On the Mumford-Narasimhan problem

Andrei Tyurin

October 11, 2002

To Prof. M. Narasimhan on his 70th birthday

Thirty years ago, after appearing of the papers [T1] David Mumford asked me about an application of the proposed technique to the ”Schottki problem for vector bundles”. This problem is a non-abelian analogy of the classical cover f: ℂ^{g} → J(C) where the target space is the place where the theta functions of an algebraic curve C live. More precisely, if S_{g}^{r} ⊂ CLRep(π_{1}(C), SL(r, ℂ)) is the subset of the representations with trivial ”a”- periods and f: S_{g}^{r} → M_{r}^{ss}(C) is the forgetful map from the space of flat bundles to the moduli space of semi-stable vector bundles on C then the question is what kind of image we have. The spectrum of the cases is the following

(1) surjectivity f(S_{g}^{r}) = M_{r}^{ss}(C) is the best answer, predicted by the classical case;

(2) for general curve f(S_{g}^{r}) contains a Zariski open set in M_{r}^{ss}(C);

(3) other possibilities are too bad to discuss them.

Many times after we discussed this problem with M. Narasimhan reducing the question to new approaches like Hitchin’s Higgs bundles [H] and so on. The point was the following: by the Narasimhan-Sesadri theorem the restriction of f to the unitary Schottki space

uS_{g}^{r} = CLRep(π_{1}(C), SU(r)) ⊂ S_{g}^{r}

Large limits... 2

is an embedding. Moreover the differential of f can be investigated. C. Florentino proved that around uS_{g}^{r} this differential is an isomorphism. Long time after we hadn’t any information about its behavior. In this paper we prove at least that the second case is realized. That is for general smooth curve C general vector bundle admits the Schotki representation. It is quite relevant to dedicate this paper to Prof. M.Narasimhan on his 70th birthday in spite of the fact that this is the first step only to an expected perfect solution to the Mumford-Narasimhan problem. The main idea is

(1) to prove the statement for special curves and

(2) to extend this statement using the standard technique of complex analysis to neighborhoods of special curves.

This problem lies on the boundary connecting algebraic geometry and complex analysis. But for our special curves the analysis can be reduce to a simple algebraic geometry. These curves are called large limit curves and we would not discuss here why. But it is quite easy to understand from section 3 which isn’t necessery for the main result proof. Such curves were used succesfully by Ciro Ciliberto, Angelo Lopez, Rick Miranda and Lucia Caporaso already in algebro-geometrical set up (see, for example [CLM]). Here we use them in the complex gauge theoretical set up. Moreover, the correlation functions of all local quantum fields in a two dimensional conformal field theory can be recovered from the partition function when all channels (tubes of the pair of pants decomposition) of the surface are constricted down to nodes. This procedure produces our reducuble curve P_{Γ} with uniquely defined complex structure. Details of such approach can be found in [T3].

1 Trinion decompositions and holomorphic flat connections on smooth algebraic curves

For an oriented compact surface Σ of genus g a trinion decomposition is given by a maximal set of disjoint, isotopy inequivalent circles

(C_{1}, ..., C_{3g−3}) ⊂ Σ  (1.1)

(see [HT]). Removing these circles we get

Σ − {C_{1}, ..., C_{3g−3}} = ∪_{i=1}^{2g−2}ṽ_{i}  (1.2)

the finite set of trinions (or ”pairs of pants”) that is a trinion decomposition of Σ. Such decomposition defines dual trivalent graph Γ such that (1) the set of its vertices

V (Γ) = {v_{i}} = {ṽ_{i}}  (1.3)

is the set of trinions;

(2) and with the set of edges

E(Γ) = {e} = {C_{e}};  (1.4)

(3) and two vertices v and v′ are joined by edge e iff there exists the cor-

responding circle

C_{e} = ∂ṽ ∩ ∂ṽ′.  (1.5)

Actualy we can start with any 3-valent graph Γ with the set of edges E(Γ), then |E(Γ)| = 3g − 3, and with the set of vertices V (Γ), |V (|Γ)| = 2g − 2 where |S| is the cardinality of a finite set S.

Topologically it is equivalent to a 3-dimensional handlebody H_{Γ} with boundary ∂H_{Γ} = Σ_{Γ} where Σ_{Γ} is the Riemann surface given by the pumping up trick (see [T2]): we pump up every edge of Γ to a tube and every vertex to a trinion (= 2-sphere with 3 holes). By the construction our Riemann surface Σ_{Γ} has a trinions (or ”pair of pants”) decomposition given by removing all tubes.

For a visualization we can input our graph

i: Γ ↪ H_{Γ}  (1.6)

in the handlebody by the natural way. Then we can see that

(1) an orientation e⃗ of an edge e ∈ E(Γ) gives the orientation C⃗_{e} of the corresponding circle from the collection (1.1);

(2) sending any 1-cycle on Σ_{Γ} to the cycle on the handlebody H_{Γ} we get an epimorphism of the fundamental groups:

r: π_{1}(Σ_{Γ}) → π_{1}(Γ) = π_{1}(H_{Γ}) → 1  (1.7)

the kernel of which is the free group with g generators

ker r = F_{g}.  (1.8)

(3) According to the corresponding exact sequence

1 → ker r → π_{1}(Σ_{Γ}) → π_{1}(Γ) = π_{1}(H_{Γ}) → 1  (1.9)

we can choose standard generators of π_{1}(Σ_{Γ})

π_{1}(Σ_{Γ}) =< a_{1}, ..., a_{g}, b_{1}, ..., b_{g}| ∏_{i=1}^{g}[a_{i}, b_{i}] = 1 >

such that

ker r = F_{g} =< a_{1}, ..., a_{g} >  (1.10)

and

π_{1}(Γ) =< r(b_{1}), ..., r(b_{g}) >;

(4) obviously whole the collection of cycles {[C_{e}]} (1.1) lies in ker r and

moreover

ker r = F_{g} =< [C_{1}], ..., [C_{3g−3}] >;  (1.11)

Recall that a path of lenth 1 on Γ is just an oriented edge e⃗. Let the set P_{1}(Γ) = E⃗(Γ) be the set of 1-paths on Γ. Every such path e⃗ has vertices of two types

v_{s}(e⃗), v_{t}(e⃗) ∈ V (Γ)  (1.12)

- the source and the target - which are equal for loops.

A path of length d in Γ is an ordered sequence (e⃗_{1}, ..., e⃗_{d}) of oriented edges such that for every i

v_{t}(e⃗_{i}) = v_{s}(e⃗_{i+1}).  (1.13)

If e⃗_{d+1} = e⃗_{1}, then our path is a loop. A path (e⃗_{1}, ..., e⃗_{d}) (or a loop) is called irreducible if e_{i} ≠ e_{i+1} for every i (including i = d + 1 for a loop).

Every path (e⃗_{1}, ..., e⃗_{d}) ∈ P_{d}(Γ) defines two vertices

v_{s}((e⃗_{1}, ..., e⃗_{d})), v_{t}((e⃗_{1}, ..., e⃗_{d})) ∈ V (Γ)  (1.14)

- the source and the target and the coincidence of which shows that our parth is a loop.

Let

L_{d}(Γ) ⊂ P_{d}(Γ)  (1.15)

be the set of oriented irreducible loops of length d. For every vertex v ∈ V (Γ) we have the set of irreducible loops marked by v

L_{d}(Γ)_{v} = {l ∈ L_{d}(Γ)|v ⊂ l}.  (1.16)

The union

L_{∞}(Γ)_{v} = ∪_{d=1}^{∞}L_{d}(Γ)_{v}  (1.17)

admits a group structure

π_{1}^{C}(Γ)_{v} = L_{∞}(Γ)_{v}.  (1.18)

if we consider only irreducible fragments of the compositions.

Obviously, this group depends on the marking point v.

Let π_{1}(Γ) be the standard fundamental group of Γ (as a 1-complex). Then the natural epimorphism r: π_{1}^{C}(Γ) → π_{1}(Γ) is an isomorphism. Obviously, if Γ is a 3-valent graph of genus g then π_{1}(Γ) = F_{g} is a free group with g generators.

Now let us fix a point p ∈ Σ_{Γ} (suppose it coincides with i(v) (1.6)) and join all circles with p by any system of paths. Then we can consider the oriented circles C⃗_{e} as elements of the group ker r (1.9) that is as elements of the fundamental group π_{1}(Σ_{Γ}).

Sending every combinatorial loop

l = (e⃗_{1}, ..., e⃗_{d}) ∈ L_{∞}(Γ)_{v}

to the ordered product

Int(l) = C⃗_{e_{1}} · ... · C⃗_{e_{d}} ∈ ker r ⊂ π_{1}(Σ_{Γ})  (1.19)

we get a homomorphism

Int: π_{1}(Γ) = π_{1}(H_{Γ}) → π_{1}(Γ)  (1.20)

Recall that both of these groups are free groups with g generators and it is easy to see that Int is an isomorphism.

Our surface Σ_{Γ} doesn’t carry any ”natural” complex structure. Moreover after the fixing of the previous basis of the fundamental group (1.9)- (1.10) the space of complex structures on Σ_{Γ} turns to be the Teichmuller space τ_{g}. If we fix such a structure I ∈ τ_{g} then we can consider the space of classes of representations

A^{na} = CLRep(π_{1}(Σ), SL(2, ℂ))  (1.21)

as the full space A^{na}(Σ_{I}) of all holomorphic flat connections on topologicaly trivial vector bundle on Σ_{I}. Any representation ρ with class

[ρ] ∈ CLRep(π_{1}(Σ), SL(2, ℂ))  (1.22)

defines a holomorphic vector bundle just by the standard construction

E = U × ℂ^{2}/(π_{1}, ρ)  (1.23)

where U is the universal cover of Σ_{I} with the natural action of the fundamental group of the base.

Of course, in the set of such bundles there are non-stable but indecomposable vector bundles and even semi-stable decomposable ones. Let us remove the representations corresponding to non-stable bundles and get the space

CLRep^{I−ss}(π_{1}(Σ_{I}), SL(2, ℂ)) ⊂ CLRep(π_{1}(Σ), SL(2, ℂ))  (1.24)

of classes of representations which give stable, semi-stable and decomposable semi-stable bundles.

Obviously this Zariski dense (=containing a Zariski open set) subspace depends on the complex structure I ∈ τ_{g}. Moreover we have the holomorphic forgetful map

f: A_{I−ss}^{na} = CLRep^{I−ss}(π_{1}(Σ), SL(2, ℂ)) → M^{ss}(Σ_{I})  (1.25)

where the target space is the moduli space of semi-stable topologicaly trivial vector bundles. This is the affine bundle over the cotangent bundle of M^{ss}(Σ_{I}). Indeed, any fiber f^{−1}(E) is the affine space of holomorphic flat connections on E and the difference of any two connections is a traceless Higgs field

ϕ: E → E(K_{Σ_{I}}) ∈ T_{Σ_{I}}^{∗}M^{ss}(Σ_{I})  (1.26)

that is a covector to M^{ss}(Σ_{I}) at E.

Every affine bundle is given by a 1-cocycle of the vector bundle (see [T1]). In our case this is a cocycle

ε_{na} ∈ H^{1}(M^{ss}, Ω)  (1.27)

and

H^{1}(M^{ss}, Ω) = H^{1,1}(M^{ss}, ℂ)

by the Dolbault theorem. In our case this class is precisely the class of the polarization that is the class of theta divisor:

Proposition 1.1 The cohomology class (1.27) is given by the formula

ε_{na} = [Θ_{na}] = [ω_{na}]  (1.28)

where ω is the standard symplectic structure on the moduli space.

Corollary 1.1 The forgetful projection f (1.25) doesn’t admit any holomorphic section.

That is, the answer is precisely the same as in the abelian case.

Remark By the Narasimhan-Sesadri theorem there exists the non-holomorphic section

M^{ss}(Σ_{Γ}) = CLRep(π_{1}(Σ), SU(2)) ↪ CLRep(π_{1}(Σ), SL(2, ℂ)) (1.29)

Now let us perform the trinion decomposition corresponding to Σ_{Γ} (1.1)- (1.2). We can construct the space of holomorphic flat bundles CLRep(π_{1}(Σ), SL(2, ℂ)) gluing the spaces of SL(2, ℂ)-flat connections over the trinions {ṽ_{i}} (1.2).

For one trinion ṽ (which is 2-sphere with 3 holes) the space of SL(2, ℂ)-flat connections is given as the classes representations space

A^{na}(ṽ) = CLRep(π_{1}(ṽ), SL(2, ℂ))  (1.30)

where π_{1}(ṽ) = F_{2} is the free group with 2 generators.

Definition 1.1 For the free group F_{g} with g generators the classes representations space

S_{g} = CLRep(F_{g}, SL(2, ℂ))  (1.31)

is called the Schottki space of genus g.

So A^{na}(ṽ) = S_{2} is the Schottki space of genus 2.

If two vertices v_{i} and v_{j} are joined by edge e_{l} that is if the circle C_{l} corresponding to e_{l} is a boundary component of trinions ṽ_{i} and ṽ_{j}:

C_{l} = ∂ṽ_{i} ∩ ∂ṽ_{j}  (1.32)

then we can glue A^{na}(ṽ_{i}) and A^{na}(ṽ_{j}) along the boudary component C_{l} by just the same way as we do this for SU(2) representations (see for example [JW]): let Conj(SL(2, ℂ)) be the set of conjugation classes of elements of SL(2, ℂ). Then we have the natural map

conj: SL(2, ℂ) → Conj(SL(2, ℂ)).  (1.33)

Two representations ρ_{i}, [ρ_{i}] ∈ A^{na}(ṽ_{i}) and ρ_{j}, [ρ_{j}] ∈ A^{na}(ṽ_{j}) are glued iff

conj(ρ_{i}([C_{l}])) = conj(ρ_{j}([C_{l}])).  (1.34)

The ambiguity of such gluing is the stabilizer Z(ρ_{i}([C_{l}])) of the monodromy around this loop. For example for a semi-simple element m ∈ SL(2, ℂ) the stabilizer Z(m) = ℂ^{∗}. The result of such gluing is

A^{na}(ṽ_{i}) ∗ A^{na}(ṽ_{j}) = S_{3}  (1.35)

if C_{l} = ∂˜(v_{i}) ⋂ ∂˜(v_{j}). Gluing all trinions ṽ_{i} we get our surface and gluing all spaces of flat bundles A^{na}(ṽ) we get the space CLRep(π_{1}(Σ), SL(2, ℂ)) of flat bundles on Σ_{Γ}.

The exact sequence (1.9) defines the canonical embedding of the Schottki space (1.31)

i: S_{g} ↪ CLRep(π_{1}(Σ), SL(2, ℂ))  (1.36)

such a way that the image is in a sense a complete intersection. Namely, consider the space of representations Rep(π_{1}(Σ), SL(2, ℂ)) ( without the diagonal adjoint factorization), such that

CLRep(π_{1}(Σ), SL(2, ℂ)) = Rep(π_{1}(Σ), SL(2, ℂ))/P GL(2, ℂ)  (1.37)

where the last action is the natural diagonal adjoint action of SL(2, ℂ) on the space of representations. So, let

p: Rep(π_{1}(Σ), SL(2, ℂ)) → CLRep(π_{1}(Σ), SL(2, ℂ))  (1.38)

be the natural projection to the factor by this action.

Geometrically this means that we fixed a point p ∈ Σ_{I} and a trivialization E_{p} = ℂ^{2} of the fiber of vector bundle E over this point. The moduli space of vector bundles with such additional structure { \atop M^{ss}(Σ_{I})} admits the structure of principal P GL(2, ℂ)-bundle over M^{ss}(Σ_{I}):

ϕ: { \atop M^{ss}(Σ_{I})} → M^{ss}(Σ_{I})  (1.39)

where the group SL(2, ℂ) modulo ±1 acts on E_{p} = ℂ^{2} as (2 X 2)-matrices with determinant 1.

The moduli space of flat vector bundles with such additional structure is Rep(π_{1}(Σ), SL(2, ℂ)) and it admits the structure p (1.38) of a principal P GL(2, ℂ)-bundle over CLRep(π_{1}(Σ), SL(2, ℂ)).

Now every element γ ∈ π_{1}(Σ) defines regular functions

c_{ij}(γ): Rep(π_{1}(Σ), SL(2, ℂ)) → ℂ  (1.40)

by the formula

c_{ij}(γ)(ρ) = (ρ(γ))_{ij}

where ij is the matrix element of the corresponding matrix.

Then the half basis (a_{1}, ..., a_{g}) (1.10) defines 3g functions

c_{11}(a_{1}), c_{12}(a_{1}), c_{21}(a_{1}), ..., c_{11}(a_{g}), c_{12}(a_{g}), c_{21}(a_{g})  (1.41)

and a system of divisors

D_{ij}(a_{l}) = {c_{ij}(a_{l}) = 0, if i ≠ j, }  (1.42)

and

D_{11}(a_{l}) = {c_{11}(a_{l}) = 1}.

These divisors are not invariant with respect to the diagonal adjoint action of SL(2, ℂ). But the complete intersection

⋂_{ij,l} D_{ij}(a_{l}) = ϕ^{−1}(i(S_{g}))  (1.43)

is the preimage of the Schottki space (1.36).

Returning to the affine bundle (1.25) consider a bundle E ∈ M^{ss}(Σ_{I}) and the fiber

f^{−1}(E) = ℂ_{E}^{3g−3}  (1.44)

Such affine subspace of CLRep(π_{1}(Σ), SL(2, ℂ)) is called a packet of classes of representations. This is an affine space over the space of Higgs fields H^{0}(Σ_{I}, adE ⊗ T^{∗}Σ_{I}).

Now consider the restriction of the principal SL(2, ℂ)-bundle (1.38) to this affine space

p: p^{−1}(ℂ_{E}^{3g−3}) → ℂ_{E}^{3g−3}.  (1.45)

Obviously the space p^{−1}(ℂ_{E}^{3g−3}) is an affine algebraic variety.

Restricting the functions (1.40) to this affine variety we get the complete intersection

ϕ^{−1}(i(S_{g})) ∩ p^{−1}(ℂ_{E}^{3g−3})  (1.46)

and E admits a Schottki representation iff this complete intersection is non empty.

Non emptyness is an (Zariski) open condition thus to solve the Mumford-Narasimhan problem for general curve and general vector bundle we need to find one curve with this property.

That is our strategy is the same as C. Ciliberto, A. Lopez and R. Miranda in [CLM] for the computation of corank of the Gaussian-Wahl map for general curve. Moreover, the probe curves are the same.

2 3-valent graphs and ll-curves

Sending every vertex v ∈ V (Γ) to the complex Riemann sphere P_{v} = ℂℙ^{1} with a triple of points (p_{e_{1}}, p_{e_{2}}, p_{e_{3}}) corresponding to edges of the star S(v) of the vertex (= the set of edges incident to v) and gluing two components P_{v} and P_{v′} such that ∂e = v, v′ along the points p_{e} ∈ P_{v}, P_{v′} we obtain a reducible algebraic curve P_{Γ} with the following properties:

(1) arithmetical genus of the connected reducible curve P_{Γ_{g}} is equal to g;

(2) for any 3-valent graph Γ_{g} the curve P_{Γ_{g}} is Deligne - Mumford stable

(3) hence the curve P_{Γ_{g}} defines a point of the Deligne-Mumford compactification \overline{M_{g}} (with the same notation);

So in the Deligne-Mumford compactification \overline{M_{g}} of the moduli space M_{g} of smooth curves of genus g we get a finite configuration of points

P ⊂ \overline{M_{g}}  (2.1)

enumerated by the set of 3-valent graphs T G_{g} - the set of large limit curves.

In the algebraic geometry framework these curves were described by Artamkin in [A]:

(1) on every curve P_{Γ} the canonical class K_{Γ} is a line bundle for which the restriction to every component P_{v}, v ∈ V (Γ) is the sheaf of meromorphic differentials ω with simple poles at p_{e_{1}}, p_{e_{2}}, p_{e_{3}} where e_{1} ⋃ e_{2} ⋃ e_{3} = S(v):

K_{Γ}|_{P_{v}} = K_{P_{v}}(p_{e_{1}} + p_{e_{2}} + p_{e_{3}}) = O_{P_{Γ}}(1);  (2.2)

(2) thus every holomorphic section s of K_{Γ} is a collection of meromorphic differentials {ω_{v}} on the components P_{v} with poles at ⋃_{e∈E(Γ)} p_{e} with constraints: for every e such that ∂e = v, v′ one has

res_{p_{e}}ω_{v} + res_{p_{v′}}ω_{v′} = 0;  (2.3)

(3) beside of these equalities we have 2g-2 linear relations: for every v ∈ V (Γ) with (e_{1}, e_{2}, e_{3}) = S(v)

res_{p_{e_{1}}}ω_{v} + res_{p_{e_{2}}}ω_{v} + res_{p_{e_{3}}}ω_{v} = 0;  (2.4)

(4) from here we have the right number g for the dimension of holomorphic differential space H^{0}(P_{Γ}), O(K_{Γ}));

(5) but the properties of the canonical complete linear system |K_{Γ}| depend on the topology of Γ.

Definition 2.1 The minimal number of edges that may be removed to make the graph disconnected is called the thickness th(Γ) of the graph.

Obviously for a 3-valent graph Γ the number th(Γ) ≤ 3.

In [A] Artamkin proved the following

Proposition 2.1 The canonical linear system |K_{P_{Γ}}| is

(1) base points free iff th(Γ) ≥ 2;

(2) very ample iff th(Γ) ≥ 3.

The double canonical systems on ll-curves are much more regular:

(1) every holomorphic section s of O(2K_{Γ}) is a collection of meromorphic quadratic differentials {ω_{v}} on the components P_{v} with poles of degree 2 at ⋃_{e∈E(Γ)} p_{e}; (2) every such quadratic differential defines biresidue bires_{p_{e}} at every pole

(Res_{p_{e}}^{2} in notations of [T1]) and

(3) the constraints in this case are the following: for every e such that

∂e = v, v′

bires_{p_{e}}ω_{v} + bires_{p_{v′}}ω_{v′} = 0;  (2.5)

(4) thus the system of nodes {p_{e}} of our ll-curve P_{Γ} defines the decompo-

sition

H^{0}(P_{Γ}, O(2K_{Γ}))^{∗} = ℂ^{E(Γ)}  (2.6)

where for every quadratic differential ω the value of the linear form H_{e} is given by the formula

H_{e}(ω) = bires_{p_{e}}ω;  (2.7)

(5) thus for a 3-valent graph Γ_{g} the large limit curve P_{Γ} ∈ \overline{M_{g}} is a smooth point of \overline{M_{g}} as orbifold.

Recall that the fiber of the tangent bundle of the moduli space at a point P_{Γ} is given by the equality

T_{P_{Γ}} \overline{M_{g}} = H^{0}(P_{Γ}, O_{P_{Γ}}(2K_{P_{Γ}})  (2.8)

and the decomposition (2.6) is given by the following geometrical way: the double canonical map of P_{Γ} given by the complete linear system |2K_{Γ}|

ϕ_{2K_{Γ}} : P_{Γ} → ℙ^{3g−4}  (2.9)

has as the target space the projectivization of the tangent space of the moduli space \overline{M_{g}} at the point P_{Γ}.

Then the images of nodes

{ϕ_{2K_{Γ}}( ⋃_{e∈E(Γ)} p_{e})}  (2.10)

define the configuration of

3g − 3 = rkH^{0}(P_{Γ}, O(2K_{P_{Γ}}))

linear independent points ( since the components are coming to irreducible conics and every conic is defined by any triple of points on it). Thus this configuration of points gives a decomposition of the tangent space

T_{P_{Γ}} \overline{M_{g}} = ⨁_{e∈E(Γ)} ℂ_{e}  (2.11)

where ℙℂ_{e} = ϕ_{2K_{Γ}}(p_{e}) coincides with the decomposition (2.6).

Let e ⊂ Γ be an edge of a 3-valent graph Γ with two vertices v, v′ = ∂e and with two stars S(v) = e, e_{1}, e_{2} and S(v′) = e, e′_{1}, e′_{2}. From this we can get a graph Γ′_{e} of genus g − 1 by the following construction:

(1) remove e and get two 2-valent vertices v and v′ with stars S(v) = e_{1}, e_{2} and S(v′) = e′_{1}, e′_{2};

(2) consider the pair e_{1}, e_{2} as the first new edge e_{new} and e′_{1}, e′_{2} as the second new edge e′_{new};

(3) so, we get a new graph Γ′ with fixed pair of (a priori) disjoint edges (e_{new} ⋃ e′_{new}) ⊂ Γ′;

(4) obviously genus of Γ′ is equal to g − 1.

Coming to ll-curves we have the correspondence:

(p_{e} ∈ P_{Γ}) ⟷ (p_{e_{new}}, p_{e′_{new}} ⊂ P_{Γ′})  (2.12)

and vice versa.

Obviously this operation corresponds to the projection of the canonical curve P_{Γ} ⊂ ℙ^{g−1} from the node p_{e} to the canonical curve P_{Γ′} ⊂ ℙ^{g−2}.

Remark This projection is an analog of the Fano double projection for Fano threefolds.

So, we have the correspondence

C ⊂ P_{g} × P_{g−1}  (2.13)

with two projections:

p_{g}: C → P_{g}  (2.14)

with the fiber

p_{g}^{−1}(Γ) = E(Γ)

and

p_{g−1} : C → P_{g−1}  (2.15)

with the fiber

p_{g−1}^{−1}(Γ′) = S^{2}(E(Γ′)).

Now we blow down this edge in our graph. We get a new graph with 4-valent vertex v_{new} = v = v′ with the star S(v_{new}) = e_{1}, e_{2}, e′_{1}, e′_{2}. There are 3 partitions of this set to pairs: the old one (e_{1}, e_{2})|(e′_{1}, e′_{2}) and two new ones:

(e_{1}, e′_{2}|e_{2}, e′_{1}) and (e_{1}, e′_{1}|e_{2}, e′_{2}).  (2.16)

Now we can blow up the vertex v_{new} to the edge e_{new} with vertices ∂e_{new} = v_{new}, v′_{new} and stars

(1) S(v_{new}) = e_{new}, e_{1}, e_{2} and S(v′_{new}) = e_{new}, e′_{1}, e′_{2}. This is our starting graph Γ.

(2) S(v_{new}) = e′_{new}, e_{1}, e′_{2} and S(v′_{new}) = e′_{new}, e′_{1}, e_{2}. This is the first new graph Γ′.

(3) S(v_{new}) = e′′_{new}, e_{1}, e′_{1} and S(v′_{new}) = e′′_{new}, e′_{2}, e′_{2}. This is the second new graph Γ′′.

As a result of this construction every obtained graph has distinguished edge that is we have a triple of flags

(e ⊂ Γ), (e′_{new} ⊂ Γ′), (e′′_{new} ⊂ Γ′′).  (2.17)

Such triple we call a nest of flags.

The correspondence (2.14) gives a nest of graphs of genus g − 1

(e_{1}, e′_{1} ⊂ Γ_{1}), (e_{2}, e′_{2} ⊂ Γ_{2}), ((e_{3}, e′_{3} ⊂ Γ_{3}))  (2.18)

Every nest is defined uniquely by any flag (e ⊂)Γ or (e_{i}, e′_{i} ⊂ Γ_{i}) from the triple.

Moreover, let S(e) be the star of an edge e that is the union of stars of the boundary v, v′ = ∂e

S(e) = S(v) ⋃ S(v′).  (2.19)

Then we have the canonical identification of graphs

Γ − S(e) = Γ′ − S(e_{new}) = Γ′′ − S(e_{new}).  (2.20)

Remark It is easy to see that if e is a loop this construction gives the same graph with the same loop e again.

3 Special 1-parameter deformations of large limit curves

Now we are ready to construct 1-parameter family of reducible curves with 2g − 3 rational components.

For our flag e ⊂ Γ consider two componets P_{v}, P_{v′} of the curve P_{Γ} with common point p_{e} where v, v′ = ∂e. Remove this point and glue P_{v} and P_{v′} by a tube that is, consider the connected sum

P_{v}#_{p(e)}P_{v′} = P_{v,v′} = S^{2}.  (3.1)

This is a 2-sphere with two pairs of points (p_{e_{1}}, p_{e_{2}}) and (p_{e′_{1}}, p_{e′_{2}}) where (e, e_{1}, e_{2}) = S(v) and (e, e′_{1}, e′_{2}) = S(v′). If we fix a complex structure on S^{2} and consider the double cover

ϕ: E → ℙ^{1}  (3.2)

with ramification points

W = p_{e_{1}} ⋃ p_{e_{2}} ⋃ p_{e′_{1}} ⋃ p_{e′_{1}}  (3.3)

we obtain an elliptic curve E with a point of second order

σ = p_{e_{1}} + p_{e_{2}} − p_{e′_{1}} − p_{e′_{2}} ∈ P ic(E)_{2}.  (3.4)

So the moduli space of complex structures on S^{2} with quadruple of points divided in two pairs is equal to M_{1}^{2} - the moduli space of smooth elliptic curves with fixed point of order 2.

Every such complex structure τ ∈ M_{1}^{2} on S^{2} with quadruple of points divided in two pairs and the standard complex structures on all others components glued as before define a stable algebraic reducuble curve P_{e⊂Γ,τ}. Thus we obtain an embedding

ψ_{e⊂Γ} : M_{1}^{2} → \overline{M_{g}}.  (3.5)

The moduli space

M_{1}^{2} = ℙ^{1} − (σ, σ′, σ′′)  (3.6)

is the projective line without 3 points. These 3 points correspond to 3 possibility to divide 4 points p_{e_{1}}, p_{e_{2}}, p_{e′_{1}}, p_{e′_{2}} in 2 pairs that is a choice of a point of order 2 on an elliptic curve.

It is easy to see that Proposition 3.1 (1) The map ψ_{e⊂Γ} (3.5) can be extended to a map

ψ_{e⊂Γ} : ℙ^{1} → \overline{M_{g}};  (3.7)

(2)

ψ_{e⊂Γ}(σ) = p_{e} ∈ P_{Γ}  (3.8)

where σ is given by (3.4) and P_{Γ} is a large limit curve with fixed node corresponding to the edge e,

(3)

ψ_{e⊂Γ}(σ′) = P_{Γ′};  (3.9)

and

(4)

ψ_{e⊂Γ}(σ′′) = P_{Γ′′};  (3.10)

where the triple

(e ⊂ Γ), (e_{new} ⊂ Γ′), (e_{new} ⊂ Γ′′)  (3.11)

is a nest of flags (2.17).

(5) Now we can identify the tangent direction

T ψ_{e⊂Γ}(ℙ^{1})_{P_{Γ}} = ℂ_{e}  (3.12)

from the decomposition (2.11) with the corresponding node of the double canonical curve. We obtain the uniquely defined 2-canonical model.

This identification of images of nodes under the double canonical embedding and the directions of special deformations cancels the projective transformation ambiguity.

Remark If edge e is a loop with a vertex v such that the star S(v) = e, e′ then P_{v} is a rational curve with one double point p_{e} and the smooth point p_{e′}. The operation of connected summing around double point p_{e} gives a smooth 2-torus T^{2} with fixed point p_{e′} and the isotopy class a ∈ H_{1}(T^{2}, ℤ) which is the class of the neck of gluing tube. The class a mod 2 gives a point of order 2 on T^{2}. The space M_{1}^{2} of complex structures on T^{2} with such additional data is ℂ^{∗} − 1 that is ℙ^{1} without 3 points again. In this case the rational curve ψ_{e⊂Γ}(M_{1}^{2}) admits the compactification by the double point corresponding to P_{Γ}.

So, the Deligne-Mumford compactification \overline{M_{g}} contains the configuration C of rational curves

{C} = C, where every C = ψ_{e⊂Γ}(ℙ^{1})  (3.13)

for some flag e ⊂ Γ. For three flags from the same nest (2.17) the curve C is the same. We can consider this configuration of rational curves as a reducible curve

C = ⋃ C  (3.14)

that is the union of all components. It is easy to see that

Proposition 3.2 (1) for every component C of C

C ⋂ P_{g} = P_{Γ} + P_{Γ′} + P_{Γ′′}  (3.15)

where (Γ, Γ′, Γ′′) is a nest of graphs (2.17);

(2) for a pair C, C′ of componets either the intersection C ⋂ C′ is empty

or it is transversal and

C ⋂ C′ ∈ P  (3.16)

(3) the set S(Γ) of components through every point P_{Γ} ∈ P is enumerated by the set E(Γ);

Now let ET G_{g} be the set of 3-valent flags of genus g with the projection to the set of graphs

γ: ET G_{g} → T G_{g}, γ^{−1}(Γ) = E(Γ)  (3.17)

and Com(C) be the set of components of the reducible curve C. Then we have 3-cover

c: ET G_{g} → Com(C)  (3.18)

with remification along flags e ⊂ Γ, where e is a loop in Γ. Let LT G_{g} be the set of such loop flags. Then one has

3 · |Com(C)| − |LT G_{g}| = 3(g − 1) · |T G_{g}|.  (3.19)

Thus, roughly speaking, the cardinality of Com(C) is equal to (g − 1) times the cardinality of P_{g}.

We saw that if an edge e ∈ E(Γ) is not a loop then the map ψ_{e⊂Γ} (3.7) is an embedding and the corresponding component C of the reducible curve S ⊂ \overline{M_{g}} is a smooth rational curve with fixed 3 points (P_{Γ}, P_{Γ′}, P_{Γ′′}) such that the corresponding 3 graphs can be equipped with flags structures forming the nest (2.17). Over every of such points the tangent space admits decomposition (2.11).

Geometry of the other curves from the family of curves parametrized by C is very near to the geometry of large limit curves: let e ∈ E(Γ), v, v′ = ∂e and P_{E,σ} is the point of C corresponding to the elliptic curve (3.2) with a point of order 2 (3.4). Then P_{E,σ} has 2g-4 old components

⋃_{v′′≠v,v′} P_{v′′}  (3.20)

with triples of points and 3g-1 nodes p_{e′}, e ≠ e′ and one new component P_{v,v′} with the quadruple of points (see (3.1). Then

(1) the canonical class K_{P_{E,σ}} is a line bundle: a restriction of it to every component P_{v′′}, v′′ ≠ v, v′ is the sheaf of meromorphic differentials ω with simple poles at p_{e}, p_{e′}, p_{e′′} where {e, e′, e′′} = S(v);

(2) the restriction of the canonical class to the component P_{v,v′} is the sheaf of meromorphic differentials ω with simple poles at p_{e_{1}}, p_{e_{2}}, p_{e′_{1}}, p_{e′′_{2}} (see (3.1) and (3.3)).

(3) Thus

c(K_{P_{E,σ}}) = (2, 1, ..., 1) ∈ NS_{P_{E,σ}}  (3.21)

where the first coordinate corresponds to P_{v,v′}.

(4) Again every holomorphic section s of the canonical class is a collection of meromorphic differentials {ω_{v′′}} on the components P_{v′′} with poles at p_{e}, p_{e′}, p_{e′′} where {e, e′, e′′} = S(v) and a meromorphic differential ω_{v,v′} on the component P_{v,v′} with poles at the quadruple with the same constraints (2.3) and (2.4).

(5) The canonical map defined by the complete linear system |K_{P_{E,σ}}|

ϕ_{K}: P_{E,σ} → ℙ^{g−1}  (3.22)

sends P_{v′′} to a configuration of lines and P_{v,v′} to a conic in ℙ^{g−1}.

(6) Again the dimension of the space of quadratic differentials on P_{E,σ} is equal to 3g-3 and this curve is an orbifold smooth point of \overline{M_{g}}.

(7) The double canonical map of P_{E,σ} given by the complete linear system |2K_{Γ}| is an embedding

ϕ_{2K_{P_{E,σ}}} : P_{E,σ} → ℙ^{3g−4} = ℙT \overline{M_{g}}_{P_{E,σ}}.  (3.23)

(see (2.11)).

(8) The images of nodes

{ϕ_{2K_{P_{E,σ}}} (p_{e′})}, e ≠ e′ ∈ E(Γ)  (3.24)

define the decomposition of the restriction of the tangent bundle

T \overline{M_{g}}|_{C} = T C ⨁( ⨁_{e≠e′∈E(Γ)} L_{e′})  (3.25)

where the fiber of the line bundle L_{e′} over a point is the component of the decomposition (2.11).

(9) Thus every line bundle L_{e′} from the previous decomposition is the tau-

tological line bundle

L_{e′} = O_{C}(−1),  (3.26)

(10) hence the previous decomposition is

T \overline{M_{g}}|_{C} = O_{C}(2) ⨁(3g − 4)O_{C}(−1).  (3.27)

(11) The restriction to C of the canonical class of \overline{M_{g}} is

K\overline{M_{g}}|_{C} = O_{C}(3(g − 2)).  (3.28)

For a singular curve the moduli spaces of vector bundles are not compact. These moduli spaces admit the compactifications by torsion free sheaves. These sheaves are not local free only over nodes and the theory of the compactification is very close to the theory for algebraic surfaces. All details of this theory can be found in forthcoming Artamkin paper [A]. All constructions are very close to the complex gauge theory on smooth compact Riemann surfaces. We will see this in the following section where we apply the main constructions of the complex gauge theory from section 1 to ll-curves.

4 The complex gauge theory on ll-curves

A vector bundle E on P_{Γ} is called topologicaly trivial if the restrictions E|_{P_{v}} are trivial for all v ∈ V (Γ). We begin with the description of the moduli spaces of topologicaly trivial

(1) line bundles P ic_{0}(P_{Γ}),

(2) rk 2 semi-stable bundles M_{vb}^{ss}.

Since each component P_{v} is projective line the restriction of any topologicaly trivial line bundle equals

L|_{P_{v}} = O_{P_{v}}.  (4.1)

To describe a line bundle on P_{Γ} consider a collection of any trivializations of L on all components P_{v} of P_{Γ} and denote the line bundle with such addition structure by L_{0}.

Now to get a line bundle L on P_{Γ} we have to concord every pair of line bundles O_{P_{v}}, O_{P_{v′}} at common point p_{e} if v, v′ = ∂e. Under our trivializations such concordance is given by a multiplicative constant a(e⃗) ∈ ℂ^{∗} of oriented edge such that under the orientation reversing involution i_{e}

a(i_{e}(e⃗)) = a(e⃗)^{−1}.  (4.2)

Thus L_{0} defines a map

a: E⃗(Γ) → ℂ^{∗}.  (4.3)

subjecting (4.2).

The changing of trivialization is given by a function

λ̃: V (Γ) → ℂ^{∗}  (4.4)

which acts on the functions a by the formula

λ̃(a(e⃗)) = λ̃(v_{s}) · a(e⃗) · λ̃(v_{t})^{−1}  (4.5)

where ∂e⃗ = v_{s} ⋃ v_{t} and v_{s} is the source of arrow and v_{t} is the target.

Geometrically the construction of a concordance a(e⃗) at a point p_{e} for ll-curve plays the role of the period (monodromy) ρ([C_{e}]) of a flat connection [ρ] ∈ CLRep(π_{1}(Σ_{Γ}), ℂ^{∗}) along the cycle C⃗_{e} for non-singular curve Σ_{Γ} from the previous section.

Hence if A_{ℂ} is the space of trivialized topologicaly trivial line bundles that is the space of functions a (4.2) subjecting (4.3) and G_{ℂ} be the group of changing of trivializations (4.4). Then the moduli space of line bundles on P_{Γ}

P ic_{0}(P_{Γ}) = Hom(ker r, ℂ^{∗}) = (ℂ^{∗})^{g}  (4.6)

is the abelian Schottky space.

For rk 2 bundle E on P_{Γ} we have the same type description: consider a collection of trivializations of restrictions E|_{P_{v}} = O ⊕ O on all components of P_{Γ} and denote the bundle with such addition structure by E_{0}.

Now we have to glue every pair of bundles O_{P_{v}} ⊕ O_{P_{v}}, O_{P_{v′}} ⊕ O_{P_{v′}} at common point p_{e} if v, v′ = ∂e. Such gluing is given by a function

a: E⃗(Γ) → SL(2, ℂ)  (4.7)

subjection to the equation

a(i_{e}(e⃗)) = a(e⃗)^{−1}.  (4.8)

The changing of the trivialization is given by the function

g̃: V (Γ) → SL(2, ℂ)  (4.9)

which acts on the functions a by the formula

g̃(a(e⃗)) = ũ(v_{s}) · a(e⃗) · g̃(v_{t})^{−1}  (4.10)

where ∂e⃗ = v_{s} ⋃ v_{t}.

We would like to emphisize again that geometrically the construction of a concordance a(e⃗) at a point p_{e} for ll-curve plays the role of the period (monodromy) ρ([C_{e}]) of a flat connection [ρ] ∈ CLRep(π_{1}(Σ_{Γ}), SL(2, ℂ)) along the cycle C⃗_{e} for non-singular curve Σ_{Γ} from the previous section.

Again if A_{ℂ} is the space of trivialized topologicaly trivial rk 2 bundles with the group of trivializations G_{ℂ} acting by the formula (4.10) then the moduli space of semi-stable topologicaly trivial vector bundles on P_{Γ} is the quotient

M_{vb}^{ss}(P_{Γ}) = A_{ℂ}/G_{ℂ}.  (4.11)

Again it is easy to see that

M_{vb}^{ss}(P_{Γ}) = CLRep(ker r, SL(2, ℂ)) = S_{g}  (4.12)

(see (1.31)) is the Schottky space of genus g.

Now we would like to describe the space of holomorphic flat connections on vector bundles over this moduli space.

To do this let us draw the trinion decomposition corresponding to Σ_{Γ} (1.1)- (1.2) again. We can construct the space of holomorphic flat bundles gluing the spaces of SL(2, ℂ)-flat connections on the every component P_{v} − ⋃_{e∈S(v)} p_{e} that is on ℙ^{1} without 3 points.

Such spaces of SL(2, ℂ)-flat connections are given as the classes of representations spaces

A^{na}(P_{v} − ⋃_{e∈S(v)} p_{e}) = CLRep(π_{1}(P_{v} − ⋃_{e∈S(v)} p_{e}), SL(2, ℂ))  (4.13)

where π_{1}(ṽ) = F_{2} is the free group with 2 generators. Thus we can identify the spaces

A^{na}(P_{v} − ⋃_{e∈S(v)} p_{e}) = A^{na}(ṽ) = S_{2}  (4.14)

(see (1.30)) where the last one is the Schottki space of genus 2.

If two vertices v and v′ are joined by the edge e that is if the components P_{v} and P_{v′} intersect at p_{e} then we can glue A^{na}(P_{v} −⋃_{e∈S(v)} p_{e}) and A^{na}(P_{v′} − ⋃_{e∈S(v′)} p_{e}) along this intersection point p_{e} just in the same way as we do this for the smooth case (1.32)- (1.35). Again for a semi-simple element m ∈ SL(2, ℂ) the stabilizer Z(m) = ℂ^{∗}. The result of such gluing is

A^{na}(P_{v} − ⋃_{e∈S(v)} p_{e}) ∗ A^{na}(P_{v′} − ⋃_{e∈S(v′)} p_{e})) = S_{3}  (4.15)

if P_{v} ⋂ P_{v′} = p_{e}. Gluing all components we get our ll-curve P_{Γ} and gluing all spaces of flat bundles A^{na}(P_{v} − ⋃_{e∈S(v)} p_{e})) we get the same space CLRep(π_{1}(Σ_{Γ}), SL(2, ℂ)) of flat bundles on P_{Γ} as for the smooth case:

A^{na}(P_{Γ}) = CLRep(π_{1}(Σ_{Γ}), SL(2, ℂ)) = A^{na}(Σ_{Γ}).  (4.16)

The direct interpretation gives the following coincidence:

Proposition 4.1 (1) The forgetful map (1.25) for ll-curve P_{Γ}

f: A^{na}(P_{Γ}) = CLRep(π_{1}(Σ_{Γ}), SL(2, ℂ)) → M_{vb}^{ss}(P_{Γ}) =  (4.17)

= CLRep(ker r, SL(2, ℂ)) = S_{g}

is the natural map induced by the restriction of representations to the kernel ker r ⊂ π_{1}(Σ_{Γ}) (see (1.7)- (1.9)).

(2) This forgetful map admits a holomorphic section

s: CLRep(ker r, SL(2, ℂ)) = S_{g} ↪ CLRep(π_{1}(Σ_{Γ}), SL(2, ℂ))

(4.18)

induced by epimorphism r (1.7) and isomorphism Int (1.20).

(3) Thus the affine bundle (4.17) is the Higgs fields vector bundle (or the Hitchin bundle from [H]), which we may consider as the cotangent bundle to the singular variety M_{vb}^{ss}(P_{Γ}) = S_{g}.

(4) For every complex structure I ∈ τ_{g} (see the text between formula (1.20) and (1.21)) we have the rational map

F_{(Γ,I)} : M_{vb}^{ss}(P_{Γ}) = S_{g} → M^{ss}(Σ_{I})  (4.19)

which is the composition of section s (4.18) and forgetful map f (1.25).

Again consider the space of representations Rep(π_{1}(Σ), SL(2, ℂ)) ( without the diagonal adjoint factorization) (1.37) with the projection p (1.38). The geometrical meaning of this procedure is almost the same as before: we fixed a component P_{v} and a trivialization E|_{P_{v}} = ℂ^{2} of the vector bundle E over this component. The moduli space of vector bundles with such additional structure

{ \atop M_{vb}^{ss}(P_{Γ})} = (SL(2, ℂ))^{g}  (4.20)

is a non singular algebraic variety with the structure of principal P GL(2, ℂ)- bundle over S_{g}:

ϕ(SL(2, ℂ))^{g} → S_{g}  (4.21)

where the group SL(2, ℂ) modulo ±1 acts on E|_{P_{v}} = ℂ^{2} as (2 X 2)-matrices with determinant 1.

The moduli space of flat vector bundles on P_{Γ} with such additional structure is Rep(π_{1}(Σ), SL(2, ℂ)) with the structure p (1.38) of a principal P GL(2, ℂ)-bundle. The moduli space of flat bundles Rep(π_{1}(Σ), SL(2, ℂ)) can be described as the preimage of 1 for the algebraic map

com: (SL(2, ℂ))^{2g} = ∏ SL(2, ℂ)_{a_{i}} × ∏ SL(2, ℂ)_{b_{i}} → SL(2, ℂ) (4.22)

com(g(a_{1}), ..., g(a_{g}), g(b_{1}), ..., g(b_{g})) = ∏_{i=1}^{g}[g(a_{i}), g(b_{i})]

(see (1.9)):

com^{−1}(1) = Rep(π_{1}(Σ), SL(2, ℂ))

which is an algebraic variety obviously.

Again regular functions (1.40)-(1.41) defines the divisors (1.42) with the intersection (1.43).

But now the affine bundle (4.17) over M_{vb}^{ss}(P_{Γ}) is a vector bundle because it admits the holomorphic section (4.18). It has as the fiber over a point E ∈ M_{vb}^{ss}(P_{Γ}) the space

f^{−1}(E) = H^{0}(P_{Γ}, adE × K_{Γ}) = ℂ_{E}^{3g−3}  (4.23)

which is a packet of classes of representations of π_{1}(Σ_{Γ}) (see (1.2), (1.6)).

Again consider the restriction of the principal SL(2, ℂ)-bundle (1.38) to this affine space

p: p^{−1}(H^{0}(P_{Γ}, adE × K_{Γ})) → H^{0}(P_{Γ}, adE × K_{Γ}).  (4.24)

Restricting the functions (1.40) to this affine variety we get the complete intersection

i(S_{g}) ∩ (H^{0}(P_{Γ}, adE × K_{Γ})) = E  (4.25)

as a point of the moduli space M_{vb}^{ss}(P_{Γ}) = S_{g}. That is for every vector bundle E on P_{Γ}

(1) this complete intersection is non empty, and

(2) there exists unique class of the Schottki representation.

From this we get immideately our final result:

Theorem 4.1 On general curve general stable vector bundle admits the Schottki representation.

Remark Of course all our constructions are valid for vector bundle of any rank. For simplicity we worked with rk 2.

In this paper we uses only direct ”classical” arguments. But obviously these results can be refined by using the Higgs bundles technique for ll-curves. In particular we know (see [H]) that the Higgs pairs space T^{∗}M^{ss}(P_{Γ}) admits a partial compactification \overline{T^{∗}M^{ss}(P_{Γ})} completing the fibers of the holomorphic moment map. This partial compactification defines a compactification

\overline{S_{g}} ⊂ \overline{T^{∗}M^{ss}(P_{Γ})}.  (4.26)

of the moduli space of vector bundles on ll-curves by torsion free sheaves. Now we postpone this beautifull subject to the next paper of the serie refining [T3].

References

[A] I. V. Artamkin ”Algebraic geometry of ll-curves.” appears in Izv. RAN,ser.Math.

[CLM] C. Ciliberto, A. Lopez and R. Miranda ”Projective degenerations of K3 surfaces, Gaussian maps and Fano threefolds”, Invent. Math. v.114, 1993, p. 641- 667.

[HT] Hatcher A. and Thurston W. ”A presentation for the mapping class group of closed oriented surface”, Topology, 19 (3),1980,221-237.

[H] Hitchin N. ”Stable bundles and integral systems.” Duke Math. Jour. ,v.54, n.1, !987, 91-114.

[JW] L. C. Jeffrey and J. Weitsman, Bohr–Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula, Commun. Math. Phys. 150 (1992), 593–630

[T1] A.N. Tyurin, Periods and principal parts on a Riemann surface with a flat structure, Izv. Akad. Nauk SSSR, Ser. Mat., 41:6, (1977), 1425-1443, Math. USSR-Izv., 11, 1979.

A.N. Tyurin, On periods of quadratic differentials. Uspekhi Mat.Nauk, 33:6, (1978), 59-88; Russian Math. Surveys, 33:6 (1979), 169-221.

[T2] A. Tyurin, “Three mathematical facets of SU(2)-spin networks”, math.DG/0011035, 1-20.

Large limits... 26

[T3] A. Tyurin,”Quantization, Classical and Quantum Field Theory and Theta-Functions”, Les publication du CRM, a paraitre 2002.