Indranil Biswas
email: indranil@math.tifr.res.in

J O Ão

Pedro Dos Santos
email: joaopedro.dossantos@yahoo.com

Sorin Dumitrescu
email: dumitres@unice.fr

AND Lynn Heller
email: lynn.heller@math.uni-hannover.de

Sebastian Heller To

Oscar García-Prada

J P Dos Santos

S Heller

ON THE MONODROMY OF HOLOMORPHIC DIFFERENTIAL SYSTEMS

Keywords: 1991 Mathematics Subject Classification. 32S40, 53B15, 34M03, 14H15 Differential system, holomorphic connection, neutral Tannakian category, iterated integral

We explain the strategy of some recent results that construct holomorphic sl(2, C)-differential systems over some Riemann surfaces Σ g of genus g ≥ 2, satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [BDHH1] or some cocompact Kleinian subgroup

As a consequence, there exist holomorphic maps from Σ g to the quotient space SL(2, C)/Γ, where Γ ⊂ SL(2, C) is a cocompact lattice, that do not factor through any elliptic curve [BDHH2]. This answers positively a question of Ghys in [Gh]; the question was also raised by Huckleberry and Winkelmann in [HW].

When M is a Riemann surface, a Torelli type theorem holds for the affine group scheme over C obtained from the category of holomorphic connections on étale trivial holomorphic bundles.

We explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle.

For a compact Kähler manifold M , we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over M . If (respectively, Θ) stands for the affine group scheme over C obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism v : O(Θ) -→ O( ) of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals.

Introduction

In recent times, there has been a resurgent interest in the theory of holomorphic differential systems, or connections on trivial bundles, in the special case of a compact ambient space. This interest has been mainly triggered by a strategy of Ghys, presented in [CDHL], to answer a question raised first by Huckleberry and Winkelmann in [HW] and later by Ghys in [Gh]. This question in [HM, Gh] inquires about the existence of holomorphic maps from compact hyperbolic Riemann surfaces to compact complex manifolds of the form SL(2, C)/Γ, with Γ a lattice in SL(2, C) (which does not factor to any elliptic curve). The suggested direction in [CDHL] towards an answer is the study of representations of surface groups into SL(2, C) giving rise to rank two free (trivial) holomorphic vector bundles over some Riemann surface of genus g ≥ 2 (see Question 2.2).

Although the theory of differential systems (holomorphic connections on free vector bundles) has always been a much studied subject, the restrictions imposed by a compact ambient space are rather pleasing and produce fruitful geometric situations. The present paper is a contribution to this theme, where we present several different backgrounds.

To motivate the reader and display some of the beautiful geometry we are set to explain, we begin by surveying some recent results on the theory of holomorphic SL(2, C)differential systems over compact Riemann surfaces and their monodromy representations. This is done in Section 2, where after reviewing Ghys' questions and its relation to the theory of differential systems (see Question 2.1 and Question 2.2), we move on to explain some local results about the Riemann-Hilbert mapping, proved in [START_REF] Calsamiglia | The Riemann-Hilbert mapping for sl(2)systems over genus two curves[END_REF][START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF]. (It is perhaps worth noting that the main results of these works were recently extended to the logarithmic case in [ABDH].) Despite the local results obtained in [CDHL], for curves of genus g = 2, and then in [START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF][START_REF] Aprodu | On the monodromy map for logarithmic systems[END_REF], Ghys' question was still open until very recently. In particular, it was not known whether one can realize given real, discrete or Zariski dense subgroups in SL(2, C) as monodromy of SL(2, C)-local systems over some Riemann surface Σ g of genus g ≥ 2.

An important step toward the understanding of the image of the monodromy homomorphism was very recently realized in [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF] where holomorphic connections with (real) Fuchsian monodromy were constructed. This completely answers a question asked in [CDHL].

Eventually Ghys strategy has been successfully implemented in the very recent work [START_REF] Biswas | On the existence of holomorphic curves in compact quotients of SL(2, C)[END_REF]. In this work it is first shown in [START_REF] Biswas | On the existence of holomorphic curves in compact quotients of SL(2, C)[END_REF] that every irreducible SL(2, R)representation with sufficient many symmetries can be realized as the monodromy of a holomorphic irreducible SL(2, C)-connection on the rank two trivial holomorphic bundle over some (very symmetric) Riemann surface Σ such that genus(Σ) ≥ 2. Then an example of such a symmetric representation contained in a cocompact lattice Γ in SL(2, C) is given and a holomorphic map Σ -→ SL(2, C)/Γ is constructed (which does not descend on any elliptic curve), answering the open question of Ghys [Gh] and Huckleberry-Winkelmann [HW].

Once light was thrown on the pertinence of the theory of differential systems on compact spaces, it became clear, see for example [BDDH], that a reasonable way to study holomorphic connections is to impose conditions on the underlying vector bundle. It is with this idea in mind that we go over to Section 3. There, we give ourselves a compact complex manifold M and study integrable connections where the underlying vector bundle enjoys the following property: it becomes holomorphically trivial after an étale covering or, it is étale trivial. (We explain at the start of Section 3 that the latter property can be recovered from the existence of a connection with finite monodromy.) We start by reviewing recent work [START_REF] Biswas | Étale triviality of finite vector bundles over compact complex manifolds[END_REF][START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF] which connects étale trivial bundles with finite [START_REF] Nori | On the representations of the fundamental group[END_REF]p. 35] and "properly" trivial ones [START_REF] Biswas | Vector bundles trivialized by proper morphisms and the fundamental group scheme[END_REF]Property (T),p. 225]. Then, we follow the path opened by Tannakian categories and define an affine group scheme whose representations amount to the aforementioned connections.

Another of the pleasant properties of differential systems (in compact ambient) is the simplicity in which differential Galois groups can be computed, as was the theme in [BHdS]. (This paper takes the matter from the purely algebro-geometric viewpoint.) Section 4 deals with an analogue of [BHdS] in a complex analytic setting by showing that the smallest reduction of structure group preserving a connection is given by the algebraic hull of the "connection matrices".

In Section 5, we study the relation between the category of integrable connections and the category of integrable connections on trivial vector bundles by the Tannakian point of view, as in [BDDH]. Moreover, this study is done by putting together Hochschild-Mostow's [HoM, p.1140] (as understood by [Sw]) and Saavedra-Grothendieck's [Sa] approach to Tannaka duality. Recall that the first one is achieved by means of "representative functions" while the second is, from the start, categorical. In explicit terms: Let M be a compact complex manifold, x 0 a point in M , C dR (M ) and T (M ) respectively the categories of integrable connections and integrable connections on trivial vector bundles. As is well-known, C dR (M ) is equivalent to the category of Cπ 1 (M, x 0 )-modules while, as argued in [BHdS], T (M ) is equivalent to the category of representations of a certain co-commutative Hopf algebra

A M =
Tensor algebra on H 0 (Ω 1 M ) * a certain ideal .

Following [Sw, Chapter VI], these categories are then, respectively, equivalent to the categories of comodules over the Hopf-algebras Cπ 1 (M, x 0 ) • and A • M . Hence, the morphism [Sa] gives rise to an arrow between certain completions:

A • M -→ Cπ 1 (M, x 0 ) • coming from the inclusion T et (M ) -→ C dR (M ) through
u : Cπ 1 (M, x 0 ) -→ A M ;
it is rendering explicit the composition Cπ 1 (M, x 0 ) -→ Cπ 1 (M, x 0 ) -→ A M in terms of iterated integrals that occupies Section 5. In doing so, we recover expressions appearing in the works of Chen, Parshin and Hain but, in our context, the source and target of the map u are bigger completions and carry more information. To wit, these are the pro-finite dimensional (as opposed to pro-finite) completions.

Curves in compact quotients of SL(2, C)

A natural class of compact complex manifolds generalizing complex tori are complex manifolds whose holomorphic tangent bundle is holomorphically trivial. By a classical result of Wang [Wa], compact complex manifolds with holomorphically trivial tangent bundle are biholomorphic to the complex manifolds of the form G/Γ, where G is a complex Lie group and Γ ⊂ G a discrete cocompact subgroup. Compact complex manifolds in this class are called (complex) parallelizable manifolds. They are in general not Kähler. More precisely, a complex parallelizable manifold G/Γ is Kähler if and only if the complex Lie group G is abelian, in which case G/Γ is biholomorphic to a compact compact complex torus [Wa].

In particular, when G is a complex semi-simple Lie group, and Γ ⊂ G is a cocompact lattice, the complex parallelizable manifold G/Γ is non-Kähler. Moreover, a theorem due to Huckleberry and Margulis [HM] says that G/Γ does not admit any complex analytic hypersurface. This implies, in particular, that G/Γ does not admit any nonconstant meromorphic function and, consequently, the algebraic dimension of the complex parallelizable manifold G/Γ is zero.

When G is a complex semi-simple Lie group with no local factor isomorphic to SL(2, C) (for example, G = SL(n, C), with n ≥ 3), a vanishing result of Raghunathan, [Ra], says that the complex structure of any compact quotient G/Γ, where Γ a discrete cocompact subgroup in G, is rigid: any complex structure on the underlying real manifold G/Γ close to the standard one (endowed by the complex structure of G) actually coincides with the standard one.

The remaining question about the rigidity (or flexibility) of the standard complex structure on the compact quotients G/Γ, with G = SL(2, C), was studied by Ghys in [Gh] (see also [Raj]). In this situation Ghys computes in [Gh] the Kuranishi space of SL(2, C)/Γ and proves that this deformation space is nontrivial for any discrete cocompact subgroup Γ ⊂ SL(2, C) with positive first Betti number. This implies that the standard complex structure of these compact quotients of SL(2, C) is flexible and admit nontrivial deformations described by Ghys.

It may be mentioned that the discrete cocompact subgroup Γ ⊂ SL(2, C) with positive first Betti number are quite abundant (see [La]); their existence is closely related to the compact quotients of the 3-hyperbolic space H 3 (i.e. the compact hyperbolic 3-manifolds). More precisely, the complex Lie group PSL(2, C) being the orientation preserving isometry group of the 3-hyperbolic space H 3 , it is identified with the oriented orthonormal frame bundle of H 3 . Consequently, the orthonormal frame bundle of any compact hyperbolic 3-manifold M (diffeomorphic to a quotient H 3 /Γ, with Γ a lattice in SL(2, C)) admits the structure of a compact complex parallelizable manifold which is biholomorphic to a quotient of SL(2, C) by its cocompact lattice Γ (isomorphic to the fundamental group of M ). One of the reasons for the complex geometry of the compact quotients of SL(2, C) to be of interest is this fact.

As mentioned before, the flexibility of the complex structure of SL(2, C)/Γ was discovered by Ghys in [Gh] (see also [Raj]) where he showed that the corresponding Kuranishi space has positive dimension for all Γ with positive first Betti number. Notice that compact hyperbolic 3-manifolds with prescribed rational cohomology ring (in particular, with arbitrarily large first Betti number) can be constructed using Thurston's hyperbolisation Theorem (see [START_REF] Ghys | Déformations des structures complexes sur les espaces homogènes de SL(2, C)[END_REF]Lemme 6.2]). It should be mentioned that the complex geometry of the orthonormal frame bundle of the compact hyperbolic 3-manifold H 3 /Γ endowed with the complex structure SL(2, C)/Γ is not well understood yet.

When studying the geometry of compact quotients of SL(2, C), Ghys asked in [Gh] the following question, also raised by Huckleberry and Winkelmann in [HW]:

Question 2.1. Does there exist a compact quotient SL(2, C)/Γ admitting a compact holomorphic curve of genus g ≥ 2?

A negative answer to this question would generalize the Huckleberry-Margulis Theorem [HM] (for G = SL(2, C)) to compact holomorphic curves of genus g ≥ 2. Notice that, in the case g = 1, elliptic curves covered by one-parameter subgroups in SL(2, C) are known to exist in certain quotients SL(2, C)/Γ.

In an attempt to answer positively Question 2.1, Ghys developed the following strategy to characterize compact holomorphic curves of genus g ≥ 2 in quotients SL(2, C)/Γ (see [CDHL] where this strategy is explained in detail).

Find a Riemann surface Σ of genus g ≥ 2 and an irreducible holomorphic SL(2, C)connection on the rank two trivial holomorphic bundle Σ × C 2 over Σ such that the image of the corresponding monodromy homomorphism lies in a cocompact lattice Γ in SL(2, C).

Recall that a holomorphic connection on a holomorphic bundle over a Riemann surface is automatically flat.

As Ghys observed, in this context, the parallel frame of the above (flat) holomorphic connection gives rise to a holomorphic map from Σ into the quotient SL(2, C)/Γ (which, due to the irreducibility of the connection, does not factor through any elliptic curve) (see [CDHL] for more details). More generally, Ghys asked: Question 2.2. Given a compact orientable surface X of genus g ≥ 2, describe the space of all irreducible group homomorphisms ρ : π 1 (X) -→ SL(2, C) for which there exists a complex structure Σ on X such that the holomorphic vector bundle over Σ defined by ρ is holomorphically trivial.

If some irreducible homomorphism ρ : π 1 (X) -→ SL(2, C) satisfying the above condition has the property that ρ(π 1 (X)) is contained in a cocompact lattice Γ ⊂ SL(2, C), then the local system on the Riemann surface Σ given by ρ produces a nontrivial holomorphic map from Σ to SL(2, C)/Γ (more details about this method can be found in [CDHL]).

A first local result in the direction of Question 2.2 was obtained in [CDHL] for the genus g = 2. Other local results generalizing the one in [CDHL] were obtained in [START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF], and then in [ABDH] for logarithmic differential systems. We briefly recall below those local results obtained in [START_REF] Calsamiglia | The Riemann-Hilbert mapping for sl(2)systems over genus two curves[END_REF][START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF].

The aim of both [START_REF] Calsamiglia | The Riemann-Hilbert mapping for sl(2)systems over genus two curves[END_REF][START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF] is to study the Riemann-Hilbert correspondence for differential systems over compact Riemann surfaces of genus g ≥ 2. Denote by X a compact connected oriented topological surface of genus g ≥ 2. We also denote by G a connected reductive affine algebraic group defined over the field of complex numbers.

Let us now fix a complex structure Σ on X (equivalently, it is the choice of an element in the Teichmüller space of X). We consider also a holomorphic connection ∇ on the trivial holomorphic principal G-bundle Σ × G over Σ. Notice that ∇ is determined by an element δ ∈ g ⊗ H 0 (Σ, K Σ ), where g is the complex Lie algebra of G and K Σ is the canonical bundle of Σ. Fix a base point x 0 ∈ X and consider the corresponding universal cover π : Σ -→ Σ of Σ (endowed with the induced complex structure). On the trivial principal G-bundle Σ × G over Σ we have the pulled back holomorphic (flat) connection π * ∇.

For any locally defined ∇-parallel section φ of Σ × G, the pulled back local section π * φ of Σ × G extends to a π * ∇-parallel section on entire Σ. The above extension of π * φ identifies with a holomorphic map Σ -→ G which is π 1 (X, x 0 )-equivariant with respect to the natural action of π 1 (X, x 0 ) on Σ through deck transformations and the action of π 1 (X, x 0 ) on G through a group homomorphism π 1 (X, x 0 ) -→ G, which is known as the monodromy (homomorphism) of the flat connection ∇.

The above monodromy homomorphism depends on the choice of the holomorphic trivialization of the principal G-bundle over Σ. Nevertheless, the corresponding element of the character variety of G-representations

Ξ := Hom(π 1 (X), G)/ /G
does not depend either on the choice of the trivialization of the principal G-bundle over Σ, or on the choice of the base point x 0 (or on the choice of the local section φ).

The character variety Ξ is known to be a singular complex analytic space of complex dimension 2((g -1)

• dim [G, G] + g • (dim G -dim [G, G]
)); see, for example, [START_REF] Sikora | Character varieties[END_REF]Proposition 49]. The dimension of Ξ equals 2(g -1)d + 2gc, with d being the complex dimension of the commutator group [G, G] for G and c being equal to dim

G -dim [G, G].
As in [START_REF] Calsamiglia | The Riemann-Hilbert mapping for sl(2)systems over genus two curves[END_REF][START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF], denote by Syst the space of all differential systems, meaning the space of the pairs (Σ, ∇), where Σ is a complex structure on X (an element of the Teichmüller space for X) and ∇ is a holomorphic connection on the trivial holomorphic principal G-bundle Σ × G over Σ. This space of differential systems on X is a singular complex analytic space of dimension (g -1)(d + 3) + gc. Associating to any pair (Σ, ∇) ∈ Syst the element in the character variety Ξ corresponding to the monodromy morphism of ∇, we obtain a holomorphic map from Syst to the character variety Ξ. This map is the restriction to Syst of the Riemann-Hilbert correspondence.

We consider now the nontrivial Zariski open subset Syst irred in Syst defined by all pairs (Σ, ∇) such that the holomorphic connection ∇ is irreducible (i.e., the monodromy homomorphism associated to ∇ does not factor through any proper parabolic subgroup of G). The Zariski open set Syst irred is a smooth complex manifold (see, for example, Corollary 50 in [Sik]). The image of Syst irred under the Riemann-Hilbert correspondence described above lies in the smooth Zariski orbifold open subset Ξ irred ⊂ Ξ defined by the irreducible homomorphisms π 1 (X) -→ G (i.e., homomorphisms that do not factor through some proper parabolic subgroup of G).

Denote by Mon (as monodromy) this restriction of the Riemann-Hilbert correspondence to Syst

irred : Mon : Syst irred -→ Ξ irred (2.1)
Then Mon is a holomorphic map between the complex manifolds Syst irred and Ξ irred .

The main result proved in [START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF] is the following: [START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF]). If the complex dimension of G is at least three, the map Mon in (2.1) is an immersion at the generic point.

Theorem 2.1 ([
If G = SL(2, C), the dimensions of Syst irred and Ξ irred are both 6g -6. Therefore, in that case Theorem 2.1 implies that Mon is a local biholomorphism at the generic point. It should be mentioned that examples constructed in [CDHL] show that for G = SL(2, C) and Σ of genus g ≥ 3, the monodromy map Mon is not always a local biholomorphism (over the entire Syst irred ).

When G = SL(2, C) and g = 2, the main result of [CDHL] says that the map Mon in (2.1) is a local biholomorphism over entire Syst irred . An alternative proof of this result of [CDHL] was also provided in [START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF]Corollary 5.6].

The main motivation for the authors of [CDHL] and [START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF] was the case G = SL(2, C) because Ghys strategy relates the monodromy of sl(2, C)-differential systems to the question of existence of holomorphic curves of genus g > 1 lying in quotients of SL(2, C) by cocompact lattices.

It was asked in [CDHL] if there exist holomorphic sl(2, C)-differential systems on Riemann surfaces of genus g > 1 having real or discrete monodromy. Indeed, despite those local results obtained in [CDHL], for curves of genus g = 2, and then in [START_REF] Biswas | Riemann-Hilbert correspondence for differential systems over Riemann surfaces[END_REF][START_REF] Aprodu | On the monodromy map for logarithmic systems[END_REF]. Ghys' question was still open until very recently. In particular, it was not known whether it is possible to realize any discrete or Zariski dense subgroup in SL(2, C) as the monodromy of a SL(2, C)-local system for some complex structure Σ on X.

A first answer to this question was given in [BDH] where the main result is a construction of an irreducible holomorphic connection with SL(2, R)-monodromy on the rank two trivial holomorphic vector bundle over a compact Riemann surface of genus g > 1.

Another important step towards realizing Ghys' strategy was very recently made in [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF] where holomorphic connections with (real) Fuchsian monodromy were constructed. The result is the following: [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF]). For every integer g ≥ 2, there exists a (hyperelliptic) Riemann surface Σ g of genus g, such that the rank two trivial holomorphic vector bundle Σ g ×C 2 over Σ g admits infinitely many holomorphic SL(2, C)-connections with Fuchsian monodromy representation.

Theorem 2.2 ([
The above result completely answers the question asked in [CDHL] by Deroin, Calsamiglia, Heu and Loray.

In this context it is also natural to ask which holomorphic rank two bundles over a given Riemann surface Σ admit holomorphic connections with Fuchsian monodromy representations. This question was indeed raised by Katz in [START_REF] Katz | An overview of Deligne's work on Hilbert's twenty-first problem, Mathematical developments arising from Hilbert problems[END_REF] (where the question is attributed to Bers) in 1978 and is still unsolved. Even when restricting to the trivial rank two holomorphic bundle, it was not known before [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF] whether a holomorphic connection ∇ with Fuchsian monodromy representation exists.

Eventually, Ghys strategy was successfully realized in the recent work [START_REF] Biswas | On the existence of holomorphic curves in compact quotients of SL(2, C)[END_REF]. In [START_REF] Biswas | On the existence of holomorphic curves in compact quotients of SL(2, C)[END_REF] it is first shown that every irreducible SL(2, R)-representation with sufficient many symmetries can be realized as the monodromy of a holomorphic irreducible SL(2, C)connection on the rank two trivial holomorphic bundle over some (very symmetric) Riemann surface Σ such that genus(Σ) ≥ 2. Then an example of such a symmetric representation with image contained in a cocompact lattice Γ in SL(2, C) is constructed, with Γ being by the dodecahedron tiling {5, 3, 4} of the hyperbolic 3-space H 3 . The results in [START_REF] Biswas | On the existence of holomorphic curves in compact quotients of SL(2, C)[END_REF] imply the following: [START_REF] Biswas | On the existence of holomorphic curves in compact quotients of SL(2, C)[END_REF]). Let Γ be the cocompact lattice in H 3 given by the dodecahedron tiling {5, 3, 4} of the hyperbolic 3-space. Then there exists a non constant holomorphic map from a compact curve Σ of genus 4 in SL(2, C)/Γ.

Theorem 2.3 ([
Remark 2.4. The holomorphic map from Σ to SL(2, C)/Γ, constructed in Theorem 2.3, has 4 simple branch points and by Riemann-Hurwitz Theorem it cannot factor through a lower genus surface.

This gives an affirmative answer to above question raised by Huckleberry and Winkelmann [HW] and by Ghys [Gh].

2.1. Strategy of the proofs of Theorem 2.2 and Theorem 2.3. For both above results, namely Theorem 2.2 and Theorem 2.3, straightforward methods such as computing the monodromy representation explicitly or using geometric existence results, such as the uniformization metric for Fuchsian representations on the Gunning bundle, is not developed or applicable. Therefore, the general philosophy is to start with a given monodromy of a known holomorphic differential system and try to deform it until the deformed monodromy has the desired properties. This leaves of course a lot of freedom for implementation. Though initial setup of both proofs are the same, Theorem 2.3 requires a completely new set of further ideas as well as some further applications of the ideas in Theorem 2.2. Hence, we begin with describing the general setup as well as the strategy for Theorem 2.2. The first step is to reduce complexity. We first impose various symmetries of the underlying Riemann surface and the considered holomorphic differential systems as follows.

For every genus g consider the complex 1-dimensional family of Riemann surfaces which admit a Z g+1 -action with 4 fixed points of order g, i.e, the Riemann surface is given by a (g + 1)-fold covering of the complex projective line totally branched over 4 points. Then the connected components of Z g+1 -equivariant connections are complex 2-dimensional subspaces of the de Rham moduli space of flat SL(2, C)-connections. The Fuchsian representations of the symmetric Riemann surfaces automatically give rise to Z g+1 -equivariant connections. By [B97], all these equivariant connections are determined by logarithmic SL(2, C)-connections on the 4-punctured sphere with prescribed local conjugacy classes depending (only) on both the genus of the surface and the connected component of Z g+1equivariant connections inside the de Rham moduli space.

For example, the eigenvalues of the residues of a logarithmic connection over the complex the projective line have to be ± g 2(g+1) , at each of the 4 singular points, in order to correspond to an equivariant Fuchsian representation, see [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF]Section 3]. Moreover, there exist exactly one affine (complex) line of logarithmic connections in each of these connected components which correspond to holomorphic systems on the compact Riemann surface of genus g, given by a specific parabolic structure (see [START_REF] Heller | Navigating the space of symmetric cmc surfaces[END_REF]Theorem 3.2 (5)] or [BDHH1, Proposition 3.1]).

By imposing compatible reality conditions on the Riemann surface and the flat connections respectively, we can reduce further such that (the smooth components of) the moduli space of real symmetric Riemann surfaces is real 1-dimensional, and the real subspace of the Betti moduli space is real 2-dimensional. In other words, the considered Riemann surfaces possess a real involution τ compatible with the Z g+1 -action (which is equivalent to the 4 branch points lying on one circle) and the considered connections ∇ lies in the same gauge class as τ * ∇, we refer to ∇ then as a τ -real connection. Considering τ -real connections on real Riemann surfaces leads to certain traces of the monodromies being automatically real. In order for the connection to be called real in following, we require all the monodromy traces along every closed curve to be real.

2.1.1. Theorem 2.2. Theorem 2.2 asserts the existence of infinitely many holomorphic systems with Fuchsian monodromy on a real symmetric Riemann surface. The first observation is that τ -real connections in the appropriate component (determined by the eigenvalues) give rise to Fuchsian representations if it is real and the traces satisfy some additional inequalities. Since the underlying Riemann surface is a 4-punctured sphere, the character variety, or its Betti moduli space, is determined by 3 traces x, y and z. For given x and y the third one must satisfy the character variety equation, which has two solutions z and z. Due to τ -symmetry z and z are automatically real and the reality of x implies then y being real as well. When working with the one-puncture torus instead of the 4-punctured sphere we can avoid dealing with further inequalities determining the connected component of Fuchsian representations. In other words, in order to find a Fuchsian representation it suffices to show that just one (additional) trace x is real on the one-punctured torus.

The trivial holomorphic structure when lifting the rank two flat bundle to the compact Riemann surface Σ g of genus g is determined by a specific parabolic structure (see [START_REF] Heller | Navigating the space of symmetric cmc surfaces[END_REF]Theorem 3.2 (5)] or [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF]Proposition 3.1]). Since the space of (τ -symmetric) Higgs fields on the other hand is only (real) one-dimensional, i.e., generated by some Ψ, it remains to show that there exist (infinitely many) parameters t ∈ R such that ∇ + tΨ has real x, where ∇ is the unique unitary connection inducing that parabolic structure. Again, computing the monodromy directly is out of reach, but its asymptotic for large t can be studied using W KB analysis. Though the method has been carried out successfully in various situations, somehow surprisingly, the existing mathematical literature did not cover the asymptotic analysis of the trace of the monodromy, at least to the best of our knowledge. It was then shown by Takuro Mochizuki in the appendix of [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF] that

lim t→∞ tr(P γ (∇ + tΨ))e t γ √ det Ψ = c,
where P γ (∇ + tΨ) is the monodromy of ∇ + tΨ along the curve γ which has to satisfy

( √ det Ψ(γ )) < 0, (2.2)
and c = 0 is a non-zero constant. As the curve γ corresponding to the trace x satisfies (2.2) and γ √ det Ψ / ∈ R we find infinitely many parameters t n accumulating at t = ∞ for which all monodromies of the connections ∇+t n Ψ are real and therefore give Fuchsian representations.

2.1.2. Theorem 2.3. While in the space of SL(2, R)-representations, the Fuchsian ones form a whole connected component, cocompact lattices in SL(2, C) are discrete. Therefore, a new set of ideas is needed to attain a specific representation with holomorphic connections rather than just reaching the component when varying a parameter. In contrast to Theorem 2.2 where we fixed the Riemann surface type and show existence of countably infinite many Fuchsian representations on the trivial holomorphic bundle, we now vary the Riemann surface type and show that the representations obtained from holomorphic system exhaust the whole (real 1-dimensional) space of symmetric and real representations. Theorem 2.3 then follows from the fact that the dodecahedron tilling has a sublattice inducing such a symmetric and real representation.

In other words, let ρ be an arbitrary real and symmetric representation and let ρ F be a suitable real, τ -real and Z g+1 -equivariant representation induced by the monodromy of a holomorphic system on a real, symmetric Riemann surface. Then the aim is to deform ρ F via deforming the underlying (real, symmetric) Riemann surface structure and the corresponding holomorphic system until we hit ρ.

The existence of the initial value ρ F is guaranteed by [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF], but the proof of 2.3 in fact gives an independent proof for the existence of the Fuchsian representations of [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF]. Local existence and uniqueness of the deformation by varying the underlying real symmetric Riemann surface is based on a technical Lemma [BDHH2, Lemma 5.2] which was motivated by the main result of [CDHL] for genus 2 surfaces and whose proof uses the detailed analysis of the isomonodromic deformations of the Lamé equation by Loray in [L]. An important observation is the following. We can squeeze a real, τreal and equivariant holomorphic system between the (orbifold) uniformization of the Riemann surface and a grafting of the (orbifold) uniformization of another real, symmetric Riemann surface inside the ordered space of real, τ -real and equivariant representations. The ordering of all three connections is preserved along the deformation, and it is shown that the orbifold uniformization and their graftings both cover the whole 1-dimensional space of real, τ -real and equivariant representations. We should emphasize that the induced deformation of the representations is not necessarily continuous introducing some technicalities to show the exhaustion property.

The last point to explain here is why and how we can squeeze the holomorphic system with real monodromy between a (orbifold) uniformization and a grafting of another uniformization. The main ingredient is the new observation that grafting changes the spin structure in a very natural way. Using this we can show the existence of a non-trivial path in the moduli space of bundles which admit a unique lift to real, τ -real and equivariant connections, which starts and ends at two different Gunning bundles (respectively maximally unstable bundles in a given component of equivariant connections) and passes through the trivial holomorphic structure. This path naturally deforms under the deformation of the underlying Riemann surface structure, and as orbifold uniformisations and their graftings exhaust the 1-dimensional space of real, τ -real equivariant connections the existence of holomorphic curves of genus g > 1 in certain quotients SL(2, C)/Γ by cocompact lattices follows.

Étale trivial bundles

Let M be a compact connected complex manifold. The tangent bundle and the iexterior product of the cotangent bundle of M will be denoted by T M and Ω i M respectively. A holomorphic connection on a holomorphic vector bundle E on M is a first order holomorphic differential operator D :

E -→ E ⊗ Ω 1
M satisfying the Leibniz identity, which says that D(f s) = f D(s) + s ⊗ df for all locally defined holomorphic section s of E and all locally defined holomorphic function f on M (see [At]). Any coherent analytic sheaf on M admitting a holomorphic connection is locally free (see [START_REF] Borel | Algebraic Dmodules[END_REF]p. 211,Proposition 1.7] and [BDDH, p. 1037

, Remark 2.1]). A holomorphic connection D is called integrable if its curvature D 2 ∈ H 0 (M, End(E) ⊗ Ω 2
M ) vanishes identically. For an integrable holomorphic connection D we have the monodromy homomorphism

Mon(D) : π 1 (M, x 0 ) -→ GL(E| x 0 ) , (3.1)
where x 0 ∈ M is any base point.

A holomorphic vector bundle E on M is called étale trivial if there is a finite étale covering ψ : M -→ M such that ψ * E is holomorphically trivial.

The following lemma is straightforward. Given any a connected finite étale covering

ψ 1 : M 1 -→ M , there is a connected finite étale covering ψ 2 : M -→ M 1 such that ψ := ψ 1 • ψ 2 is Galois. If ψ *
1 E is holomorphically trivial, then ψ * E is also holomorphically trivial. So if E is étale trivial, then we may assume that the trivializing connected finite étale covering ψ is Galois. If ψ * E is holomorphically trivial, then the natural evaluation map

M × H 0 ( M , ψ * E) -→ ψ * E
is a holomorphic isomorphism. Using this isomorphism, the trivial holomorphic connection on M × H 0 ( M , ψ * E) produces an integrable holomorphic connection D on ψ * E with trivial monodromy. This holomorphic connection D is preserved by the natural action of Gal(ψ) on ψ * E, and hence it descends to a holomorphic connection on E. This descended holomorphic connection on E is evidently integrable and its monodromy homomorphism has finite image.

To verify the last property, we observe the following. Let f : E -→ F be an arrow between holomorphic vector bundles and ψ : M -→ M an étale covering such that ψ * E and ψ * F are trivial vector bundles. Suppose that ∂ E and ∂ F are integrable connections on E and F , respectively, such that

ψ * (f ) : (ψ * E, ψ * ∂ E ) -→ (ψ * F, ψ * ∂ F ) is horizontal. It then follows that f is horizontal. Take any polynomial p(x) = d i=0 a i x i
, where a i are nonnegative integers. For any holomorphic vector bundle E on M , define the holomorphic vector bundle

p(E) := E ⊗i ⊕a i ,
where E ⊗0 is the trivial holomorphic line bundle O M , and F ⊕0 = 0 for any vector bundle F . A holomorphic vector bundle E is called finite if there are two polynomials p and q as above, with p = q, such that the two holomorphic vector bundles p(E) and q(E) are holomorphically isomorphic [START_REF] Nori | On the representations of the fundamental group[END_REF]p. 35], [START_REF] Nori | The fundamental group-scheme[END_REF].

The following is a criterion for a vector bundle to be finite.

Theorem 3.2 ([Bi, Theorem 1.1]). A holomorphic vector bundle E on M is finite if and only if it is étale trivial.

In [START_REF] Nori | On the representations of the fundamental group[END_REF], [START_REF] Nori | The fundamental group-scheme[END_REF], Theorem 3.2 was proved in the algebro-geometric set-up. Theorem 3.2 is a key ingredient in the proof of the following result (whose algebro-geometric counterpart appears in [START_REF] Biswas | Vector bundles trivialized by proper morphisms and the fundamental group scheme[END_REF]Remarks,.) Theorem 3.3 ([BD, Theorem 1.1]). Let f : X -→ M be a surjective holomorphic maps between compact connected complex manifolds and E -→ M a holomorphic vector bundle, such that f * E is holomorphically trivial. Then E is étale trivial.

3.1.

A neutral Tannakian category. Let M be a compact, connected complex Kähler manifold. Fix a base point x 0 ∈ M . In [BDDH], the following was studied: The category T (M ) consisting of trivial vector bundles endowed with an integrable connection. Using the fiber functor E -→ E| x 0 , it was deduced that T (M ) is equivalent to the category of representations of an affine group scheme Θ(M, x 0 ).

A basic property of Θ(M, x 0 ) is that it is pro-connected [START_REF] Biswas | On certain Tannakian categories of integrable connections over Kähler manifolds[END_REF]Proposition 3.7].

Let C dR (M ) denote the category whose objects are pairs of the form (E, D), where E is a holomorphic vector bundle on M and D is an integrable holomorphic connection on E. As before, morphisms from (E, D) to (E , D ) are all holomorphic homomorphisms of vector bundles h :

E -→ E satisfying D • h = (h ⊗ Id Ω 1 M ) • D as differential operators from E to E ⊗ Ω 1
M . This category is equipped with the operators of direct sum, tensor product and dualization. More precisely, C dR (M ) is a rigid abelian tensor category (see [START_REF] Deligne | Hodge cycles, motives, and Shimura varieties[END_REF]p. 118,definition 1.14] for rigid abelian tensor categories). This category C dR (M ), endowed with the faithful fiber functor that sends any object (E, D) to the fiber E| x 0 defines a neutral Tannakian category [START_REF] Simpson | Higgs bundles and local systems[END_REF]p. 67]. The corresponding pro-algebraic affine group scheme over C will be denoted by (M, x 0 ) (see [START_REF] Simpson | Higgs bundles and local systems[END_REF]p. 69], [START_REF] Biswas | On certain Tannakian categories of integrable connections over Kähler manifolds[END_REF]Section 2]). Concerning the relation between C dR (M ) and T (M ), we can say that the morphism of group schemes

q : (M, x 0 ) -→ Θ(M , x 0 ) (3.2)
induced by the inclusion T (M ) -→ C dR (M ) is faithfully flat, or a quotient morphism. See Proposition 3.1 in [START_REF] Biswas | Fuchsian sl(2, C)-systems of compact Riemann surfaces (with an appendix by Takuro Mochizuki)[END_REF].

Let us now modify our point of view and enlarge the category of vector bundles on which we allow a connection. Let T et (M ) denote the category of all pairs of the form (E, D), where E is an étale trivial vector bundle on M and D is a holomorphic connection on E. This T et (M ) is evidently closed under the operations of direct sum and tensor product, just as it is closed under the operation of taking dual. For any holomorphic homomorphism between two holomorphically trivial vector bundles over a compact connected complex manifold, both kernel and cokernel are holomorphically trivializable because the homomorphism is given a constant matrix. From this it follows that for any two pairs (E 1 , D 1 ) and (E 2 , D 2 ) in T et (M ), any homomorphism ρ : E 1 -→ E 2 , satisfying the condition that

D 2 • ρ = (ρ ⊗ Id Ω 1 M ) • D 1
, has the property that both kernel(ρ) and cokernel(ρ) are étale trivial bundles, and furthermore, D 1 (respectively, D 2 ) induces a holomorphic connection on kernel(ρ) (respectively, cokernel(ρ)).

Therefore, T et (M ) together with the faithful fiber functor on it that sends any (E, D) to the fiber E x 0 produce a neutral Tannakian category (see [START_REF] Deligne | Hodge cycles, motives, and Shimura varieties[END_REF]p. 138,Definition 2.19], [Sa], [START_REF] Nori | The fundamental group-scheme[END_REF]p. 76] for neutral Tannakian category). For any neutral Tannakian category there is a naturally associated pro-algebraic affine group scheme over C [START_REF] Deligne | Hodge cycles, motives, and Shimura varieties[END_REF]p. 130,Theorem 2.11] (and the remark following [START_REF] Deligne | Hodge cycles, motives, and Shimura varieties[END_REF]p. 138,Definition 2.19]), [Sa], [START_REF] Nori | The fundamental group-scheme[END_REF]p. 77,Theorem 1.1], [START_REF] Simpson | Higgs bundles and local systems[END_REF]p. 69]. Consequently, the neutral Tannakian category T et (M ) produces a pro-algebraic group (= affine group scheme)

E(M, x 0 )
over C such that by means of the functor •| x 0 , the category T et (M ) becomes equivalent to the category of representations of E(M, x 0 ) on finite dimensional complex vector spaces.

The tautological functor

T et (M ) -→ C dR (M ) produces a homomorphism r : (M, x 0 ) -→ E(M, x 0 ) . (3.3) Lemma 3.4. The homomorphism r in (3.3) is faithfully flat (in other words, it is sur- jective).
Proof. This can be proved by the same method employed to verify Proposition 3.1 in [BDDH].

Let π et (M, x 0 ) be the pro-finite completion of the fundamental group of M at x 0 ; it carries a natural structure of pro-algebraic group, or affine group scheme, once regarded as a projective limit of finite groups. This is commonly known as the étale fundamental group scheme.

Lemma 3.5. There is a natural faithfully flat morphism (same as surjective morphism)

γ : E(M, x 0 ) -→ π et (M, x 0 )
to the étale fundamental group scheme π et (M, x 0 ).

Proof. The étale fundamental group π et (M, x 0 ) corresponds to the neutral Tannakian category defined by the étale trivial holomorphic vector bundles on M . In the proof of Lemma 3.1 it was shown that any étale trivial vector bundle has a canonical integrable holomorphic connection whose monodromy is finite. Any homomorphism between two étale trivial holomorphic vector bundles is flat with respect to this canonical connection. Indeed, write d E for the canonical connection on an étale trivial bundle E. Then, if ψ : N -→ M is a finite étale covering such that ψ * E O r M , then the pulled-back connection ψ * d E corresponds to the trivial connection on O r M . (Note that this notion is independent of the isomorphism ψ * E O r M .) This being so, if f : E -→ F is an arrow between vector bundles which become trivial on the étale covering ψ : M -→ M , then the arrow ψ * f is also horizontal with respect to the connections ψ * (d E ) and ψ * (d F ). Consequently, surjectivity of ψ assures that h is horizontal.

The lemma is a straightforward consequence of these.

The group of multiplicative characters G -→ G m = C * of an affine group scheme G will be denoted by X(G).

Let

γ * : X(π et (M, x 0 )) -→ X(E(M, x 0 )) (3.4) be the homomorphism induced by γ in Lemma 3.5.

Proposition 3.6. The image of the homomorphism γ in (3.4) is the subgroup consisting of all characters of finite order.

Proof. Any character in X(π et (M, x 0 )) is of finite order, and hence the image of γ * is contained in the subgroup of X(E(M, x 0 )) consisting of all characters of finite order.

Let L be an étale trivial line bundle on M and D a holomorphic connection on L. If the curvature of D is R ∈ H 0 (M, Ω 2 M ), then the curvature of the connection on L ⊗n induced by D is n • R. Therefore, if (L, D) corresponds to a multiplicative character of E(M, x 0 ) of finite order, then D is integrable.

To prove the proposition we need to show the following: Let D be an integrable holomorphic connection on an étale trivial line bundle L on M . If the monodromy homomorphism for D has finite image, then D is the canonical connection on L constructed in Lemma 3.1.

Since the monodromy homomorphism for D has finite image, there is a connected étale Galois covering ψ : M -→ M such that ψ * L is the trivial holomorphic line bundle and ψ * D is the trivial connection on ψ * L. From the construction of the connection in the proof of Lemma 3.1 it follows immediately that D coincides with the canonical connection on L in Lemma 3.1. This completes the proof.

Let is end this section by rendering explicit the group of additive characters of E(M, x 0 ); this is a key observation in [BDDH]. The natural functor T (M ) -→ T et (M ) defines a morphism of group schemes

p : E(M, x 0 ) -→ Θ(M, x 0 )
which must be automatically surjective in view of the fact that q :

(M, x 0 ) -→ Θ(M, x 0 ) is. For an affine group scheme, let X a (G) stand for the group of homomorphisms G -→ G a . We then obtain an injection

X a (Θ(M, x 0 )) -→ X a (E(M, x 0 )) (3.5)
Lemma 3.7. The arrow in eq. (3.5) is a bijection. In particular, the group of additive characters of E(M,

x 0 ) is isomorphic to H 0 (M, Ω 1 M ).
Proof. Let G be an affine group scheme. We first note that X a (G) can be canonically identified with the group Ext 1 G (C, C) of extensions of the trivial representation by itself. On the side of vector bundles, we are then required to show that given an extension

0 -→ (O M , d) -→ (E, D) -→ (O M , d) -→ 0, where E ∈ T et (M ), we must immediately have E O 2 M .
Let ψ : M -→ M be the an étale Galois covering with group G trivializing E and let y 0 ∈ ψ -1 (x 0 ). Giving E the canonical connection (see Lemma 3.1), call it d E , the sequence 3.2. The special case of Riemann surfaces. In this subsection assume that dim C M = 1. Since Ω 2 M = 0, any holomorphic connection on M is integrable, Let C B (M ) denote the category whose objects are all finite dimensional complex representations of the topological fundamental group π 1 (M, x 0 ). Sending a holomorphic connection (E, D) to its monodromy homomorphism π 1 (M, x 0 ) -→ GL(E| x 0 ) we obtain an equivalence between the two categories C dR (M ) and C B (M ). Using the tautological fiber functor C B (M ) defines a neutral Tannakian category. In view of the above equivalence of categories, the pro-algebraic affine group scheme over C corresponding to C B (X) is the group scheme (X, x 0 ) [START_REF] Simpson | Higgs bundles and local systems[END_REF]p. 69,Lemma 6.1]. In other words, (M, x 0 ) is the pro-algebraic completion of π 1 (M, x 0 ).

1 -→ (O M , d) -→ (E, d E ) -→ (O M , d) -→ 0 is also exact. Since ψ * E is trivial, it is the case that π 1 ( M , y 0 ) → GL(E| x 0 ) = GL 2 (C) is trivial and hence π 1 (M, x 0 ) → GL 2 (C)
Let N be a compact connected Riemann surface and

φ : M -→ N (3.6)
an orientation preserving map. The corresponding homomorphism of fundamental groups

φ * : π 1 (M, x 0 ) -→ π 1 (N, φ(x 0 )) produces a homomorphism φ * : (M, x 0 ) -→ (N, φ(x 0 )) (3.7)
between the pro-algebraic completions.

Proposition 3.8. Take any φ as in (3.6) which is a homeomorphism. If the homomorphism φ * is (3.7) descends to a homomorphism

E(M, x 0 ) -→ E(N, φ(x 0 )) .
of quotients (see Lemma 3.4), then the two Riemann surfaces M and N are isomorphic.

Proof. Its proof is very similar to the proof of Theorem 4.1 of [BDDH, p. 1050]; the key point is the existence of an arrow E(M, x 0 ) -→ E(N, φ(x 0 )) which then assures, through Lemma 3.7, the existence of an arrow

H 0 (N, Ω 1 N ) -→ H 0 (M, Ω 1 M ) rendering H 1 dR (N, C) φ * / / H 1 dR (N, C) H 0 (N, Ω 1 N ) O O / / H 0 (M, Ω 1 M ) O O commutative.

Holonomy of connection on trivial bundle

Let M be a compact connected complex manifold and G a complex connected Lie group. The Lie algebra of G will be denoted by g. Consider the trivial holomorphic principal

G-bundle E := M × G -→ M
on M . Take a holomorphic connection D on E. Denoting the trivial connection on E by D 0 , we have

D -D 0 ∈ H 0 (M, Ω 1 M ⊗ C g) = H 0 (M, Ω 1 M ) ⊗ C g . Let D : H 0 (M, Ω 1
M ) * -→ g be the homomorphism given by D -D 0 . The complex Lie subalgebra of g generated by the image of the homomorphism D will be denoted by g D . Let G D denote the unique smallest connected closed complex Lie subgroup of G such that the Lie algebra of G D contains g D .

Consider the holomorphic reduction of structure group of E

E D := M × G D ⊂ M × G = E (4.1) to the subgroup G D . Proposition 4.1. The connection D preserves E D in (4.1) (equivalently, D is induced by a connection on E D ). If E H ⊂ E is a holomorphic reduction of structure of of E to a complex Lie subgroup H ⊂ G such that
• the connection D preserves E H , and • there is a point x 0 ∈ M such that (x 0 , e) ∈ E H , where e ∈ G is the identity element,

then G D ⊂ H and E D ⊂ E H . Proof. Since D(H 0 (M, Ω 1 M ) * )) ⊂ g D ⊂ Lie(G D )
, it follows that the connection D preserves the reduction E D in (4.1). The second statement is equally straightforward.

Expressing the arrow → Θ with the aid of iterated integral

Let M be a complex compact Kähler manifold and x 0 a point of it. Recall that = (M, x 0 ) (respectively, Θ = Θ(M, x 0 )) is the Tannakian group associated to the category of integrable holomorphic connections C dR (M ) (respectively, integrable holomorphic connections on trivial vector bundles T (M )) over M . And our objective in this section is to render the arrow q of eq. (3.2) more explicit by using iterated integrals, therefore linking it to other works like [Pa] and [Ha].

It is showed in [BHdS] that Θ is the Tannakian envelope of a co-commutative Hopf algebra A constructed via the free algebra on the vector space H 0 (M, Ω 1 M ) * . Constructing A is quite simple: letting {β i } be a basis of H 0 (M, Ω 2 M ), define

B i ∈ H 0 (M, Ω 1 M ) * ⊗ H 0 (M, Ω 1 M ) * , ω ⊗ ω -→ β i (ω ∧ ω ). Then A = T(H 0 (M, Ω 1 M ) * )/(B i ).
See [START_REF] Biswas | Connections on Trivial Vector Bundles over Projective Schemes[END_REF]Section 2] for details about A. Concerning the relation between Θ and A, we can say, in a nutshell, that the restricted dual of A [Sw, VI],

A • = lim -→ (A/I) * , I an ideal of finite codimension, is the C-algebra of an affine group scheme isomorphic to Θ. In fact, we have natural tensor equivalences

comod-A • ∼ -→ A-mod V -→ T (M ),
where V is constructed as follows (see [START_REF] Biswas | Connections on Trivial Vector Bundles over Projective Schemes[END_REF]Section 3] for more). Given

f : A -→ End(E)
an object of A-mod, we associate to it the connection d f on O M ⊗ E defined by the End(E)-valued form derived from the composition

H 0 (Ω 1 M ) * -→ A -→ End(E).
(5.1)

Under this equivalence, the forgetful functor is naturally isomorphic to •| x 0 . Now, C dR (M ) is also equivalent to the category of representations of the topological fundamental group Γ = π 1 (M, x 0 ) by means of the monodromy, and the latter category can be shown to be isomorphic to the category of "continuous" modules over a certain topological C-algebra, CΓ , associated to CΓ. Thus, in order to render q explicit, we shall elaborate on the expression of the monodromy representation for objects in T (M ). 5.1. Parallel transport and monodromy. Let Γ be the fundamental group of M based at x 0 . Given (E, ∇) ∈ C dR (M ), the monodromy representation Mon x 0 ∇ : Γ -→ GL(E| x 0 ) is obtained via parallel transport. Write E for the (geometric) holomorphic vector bundle whose sheaf of sections is E. Given a smooth loop γ : [0, 1] -→ M based at x 0 , let

P γ : E| γ(0) ∼ -→ E| γ(1)
be the isomorphism of complex vector spaces obtained from parallel transport

P γ : E| γ(0) -→ E| γ(1)
via the canonical identification E| x ∼ -→ E| x . In other words, we extend ∇ to a C ∞ connection, find a C ∞ lift γ : [0, 1] -→ E of γ such that γ(0) = e and ∇ γ γ = 0, and then define P γ (e) = γ(1).

As the composition γ 1 * γ 2 in Γ is the path γ 1 followed by γ 2 , we must write Mon x 0 ∇ (γ) = P γ -1 .

5.2. Parallel transport, monodromy in T (M ) and iterated integrals. In this section we explain how to write the monodromy representation in terms of iterated integrals. This is explained in [START_REF] Hain | On a generalization of Hilbert's 21st problem[END_REF]§ 2], and attributed to Chen, but Hain adopts different conventions (like letting parallel transport "act on the right" and working with connections defined by the negative of a form, see (2.1) and line -11 on p. 614 of [Ha]), so that it is worth explaining it in full here. Let E be a vector space of finite dimension and

ω ∈ H 0 (Ω 1 ) ⊗ End(E)
a differential form with values on End(E). Write d ω for the connection on the trivial vector bundle M × E -→ M whose associated 1-form is ω. By definition, if s = (id, e) is a section on some open set then, for any C ∞ vector field v,

(d ω ) v (s) = (id, v(e) + ω(v) • e).
Given a smooth loop γ : [0, 1] -→ M about x 0 , a lift t -→ (γ(t), e(t)) is parallel if and only if e (t) + A(t)e(t) = 0, where A(t) = ω(γ (t)). It then follows that the parallel transport of (x 0 , e 0 ) ∈ (M ×E)| x 0 along γ, call it P γ (x 0 , e 0 )

is the value y(1) of a solution to

y (t) = -A(t)y(t) y(0) = e 0 .
Solutions to this sort of equation are described concisely using Picard iteration.

Recall the following fundamental result. (A proof can be adapted from the proofs of the Picard-Lindelöf Theorem, which is to be found in almost all texts on ordinary differential equations, e.g. [Co], Chapter 6, Theorem 4 and Theorem 6.) Theorem 5.1 (Picard-Lindelöf). Let R be a finite dimensional associative C-algebra and F : [0, 1] -→ R be a continuous map. Given f 0 ∈ R, define inductively continuous functions [0, 1] -→ R by

f k+1 (t) = f 0 + t 0 F (s) • f k (s) ds. (1) The limit f (t) = lim k f k (t) exists for each t ∈ [0, 1] and f = F • f . (2) Let δ k = f k -f k-1 , for each k ≥ 1. Then, δ 1 (t) = t 0 F (s) • f 0 ds, δ k+1 (t) = t 0 F (s)δ k (s) ds, for k ≥ 1 and f (t) = f 0 + ∞ k=1 δ k (t).
The functions {δ k } appearing in the above statement have a well-known expression in terms of iterated integrals. Let R be as in Theorem 5.1 and consider [START_REF] Chen | Algebras of Iterated Path Integrals and the Fundamental Group[END_REF]p. 359] and [Ha, § 1], among others.) For example,

R-valued C ∞ differential forms ρ 1 , • • • , ρ m . Given a smooth path c : [0, 1] -→ M , let F i = ρ i (c ) and define G 1 (t) = t 0 F 1 (s) ds ∈R , G i+1 (t) = t 0 G i (s) • F i+1 (s) ds ∈R and c ρ 1 • • • ρ m = G m (1) ∈R (See
c ρ 1 ρ 2 = 1 0 s 2 0 F 1 (s 1 ) ds 1 • F 2 (s 2 ) ds 2 .
Alternatively, it is sometimes convenient to think of the iterated integral as an integral over a simplex:

c ρ 1 • • • ρ m = 0≤s 1 ≤•••≤sn≤1 F 1 (s 1 ) • • • F m (s m ) ds 1 • • • ds m .
Returning to the problem of expressing the parallel transport of (x 0 , e 0 ) ∈ (M × E)| x 0 along the path γ with respect to d ω , the above terminology and Theorem 5.1 allow us to write

P γ (x 0 , e 0 ) = x 0 , e 0 - 1 0 A(s 1 ) ds 1 • e 0 + 0≤s 1 ≤s 2 ≤1 A(s 1 )A(s 2 ) ds 1 ds 2 • e 0 -• • • . A simple change of variables shows that γ -1 ω 1 • • • ω m = (-1) m γ ω m • • • ω 1 and hence Mon x 0 dω (γ) : (x 0 , e 0 ) -→ x 0 , e 0 + γ ω • e 0 + γ ω 2 • e 0 + γ ω 3 • e 0 + • • • . (5.2) (Here, obviously, ω n stands for ω • • • ω n .)
5.3. Algebraic interlude. To sustain the arguments concerning the relation between A and CΓ necessary in explaining the morphism q, we shall require some material on completions of rings and Hopf duals. We fix a field K.

Completions. Let A be an associative and unital K-algebra.

, • • • , I r ∈ S A be such that V = {(y I ) ∈ A : y I j = x I j , ∀ j} is contained in U . Let J = I 1 ∩ • • • ∩ I r ∩ a; it clearly belongs to S A . Then, if ξ ∈ A is such that ξ + J = x J , we conclude that ιξ ∈ V ⊂ U and ξ ∈ a.
Example 5.3. Suppose that K is algebraically closed and A is a commutative K-algebra of finite type. Then A is m A m , where m ranges over the maximal ideals.

Let A-mod be the category of A-modules which have finite dimension as K-spaces. Analogously, let A-modc be the category of continuous A-modules, i.e., those (left) Amodules of finite dimension over K which are annihilated by some a with a ∈ S A . (Note that, although any A-module of finite dimension is annihilated by an ideal I ∈ S A , we cannot assure a priori that such an ideal contains a certain I.) Using the natural morphism A -→ A, we obtain a functor

A-modc -→ A-mod (Note that ϕ ij ∈ (A/a) * ⊂ A • .) Consequently, the coefficients of the B • -comodule E coming from the composition via v # are v(ϕ ij ). By construction, v(ϕ ij ) ∈ (B/a v ) * .
We now examine the structure of B • -comodule on E obtained through the composition via u # . Clearly, a v ⊂ B annihilates E. For each α ∈ (A/a) * and b ∈ B/a v , we have, as

u a = v * , that α • u a (b) = v(α)(b).
This shows that v(ϕ ij ) = ϕ ij • u a . By definition of u # and (5.6) we have Define a tensor functor

V : comod-A • -→ comod-(CΓ) •
by means of a composition as indicated in

T (M ) ι / / C dR (M ) ∼ Mon x 0 A-mod ∼ V O O CΓ-mod ∼ τ comod-A • V / / ∼ σ O O comod-(CΓ) • .
Here, τ and σ are the equivalences mentioned in (5.3). Letting ω denote "the" forgetful functor, we have a natural tensor isomorphism

η : ω A • ∼ -→ ω (CΓ) • • V constructed from the canonical isomorphisms ω A ∼ -→ •| x 0 • V and ω CΓ • Mon x 0 ∼ -→ •| x 0 .
(It should be noted that ω A • σ = ω A • M and ω (CΓ) • • τ = ω CΓ .) In simpler terms, given E ∈ comod-A • , the vector space

V (E) is V(E)| x 0 = (O M ⊗ E)| x 0 and η E is the natural isomorphism E ∼ -→ V(E)| x 0 .
We obtain an arrow of Hopf algebras v : A • -→ (CΓ) • jointly with an isomorphism of tensor functors η : v # -→ V determined by η [START_REF] Rivano | Catégories Tannakiennes[END_REF]p.116]. That is, η equals the composition

ω A • = ω (CΓ) • • v # ω (CΓ) • η / / ω (CΓ) • • V.
Again, in simpler terms, given E ∈ comod-A • , the canonical isomorphism of vector spaces η E : E ∼ -→ V(E)| x 0 is an isomorphism of (CΓ) • -comodules, where V(E)| x 0 carries the co-action coming from the monodromy and E the co-action coming from v.

Applying the results of Section 5.3 to v, we derive a continuous morphism of topological C-algebras u : CΓ -→ A which makes the analogue diagram in Lemma 5.4 commute strictly. We can then say the following. Let E ∈ comod-A • be given. Endow E, respectively V(E)| x 0 , with its CΓ-module structure coming from u, respectively from the monodromy. Then η : E -→ V(E)| x 0 is an arrow of CΓ-modules.

5.5. Description of u : CΓ → A in terms of iterated integrals. We can now explicitly describe the action of Γ on A-modules and hence the morphism of C-algebras u : CΓ -→ A.

Let R be a finite dimensional associative C-algebra and f : A -→ R a surjective morphism. Regarding R as a left A-module, let d f be the connection on O M ⊗ R defining V(R) and ϕ f = d f (1 ⊗ 1) ∈ H 0 (Ω 1 M ) ⊗ R.

Proposition 5.5. For any smooth loop γ based at x 0 , the series

1 + γ ϕ f + γ ϕ 2 f + . . .
in R converges, with respect to the Euclidean topology, to f (u(γ)).

Proof. Let λ : R -→ End(R) be the morphism of left multiplication. With this notation, As a consequence, we have the following. Let γ be a smooth loop based at x 0 and define, for every n ∈ N \ {0}, the linear functional J γ,n : H 0 (Ω 1 ) ⊗n -→ C by

Φ f := (id ⊗ λ)(ϕ f ) ∈ H 0 (Ω 1 M ) ⊗ End(R)
ϕ 1 ⊗ • • • ⊗ ϕ n -→ γ ϕ 1 • • • ϕ n .
This gives rise to an element of [H 0 (Ω 1 ) * ] ⊗n , then to an element of A and hence to an element J γ,n ∈ A. For convenience, we decree that J γ,0 = 1. By fixing a basis {θ i } g i=1 of H 0 (Ω 1 ), a dual basis {t i } g i=1 and writing t i 1 ,...,in for t i 1 ⊗ • • • ⊗ t in , we have

J γ,n = 1≤i 1 ,...,in≤g γ θ i 1 • • • θ in • t i 1 ,.
..,in modulo integrability relations.

These elements were studied firstly by Chen and Parshin [Pa] and hence we call the arrow For each I ∈ S A , endow the complex vector space A/I with the canonical Euclidean topology and give A the projective limit (product) topology. This topology shall be called the PE-topology (projective-Euclidean). Hence, a sequence (α n ) in A converges to α ∈ A if and only if, for each I ∈ S A , the induced sequence (α n,I ) in A/I converges α I [START_REF] Kelley | General Topology[END_REF]4 Theorem,p .91].

Corollary 5.7. In A, we have the equality f converges (Proposition 5.5), the definition of the product topology proves that n≥0 J γ,n converges in A. The limit is u(γ).

  Lemma 3.1. A holomorphic vector bundle E on M is étale trivial if and only if E admits an integrable holomorphic connection D such the image of the monodromy homomorphism Mon(D) in (3.1) is a finite subgroup of GL(E| x 0 ). Such a connection is uniquely determined by the following property. If ψ : M -→ M is a finite Galois covering for which there exists an isomorphism h : ψ * E -→ O r M , then ψ * D corresponds, under h, to the trivial connection on O r M . Proof. If the image of Mon(D) is finite, then consider the étale Galois covering ψ : M -→ M corresponding to kernel(Mon(D)) ⊂ π 1 (M, x 0 ). The pulled back holomorphic connection ψ * D on ψ * E has trivial monodromy, and hence ψ * E is holomorphically trivial.

  factors through the finite quotient G. Now Mon(d E ) has image in the subgroup of strict upper triangular matrices, which is isomorphic to C. Since C does not have any finite subgroup other than the trivial, we conclude that Mon(d E ) is trivial. Hence, E is holomorphically trivial.The last claim follows from[START_REF] Biswas | On certain Tannakian categories of integrable connections over Kähler manifolds[END_REF] Lemma 3.5].

  Given a ∈ S A , let p a : A -→ A/a be the canonical map and denote by a the open and closed ideal Ker(p a ). Note that A/a ∼ -→ A/ a. Lemma 5.2. The family { a} a∈S A is a local base at 0. The ideal a is the closure of ιa in A.Proof. We only prove the second statement. Clearly ι(a) ⊂ a. Let x = (x I ) ∈ a.Let U be an open neighbourhood of x and let I 1

  be j = u a (b)e j = i ϕ ij (u a (b))e i and the coefficients of the B • -module E are ϕ ij • u a . This shows commutativity of the diagram on the level of object. The case of arrows being trivially verified, we conclude our proof. 5.4. Fiber functor and the Tannakian picture. Using the inclusion T (M ) -→ C dR (M ) and Tannakian theory, we shall construct an arrow of Hopf algebras A • -→ (CΓ) • and from it define, using Section 5.3, an arrow of C-algebras u : A -→ CΓ . In what follows, let us denote by an upper hashtag the functor between categories of modules or comodules obtained from a morphism of algebras or co-algebras.

  is the 1-form associated to the connection d f (see (5.1)). As seen in (5γ ∈ End(R). (We write Mon γ instead of Monx 0 d Φ f (γ).) Now, for each n ∈ N, we have in End(R) that λ γ ϕ f + γ ϕ 2 f + . . . converges to an element m γ which satisfies λ(m γ ) = Mon γ .In particular, Mon γ (1 R ) = m γ . But, by construction of u : CΓ -→ A (see (5.5) and Section 5.4), we know that Mon γ (1 R ) = f (u(γ)) • 1 R , so that m γ = f (u(γ)).

  maps of order N (as of now, these are maps of sets) and (cp N ) the Chen-Parshin sequence.Lemma 5.6. Let R be a finite dimensional associative C-algebra and f : A -→ R a surjection. Let d f stand for the connection on O M ⊗ R associated to the left A-module structure on R, and letϕ f ∈ R ⊗ H 0 (Ω 1 M ) be the 1-form d f (1 ⊗ 1). Then, for each n ∈ N, we have f (J γ,n ) = γ ϕ n f . Sketch proof. Let ϕ f = g i=1 e i ⊗ θ i and denote the function θ i , γ by a i . It follows that ϕ f , γ = i e i a i ∈ C ∞ ([0, 1], R). Consequently, e i 1 a i 1 (s 1 ) • • • 1≤in≤g e i 1 a in (s n ) ds 1 • • • ds n = 1≤i 1 ,...,in≤g 0≤s 1 ≤•••≤sn≤1 e i 1 • • • e in a i 1 (s 1 ) • • • a in (s n ) ds 1 . . .ds n = 1≤i 1 ,...,in≤g e i 1 ,...,in γ θ i 1 • • • θ in .

  n = u(γ),where convergence is taken in the PE-topology. In other words, u(γ) is the limit of the Chen-Parshin sequence (cp N (γ)) in the PE topology.Proof. Let I ∈ S A and denote by f : A -→ R the quotient morphism. As in Lemma 5.6, we write ϕ f = d f (1 ⊗ 1). Then,

  ≤ J if I ⊃ J. Consider the subset of 2 A defined by {a + I} a∈A, I∈S A and let O be the unique topology on A having this set as a sub-base [Ke, p. 47-48]. Of course, S A is local base at 0 [Ke, p. 50] and all ideals of S A are open, hence closed.

				Write
	S A =	two sided ideals of finite codimension in A	;
	this set it partially ordered by decreeing I Put
		A = lim I∈S A ← -	A/I;

endowing each A/I with the discrete topology and A with the projective limit topology, A becomes a topological ring. (Rings as A are called pseudo-compact [SGA3, VIIB, § 0].) Let ι : A -→ A be the evident map; according to

[START_REF] Bourbaki | Elements of Mathematics[END_REF] III.7.3, Cor. 1]

, ι is a completion of A for O.

which is easily seen to be an isomorphism of categories. Indeed, for each E ∈ A-mod, the ideal a = Ann(E) belongs to S A , and hence we can define a structure of A-module by (x I ) • e = x a • e, (x I ) ∈ A, e ∈ E.

The restricted dual. Let A be an associative and unital K-algebra and define, as in [Sw, Ch. VI], the restricted dual

This is a co-algebra [START_REF] Sweedler | Hopf algebras[END_REF]6.0.2,p. 113] and

since dualization takes direct limits into projective limits. Letting comod-A • be the category of (right) comodules over A • having finite dimension as K-spaces, we have functors

which together define an isomorphism of categories, cf. [START_REF] Sweedler | Hopf algebras[END_REF]Theorem 2.1.3] [Mo,1.6.4,p. 11]. For future application, note that if

Conversely, if F ∈ A-mod has basis {f i } n i=1 , is annihilated by a, and the action is given by

then the co-module structure of F is defined by the matrix (ρ ij ) ∈ Mat n ((A/a) * ).

Note that (A • ) * is now an algebra [START_REF] Sweedler | Hopf algebras[END_REF]Proposition 1.1.1,p. 9] which is easily seen to be the completion A studied above.

Let now B be another algebra and

an arrow of co-algebras. Taking duals, we obtain an arrow of K-algebras u : A -→ B. Alternatively, u can be described as follows. For each I ∈ S A , there exists J ∈ S B such that v((A/I) * ) ⊂ (B/J) * . In addition, let I v ∈ S B be the largest ideal with this property. We arrive at an order preserving map (-) v : S A -→ S B and to an arrow of K-spaces u I : B/I v -→ A/I whose transpose is v| (A/I) * . Note that u I is a morphism of algebras since v| (A/I) * is a morphism of co-algebras. A moment's thought shows that for J ⊂ I in S A , the diagram

commutes and we obtain an arrow of K-algebras u : lim ← -

(5.4)

For each I ∈ S A (respectively, J ∈ S B ), endow A/I, (respectively, B/J) with the discrete topology and give the above projective limits the product topology. This being so, general topology assures that u is continuous.

We then obtain from eq. ( 5.4) a morphism of topological K-algebras u : B -→ A and consequently a functor

For the sake of communication, the morphism u shall be called the continuous transpose of v.

Lemma 5.4. The following diagram of categories Proof. We require the constructions made in the beginning of the section. Let E ∈ A-modc and pick a basis {e i }. Let a ∈ S A be such that a annihilates E. Now, the coefficients {ϕ ij } determining the A • -comodule structure of E with respect to the basis {e i } are determined by ae j = i ϕ ij (a) • e i .

(5.6)


This is the peer reviewed version of the following article: Energetics of Sulfur-Carbon Interaction by Marie-Vanessa Coulet, Loïc Gourmellen, Renaud Denoyel, which has been published in final form at https://doi.org/10.1002/cphc.202200416.

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Energetics of sulfur-carbon interaction

Marie-Vanessa Coulet*, Loïc Gourmellen, Renaud Denoyel Aix-Marseille Univ and CNRS, MADIREL(UMR7246) Campus de St Jérôme, 13013 Marseille, France Abstract: The energetics of sulfur-carbon interaction are studied using thermo-desorption and immersion microcalorimetry experiments. Sulfur is incorporated in meso-and microporous carbons by impregnation either from the liquid phase or the vapor phase. Varying the temperature of impregnation enables to fill preferentially microporous domains (vapor impregnation) or both micro-meso-macro domains (liquid impregnation). The three carbons lead to similar immersion enthalpies per unit area for liquid sulfur. This suggests that they possess similar surface-liquid interactions and that liquid sulfur, below the polymerization temperature, wets the whole surface accessible to nitrogen.

The confinement of sulfur in porous matrices is an active field of research for many years. The first works were essentially fundamental and concerned its confinement in zeolites. At that time, the aim was to identify, essentially by spectroscopic methods, the allotropes (rings or chains) that can be stabilized in the pores [START_REF] Poborchii | [END_REF] . More recently the interest for confining sulfur has increased with the development of lithiumsulfur batteries [2] . For those applications, sulfur is confined porous carbon matrices to realize conductive electrodes. The confinement is reported to reduce the formation and the diffusion of lithium polysulfides that are at the origin of the so-called shuttle effect [3] . The efficiency of sulfur-based electrodes depends on (i) the pore size distribution and organization of the carbon host, which affects the diffusion of electrochemically active species, (ii) the interaction of sulfur with carbon, which has an influence on the long-term stability of the electrode and (iii) the extent of surface area that is in direct contact with sulfur.

It is then clear that it is of utmost importance to choose materials with suited pore size distributions and to control as much as possible the procedure of pore filling since the amount of sulfur in the composite has direct consequences on the gravimetric energy density of the electrode. In terms of filling procedures, sulfur is generally introduced via liquid impregnation [4] , vapor infiltration [5] or even from the solid state via the so-called spillover phenomenon [6] . Depending on the chosen temperature, it is excepted that different allotropes are stabilized in the pores which might influence the amount of sulfur that can be encapsulated.

In terms of electrochemical efficiency, the sulfur confined into (ultra) microporous carbon becomes inaccessible to carbonate solvent from the electrolyte which reduces considerably the formation and the dissolution of non-wanted species [7] . The present study focusses on the energetics of sulfur-carbon interaction, which is not explicitly addressed in literature. This is done here by analyzing thermodesorption curves of samples impregnated by sulfur and by using a new approach based on immersion microcalorimetry. Indeed, immersion calorimetry of porous carbon in water or organic liquids, 3 has up to now only be used to improve their pore size characterization [8][9][10] . It is used here for the first time at high temperature and with liquid sulfur in order to assess sulfur accessibility to the pore network.

Three commercial porous carbons are used: a mesoporous carbon (C1) and two microporous carbon labeled C2 and C3. More details are given in SI (section 1). Thermogravimetric analyses and nitrogen sorption isotherms of the carbon matrices before impregnation are shown in Fig. 1. pore volume (open symbols) calculated using NLDFT (C1) and using QDFT (C2, C3) on carbon slit pore kernels [11] .

The two microporous carbons (C2 and C3) loose around 10% of their mass between 300 K and 423 K (Fig. 1a). This loss can be attributed to the elimination of residual water physisorbed inside the pores. In contrast, the mass remains constant for mesoporous C1 sample suggesting that this carbon is hydrophobic. Water adsorption isotherms (Fig. S1 in SI) confirm their different affinity towards water. In the case of nitrogen adsorption, the type-IV isotherm (Fig. 1b) obtained for C1 carbon evidences its mesoporous character. The hysteresis can be assigned to the H1 type [12] since the adsorption and desorption branches are nearly parallel. The adsorption isotherms of C2 and C3 are of type I, typical for microporous samples with high affinity at low pressure. In the case of C2 the closure point of the hysteresis loop is close to a relative pressure of 0.4, indicating that cavitation may occur during nitrogen desorption. This shape of the hysteresis loop is characteristic for samples with ill-defined pores between plates (H4 type) [12] . For C3 sample, a hysteresis is observed at high relative pressures. This is consistent with the co-presence of large mesopores and confirmed by mercury porosimetry (Fig. S2 in SI). In the inset of Fig. 1b, the adsorption isotherms are plotted in log scale. The one of C3 starts at slightly lower pressures than that of C2 which means that the distribution of micropore sizes is shifted towards smaller pores if the same surface chemistry is assumed. This is confirmed by the pore size distributions (Fig. 1c): the pore size distributions for C2 and C3 are very similar and only differ slightly by the minimum pore width, which is shifted towards smaller values for C3 carbon (see vertical lines in Fig. 1c). Mesoporous C1 sample shows a wider pore size distribution with a main contribution around 6 nm. The small contributions at 1 nm and 3nm are related to the graphitization of the surface which is also responsible for the inflexion points around p/p 0 =8.10 -4 and p/p°=0.3 on the adsorption isotherm. The equivalent BET surface areas and pore volumes are summarized in Table S1. Fig. 2 presents high-resolution TEM images and the corresponding EELS spectra for the three carbon matrices. The size and the number of stacked graphene layers are the highest for the C1 mesoporous carbon and the smallest for the C3 microporous carbon (Fig. 2c). This confirms that the organization of graphene layers is directly linked to the specific surface area [13,14] . The EELS spectra recorded in the core loss region all possess a peak around 285 eV, related to the * antibonding orbitals and typical for sp 2 -hybridized carbon [15] . Nevertheless, the fraction of sp 2 -bonded carbon is lower in C2 and C3 carbon since the peak is much less intense in those samples. At last, only the mesoporous C1 presents a clear feature around 292 eV which is characteristic of the unoccupied * states. These observations confirm that mesoporous C1 sample is a graphitized carbon while the microporous C2 and C3 are mainly amorphous carbons with a low degree of graphitization.

Figure 2. TEM-EELS characterization of the carbon matrices. The high resolution TEM images (left) are accompanied by the corresponding EEL spectra (right).

The carbon matrices were impregnated either from the liquid phase (at 413 K, well below the polymerization temperature) or from the vapor phase (at 873 K). All the details of the procedure are given in SI (section 4). The composites are denoted SL@Cx or SV@Cx, where x is 1, 2 or 3 depending on the porous carbon matrix and L and V stand for liquid and vapor. As evidenced in Fig. 3a and3b, after impregnation the amount of adsorbed nitrogen is very low for all samples. This indicates that sulfur has either filled or blocked the pores, whatever the sample. As seen in Fig. 3b, the mesoporous sample C1 keeps a number of open pores in the case of vapor impregnation, whereas for liquid impregnation the hysteresis loop has disappeared. Since the initial proportion between sulfur and carbon was the same within 1% (see SI, Table S2), this indicates that the pore filling may be less efficient in the case of vapor impregnation. This is probably linked to an unwanted temperature gradient during the experiment.

Figure 3. Nitrogen sorption isotherms and thermo-desorption analyses of the S@C composites synthetized by liquid impregnation (a,a') or by vapor impregnation (b,b').

For microporous samples, the onset of a hysteresis loop at p/p o values close to 1 suggests the presence of mesopores. They probably have their origin in the partial filling of macropores by sulfur. This suggests that micropores are preferentially filled and it indicates that, like for any fluid, the affinity of sulfur is larger when the pore size is smaller. The BET surface areas and total pore volumes obtained after impregnation (liquid or vapor) are given in SI (Table S3).

The filling of the pores and the influence of pore size on sulfur affinity for the surface are confirmed by the thermo-desorption analyses. The desorption curves of pure sulfur are compared with the ones of liquid (Fig. 3a') and vapor (Fig. 3b') impregnated samples. For pure sulfur the temperature range of evaporation is narrow since it is a pure element, even if it may contain several types of molecules (rings, polymers). For impregnated samples, it is observed that the temperature of sulfur desorption is shifted towards higher temperatures when the sulfur is confined inside the pores and this shift is higher as the pore size decreases. The desorption of sulfur is observed between 430 K and 633 K for C1 and between 440 K and 795 K for C2 and C3. A narrow range of temperature departure is observed for C1 which possesses mainly mesopores. In contrast, C2 and C3 have larger pore size distributions since they contain micropores, mesopores and even macropores. These results confirm that sulfur really fills the pores for both impregnation methods. Indeed, if it was outside the pores and simply blocking them, the temperature of desorption would be more or less the same as the one of pure sulfur evaporation. The weight losses obtained by thermo-desorption (SI, Table S2) give systematically lower pore filling than the one initially targeted, particularly for the impregnation from the vapor phase which confirms the results obtained by adsorption. For mesoporous carbon C1, both liquid and vapor impregnation lead to a similar desorption temperature, whatever the pore filling. This tendency is not the same for microporous carbons.

The impregnation from vapor at 873 K leads to an important increase of the desorption temperature. This could mean that under those conditions, smaller molecules can be formed and they are preferentially adsorbed in the micropores. In the case of liquid impregnation, the range of temperature desorption is larger. For instance, in the case of C3, the desorption curves follow initially the one of C1 and then join the temperature range observed for vapor impregnation. This two-regimes behavior may indicate that both micropores and larger pores (meso or macro) are filled by liquid sulfur. Finally, one can conclude that (i) sulfur confined in micropores is more strongly retained than in the one in mesopores, (ii) it is possible to fill the samples either from the liquid or the vapor phase and (iii) vapor impregnation leads to a more homogeneous filling in the micropores. Nevertheless, these observations are based on ex-situ experiments where the dynamic of filling and the interaction energy between sulfur and surface cannot be quantified since thermo-desorption curves are a combination of kinetics and thermodynamics aspects.

To the best of our knowledge, there is no attempt in the past to evaluate experimentally the interaction between sulfur and carbon surfaces. The details of the experimental method are given in SI (section 5).

The calorimeter temperature was set to 413 K i.e. at a temperature above the melting point of sulfur but below its polymerization transition. At this temperature, liquid sulfur mainly consists in S8 rings and has a low viscosity. A first interesting result of the calorimetry approach, directly accessible from the raw data, is the possibility to follow the kinetics of pore filling. As seen on the thermograms in Fig. 4, the phenomenon is relatively fast since the baseline of the calorimetric peak is recovered after 1500 s whatever the sample. This has to be compared to the 1000 s needed for a simple heat effect such as the introduction of a piece of alumina to calibrate the system (see inset in Fig. 4a). This means that the wetting of carbon by liquid sulfur is achieved within a few minutes at this temperature where the viscosity of sulfur is low (~6.10 -2 Pa. s [16] ). By contact angle measurements, the spontaneous wettability of carbon by sulfur was already studied using sessile drop method [17] but the observed time for complete infiltration of sulfur is larger than the one obtained here by the immersion method. Our measurements also demonstrate that the pore size range of the tested samples does not modify strongly the kinetics of wetting. This is an important result since all the S@C composites synthesized up to now by liquid impregnation are generally heated for 6h up to 24h. The immersion enthalpies in J per gram of sample are given in Fig. 4b. The values increase with the specific area of the carbon. This is a first indication that liquid sulfur, that mainly consists of rings and short chains at 413 K, is able to enter in the microporous volume. This is consistent with the molecular sizes calculated by DFT [18] . Once renormalized to the BET surface area, the areal immersion enthalpies (in J per m 2 ) are very similar for the three carbons (Fig. 4b, Table S4). This indicates that dispersive forces between sulfur and the surface are the main driving force for the filling of pores. Indeed, if only London additive dispersion forces are considered, the enthalpy of immersion of a porous solid into a liquid is at first order proportional to the wetted surface area [9] . This is easy to understand if one considers large pores: the enthalpy of immersion is the product of the surface area by the immersion enthalpy per unit area. If the pores are narrow, the situation may appear at first sight different since a molecule is interacting with more than one wall (for instance two in a slit shaped pore geometry).However, due to the additivity of London dispersion forces, it has been shown that when a molecule has a size close to the pore width, its interaction energy is twice the one obtained on an open surface but the molecule also wets twice as much surface area [9] . In the case of a cylinder, it was also shown that the ratio of energies is close to the ratio of wetted surfaces [9] . This simplified analysis was confirmed by DFT calculations on a large range of pore sizes [START_REF] Denoyel | Series on Chemical Engineering[END_REF] . Finally, since the three types of carbon lead to the same immersion enthalpy per unit area for liquid sulfur, they have probably similar surface-liquid interactions and the liquid wets the whole surface accessible to nitrogen. This is confirmed by the estimation of the accessible surface area to liquid sulfur (SI Table S4) using immersion enthalpy of C1 as reference.

It has been recently reported that chemical interactions between sulfur and carbon could be present [START_REF] Perez Beltran | [END_REF] .

They do not seem to be predominant here, otherwise the areal immersion enthalpy of microporous carbons would have been different because they have a higher density of graphene sheet defects (Fig. 2).

The value of immersion enthalpy per unit area is close to the work of adhesion -around 0.1 J/m 2 at 393 K derived from contact angle measurements [17] .

This study opens a path to the rationale of the impregnation of porous carbon by sulfur. Depending on the temperature of impregnation, it is possible to fill preferentially microporous domains via vapor impregnation, or both micro-meso-macro domains via liquid impregnation. Immersion calorimetry, allows the evaluation of the energy of interaction of sulfur with the porous material and also to follow the kinetic of wetting, which is quite fast for the samples studied here. At this temperature, where mainly S8 rings are present in the liquid, micropores can be filled by sulfur and the total accessible surface area to liquid sulfur is similar to the one determined by nitrogen adsorption at 77 K.

Experimental section

Transmission electron microscopy (TEM) coupled with electron energy loss spectroscopy (EELS) analyses were performed in order to monitor the atomic and electronic structures of the porous carbon matrices. The measurements were performed using a FEI TITAN 80-300 TEM equipped with an image corrector and a GATAN TRIDIEM imaging filter. The carbon K-edge spectra were acquired with an energy resolution of 0.8 eV.

Thermogravimetric analyses (TGA) were done using a TGA/DSC 1 STAR apparatus. All measurements were performed under Nitrogen flow (30 ml/min) using a heating rate of 5 K/min from room temperature up to 1173 K. Firstly, TGA is used to determine the temperature at which the porous carbon matrices must be treated to ensure complete removal of residual water and adsorbed organic species. Secondly, it was used to perform thermo-desorption on the S@C composite and thus to quantify the amount of sulfur in the composite and to highlight the specific interaction of sulfur with carbon.

Nitrogen adsorption at 77 K analyses were done in order to access the total porous volume of the carbon matrices before and after impregnation by sulfur in order to follow the accessibility of the pores.

Porous carbon matrices were treated at 423 K under a residual pressure lower than 1Pa before adsorption during 15 hours. After impregnation, the samples were kept under vacuum at room temperature before the adsorption measurement. Experiments were carried out with a BELMax from BEL Instrument for empty matrices and using a GEMINI apparatus from Micromeritics for the composite since the amount adsorbed is quite low.

Immersion microcalorimetry measurements were done using a Tian-Calvet type isothermal microcalorimeter. The calorimeter was set at a temperature equal to 413 K in order to ensure that the sulfur is in the liquid state but has not yet polymerized. The details of the experiment are given in SI (section 5).

Figure 1 .

 1 Figure 1. Porous carbons characterization. (a) TGA analyses realized under nitrogen atmosphere. (b) Nitrogen sorption isotherms performed at 77 K. Full symbols are used for the adsorption branch and open symbols are used for the desorption branch. The inset show the isotherm in logarithm scale in order to make more visible the inflexion points. (c-e) Pore size distribution (full symbols) and cumulative

Figure 4 .

 4 Figure 4. (a) Thermograms representing the heat flow as a function of time associated to the immersion phenomenon for the three porous carbons. For better legibility, the heat flow has been normalized to the mass of porous carbon. The immersion enthalpy is the integral of this curve. Inset: calibration by introducing a small piece of alumina at a different temperature from the cell. (b) Immersion enthalpy (J/g) and areal immersion enthalpy (mJ/m 2 ) deduced using BET area.

Acknowledgements

Région Sud is acknowledged for funding (AAP Recherche 2020 CONSSON Project). Transmission electron microscopy measurements were done at the CP2M (FR1739 CNRS-AMU). The authors thank F. Berger for her glass work and constant availability. M. Neisius is acknowledged for his artistic vision of science and for the realization of the TOC.