R(ˇν×1Jac)∗(π0
1)∗(Ln)⊗ PCar
∼
=Rπ2,∗Rˇν×1Jac )∗((π0
1)∗(Ln)⊗(ˇν×1Jac)∗PCar
∼
=Rπ2,∗R(ˇν×1Jac)∗(π0
1)∗(Ln)⊗(1g
Jac ׈ν)∗e
P
∼
=Rπ0
2,∗(π0
1)∗(Ln)⊗(1g
Jac ׈ν)∗e
P
∼
=Rπ0
2,∗(1g
Jac ׈ν)∗eπ∗
1(Ln)⊗(1g
Jac ׈ν)∗e
P
∼
=Rπ0
2,∗(1g
Jac ׈ν)∗eπ∗
1(Ln)⊗e
P
∼
=ˆν∗Reπ2,∗eπ∗
1(Ln)⊗e
P
∼
=ˆν∗e
Φ(Ln).
Finally, recalling that the usual Fourier–Mukai transform on Jac0(X)×Jacδ /n(X)
sends the line bundle Lto the (complex supported on degree ggiven by) sky-scraper
sheaf Oˆ
L, we have that ΦCar(ˇν∗Ln) is (the complex supported on degree ggiven
by)
ˆν∗e
Φ(Ln)∼
=ˆν∗O(ˆ
Ln),
and the proof is complete.
Recalling Proposition 4.5, we arrive to the main result of the section, which shows
that our (BBB)-brane Car(L) and our (BAA)-brane Uni(L) are related under the
Fourier–Mukai integral functor ΦCar.
Corollary 5.4. For every v∈Vnod, the support of the image under ΦCar of the
(BBB)-brane Car(L)restricted to a Hitchin fibre h−1(v), is the support of our
(BAA)-brane Uni(L)restricted to the open subset of the (dual) Hitchin fibre given
by the locus of invertible sheaves,
supp ΦCar ˇν∗(Ln)= Uni(L)∩Jacδ(Xv).
In view of Corollary 5.4 we conjecture the following.
THE BOREL SUBGROUP AND BRANES 25
Conjecture 1. The branes Car(L)and Uni(L)are dual under mirror symmetry.
6. Parabolic subgroups and branes on the singular locus
Cartan branes are the simplest example of branes supported on the singular
locus of the moduli space Msing
n. In this section we construct hyperholomorphic
and Lagrangian subvarieties covering the singular locus, and study their spectral
data.
6.1. Levi subgroups and the singular locus. Let L <GL(n, C) be a maximal
rank reductive subgroup. Then L is conjugate to Lr:= GL(r1,C)× · · · × GL(rs,C)
where Ps
i=1 ri=n, 0 < r1≤ · · · ≤ rs. Denote by Mr⊂Mnthe image of the
moduli space MLrof Lr-Higgs bundles. The same arguments as in the case of
Cartan subgroups shows that this is a complex subscheme in all three complex
structures of Mn.
Remark 6.1.In particular, Car = M(1,...,1).
Note that
Msing
n⊂[
Pri=n
Mr,
as the singular locus lies in the locus of strictly semistable bundles [Si2, Section 11].
Now, if (r1, . . . , rs) and (l1, . . . , lm) are such that for all j= 1, . . . , m there is an
njsuch that Pnj
i=1 ri=Pj
k=1 lj, then Mr⊂Ml. In particular
Msing
n⊂[
Pri=n
Mr=G
r1≤r2,r1+r2=n
M(r1,r2).
Fix r, and consider Mr⊂Mn. Observe that this manifold is complex in all three
complex structures Γ1,Γ2,Γ3, and therefore hyperholomorphic.
Consider the restriction of the Hitchin map
hr: Mr−→ Hr,
where Hr⊂H =: Hnis the image under the Hitchin map of Mr. Note that Hris
the image of the morphism
Hr= Hr1× · · · × Hrs−→ Hn
(b1, . . . , bs)7→ b,
where Hrdenotes the Hitchin base for GL(r, C)-Higgs bundles, and if bi= (bi1, . . . , biri),
bij ∈H0(X, K j), then
b=
s
X
i=1
bi1,..., X
Pkjk=iY
k∈I
bkjk,...,Y
i
bi,ri
.
This implies that for b= (b1, . . . , bs)∈Hr, the corresponding spectral curve Xbhas
at least sirreducible components Xb1, . . . , Xbs, which are in turn spectral curves
for bi∈Hri. Denote by
Xbi
ιi//
πbi!!
Xb
πb
X.
We consider the nodal locus Hno d
r⊂Hr, consisting of generic points corresponding
to spectral curves with exaclty sirreducible components and nodal singularities.
26 EMILIO FRANCO AND ANA PE ´
ON-NIETO
Lemma 6.2. Let b∈Hr. Then Dij =Xbi∩Xbjis a divisor of degree 2rirj(g−1).
If moreover b∈Hnod
r, the divisor consists of simple points and the normalization
of Xbis νb:e
Xb=Xb1t · · · t Xbs→Xb.
Proof. To see the first statement, deform the plane curve Xbito λri= 0. Then,
the intersection with Xbjis the vanisihing locus of a section of π∗
bKrj.
The second statement is obvious.
The following proposition is proved as Proposition 3.7
Proposition 6.3. Let b∈Hnod
r, and let δi= (r2
i−ri)(g−1). Then
h−1
r(b) = ν∗Jacδ(e
Xb),
where δ= (δ1, . . . , δs).
6.2. Parabolic subgroups and complex Lagrangian subvarieties. In this
section we define another submanifold Unir(m)⊂Mnassociated to the choice of a
a parabolic subgroup P whose Levi subgroup is L, as well as some suitable integers
m= (m1, . . . , ms).
Let P = L nU be a parabolic subgroup containg L, where U = [P,P] is its
unipotent radical.
Assume there exist natural numbers misuch that for all I({1, . . . , s},
(6.1) X
i∈I
rimi>(r2
I−rI)(g−1),
s
X
i=1
rimi= (n2
I−nI)(g−1).
where rI=Pi∈Iri.
For example, if ris|n(n−1)(g−1), one could take
mi=n(n−1)(g−1)
sri
.
Let
(6.2) Unir(m) =
(E, ϕ)
∃σ∈H0(X, E/P) :
ϕ∈H0(X, Eσ(p)⊗K);
Eσ/U := EL∼
=Ls
i=1 Ei.
,
where
Ei=K−Pn
j=i+1 rirjO(mi)(O ⊕ K−1⊕. . . K −ri+1).
In what follows we prove that Unir(m) is a Lagrangian submanifold, for what we
need to study the associated spectral data. Consider restriction of hnto Unir(m),
hr(m) : Unir(m)−→ Hr.
The fact that the image is Hrcan be argued as in Theorem 4.4. Given b=
(b1, . . . , bs)∈Hnod
r, define
(6.3) ˆ
Lbi=π∗
biO(mi)−→ Xbi.
Then, analogously to Proposition 4.5, we have
Proposition 6.4. Assume that msatisfies (6.1). Let b∈Hnod
r. Then
hr(m)−1(b)∩Jacδ= ˆν−1(ˆ
L1,..., ˆ
Ls)
where we identify Jacδ(Xb)with an open subset of h−1(b)and define
ˆν: Jac(Xb)−→ Jac( e
Xb)
to be the pullback map.
THE BOREL SUBGROUP AND BRANES 27
Proof. After checking that (6.1) ensures the stability of the points of hr(m)−1(b),
the proof follows as in Proposition 4.5.
Continuing the parallelism with Uni(L), we next prove Lagrangianity of the
submanifold Unir(m).
Theorem 6.5. The subscheme Unir(m)is Lagrangian.
Proof. Isotropicity is proved as we did in Proposition 4.2.
By Lemma 6.2, there is an exact sequence
0−→ C×−→ (C×)s−→ (C×)δr−→ Jac(Xb)−→ Jac( e
Xb)−→ 0
where δr=P1≤i<j≤s2rirj(g−1). By Proposition 6.4, there are smooth points in
Unir(m), so the dimension is
dim Unir(m) = δ−s+ 1 + dim Hr=δ−s+1+X
i
(r2
i(g−1) + 1)
=n2(g−1) + 1,
which is half of the dimension of Mn.
Remark 6.6.The bundles ˆ
Lbiare built in a similar way to ˆ
Lfrom (4.2). Indeed,
the restriction of Oto Xbiis again O, which we twist by π∗
biO(mi). The fact that
ˆ
Lbiis a pullback from a line bundle on Xensures isotropicity of Mr(mi), as the
πbi,∗(ˆ
Lbi) has underlying vector bundle independent of bi. On the other hand, the
fact that there is a spectral datum over each point of the Hitchin base, ensures
Lagrangianity.
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Adv. Theo. Math. Phys. 20 (2016), 525–551.
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thogonal Higgs bundles. Preprint arXiv:1708.08828.
[BNR] A. Beauville, M. S. Narasimhan and S. Ramanan, Spectral curves and the generalised
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Emilio Franco, CMUP (Centro de Matem´
atica da Universidade do Porto), Universi-
dade do Porto, Rua do Campo Alegre 1021/1055, 4169-007, Porto (Portugal)
E-mail address:emilio.franco@fc.up.pt
Ana Pe´
on-Nieto, Laboratoire de Math´
ematiques J.A. Dieudonn´
e, UMR 7351 CNRS,
Universit´
e de Nice Sophia-Antipolis, 06108 Nice Cedex 02, France
E-mail address:ana.peon-nieto@unice.fr
