arxiv:2003.06997v1 [math.ag] 16 mar 2020indranil biswas, sorin dumitrescu, and sebastian heller...

IRREDUCIBLE FLAT SL(2,R)-CONNECTIONS ON THE TRIVIALHOLOMORPHIC BUNDLE

INDRANIL BISWAS, SORIN DUMITRESCU, AND SEBASTIAN HELLER

Abstract. We construct an irreducible holomorphic connection with SL(2,R)–monodromyon the trivial holomorphic vector bundle of rank two over a compact Riemann surface.This answers a question of Calsamiglia, Deroin, Heu and Loray in [CDHL].

Résumé. Dans cet article nous munissons le fibré vectoriel holomorphe trivial de rangdeux au-dessus d’une surface de Riemann compacte de genre g ≥ 2, d’une connexionholomorphe irréductible dont la monodromie est contenue dans SL(2,R). Ceci répond àune question posée par Calsamiglia, Deroin, Heu et Loray dans [CDHL].

1. Introduction

Take a compact connected oriented topological surface S of genus g, with g ≥ 2. Thereis a natural bijection between the isomorphism classes of flat SL(2,C)–connections overS and the conjugacy classes of group homomorphisms from the fundamental group of Sinto SL(2,C) (two such homomorphisms are called conjugate if they differ by an innerautomorphism of SL(2,C)). This bijection sends a flat connection to its monodromy rep-resentation. When S is equipped with a complex structure, a flat SL(2,C)–connectionon S produces a holomorphic vector bundle of rank two and trivial determinant on theRiemann surface defined by the complex structure on S; this is because locally constanttransition functions producing the vector bundle are holomorphic. In fact, since a holo-morphic connection on a compact Riemann surface Σ is automatically flat, there is anatural bijection between the following two:

(1) isomorphism classes of flat SL(2,C)–connections on a compact Riemann surfaceΣ;

(2) isomorphism classes of pairs of the form (E, D), where E is a holomorphic vector

bundle of rank two on Σ with∧2E holomorphically trivial, and D is a holomorphic

connection on E that induces the trivial connection on∧2E.

The above bijection is a special case of the Riemann–Hilbert correspondence [De].

Consider the flat SL(2,C)–connections on a compact Riemann surface Σ satisfyingthe extra condition that the corresponding holomorphic vector bundle of rank two onΣ is holomorphically trivial; they are known as differential sl(2,C)–systems on Σ (see[CDHL]), where sl(2,C) is the Lie algebra of SL(2,C). The differential sl(2,C)–systemson Σ are parametrized by the complex vector space sl(2,C)⊗C H0(Σ, KΣ), where KΣ isthe holomorphic cotangent bundle of Σ. A differential sl(2,C)–system is called irreducible

Date: October 20, 2020.2020 Mathematics Subject Classification. 34M03, 34M56, 14H15, 53A55.Key words and phrases. Local system, character variety, holomorphic connection, monodromy.

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2 I. BISWAS, S. DUMITRESCU, AND S. HELLER

if the monodromy representation of the corresponding flat connection is irreducible. Weshall now describe a context in which irreducible differential sl(2,C)–systems appear.

For any cocompact lattice Γ ⊂ SL(2,C), the compact complex threefold SL(2,C)/Γdoes not admit any compact complex hypersurface [HM, p. 239, Theorem 2]. WhileSL(2,C)/Γ does not contain a CP1, it may contain some elliptic curves. A questionof Margulis asks whether SL(2,C)/Γ can contain a compact Riemann surface of genusbigger than one. Ghys has the following reformulation of Margulis’ question: Is there apair (Σ, D), where D is a differential sl(2,C)–system on a compact Riemann surface Σof genus at least two, such that the image of the monodromy homomorphism π1(Σ) −→SL(2,C) for D is a conjugate of Γ? Existence of such a pair (Σ, D) is equivalent to theexistence of a holomorphic map ψ : Σ −→ SL(2,C)/Γ such that the homomorphismψ∗ : π1(Σ) −→ π1(SL(2,C)/Γ) is surjective.

Being inspired by Ghys’ strategy, the authors of [CDHL] study the Riemann–Hilbertmapping for the irreducible differential sl(2,C)–systems (see also [BD]). Although some(local) results were obtained in [CDHL] and [BD], the question of Ghys is still open. In thisdirection, it was asked in [CDHL] (p. 161) whether discrete or real subgroups of SL(2,C)can be realized as the monodromy of some irreducible differential sl(2,C)–system on somecompact Riemann surface. Note that if the flat connection on a compact Riemann surfaceΣ corresponding to a homomorphism π1(Σ) −→ SL(2,C) with finite image is irreducible,then the underlying holomorphic vector bundle is stable [NS], in particular, the underlyingholomorphic vector bundle is not holomorphically trivial.

Our main result (Theorem 4.3) is the construction of a pair (Σ, D), where Σ is acompact Riemann surface of genus bigger than one and D is an irreducible differentialsl(2,C)–system on Σ, such that the image of the monodromy representation for D iscontained in SL(2,R).

In Section 2 we collect preliminaries about moduli spaces and parabolic bundles. InSection 3 we construct flat connections ∇ with SL(2,R)-monodromy on prescribed par-abolic bundles over a 4-punctured torus, and in Section 4 we show how ∇ gives rise toan irreducible sl(2,C)–system with real monodromy on certain ramified coverings of thetorus, such that the underlying rank two holomorphic bundle is trivial.

2. Preliminaries

2.1. The Betti moduli space of a 1-punctured torus. Let Γ = Z +√−1Z ⊂ C be

the standard lattice. Define the elliptic curve T 2 := C/Γ, and fix the point

o =[

1+√−1

2

]∈ T 2 . (2.1)

For a fixed ρ ∈ [0, 12 [, we are interested in the Betti moduli space Mρ1,1 parametrizing

flat SL(2,C)–connections on the complement T 2 \ {o} whose local monodromy around olies in the conjugacy class of(

exp(2π√−1ρ) 0

0 exp(−2π√−1ρ)

)∈ SL(2,C) . (2.2)

This Betti moduli space Mρ1,1 does not depend on the complex structure of T 2. Whenρ = 0, it is the moduli space of flat SL(2,C)–connections on T 2; in that case Mρ1,1 is asingular affine variety. However, for every 0 < ρ < 12 , the space M

ρ1,1 is a nonsingular

affine variety. We shall recall an explicit description of this affine variety.

IRREDUCIBLE FLAT SL(2,R)-CONNECTIONS ON TRIVIAL BUNDLE 3

Let x, y, z be the algebraic functions onMρ1,1 defined as follows: for any homomorphism

h : π1(T2 \ {o}, q) −→ SL(2,C)

representing [h] ∈ Mρ1,1, where q = [0],

x([h]) = tr(h(α)), y([h]) = tr(h(β)), z([h]) = tr(h(βα)), (2.3)

where α, β are the standard generators of π1(T2 \ {o}, q), represented by the curves

t ∈ [0, 1] 7−→ α(t) = t mod Γ (2.4)and

t ∈ [0, 1] 7−→ β(t) = t√−1 mod Γ , (2.5)

respectively.

The variety Mρ1,1 is defined by the equation

Mρ1,1 = {(x, y, z) ∈ C3 | x2 + y2 + z2 − xyz − 2− 2 cos(2πρ) = 0} ; (2.6)

the details can be found in [Go], [Ma].

Lemma 2.1. Take any ρ ∈ ]0, 12 [, and consider a representation

h : π1(T2 \ {o}, q) −→ SL(2,C) ,

with [h] ∈ Mρ1,1. Then, the representation of the free group F (s, t), with generators s andt, defined by

s 7−→ X := h(α)h(α) and t 7−→ Y := h(β)h(β)is reducible if and only if x([h])y([h]) = 0, where x, y are the functions in (2.3).

Proof. It is known that, up to conjugation, we have

h(α) =

(x([h]) 1−1 0

), h(β) =

(0 −ζζ−1 y([h])

), (2.7)

where

ζ + ζ−1 = z([h]) (2.8)

(see [Go]). The lemma follows from a direct computation by noting that a representationgenerated by two SL(2,C) matrices A, B is reducible if and only if AB −BA has a non-trivial kernel. If AB−BA has a non-trivial kernel, then A, B and AB−BA lie in a Borelsubalgebra of sl(2,C). �

2.2. Parabolic bundles and holomorphic connections.

2.2.1. Parabolic bundles. We briefly recall the notion of a parabolic structure, mainly forthe purpose of fixing the notation. We are only concerned with the SL(2,C)–case, so ournotation differs from the standard references, e.g., [MS, Biq, Bis]. Instead, up to a factor2, we follow the notation of [Pi]; see also [HH] for this notation.

Let V −→ Σ be a holomorphic vector bundle of rank two with trivial determinantbundle over a compact Riemann surface Σ. Let p1, · · · , pn ∈ Σ be pairwise distinct npoints, and set the divisor

D = p1 + . . .+ pn .

For every k ∈ {1, · · · , n}, let Lk ⊂ Vpk be a line in the fiber of V at pk, and also take

ρk ∈ ]0, 12 [ .


Definition 2.2. A parabolic structure on V is given by the data

P := (D, {L1, · · · , Ln}, {ρ1, · · · , ρn}) ;we call {Lk}nk=1 the quasiparabolic structure, and ρk the parabolic weights. A parabolicbundle over Σ is given by a rank two holomorphic vector bundle V , with

∧2 V = OΣ,together with a parabolic structure P on V .

It should be emphasized that Definition 2.2 is very specific to the case of parabolicSL(2,C)–bundles. The parabolic degree of a holomorphic line subbundle F ⊂ V is

par-deg(F ) := degree(F ) +

n∑k=1

ρFk ,

where ρFk = ρk if Fpk = Lk and ρFk = −ρk if Fpk 6= Lk.

Definition 2.3. A parabolic bundle (V, P) is called stable ifpar-deg(F ) < 0

for every holomorphic line subbundle F ⊂ V .

As before, P = (D = p1 + . . .+ pn, {L1, · · · , Ln}, {ρ1, · · · , ρn}) is a parabolic struc-ture on a rank two holomorphic vector bundle V of trivial determinant.

A strongly parabolic Higgs field on a parabolic bundle (V, P) is a holomorphic section

Θ ∈ H0(Σ, End(V )⊗KΣ ⊗OΣ(D))such that trace(Θ) = 0 and Lk ⊂ kernel(Θ(pk)) for all 1 ≤ k ≤ n. These two conditionstogether imply that all the residues of a strongly parabolic Higgs field are nilpotent.

2.2.2. Deligne extension. Take a flat SL(2,C)–connection ∇ on a holomorphic vector bun-dle E0 over T

2 \ {o} (see (2.1)), corresponding to a point of Mρ1,1. Then locally aroundo ∈ T 2, the connection ∇ is holomorphically SL(2,C)–gauge equivalent to the connection

d+

(ρ 00 −ρ

)dw

w(2.9)

on the trivial holomorphic bundle of rank two, where w is a holomorphic coordinatefunction on T 2 defined around o with w(o) = 0. Take such a neighborhood Uo of o and aholomorphic coordinate function w. Consider the trivial holomorphic bundle Uo×C2 −→Uo equipped with the logarithmic connection in (2.9). Now glue the two holomorphicvector bundles, namely Uo × C2 and E0, over the common open subset Uo \ {o} suchthat the connection ∇|Uo\{o} is taken to the restriction of the logarithmic connection in(2.9) to Uo \ {o}. This gluing is holomorphic because it takes one holomorphic connectionto another holomorphic connection. Consequently, this gluing produces a holomorphicvector bundle

V −→ T 2 (2.10)of rank 2. Furthermore, the connection ∇ on E0 −→ T 2 \ {o} extends to a logarithmicconnection on V over T 2, because the meromorphic connection in (2.9) is a logarithmicconnection on Uo × C2. This resulting logarithmic connection on V will also be denotedby ∇ (see [De] for details). The logarithmic connection on the exterior product

Uo ×∧2

C2 = Uo × C −→ Uo


induced by the logarithmic connection on Uo × C2 −→ Uo in (2.9) is actually a regularconnection; in fact, it coincides with the trivial connection on Uo×C. On the other hand,the connection on

∧2E0 = OT 2\{o} induced by the connection ∇ on E0 coincides withthe trivial connection (recall that ∇ is a SL(2,C)–connection on E0). Consequently,

(1)∧2 V = OT 2 , where V is the vector bundle in (2.10), and

(2) the logarithmic connection on∧2 V induced by the logarithmic connection ∇ on

V coincides with the trivial holomorphic connection on OT 2 induced by the deRham differential d.

In particular, we have degree(V ) = 0.

From Atiyah’s classification of holomorphic vector bundles over any elliptic curve [At],the possible types of the vector bundle V in (2.10) are:

(1) V = L⊕ L∗ , with degree(L) = 0;(2) there is a spin bundle S on T 2 (meaning a holomorphic line bundle of order two),

such that V is a nontrivial extension of S by itself; and(3) V = L⊕ L∗ , with degree(L) > 0.

Lemma 2.4. Consider the vector bundle V in (2.10) for 12 > ρ > 0. Then the last oneof the above three cases, as well as the special situation of the first case where L = S isa holomorphic line bundle with S⊗2 = OT 2, cannot occur.

Proof. First assume that case (3) occurs. So V = L⊕ L∗ , with degree(L) > 0. We have

degree(Hom(L, L∗)⊗KT 2 ⊗OT 2(o)) = 1− 2degree(L) < 0 ,

where KT 2 is the holomorphic cotangent bundle. So the second fundamental form of Lfor ∇, which is a holomorphic section of Hom(L, L∗)⊗KT 2⊗OT 2(o), vanishes identically.Consequently, the logarithmic connection ∇ on V preserves the line subbundle L. SinceL admits a logarithmic connection, with residue r ∈ {ρ, −ρ} at o, we have

degree(L) + r = 0

[Oh, p. 16, Theorem 3]. But this contradicts the fact that ρ ∈ ]0, 12 [. So case (3) does notoccur.

Next assume that V = S ⊕ S∗ = S ⊗C C2, where S is a holomorphic line bundle withS⊗2 = OT 2 . Then V admits a holomorphic connection, and moreover

H0(T 2, End(V )⊗KT 2 ⊗OT 2(o)) = H0(T 2, End(V )⊗KT 2) = End(C2) .

So V does not admit a logarithmic connection singular exactly over o ∈ T 2, because suchlogarithmic connections form an affine space for the vector space H0(T 2, End(V )⊗KT 2⊗OT 2(o)). �

2.2.3. Parabolic structure from a logarithmic connection. Consider a logarithmic connec-tion∇ on a holomorphic bundle V of rank two and with trivial determinant over a compactRiemann surface Σ. We assume that ∇ is a SL(2,C)–connection, meaning the logarithmicconnection on

∧2 V induced by ∇ is a holomorphic connection with trivial monodromy;note that this implies that

∧2 V = OΣ. Let p1, · · · , pn ∈ Σ be the singular points of ∇.We also assume that the residue

Respk(∇) ∈ End0(Vpk)


of the connection ∇ at every point pk, 1 ≤ k ≤ n, has two real eigenvalues ±ρk, withρk ∈ ]0, 12 [. For every 1 ≤ k ≤ n, let

Lk := Eig(Respk(∇), ρk) ⊂ Vpkbe the eigenline of the residue of ∇ at pk for the eigenvalue ρk.

The logarithmic connection ∇ gives rise to the parabolic structureP = (D = p1 + . . .+ pn, {L1, · · · , Ln}, {ρ1, · · · , ρn}) .

It is straightforward to check that another such logarithmic connections ∇1 on V inducesthe same parabolic structure P if and only if ∇ − ∇1 is a strongly parabolic Higgs fieldon (V, P), in the sense of Section 2.2.1.

It should be mentioned that in [MS] a different local form of the connection is used(instead of the local form in (2.9)). In that case the Deligne extension gives a rank

two holomorphic vector bundle W (instead of V ) with∧2W = OΣ(−D) (instead of∧2 V = OΣ), while the parabolic weights at pk become ρk, 1− ρk (instead of ρk, −ρk).

A theorem of Mehta and Seshadri [MS, p. 226, Theorem 4.1(2)], and Biquard [Biq,p. 246, Théorème 2.5] says that the above construction of a parabolic bundle (V, P) froma logarithmic connection ∇ produces a bijection between the stable parabolic bundles (inthe sense of Section 2.2.1) on (Σ, D) and the space of isomorphism classes of irreducibleflat SU(2)–connections on the complement Σ\D. See, for example, [Pi, Theorem 3.2.2] forour specific situation. As a consequence of the above theorem of [MS] and [Biq], for everylogarithmic connection ∇ on V which produces a stable parabolic structure P, there existsa unique strongly parabolic Higgs field Θ on (V, P) such that the holonomy of the flatconnection ∇+ Θ is contained in SU(2). This flat SU(2)–connection ∇+ Θ is irreducible,because the parabolic bundle is stable.

2.3. Abelianization. Take o ∈ T 2 as in (2.1). In [He], representatives ∇ for each gaugeclass in Mρ1,1 are computed for the special case where ρ = 16 and L ∈ Jac(T

2) \ {S |S⊗2 = KT 2}. We shall show (see Proposition 2.5) that for general ρ and

L ∈ Jac(T 2) \ {S | S⊗2 = OT 2} ,the corresponding connection ∇ is of the form

∇ = ∇a,χ,ρ =(∇L γ+χγ−χ ∇L

∗

), (2.11)

where a, χ ∈ C, and∇L = d+ a · dw + χ · dw

is a holomorphic connection on L with ∇L∗ being the dual connection on L∗ ; here wdenotes a complex affine coordinate on T 2 = C/Γ. The off–diagonal terms in (2.11) canbe described explicitly in terms of the theta functions as explained below.

Before doing so, we briefly describe both the Jacobian and the rank one de Rham modulispace for T 2 in terms of some useful coordinates. Let d = ∂ + ∂ be the decomposition ofthe de Rham differential d on T 2 into its (1, 0)–part ∂ and (0, 1)–part ∂. It is well–knownthat every holomorphic line bundle of degree zero on T 2 is given by a Dolbeault operator

∂χ

= ∂ + χ · dwon the C∞ trivial line bundle T 2 × C −→ T 2, for some χ ∈ C, where w is an affinecoordinate function on C/(Z+

√−1Z) = T 2 (note that dw does not depend on the choice


of the affine function w). So the operator ∂χ

sends a locally defined C∞ function f to the(0, 1)-form ∂f + χf · dw = ∂f + χf · ∂w. Two such differential operators

∂χ1

and ∂χ2

determine isomorphic holomorphic line bundles if and only if ∂χ1

and ∂χ2

are gaugeequivalent. They are gauge equivalent if and only if

χ2 − χ1 ∈ Γ∗ := πZ + π√−1Z (2.12)

Similarly, flat line bundles over T 2 are given by the connection operator

da,χ = d+ a · dw + χ · dwon the C∞ trivial line bundle T 2 ×C −→ T 2, for some a, χ ∈ C. Moreover two connec-tions da1,χ1 and da2,χ2 are isomorphic if and only if

(a2 − a1) + (χ2 − χ1) ∈ 2π√−1Z and (a2 − a1)− (χ2 − χ1) ∈ 2π

√−1Z .

The (shifted) theta function for Γ = Z + Z√−1 will be denoted by ϑ. In other words,

ϑ is the unique (up to a multiplicative constant) entire function satisfying ϑ(0) = 0 and

ϑ(w + 1) = ϑ(w), ϑ(w +√−1) = −ϑ(w) exp(−2π

√−1w) .

Then the function

tx(w − 1+√−1

2 ) :=ϑ(w − x)ϑ(w)

exp(−πx(w − w))

is doubly periodic on C \ (1+√−1

2 + Γ) with respect to Γ and satisfies the equation

(∂−πxdw)tx = 0 .

Thus tx is a meromorphic section of the holomorphic bundle L(∂−πx

) := [∂−πx

] (it isthe holomorphic line bundle given by the Dolbeault operator ∂−πxdw). Notice that forx /∈ Γ, the section tx has a simple zero at w = x and a first order pole at w = o (see(2.1)). Moreover, up to scaling by a complex number, this tx is the unique meromorphic

section of L(∂−πx

) := [∂−πx

] with a simple pole at o.

For 12 > ρ > 0, if V in (2.10) is of the form V = L ⊕ L∗, then from Lemma 2.4 it

follows that degree(L) = 0 and L is not a spin bundle. In other words,

L = L(∂ + χ · dw)for some χ ∈ C \ 12Γ

∗; compare with (2.12).

Proposition 2.5. For any ρ ∈ [0, 12 [, take [∇] ∈ Mρ1,1 such that its Deligne extension

is given by the holomorphic vector bundle V = L ⊕ L∗, where L = L(∂ + χdw) is aholomorphic line bundle on T 2 of degree zero such that L⊗2 6= OT 2. Set x = − 1πχ, sox /∈ 12Γ. Then, there exists

a ∈ Csuch that one representative of [∇] is given by

∇a,χ,ρ

as in (2.11), where the second fundamental forms γ+χ and γ−χ in (2.11) are given by the

meromorphic 1–forms

γ+χ ([w]) = ρϑ′(0)ϑ(−2x) t2x(w)dw and γ

−χ ([w]) = ρ

ϑ′(0)ϑ(2x) t−2x(w)dw (2.13)

with values in the holomorphic line bundles L(∂+2χdw) and L(∂−2χdw) respectively.


Proof. Using Section 2.2.2 we know that there exists a representative ∇ of [∇] such thatits (0, 1)–part ∂

∇is given by

∂∇

= ∂ +

(χdw 0

0 −χdw

).

The (1, 0)–part ∂∇ is given by ∂∇ = ∂ + Ψ, where Ψ =

(A BC −A

)is an End(V )–valued

meromorphic 1–form on T 2, with respect to the holomorphic structure ∂∇

, such that Ψhas a simple pole at o and Ψ is holomorphic elsewhere. In particular, A is a meromorphic1–form on T 2 with simple pole at o (see (2.1)), and hence by the residue theorem it is infact holomorphic, i.e.,

A = adw

for some a ∈ C. Furthermore, B and C are meromorphic 1–forms with values in theholomorphic bundles L(∂+2χdw) and L(∂−2χdw), respectively. Note that for

x = − 1πχ ∈ 12Γ ,

the holomorphic line bundle L(∂+2χdw) would be the trivial and B and C cannot havenon-trivial residues at o by the residue theorem. The determinant of the residue of Ψ at ois −ρ2 by (2.9). Therefore, from the holomorphicity of A we conclude that the quadraticresidue of the meromorphic quadratic differential BC is

qreso(BC) = ρ2 .

From the discussion prior to this proposition it follows that there is a unique meromorphicsection of L(∂±2χdw) with a simple pole at o. Thus, after a possible constant diagonalgauge transformation, from the uniqueness, up to scaling, of the meromorphic section ofL(∂±2χdw) with simple pole at o, it follows that

B = γ+χ and C = γ−χ ,

where γ+χ and γ−χ are the second fundamental forms (2.11). This completes the proof. �

Proposition 2.6. Assume that ρ ∈ ]0, 12 [. Take [∇] ∈ Mρ1,1 such that the corresponding

bundle V in (2.10) is of the form L⊕ L∗ (so L is not a spin bundle but its degree is zeroby Lemma 2.4). Then, the rank two parabolic bundle corresponding to [∇] (see Section2.2.3) is parabolic stable.

Proof. The two holomorphic line bundles L and L∗

are not isomorphic, because L is nota spin bundle. From this it can be shown that any holomorphic subbundle of degree zero

ξ ⊂ V = L⊕ L∗

is either L or L∗. Indeed, if ξ is a degree zero holomorphic line bundle different from both

L and L∗, then

H0(T 2, Hom(ξ, V )) = H0(T 2, Hom(ξ, L))⊕H0(T 2, Hom(ξ, L∗)) = 0 .

As the residue in (2.11) is off–diagonal (with respect to the holomorphic decompositionV = L ⊕ L∗), the above observation implies that every holomorphic line subbundleξ ⊂ V of degree zero has parabolic degree −ρ. On the other hand, the parabolic degreeof a holomorphic line subbundle of negative degree is less than or equal to −1 + ρ < 0 .Consequently, the parabolic bundle corresponding to [∇] is stable. �


3. Flat connections on the 4-punctured torus

Consider

T̂ 2 := C/(2Z + 2√−1Z)

and the 4–fold covering

Π : T̂ 2 −→ T 2 := C/(Z +√−1Z) (3.1)

produced by the identity map of C. Let

{p1, p2, p3, p4} := Π−1(o) ⊂ T̂

be the preimage of o =[

1+√−1

2

]∈ T 2 (see (2.1)). Fix

ρ = 0 ,

and consider the corresponding connection ∇ = ∇a,χ,0 in (2.11). We use Π in (3.1) topull back this connection to T̂ 2.

Let

h : π1(T̂2, q) −→ SL(2,C)

be the monodromy representation for Π∗∇a,χ,0, where q = [0] ∈ T̂ 2.The traces

T1(χ, a) = tr(h(α̂)) and T2(χ, a) = tr(h(β̂))

along

α̂ , β̂ ∈ π1(T̂ 2 \ {p1, · · · , p4}, q) (3.2)with representatives

α̂ : [0, 2] 7−→ 2t mod 2Γ ∈ T̂ 2

and

β̂ : [0, 2] 7−→ 2t√−1 mod 2Γ ∈ T̂ 2

(see Figure 1a, Figure 1b) are given by

T1(χ, a) = exp(−2(a+ χ)) + exp(2(a+ χ))and

T2(χ, a) = exp(2√−1(−a+ χ)) + exp(2

√−1(a− χ))

respectively, while the local monodromy of Π∗∇a,χ,0 around each of p1, · · · , p4 is trivial,because we have ρ = 0.

In the following, set

χ =π

4(1−

√−1) and ak = −

π

4(1 +

√−1) + kπ(1 +

√−1) (3.3)

for all k ∈ Z. Then we haveT1(χ, ak) = −(exp(−2kπ) + exp(2kπ)) ∈ R (3.4)

T2(χ, ak) = −(exp(−2kπ) + exp(2kπ)) ∈ R ; (3.5)as before, T1(χ, ak) and T2(χ, ak) are the traces of holonomies of Π

∗∇ak,χ,0 (see (2.11) and(3.1)) along α̂ and β̂ respectively (see (3.2)). Moreover, a direct computation shows that

(s, t) 7−→ (=(T1(χ, ak + s+√−1t)), =(T2(χ, ak + s+

√−1t))) (3.6)

is a local diffeomorphism at (s, t) = (0, 0) by the implicit function theorem.


Theorem 3.1. Let k ∈ Z\{0}, χ = π4 (1−√−1) and ak = −π4 (1+

√−1)+kπ(1+

√−1).

Then there exists � > 0 such that for each ρ ∈ ]0, �[, there is a unique number a ∈ Cnear ak satisfying the condition that the monodromy of the flat connection

Π∗∇a,χ,ρ

on T̂ 2 \{p1, · · · , p4} (see (2.11) and (3.1)) is irreducible and furthermore the image of themonodromy homomorphism is conjugate to a subgroup of SL(2,R).

Proof. Using the fact that the map in (3.6) is a local diffeomorphism, there exists foreach sufficiently small ρ a unique complex number a such that the traces T1 and T2, of

holonomies of ∇a,χ,ρ along α̂ and β̂ respectively (see (3.2)), are real. Because k 6= 0, andρ is small, we obtain from (3.4) and (3.5) that these traces satisfy

T1 < −2 and T2 < −2 .

Recall the general formula

tr(X)tr(Y ) = tr(XY ) + tr(XY −1) (3.7)

for X, Y ∈ SL(2,C). Let

x = tr(h(α)) and y = tr(h(β)) (3.8)

be the traces of the monodromy homomorphism h of the connection ∇a,χ,ρ on T 2 \ {0}along α and β defined in (2.4) and (2.5) respectively.

Applying (3.7) to

X = h(α) = Y ( respectively, X = h(β) = Y )

we obtain that x (respectively, y) in (3.8) must be purely imaginary. Then it can bechecked directly that the trace along any closed curve in the 4–punctured torus is real.The fact that

z = tr(h(α ◦ β))is real is a direct consequence of (2.6) combined with the above observation that x, y ∈√−1R. Using (3.7) repeatedly (compare with [Go]) it is deduced that the trace of the

monodromy along any closed curve on T̂ 2 is real.

For ρ 6= 0 sufficiently small, the connection Π∗∇a,χ,ρ on T̂ 2 is irreducible as a conse-quence of Lemma 2.1 — note that the condition xy 6= 0 follows directly from the factthat ρ 6= 0 — applied to h(α̂) and h(β̂) (see (3.2)).

We will show that the image of the monodromy homomorphism h is conjugate to asubgroup of SL(2,R).

To prove the above statement, first note that since the monodromy h is irreducible andhas all traces real, the homomorphism h is in fact conjugate to its complex conjugaterepresentation h, meaning there exists C ∈ SL(2,C) such that

C−1hC = h .

Applying this equation twice we get that

CC = ±Id

because h is irreducible. If

CC = −Id ,


a straightforward computation shows that h is conjugate to a unitary representation. Sincethe traces of some elements in the image of the monodromy are not contained in [−2, 2],we are led to a contradiction.

Thus, we have

CC = Id .

A direct computation gives that

C = D−1D

for some D ∈ SL(2,C). Consequently, we have

DhD−1 = DhD−1.

Hence the image of the monodromy homomorphism h is conjugate to a subgroup ofSL(2,R). �

We shall use the following theorem.

Theorem 3.2. Let χ = π4 (1−√−1). For every ρ ∈ [0, 12 [, there exists a

u ∈ C such that

Π∗∇au,χ,ρ

is a reducible unitary connection satisfying the following condition: the monodromies ofΠ∗∇au,χ,ρ along

α̂ ∈ π1(T̂ 2 \ {p1, · · · , p4}, q) and β̂ ∈ π1(T̂ 2 \ {p1, · · · , p4}, q)

(see (3.2)) are both −Id. Moreover, the monodromies around the points p1, · · · , p4 are(after simultaneous conjugation) given by(

exp(2π√−1ρ) 0

0 exp(−2π√−1ρ)

),

(exp(−2π

√−1ρ) 0

0 exp(2π√−1ρ)

),(

exp(2π√−1ρ) 0

0 exp(−2π√−1ρ)

),

(exp(−2π

√−1ρ) 0

0 exp(2π√−1ρ)

)respectively.

Proof. First, for any ρ ∈ ]0, 12 [ and a ∈ C, the parabolic bundle on T2 determined by

∇a,χ,ρ with χ = π4 (1−√−1) is stable (see Proposition 2.6); this stable parabolic bundle

on T 2 will be denoted by W∗. Note that all the strongly parabolic Higgs fields on thisparabolic bundle are given by constant multiples scalar of(

dw 00 −dw

).

In view of the theorem of Mehta–Seshadri and Biquard ([MS], [Biq]) mentioned in Section2.2.3, there exists au ∈ C such that

∇au,χ,ρ

has unitary monodromy on T 2. Then, the flat connection Π∗∇au,χ,ρ on T̂ 2 has unitarymonodromy as well, where Π is the projection in (3.1).

On the other hand, the pulled back parabolic bundle Π∗W∗ on T̂2 is strictly semi-stable,

because χ = π4 (1 −√−1) and T̂ 2 = C/(2Γ) for the specific lattice 2Γ = 2Z + 2

√−1Z

(it can be proved by a direct computation, but it also follows from [HH, Theorem 3.5 andSection 2.4]), so that the unitary connection Π∗∇au,χ,ρ is automatically reducible.


In order to compute the entire monodromy representation, set x = y = 0, and considerthe unique positive solution of z in (2.6). Note, that if ρ = 0, then z = 2 and au = −χ,with χ given by (3.3). Now for a general real ρ, using (2.7) after setting ζ = exp(π

√−1ρ)

there, we see that the representation h of the fundamental group of the 1–punctured torusgiven by x(h) = 0 = y(h) and z(h) = z induces a unitary reducible representation ofthe fundamental group of the 4–punctured torus.

To identify the representation h with the monodromy representation of ∇au,χ,ρ, we notethat, for ρ < 14 (it suffices to consider this case for our proof), it can be shown that theparabolic structure on the holomorphic vector bundle

L⊕ L∗ −→ T̂ 2 (3.9)

cannot be strictly semi-stable if L⊗2 is not trivial. Indeed, the lines giving the quasi-parabolic structure are not contained in L or L∗ by (2.11). On the other hand, these twosubbundles, namely L and L

∗, are the only holomorphic subbundles of degree zero; this

follows from the assumption that L⊗2 6= OT̂ 2

, because H0(T̂ 2, Hom(ξ, L ⊕ L∗)) = 0,if ξ is a holomorphic line bundle of degree zero which is different from both L and L∗.Hence the parabolic structure on the holomorphic vector bundle in (3.9) cannot be strictlysemi-stable if L⊗2 6= O

T̂ 2.

By continuity of the monodromy representation of Π∗∇au,χ,ρ with respect to the param-eters (au, χ, ρ), the representation of ∇au,χ,ρ must be the unitary reducible representationh with x(h) = 0 = y(h) and positive z(h) = z.

Finally, the corresponding monodromies of Π∗∇au,χ,ρ can be computed using (2.7),where ζ = exp(π

√−1ρ) : the monodromies along α̂ and β̂ (see (3.2)) are given by

h(α)h(α) and h(β)h(β) respectively, and both are equal to −Id by (2.7), and the mon-odromies (based at q = [0]) around p1, · · · , p4 are given by

h(β)−1h(α)−1h(β)h(α) =

(exp(2π

√−1ρ) 0

0 exp(−2π√−1ρ)

),

h(α)h(β)−1h(α)−1h(β) =

(exp(−2π

√−1ρ) 0

0 exp(2π√−1ρ)

),

h(β)h(α)h(β)−1h(α)−1 =

(exp(2π

√−1ρ) 0

0 exp(−2π√−1ρ)

),

h(α)−1h(β)h(α)h(β)−1 =

(exp(−2π

√−1ρ) 0

0 exp(2π√−1ρ)

)(3.10)

respectively; compare with Figure 1a, Figure 1b. �

4. Flat irreducible SL(2,R)–connections on compact surfaces

We assume that

ρ =1

2p,

for some p ∈ N odd, with ρ being small enough so that Theorem 3.1 is applicable.

The torus T̂ 2 in (3.1) is of square conformal type, and it is given by the algebraicequation

y2 =z2 − 1z2 + 1

. (4.1)


Without loss of any generality, we can assume that the four points

{p1, · · · , p4} = Π−1({o}) ,

where Π is the map in (3.1) and o is the point in (2.1), are the branch points of thefunction z. With the labelling of the points as in Figure 1a, Figure 1b, i.e.,

p1 =

[1 +√−1

2

], p2 =

[3 +√−1

2

], p3 =

[3 + 3

√−1

2

],

p4 =

[1 + 3

√−1

2

]∈ C/(2Z + 2

√−1Z) = T̂ 2 ,

it can be shown (for example using the Weierstrass ℘-function) that the (y, z) coordinatesof p1, · · · , p4 and q = [0] can be chosen to be

p1 = (0, 1), p2 = (∞,√−1), p3 = (0, −1), p4 = (∞, −

√−1), q = (

√−1, 0) .

(4.2)

Define the compact Riemann surface Σ by the algebraic equation

x2p =z2 − 1z2 + 1

. (4.3)

Consider the p–fold covering

Φp : Σ −→ T̂ 2 , (x, z) 7−→ (xp, z) , (4.4)

which is totally branched over p1, · · · , p4. Denote the inverse image Φ−1p (pi), 1 ≤ i ≤ 4,by Pi (see Figure 1c).

For a connection ∇A (respectively, ∇B) on a vector bundle A (respectively, B), theinduced connection (∇A ⊗ IdB)⊕ (IdA ⊗∇B) on A⊗B will be denoted by ∇A ⊗∇B fornotational convenience.

There are holomorphic line bundles

S −→ Σ

of degree −2 such that

S ⊗ S = OΣ(−P1 − P2 − P3 − P4) .

For every such S, there is a unique logarithmic connection ∇S on S with the propertythat

(∇S ⊗∇S)(s−P1−P2−P3−P4) = 0 ,where s−P1−P2−P3−P4 is the meromorphic section of OΣ(−P1−P2−P3−P4) given by theconstant function 1 on Σ (this section has simple poles at P1, · · · , P4). The residue of ∇Sat Pj , 1 ≤ j ≤ 4, is 12 . Observe that the monodromy representation of ∇

S takes values

in Z/2Z. Also, note that (S, ∇S) is unique up to tensoring with an order two holomorphicline bundle ξ equipped with the (unique) canonical connection that induces the trivialconnection on ξ ⊗ ξ.

Lemma 4.1. For given ρ = 12p and Σ (see (4.3)), consider au and χ as in Theorem 3.2.

There exists a unique pair (S, ∇S) such that the monodromy of the connection

∇S ⊗ (Π ◦ Φp)∗∇au,χ,ρ

is trivial (see (2.11), (3.1) and (4.4)).


Proof. Since p is odd, ρ = 12p , and Φp in (4.4) is a totally branched covering, the local

monodromies of(Π ◦ Φp)∗∇a

u,χ,ρ

around the points of Pi, 1 ≤ i ≤ 4, are all −Id.The totally branched covering Σ −→ CP 1 in (4.3) is determined by its monodromy

representationM : π1(CP 1 \ {±1, ±

√−1}, 0) −→ S2p

into the permutation group S2p of the 2p points over z = 0 which we label by

x ∈{√−1,√−1 exp

(π√−1p

), · · · ,

√−1 exp

(π√−1jp

), · · · ,

√−1 exp

(π√−1(2p− 1)

p

)}.

The monodromy representation of the cyclic covering Σ −→ CP 1 in (4.3) is abelian, andthe local monodromies around the 4 punctures are given by

M1 = M−1 = exp(π√−1p

), M√−1 = M−√−1 = exp(−

π√−1p

) . (4.5)

The later can be computed via the logarithmic monodromy of log x by integrating

1

2p

dx2p

x2p=

1

2p

d z2−1z2+1

z2−1z2+1

using the residue theorem.

The p–fold cyclic covering Φp in (4.4) is also determined by its monodromy representa-tion

m : π1(T̂2 \ {p1, · · · , p4}, [0]) −→ Sp .

As in (4.2), [0] ∈ T̂ 2 = C/(2Z + 2√−1Z) is a point lying over z = 0 with respect to

(4.1). Again, the image m(π1(T̂2 \ {p1, · · · , p4}, [0])) is abelian, and we claim that it is

given by

mp1 = mp3 = exp

(2π√−1p

), mp2 = mp4 = exp

(−2π√−1p

), mα̂ = 1, mβ̂ = 1 ;

(4.6)here mpk are the local monodromies around pk.

The above claim simply follows by describing closed loops on the 4-punctured torus asspecial closed loops on the 4-punctured sphere and using (4.5).

Consider the unitary abelian monodromy representation

R : π1(T̂2 \ {p1, · · · , p4}, [0]) −→ SU(2)

of the connection Π∗∇au,χ,ρ on T̂ 2\{p1, · · · , p4} (see (2.11) and (3.1)). Using the diagonalrepresentation

u : Z/pZ −→ SU(2), exp(

2π√−1p

)7−→

(exp(2π

√−1(k+1)p ) 0

0 exp(−2π√−1(k+1)p )

)it follows from Theorem 3.2 and (4.6) that the two homomorphisms u ◦ m and R fromπ1(T

2 \ {p1, · · · , p4}, [0]) to SU(2) differ only by a Z/2Z-representation (with values in{±Id} ⊂ SU(2)). Note that the local monodromies of this Z/2Z-representation are −Id,and, as p is odd, the same holds for the corresponding Z/2Z-representation µ of thefundamental group of the 4-punctured covering Σ \ Φ−1p {p1, · · · , p4}.


The spin bundle S is then chosen to give the aforementioned Z/2Z-representation µ ofthe fundamental group of the 4-punctured surface Σ\Φ−1p {p1, · · · , p4}. Finally, the lemmathen follows from the fact that the representation m induces the trivial representation onΣ \ Φ−1p {p1, · · · , p4} by the standard property of the monodromy on a covering that it isthe pullback of the monodromy. �

Henceforth, we always assume that

ρ = 12p .

The connection in Lemma 4.1

∇S ⊗ (Π ◦ Φp)∗∇au,χ,ρ = ∇S ⊗ (Π ◦ Φp)∗∇

au,χ,12p

is defined on the vector bundle

S ⊗ (L⊕ L∗) −→ Σ ,where L is the pull-back, by Π◦Φp, of the C∞ trivial line bundle T 2×C −→ T 2 equippedwith Dolbeault operator

∂ + χdw = ∂ + χ · ∂w .For each 1 ≤ i ≤ 4, the residues of the connection ∇S ⊗ (Π ◦Φp)∗∇a

u,χ,ρ at the point ofPi = Φ

−1p (pi) is

12

(1 −1−1 1

)(4.7)

with respect to a suitable frame at the points Pi compatible with the decompositionS ⊗ (L⊕ L∗) = (S ⊗ L)⊕ (S ⊗ L∗); compare with Proposition 2.5 and its proof.

As in [He, § 3], there exists a holomorphic rank two vector bundle V on Σ with trivialdeterminant, equipped with a holomorphic connection D, together with a holomorphicbundle map

F : S ⊗ (L⊕ L∗) −→ V (4.8)which is an isomorphism away from P1, · · · , P4, such that

∇S ⊗ (Π ◦ Φp)∗∇au,χ,ρ = F−1 ◦D ◦ F . (4.9)

From Lemma 4.1 we know that (V, D) is trivial.

Lemma 4.2. Assume p ≥ 3. Consider the strongly parabolic Higgs field

Ψ =

(dw 00 −dw

)with respect to the parabolic structure induced by ∇au,χ,ρ. Then,

Θ = F ◦ (Π ◦ Φp)∗Ψ ◦ F−1

is a holomorphic Higgs field on the trivial holomorphic vector bundle (V, D0,1) = (V, D′′)(here the Dolbeault operator for the trivial holomorphic structure is denoted by D′′).

Proof. Consider the holomorphic Higgs field

(Π ◦ Φp)∗Ψ : S ⊗ (L⊕ L∗) −→ KΣ ⊗ S ⊗ (L⊕ L∗)on the rank two holomorphic bundle S ⊗ (L⊕ L∗). It vanishes of order p− 1 ≥ 2 at thesingular points P1, · · · , P4. Performing the local analysis (as in [He, § 3.2]) near Pk of thenormal form of the homomorphism F in (4.8), we see that

Θ = F ◦ (Π ◦ Φp)∗Ψ ◦ F−1


has no singularities, i.e., it is a holomorphic Higgs field on the trivial holomorphic vectorbundle (V, D′′). �

Theorem 4.3. There exists a compact Riemann surface Σ of genus g > 1 with a irre-ducible holomorphic connection ∇ on the trivial holomorphic rank two vector bundle O⊕2Σsuch that the image of the monodromy homomorphism for ∇ is contained in SL(2,R).

Proof. For ρ = 12p , with p being an odd integer, consider the connection ∇a,χ,ρ, over

rank two vector bundle on T 2, given by Theorem 3.1. Since the image of the monodromyhomomorphism for Π∗∇a,χ,ρ is conjugate to a subgroup of SL(2,R), and ∇S has Z/2Z–monodromy, the image of the monodromy homomorphism for the connection

∇S ⊗ (Π ◦ Φp)∗∇a,χ,ρ

can be conjugated into SL(2,R) as well. The same holds for the connection

∇ := F ◦ (∇S ⊗ (Π ◦ Φp)∗∇a,χ,ρ) ◦ F−1

because F is a (singular) gauge transformation. From Lemma 4.2 we know that ∇−D isa holomorphic Higgs field on the trivial holomorphic vector bundle (V, D′′), where D isthe trivial connection in (4.9).

It remains to show that the monodromy homomorphism for ∇ is an irreducible rep-resentation of the fundamental group. Since ρ 6= 0 is small, this follows from Lemma2.1. Indeed, observe that there exists α̃, β̃ ∈ π1(Σ, q) (see Figure 1c) along which themonodromies of ∇ are given by

h(α)h(α) and h(β)h(β)

up to a possible sign. For example, representatives of α̃, β̃ are given by a connected

component of the preimage of α̂ and β̂ respectively. Because xy 6= 0, in view of (3.4) and(3.5) and continuity in ρ, the monodromy representation must be irreducible by Lemma2.1. �

5. Figures

origins

hat alpha

hat beta

R,

B

B

p, Pz

n

•

a 9-

(a) The 4-puncturedtorus.

f

pg

EFFI,

Efim ,••--→ME⇐EqiE¥E*.. . .waifE. sweettwo••

R,

(b) A second view of the 4-puncturedtorus.

••

P3

Fogo#

QanatBeg

array,

f.

B-4!hqasmmmf.Pd9-

Madof;He...dSkramstad

P4•

(c) The Riemann surface Σfor p = 3, shown with verti-cal and horizontal trajecto-ries of (Π◦Φ3)∗(dw)2. Pic-ture by Nick Schmitt.

Figure 1


Acknowledgements

We are grateful to the referees for helpful comments. This work has been supported bythe French government through the UCAJEDI Investments in the Future project managedby the National Research Agency (ANR) with the reference number ANR2152IDEX201.The first-named author is partially supported by a J. C. Bose Fellowship, and school ofmathematics, TIFR, is supported by 12-R&D-TFR-5.01-0500.

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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,Mumbai 400005, India

Email address: [email protected]

Université Côte d’Azur, CNRS, LJAD, France


Institute of Differential Geometry, Leibniz Universität Hannover, Welfengarten 1,30167 Hannover, Germany


1. Introduction2. Preliminaries2.1. The Betti moduli space of a 1-punctured torus2.2. Parabolic bundles and holomorphic connections2.3. Abelianization

3. Flat connections on the 4-punctured torus4. Flat irreducible `39`42`"613A``45`47`"603ASL(2,R)–connections on compact surfaces5. FiguresAcknowledgementsReferences

arxiv:2003.06997v1 [math.ag] 16 mar 2020indranil biswas, sorin dumitrescu, and sebastian heller...

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