BENOÎT CLAUDON Dedicated to Fréderic Campana

V Shokurov

ABELIANITY CONJECTURE FOR SPECIAL COMPACT KÄHLER THREEFOLDS

, where the notion of specialness was introduced. We briefly recall below the definition, basic properties, and the role of special manifolds in classification theory.

Introduction

We denote here by X an n-dimensional compact connected Kähler manifold. The motivations for the following shortest, but non-transparent, definition of specialness will be explained below. See [START_REF]Orbifolds, special varieties and classification theory[END_REF] where this notion was introduced for more details.

Definition 1.1. A compact Kähler manifold X is said to be special if, for any p > 0, any rank-one coherent subsheaf L ⊂ Ω p X and any positive integer N , the natural meromorphic map Φ N.L : X P(V * N,L ) has an image of dimension at most (p -1) 1 . Here V * N,L denotes the dual of the complex vector space of sections of Sym N (Ω p X ) which take values in L ⊗N ⊂ Sym N (Ω p X ) at the generic point of X. In particular, X has then no surjective meromorphic map f : X Y onto a manifold Y of general type and dimension p > 0, since otherwise L := f * (K Y ) ⊂ Ω p X would contradict the bound (p -1) above. On the non-algebraic side, if X has no map onto a positive dimensional-projective manifold, it is obviously special too. An equivalent geometric definition of specialness of X actually requires that X satisfies the more restrictive condition of having no meromorphic map onto an orbifold of general type (see 3.6 for the precise definition and the relevant notions concerning orbifolds).

Specialness is preserved by bimeromorphic maps and finite élale covers, this last assertion being surprisingly difficult to show. Special manifolds generalise in higher dimensions the rational and elliptic curves, which are obviously exactly the special curves. The next fundamental examples of special manifolds are indeed those which are either rationally connected or whith zero canonical dimension (i.e. with κ = 0, see example 3.2 below). However the class of special manifolds is much larger than the union of these two classes, since one shows by classification that a compact Kähler surface X is special if and only if κ(X) ≤ 1 and π 1 (X) is almost abelian. In particular, ruled elliptic surfaces are special and surfaces with κ(X) = 1 are special if and only if they do not map onto any hyperbolic curve after some finite étale cover. No such simple characterisation is true when n ≥ 3.

The central role of special manifolds in classification theory comes from the fact that, as shown in [START_REF]Orbifolds, special varieties and classification theory[END_REF]5.8], any compact Kähler manifold is canonically and functorially decomposed by its core fibration c X : X → C(X) into its special part (the fibres of c X ) and its general type part (the orbifold base (C(X), ∆(c X )), which is its usual base C(X) together with a ramification divisor ∆(c X ) on C(X) encoding the multiple fibres of c X ).

Special manifolds and general type orbifolds are thus the two antithetical building blocks from which arbitrary compact Kähler manifolds are built in one single step. In contrast to general type manifolds, for which no classification scheme seems to be known or even expected, special manifolds are conjectured to have many fundamental properties in common with rational and elliptic curves.

Conjecturally indeed, an orbifold version of the C n,m conjecture implies that any special manifold is canonically and functorially decomposed, by means of orbifold versions of the rational quotient and of the Iitaka-Moishezon fibration, as a tower of fibrations whose orbifold fibres have either κ = 0, or κ + = -∞ (a weak version of rational-connectedness, see [START_REF]Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes[END_REF]). Let us stress that the orbifold considerations are essential here (as in the LMMP, but for different reasons), and that apparently there is no possibility to work in the category of varieties without additional structure. This tower decomposition permits to lift (conditionally) to special manifolds2 properties which are expected to be common to manifolds which are either rationally connected or with κ = 0 and naturally leads to the following conjectures:

Conjecture 1.1 ( [START_REF]Orbifolds, special varieties and classification theory[END_REF][START_REF]Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes[END_REF]). 1. Abelianity Conjecture: A special compact Kähler manifold has an almost abelian fundamental group (i.e: π 1 (X) has an abelian subgroup of finite index). 2. A compact Kähler manifold (resp. a projective manifold defined over a number field) has an identically vanishing Kobayashi pseudometric (resp. is potentially dense) if and only if it is special (this last statement is inspired by Lang's conjectures).

For example, rationally connected manifold are simply connected [START_REF] Campana | Connexité rationnelle des variétés de Fano[END_REF] and compact Kähler manifolds X with c 1 (X) = 0 have an almost abelian fundamental group by [START_REF] Shing | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF] and [START_REF] Beauville | Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF].

In this article we prove the Abelianity conjecture above for threefolds (as said above, this is known for surfaces by classification, see Proposition 3.1).

Theorem 1.1. Let X be a compact Kähler threefold. If X is special, its fundamental group is almost abelian. This immediately implies, among several other things, a precise solution of Shafarevich's conjecture in this case (and for all special manifolds if the Abelianity Conjecture holds):

Corollary 1.1. If X is a special compact Kähler threefold with universal cover X, let X ′ be any finite étale cover of X with abelian torsionfree fundamental group and Albanese variety Alb(X ′ ) of dimension d := q(X ′ ).Then X = X ′ × Alb(X ′ ) Alb(X ′ ) is holomorphically convex, the universal cover Alb(X ′ ) of Alb(X ′ ) being Stein, since isomorphic to C d .

Reduction to the 2-dimensional orbifold case

We shall prove the theorem in this section only in the cases where no orbifold structure is needed, that is except when either X is projective and κ(X) = 2, or when a(X) = 2. The treatment of these two residual cases needs the consideration of two-dimensional projective special orbifolds and will be the subject of the subsequent sections.

Because of the lack of a Minimal Model program in the Kähler non-projective case, we need to treat it differently from the projective case. So assume first that X is projective. We work according to the value of κ(X) ≤ 2. κ(X) = -∞ : By Miyaoka's theorem, X is uniruled. Let r X : X → R(X) be its rational quotient (also known as its MRC fibration). Then R(X) is special too, with dim(R(X)) ≤ 2, and π 1 (X) ≃ π 1 (R(X)), since the fibres of r X are rationally connected. Since π 1 (R(X)) is almost abelian, so is π 1 (X).

κ(X) = 0 : If c 1 (X) = 0, the theorem is true, by [START_REF] Shing | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF] and [START_REF] Beauville | Variétés Kähleriennes dont la première classe de Chern est nulle[END_REF]. One reduces to this case by the Minimal Model Program and [START_REF] Namikawa | Global smoothing of Calabi-Yau threefolds[END_REF] (see [START_REF] Kollár | Shafarevich maps and automorphic forms[END_REF](4.17.3)]). The details are as follows: there exists a terminal model X ′ birational to X such that K X ′ is torsion, hence trivial after finite étale (in codimension 1) cover. Then X ′ has only cDV singularities. It is thus smoothable in the projective category by [START_REF] Namikawa | Global smoothing of Calabi-Yau threefolds[END_REF]. The conclusion follows, since π 1 (X) drops by specialisation.

κ(X) = 1 : Let J X : X → Y be the Moishezon-Iitaka fibration. Because its base is a curve, there exists a finite étale cover of X (still written X) such that J X has no multiple fibre3 . Then π 1 (X) is an extension of π 1 (Y ) (Y being a rational or elliptic curve, since special) by a quotient of π 1 (X y ), X y being the generic smooth fibre of J X , which is a surface with κ = 0. Because these two groups are almost abelian, π 1 (X) is a priori only polycyclic. The following result however implies the conclusion. Theorem 2.1 ( [START_REF]Ensembles de Green-Lazarsfeld et quotients résolubles des groupes de Kähler[END_REF]). Let f : X → Y be a fibration without multiple fibres in codimension 1 on Y from a compact Kähler manifold onto a manifold Y . Assume that Y and the generic fibre X y of f both have an almost abelian fundamental group. Then X also has an almost abelian fundamental group.

The proof of this theorem rests on two deep results of Hodge theory: Deligne's strictness theorem for morphisms of MHS, and Hain's functorial MHS on the Malčev completion of π 1 (X) when X is compact Kähler. κ(X) = 2 : When X is projective, this is the only remaining case. Notice that when κ(X) = 1, we could remove the multiple fibres of J X by making a suitable finite étale cover of X because Y was a curve. When κ(X) = 2 this is no longer possible in general. This is the reason why the notion of orbifold base, which virtually removes the multiple fibres, is introduced in this case, and why the geometry of such orbifolds needs to be developed. But once this is done, and the corresponding properties established, the proof is entirely parallel.

We now deal with the case when X is not projective. We work this time according to the algebraic dimension a(X) ∈ {0, 1, 2} of X. a(X) = 0 : The irregularity of X can take the values q(X) = 0, 2 or 3 (since q(X) = 1 would imply a(X) ≥ 1). The assumption a(X) = q(X) = 0 leads to the finiteness of π 1 (X) according to [START_REF]Fundamental group and positivity of cotangent bundles of compact Kähler manifolds[END_REF]Cor. 5.7]. Recall that the Albanese map is connected and surjective when a(X) = 0. Thus, If q(X) = 3, then X is birational to its Albanese variety, and π 1 (X) is Abelian. When q(X) = 2, the Albanese fibration α X : X -→ Alb(X) has no multiple fibre in codimension one by [START_REF]Orbifolds, special varieties and classification theory[END_REF]Prop. 5.3]. Moreover, the general fibre of α X is elliptic or rational, by [START_REF] Ueno | Classification theory of algebraic varieties and compact complex spaces[END_REF]Th. 13.8]. From Theorem 2.1, we conclude that π 1 (X) is almost abelian. a(X) = 1 : The algebraic reduction a X : X → A(X) is then a fibration onto a curve A(X) with general fibre special [Cam04b, Th. 2.39]; the fundamental groups of the base and of the fibre are then almost abelian (see Theorem 3.1). As in the projective case with κ(X) = 1, we may assume that there are no multiple fibres. Then a final application of Theorem 2.1 implies the result.

We are thus left with the following two cases:

(1) X is projective and κ(X) = 2, (2) X is a compact Kähler threefold with a(X) = 2. Let us briefly explain how the conclusion is obtained there. See next sections for the notions which are being used. In both cases we have a fibration (which may be assumed to be neat after suitable modifications of X and S) f : X → S on a smooth projective surface S, with generic fibres elliptic curves. This defines a smooth (hence klt) orbifold base (S, ∆ f ). Because X is special, so is (S, ∆ f ). As above, for such a fibration, π 1 (X) is now an extension of π 1 (S, ∆ f ) by π 1 (X s ). We show below:

Theorem 2.2. The fundamental group of a special geometric orbifold (S, ∆ S ) of dimension 2 is almost abelian of even rank at most 4.

The conclusion now follows from the orbifold version of theorem 2.1 above.

Let us stress that our proof of theorem 2.2 depends in an essential way on the fact that integral klt pairs are locally uniformisable by smooth germs of surfaces, because we are using the existence of Ricci-flat orbifold metrics when c 1 (S, ∆) = 0. This is the main reason why we cannot extend the above theorem 2.2 to any higher dimension. Otherwise, using the Abundance Conjecture, it seems that one could derive the Abelianity Conjecture in the projective case for klt orbifolds in any dimension using inductively on the dimension essentially the same arguments as the ones presented below.

3. Some basic facts on orbifolds 3.1. Notion of orbifold. The orbifolds considered here are spaces with local smooth uniformisations under the action of some finite groups. To deal with fixed point in codimension one, the structure is enriched with a Q-divisor of a specific form. The definition below is taken from [START_REF] Ghigi | Kähler-Einstein metrics on orbifolds and Einstein metrics on spheres[END_REF].

Definition 3.1. An orbifold is a pair (X, ∆) where X is a normal variety and ∆ a Q-Weil divisor of the following form:

∆ = i∈I (1 - 1 m i )∆ i ,
where the m i ≥ 2 are integers, and the pair (X, ∆) is locally uniformizable in the following sense: X is covered by the domains U of finite maps ϕ : U -→ X satisfying the following properties:

(i) ϕ(U ) is open in X, (ii) ϕ : U -→ ϕ(U ) is a Galois cover whose branching divisor is exactly ∆ |ϕ(U) .
The support of ∆ is then:

|∆| = i ∆ i .
Terminological remark.

The orbifolds we consider are compatible with all situations in which this term is used: they are particular cases of the ones in [START_REF]Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes[END_REF] and the special manifolds defined below are actually the classical special ones in ibid. They are smooth Deligne-Mumford stacks too and also klt pairs of the LMMP. Because they are locally smoothly uniformised, we can attach to them fundamental groups and differentialgeometric notions such as metrics and differential forms.

Definition 3.2. The canonical divisor of such a pair is the (Weil) Q-divisor K X + ∆.

Example 3.1. Let X be a smooth variety, and |∆| = i ∆ i be a normal crossing divisor; the choice of multiplicities m i ≥ 2 on each component of the divisor defines a canonical orbifold structure on (X, ∆). Since |∆| is locally given by the equation

z 1 • • • z k = 0 in suitable coordinates (z 1 , . . . , z n ), the map (z 1 , . . . , z n ) → (z m1 1 , . . . , z m k k , z k+1 , .
. . , z n ) gives a local uniformization. These orbifolds are said to be smooth and integral in [START_REF]Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes[END_REF].

In particular, an orbifold curve is simply a smooth curve with a finite set of points marked with integral multiplicities at least 2.

Since we consider only integral and finite multiplicities here, we define: Definition 3.3. A geometric orbifold is an orbifold (X, ∆) with X smooth and |∆| of normal crossings.

Orbifold base of a fibration and special manifolds.

Definition 3.4 (Def. 1.2, [START_REF]Orbifolds, special varieties and classification theory[END_REF]). Let f 0 : X 0 -→ Y 0 be a fibration (surjective morphism with connected fibres) between compact complex manifolds. A neat model of f consists in a commutative diagram:

X f u / / X 0 f0 Y v / / Y 0
where 1. X and Y are smooth, 2. u and v are bimeromorphic morphisms, 3. the locus of singular fibres of f is a normal crossing divisor of Y , 4. every f -exceptional divisor is u-exceptional too.

Such models actually exist birationally for any fibration; this can be proven using Hironaka desingularization and Raynaud flattening's theorems [Cam04b, Lemma 1.3]. Notice that any fibration is neat when Y is a curve. Given a neat fibration f : X -→ Y , we can naturally associate a Q-divisor ∆ * (f ) on the base Y of f . This divisor will be supported on the singular locus of f and will thus be normal crossings: the pair (Y, ∆ * (f )) will be a geometric orbifold.

The construction goes as follow (see also [START_REF]Orbifolds, special varieties and classification theory[END_REF]1.1.4]): if ∆ i is any component of the singular locus of f , its pull-back can be written

f * (∆ i ) = j m j D j + R
where R is f -exceptional and D j is mapped surjectively onto ∆ i . The multiplicity of f along ∆ i is defined by: m i = m(f, ∆ i ) = gcd j (m j ).

Definition 3.5. The pair (Y, ∆ * (f )), where:

∆ * (f ) = i (1 -1 mi )∆ i
, is called the orbifold base of the fibration f 0 (notations as above). A fibration f 0 is said to be of general type if the canonical divisor of the orbifold base

(Y, ∆ * (f )) is big: κ(Y, K Y + ∆ * (f )) = dim(Y ) > 0.
Definition 3.6 ([Cam04b], Def. 2.1). A compact Kähler manifold X is said to be (classically) special if it does not admit any fibration of general type.

Example 3.2. The main examples of special manifolds are given by the following classes [Cam04b, 3.22, 5.1, 2.39]: ⋆ rationally connected manifolds, ⋆ compact Kähler manifolds X with κ(X) = 0. This is a consequence of the additivity of canonical dimensions in general type fibrations ([Cam04b, Th. 4.2], an orbifold version of Viehweg's theorem). ⋆ The special curves are thus just the rational or elliptic ones. ⋆ compact Kähler manifolds of algebraic dimension zero and more generally fibres of algebraic reductions.

Conjecturally in an orbifold version of Iitaka's C n,m -conjecture, special manifolds can be reconstructed as a tower (in a suitable sense) of fibrations with fibres belonging to the classes above. This reduces Abelianity Conjecture 1.1 to the case of orbifolds with either κ = 0 or κ + = -∞ (see [START_REF]Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes[END_REF]13.10]).

In dimension 2 only, we still have a simple topological characterisation of specialness:

Proposition 3.1 (Prop. 3.32 ,[Cam04b]). A compact Kähler surface X is special if and only if: κ(X) ≤ 1 and if π 1 (X) is almost abelian.
We shall need also the notion of orbifold base of a fibration f : X → Y also when X is equipped with an orbifold divisor ∆ X , at least when Y is a curve and X is a surface (for the general case, see the definition in [Cam07, Def. 4.2]). Definition 3.7. Let (X, ∆ X ) be a geometric orbifold of dimension 2 and f : X -→ C a fibration onto a curve. Let us define the multiplicity of a point y ∈ C (relatively to f and ∆ X ) by the following formula: m y (f, ∆ X ) := gcd i (m i .mult ∆X (F i )) , where f * (y) = i m i F i . The orbifold base is then the pair (C, ∆ * (f, ∆ X )) where:

∆ * (f, ∆ X ) = y (1 - 1 m y (f, ∆ X ) ). {y} We say that f is of general type if: 2g(C) -2 + deg (∆ * (f, ∆ X )) > 0.
The 2-dimensional orbifold (X, ∆ X ) is then said to be special if (X, ∆ X ) does not admit any general type fibration onto a curve and if κ(X, K X + ∆ X ) < 2.

Fundamental groups and fibrations

4.1. An orbifold exact sequence. In this paragraph, we define fundamental groups of orbifolds and study the morphisms induced at the level of fundamental groups by classical orbifold morphisms.

Definition 4.1. Let (X, ∆) be an orbifold; its fundamental group π 1 (X, ∆) is the quotient of the group π 1 (X * \ |∆|) by the normal subgroup generated by the loops γ mj j

where γ j is a small loop around the component ∆ j of multiplicity m j (where X * denote the smooth locus of X).

This definition is derived from the local models. If X = C n /G where G is a finite subgroup of GL n (C) and ∆ is the branching divisor of the projection π : C n -→ X, we recover the group G as an orbifold fundamental group 4 :

π 1 (X, ∆) ≃ G.
Example 4.1. The case of orbifold curves is quite classical. The structure of the fundamental group of such an orbifold curve (C, ∆ = (m 1 , . . . , m n )) (we just keep in mind the deformation invariant, that is: the multiplicities, not the marked points) is determined by the sign of its canonical bundle:

deg (K C + ∆) = 2g(C) -2 + n j=1 (1 - 1 m j ).
When this quantity is positive (resp. zero, resp. negative), the fundamental group of (C, ∆) is commensurable to the fundamental group of an hyperbolic curve (resp. commensurable to Z 2 , resp. finite and explicitely known).

When dealing with fibrations having multiple fibres, we need to consider orbifold fundamental groups. A fibration f : X -→ Y with general fibre X y gives rise to a natural sequence of fundamental groups:

π 1 (X y ) i * -→ π 1 (X) f * -→ π 1 (Y ) -→ 1.
Although f * is surjective since f has connected fibres, this sequence is in general not exact in the middle. Multiple fibres are responsible for this failure, remedied by the orbifold fundamental group.

We will need the case where X has an orbifold structure and so a slightly more general version. For the definition (of a neat fibration in the orbifold setting), we refer to [Cam07, Def. 4.8].

Proposition 4.1 ([Cam07], Cor. 12.10). Let f : (X, ∆ X ) -→ Y be a neat fibration. If X y denote a general fibre of f and ∆ y the restriction of the orbifold structure to X y , the sequence

π 1 (X y , ∆ y ) -→ π 1 (X, ∆ X ) f * -→ π 1 (Y, ∆ * (f, ∆ X )) -→ 1 is exact.
Remark 4.1. We omit the detailled definition of neatness required for the preceding statement because in the sequel we will only use it in the two following quite simple situations:

(a) when ∆ X is empty, this neatness assumption has been already encountered (cf. Definition 3.4). The content of the previous proposition is then that π 1 (X) sits then in the middle of a short exact sequence (with the fundamental group of the orbifold base on the right hand). (b) when Y is a curve: the fibration f : (X, ∆ X ) -→ Y is always neat.

Nilpotency class in fibrations.

To complete this study of the behaviour of fundamental groups in fibrations, we prove that nilpotency conditions on the fundamental groups are preserved in fibrations between Kähler orbifolds. This remarkable fact (obviously false even for submersions between complex manifolds as the Iwasawa manifold shows) is a consequence of deep results in Mixed Hodge Theory 5 . 4 The orbifold divisor determines the structure of the smooth Deligne-Mumford stack X associated with (X, ∆) and called the root stack by Abramovich-Vistoli. 5 The key ingredients are: existence of a mhs on the Malčev completion of the fundamental group of a compact Kähler manifold and the strictness of morphisms of mhs.

Theorem 4.1 (Cor. 7.6, [START_REF]Quotients résolubles ou nilpotents des groupes de Kähler orbifolde[END_REF]). Let f : (X, ∆ X ) -→ (Y, ∆ Y ) be a neat fibration (see Remark 4.1) between smooth compact Kähler orbifolds. If the groups π 1 (X y , ∆ Xy ) and π 1 (Y, ∆ Y ) are almost abelian 6 , then π 1 (X, ∆ X ) is almost abelian as well.

Proof: It is a reduction to the case when ∆ X = 0. Since f is assumed to be neat, the sequence

π 1 (X y , ∆ Xy ) -→ π 1 (X, ∆ X ) f * -→ π 1 (Y, ∆ Y ) -→ 1 is exact. Since π 1 (X y , ∆ Xy ) and π 1 (Y, ∆ Y ) are almost abelian groups (of finite type), G = π 1 (X, ∆ X )
is then an almost polycyclic group; in particular, the group G is linear. A famous result of Selberg asserts that G has then a finite index subgroup G ′ ≤ G which is torsion free. Consider X ′ the orbifold cover of (X, ∆ X ) associated with G ′ ; the latter being torsion free, X ′ is a normal variety with no orbifold structure (i.e. ∆ ′ = 0). Since X has only quotient singularities, its fundamental group is isomorphic to the one of X, a desingularisation of X ′ [Kol93, Th. 7.5]. To conclude, we consider (a neat model of) the Stein factorization of

X -→ X ′ -→ X -→ Y.
Indeed, taking further blow-up of X, we can complete the picture:

X / / f ' ' X ′ / / X f / / Y Ỹ g ? ? Ð Ð Ð Ð Ð Ð Ð Ð
with f a neat fibration and g generically finite. It is then easy to see that we have an exact subsequence (with vertical maps having finite index images):

π 1 ( Xy ) / / _ π 1 ( X) f * / / _ π 1 ( Ỹ , ∆ * ( f )) / / _ 1 π 1 (X y , ∆ Xy ) / / π 1 (X, ∆ X ) f * / / π 1 (Y, ∆ Y ) / / 1
So we are reduced to the same situation with ∆ X = 0 and we can apply [Cam10, Cor. 7.6].

Fundamental groups of orbifolds with non-positive canonical bundle.

As for of compact Kähler manifolds, when the first Chern class is zero or positive, the now usual differential-geometric methods can then be applied to construct orbifold Kähler metrics with Ricci-curvature of the corresponding sign (using the local smooth uniformisations).

Theorem 4.2. Let X = (X, ∆) be an orbifold with X compact Kähler. If the first Chern class of X is non negative, the group π 1 (X ) is almost abelian; more precisely:

⋆ if c 1 (K X + ∆) = 0, then π 1 (X, ∆) is almost abelian of even rank 7 bounded by 2dim(X). ⋆ if c 1 (K X + ∆) > 0, then π 1 (X, ∆) is finite.
6 More generally, if these two groups are torsionfree nilpotent of nilpotency class at most ν, then π 1 (X, ∆ X ) is also nilpotent of nilpotency class at most ν. 7 The rank of an almost abelian group G of finite type is the maximum rank of an abelian subgroup of finite index of G.

To begin with, let us recall some basic facts on differential calculus on orbifolds. The smooth functions (differential forms, hermitian metrics,. . . ) on an orbifold (X, ∆) are the smooth functions on X * \ |∆| which can be smoothly extended (as the usual objects, after taking inverse images) in local uniformizations. For instance, if (X, ∆) is a geometric orbifold, a Kähler metric has the following form in coordinates charts adapted to ∆:

ω ∆ = ω eucl + n j=1 i∂∂ |z j | 2/mj = ω eucl + n j=1 idz j ∧ dz j m 2 j |z j | 2(1-1/mj ) .
Local uniformizations can also be used to compute integrals of forms of maximal degree. If ϕ : U -→ ϕ(U ) ⊂ X is such a local cover and α an orbifold top-form on X, define

ϕ(U) α = 1 deg(ϕ) U ϕ * α.
This local computation can then easily be globalized using partitions of unity.

The canonical divisor has been defined in 3.2 as a Q-divisor. It should be noted here that (X being compact and the uniformizations being finite) some integral multiple of this divisor defines a line bundle on X and it can be used to compute the first Chern class of X. As a byproduct, the Ricci form of any orbifold volume form is a (1,1)-orbifold form whose cohomology class coincides with c 1 (X ).

As in the manifold case, every invariant form whose cohomology class coincides with c 1 (X ) is the Ricci curvature of an orbifold Kähler metric. This fact has been already noticed several times in the literature (see for instance [START_REF] Joyce | Compact manifolds with special holonomy[END_REF][START_REF]Orbifoldes à première classe de Chern nulle[END_REF] when c 1 (X ) = 0, and [DK01] when c 1 (X ) > 0, and the references therein). For sake of completeness, let us recall the statement.

Theorem 4.3 (Calabi-Yau). Let X be a Kähler orbifold whose underlying space is compact and let us fix ω 0 an orbifold Kähler metric. For any representative ( i.e. smooth invariant (1, 1) form) α of c 1 (X ), there exists a unique orbifold Kähler metric ω in the Kähler class [ω 0 ] such that: Ricci(ω) = α.

Proof: We reproduce the arguments given in [START_REF]Orbifoldes à première classe de Chern nulle[END_REF]Th. 4.1]. This is a simple adaptation of the proof exposed in [START_REF] Tong | Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics[END_REF]. Since Ricci(ω 0 ) and α define the same class, the orbifold ∂∂-lemma provides us a smooth (orbifold) function f such that Ricci(ω 0 ) = α+i∂∂f . We normalize f (by addition of a suitable constant) in such a way that

X (e f -1)ω n 0 = 0.
The problem is then reduced to solving the Monge-Ampère equation

(MA) ω 0 + i∂∂ϕ n = e f ω n 0 .
To do so, we applied the continuity method as in [Siu87, p. 85-113] and consider the set T of t ∈ [0, 1] for which the equation (MA) t ω 0 + i∂∂ϕ t n = C t e tf ω n 0 . has a solution where

C t = X ω n 0 X e tf ω n 0 .
When t = 0, ϕ 0 = 0 is an obvious solution for (MA) 0 . The set T is open by linearization of the problem and implicit functions theorem (applied in local uniformizations). To see that T is closed we need a priori estimates on the solution of (MA) t which are established using two types of arguments: maximum principle and Nash-Moser iteration (integral inequalities). These can be used in the orbifold setting as well: maximum principle is applied in local uniformizations and integral inequalities are also valid in this formalism (see above). We proceed as in the manifold case to deduce Theorem 4.2 from Theorem 4.3. The positivity assumptions can be translated in the following way: in the Fano case (c 1 (X ) > 0) there exists a positive (1, 1) form α representing c 1 (X ); in the Ricci flat case (c 1 (X ) = 0) we choose α = 0. We can now apply Theorem 4.3: X has an orbifold Kähler metric ω with prescribed Ricci curvature: Ricci(ω) = α.

In the Fano case, by compactness of X, we get: Ricci(ω) ≥ ǫω, where ǫ > 0. The following orbifold version of Myer's theorem applied to the orbifold universal cover 8 of X implies the finiteness of the orbifold fundamental group: [START_REF] Borzellino | Orbifolds of maximal diameter[END_REF]). Let (Y, g) be a complete Riemannian orbifold (of dimension n). If the Ricci curvature satisfies the inequality Ricci(g) ≥ (n -1)k for some k > 0, the underlying space Y of Y is then compact with diameter bounded above:

Theorem 4.4 ([
diam(Y ) ≤ π √ k .
When c 1 (X ) = 0, the following orbifold splitting theorem 4.5 applied also to the orbifold universal cover of X also implies the claim.

Theorem 4.5 ([BZ94]

). Let (Y, g) be a compact Riemannian orbifold (of dimension n). If the Ricci curvature is everywhere non negative, the orbifold universal cover Y admits a (metric) splitting Y ≃ N × R m where N is a compact orbifold. The orbifold fundamental group of Y is also an extension of a crystallographic group by a finite group and is in particular almost abelian.

From the arguments given in [Cam04a, 5.4, 6.3] we deduce that the rank of the almost group π 1 (X, ∆) is even and bouded by 2dim(X).

Minimal Model Program for klt pairs in dimension 2

5.1. klt pairs as orbifolds in dimension 2. In this short paragraph, we gather from [KM98] some well known facts on the MMP for log pairs in dimension 2. Starting with a pair (X, ∆) where X is a smooth surface and ∆ = j b j ∆ j a Q-Weil effective divisor, we can perform a sequence of divisorial contractions which ends with a birational model of the initial pair and whose geometry is simplest, according to the sign of the canonical bundle. In this process, however, (K X + ∆)negative curves are being contracted, and the resulting surface is no longer smooth in general. The relevant preserved category of singularities is then described as follows.

Definition 5.1. A pair (X, ∆) where X is a Q-factorial normal variety is said to have only klt singularities if (i) ∀ j, 0 < b j < 1 (ii) For any (or equivalently, one) log-resolution f : Y -→ X of (X, ∆), in the following equality, we have a i > -1, for all i ′ s:

K Y + ∆ = f * (K X + ∆) + i a i E i ,
∆ being the strict transform of ∆, and E i the exceptional divisors of f .

A smooth (integral) orbifold is of course a klt pair. These singularities are preserved in the LMMP: Theorem 5.1 (Th. 3.47, [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]). Let (X, ∆) be a klt surface. There exists a birational morphism f : X -→ S such that the resulting pair (S, ∆ S = f * (∆)) is still klt and satisfies (exactly) one of the following properties :

(1) K S + ∆ S is nef, (2) S admits a fibration π : S -→ C onto a (smooth) curve C, the general fibre of π being a smooth K S + ∆ S -negative rational curve, (3) (S, ∆ S ) is Del Pezzo : ρ(S) = 1 and -(K S + ∆ S ) is ample.

When ∆ = 0, it is well know that klt singularities coincide with quotient singularities in dimension 2 [KM98, Prop. 4.18]. This property still holds for ∆ = 0 with integral multiplicities (i.e. coefficients b j of the form 1 -1 m ). Theorem 5.2. Let (X, ∆) be a pair where X is a surface and ∆ has integral multiplicities. The following conditions are equivalent near each point of X:

(1) (X, ∆) is klt, (2) (X, ∆) has a finite local fundamental group, (3) (X, ∆) is locally presented as a quotient:

π : C 2 -→ C 2 /G ≃ X,
where G is a finite group acting linearly on C 2 and ∆ is the unique Q-divisor on X such that π * (K

X + ∆) = K C 2 .
This statement is found and used in several places [START_REF] Kobayashi | Uniformization of complex surfaces, Kähler metric and moduli spaces[END_REF][START_REF] Megyesi | Generalisation of the Bogomolov-Miyaoka-Yau inequality to singular surfaces[END_REF], but never with a complete accessible proof: the appendix of [START_REF] Kobayashi | Uniformization of complex surfaces, Kähler metric and moduli spaces[END_REF] consists in a list of the possible cases, and refers to the thesis [START_REF] Nakamura | Classification and uniformisation of log-canonical singularities in the presence of branch loci[END_REF] for the proof. For this reason, we give a proof of Theorem 5.2 at the end of the present text (cf. Appendix).

Fundamental groups and Mori contractions.

We now relate the fundamental groups of a geometric orbifold (X, ∆) of dimension 2 and of its minimal model (S, D = ∆ S = f * (∆)) as in theorem 5.1. Proposition 5.1. Let (X, ∆) be a geometric orbifold of dimension 2 and f : (X, ∆) -→ (S, D) its minimal model. There is a natural surjective morphism of groups:

f ♯ : π 1 (S, D) -→ π 1 (X, ∆). In particular, if π 1 (S, D) is almost abelian so is π 1 (X, ∆).
Proof: Let us call E the union of the curves contracted by f ; f being an isomorphism away from E we have natural maps, S * being the smooth locus of S:

S * \ |D| f -1 -→ X\ (|∆| ∪ E) ֒→ X\ |∆|
(note that |∆| and E may have common components). At the level of fundamental groups, we get natural morphisms:

π 1 (S * \ |D|) ∼ -→ π 1 (X\(|∆| ∪ E)) ։ π 1 (X\ |∆|) ։ π 1 (X, ∆).
To get the morphism f ♯ we need only to remark that the loops around the components of D are sent onto loops around corresponding components in ∆ with the same multiplicities, since f is an isomorphism between S * and X\E.

Example 5.1. In general, the kernel of this morphism can however be very big as shown by the following example.

Let C be an elliptic curve, and let X be the blow-up of S = C ×C at a point (c, c). Let F ⊂ S the fibre of the first projection through this point and let G be its strict transform in X. If m > 1 is an integer, the minimal model of (X, ∆ = (1

-1 m )G) is (S, D = (1 -1 m )F
) and the exceptional divisor E of the blow-up is the only (K X + ∆)-negative curve on X. Since the orbifold base of the first projection is (C, (1-1 m ) {c}), the fundamental group of (S, D) has (a finite index subgroup which have) a surjective morphism onto a non-abelian free group. On the other hand the orbifold base of g : (X, ∆) → C, g being the composition of the first projection and the blow-up, is merely C since the fibre over c has E as component and inherits from it the multiplicity 1. It is then easy to see that (X, ∆) is special and that its fundamental group is isomorphic to the fundamental group of X and thus abelian.

Remark 5.1. The direction of the arrow f ♯ in proposition 5.1 does not look to be functorial. So we explain its construction differently. The map f : (X, ∆) → (S, D) = (X, ∆) min is in general not a 'classical' orbifold morphism (in the sense of [START_REF]Orbifoldes spéciales et classifications biméromorphes des variétés kählériennes compactes[END_REF]) and thus does not induce a functorial morphism of groups. The map f induces an orbifold morphism, and thus a morphism of groups, only when the multiplicities on the exceptional divisors of f are sufficiently divisible (for example by the order of the local fundamental group of (S, D) at the point under consideration). In our example 5.1 above, equipping E with a multiplicity divisible by m is the right condition.

Let thus ∆ + be an orbifold divisor on X such that f * (∆ + ) = f * (∆) = D, and such that f : (X, ∆ + ) → (S, ∆) and id X : (X, ∆ + ) → (X, ∆) are orbifold morphisms (this means for id X that the multiplicity of any component of ∆ divides the corresponding multiplicity in ∆ + ).

We then get two functorial group morphisms: f ♯ + : π 1 (X, ∆ + ) → π 1 (S, D), which is an isomorphism, and id ♯ X : π 1 (X, ∆ + ) → π 1 (X, ∆), which is surjective (as above). Our initial f ♯ was nothing but (id X ) ♯ • (f ♯ + ) -1 . 5.3. Abelianity for special klt pairs and proof of the main theorem. To conclude this section, we shall use the preceding proposition to prove Abelianity Conjecture for geometric orbifolds of dimension 2.

Proof of Theorem 2.2: Let (S, D) the minimal model of the pair (X, ∆ X ). We argue according to the values of κ = κ(X, ∆) = κ(S, D) ∈ {-∞, 0, 1}.

If κ = 1, then X admits a fibration f onto a smooth curve C whose general fiber F satisfies F • (K X + ∆) = 0; the general orbifold fibre (F, ∆ F ) is then orbifoldelliptic and its fundamental group is almost abelian (example 4.1). Adding to C the orbifold divisor ∆ * = ∆ * (f, ∆ X ) of classical multiplicities, we get an exact sequence

π 1 (F, ∆ F ) -→ π 1 (X, ∆ X ) -→ π 1 (C, ∆ * ) -→ 1.
The orbifold curve (C, ∆ * ) is special since so is (X, ∆) and Theorem 4.1 shows that π 1 (X, ∆) is almost abelian.

When (S, D) admits a structure of Mori fibre space over a (special) curve (in particular κ = -∞), we still have a fibration on X whose fibres are special (in fact rational) and we can proceed as in the lines above.

There are two cases left:

(1) κ(S, D) = 0, (2) or (S, D) is log Del Pezzo. In the first case, K S + D being nef it is semi-ample (log abundance for surfaces, see [START_REF] Fong | Log abundance for surfaces, Flips and Abundance for Algebraic Threefolds[END_REF]) and then torsion. In particular, the orbifold (S, D) is Ricci-flat. In the second case, (S, D) is Fano. In both cases, we can apply Theorem 4.2 to conclude that the fundamental group of (S, D) is almost abelian. Finally, Proposition 5.1 can be applied (it is only here we need it) to show that π 1 (X, ∆ X ) is almost abelian as well.

Appendix A. Classification of klt singularities of surfaces

In this appendix we give a proof of the classification of klt singularities for integral pairs (Theorem 5.2). The method is first to treat the singular case (using the method of [KM98, Th. 4.7]) and then to reduce the smooth case to the preceding one using orbifold étale covers. Let then (X, ∆), ∆ = j (1 -1 mj )D j be a germ of klt pair.

A.1. The singular case. To begin with we quote a useful (negativity) lemma on connected quadratic forms.

Lemma A.1 (Cor. 4.2, [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]). Let E := ∪ j E j be a connected exceptional curve on a smooth complex surface. Let A := j a j E j and B := j b j E j , with a j , b j ∈ R. Assume that A • E j ≥ B • E j for any j. Then either A = B or a j < b j for any j.

We first consider the case in which the germ X is singular.

Lemma A.2. Assume that X is singular, and that ∆ = 0. Let f : X ′ → X be the minimal resolution of the germ X,

∆ ′ = k (1-1 m k )D ′ k being the strict transform of ∆ in X ′ and E = ∪ j E j the exceptional divisor. The extended dual graph of f * (∆)
is then one of the following ones below, in which the (non-compact) components of ∆ ′ are indicated by black dots. Moreover, all intersections are transversal and all white dots are smooth rational curves. These are (-2)-curves, except possibly in the first case (see remark A.1 below for additional constraints in the first case).

v f f f v • • • v f f f f • • • v f f f f f f • • • Figure 1. Dual graphs when ∆ = 0
Proof: We assume the knowledge of the classification of Duval (i.e. canonical) and of klt germs of 2-dimensional singularities, and more precisely the fact that the dual graphs of their minimal resolutions are given either by Dynkin diagrams of (-2)-curves of type A n , D n , E k (for k = 6, 7, 8) or by Hirzebruch-Jung chains with transversal intersections ([KM98, Th. 4.7], the method of proof we shall now adapt). From negativity, rational numbers a i are uniquely determined by the numerical equalities:

(⋆) ∀j, d j := (K X ′ + ∆ ′ ) • E j = A • E j with A := i a i E i . Write also: e i := -E 2
i . Assume first that there are (at least) two components D 1 and D 2 in ∆ ′ , of multiplicities m 1 , m 2 . Consider a shortest chain E 1 , . . . , E n of components of E joining them, such that D 1 (resp. D 2 ) meets E 1 (resp. E n ), but no other E ℓ . An easy computation gives:

d j =    (e 1 -2) + (1 -1 m1 )(D 1 • E 1 ) if j = 1, (e n -2) + (1 -1 mn )(D 2 • E n ) if j = n, e j -2 otherwise.
Consider now the curve B := -β( n 1 E ℓ ), for some β > 0 to be chosen latter. We assume first that n ≥ 2. The intersection numbers are then given by:

B • E j =
β(e j -1), j = 1, n, β(e j -2), 1 < j < n.

We choose now the value of β according to the self-intersection of the E j :

(1) if e i = 2, for some i = 1, n, we set

β = inf{1, d j e j -1 , j = 1, n} = inf{1, 1 + (1 -1 m1 )(E 1 • D 1 ) -1 e 1 -1 , 1 + (1 -1 m2 )(E n • D 2 ) -1 e n -1 } (2) and choose β = inf{ dj ej -1 , j = 1, n} if e i = 2, ∀i = 1, n. We thus have A • E j ≥ B • E j
for any j, and equality for some j. From Lemma A.1 we get A = B and a j = -β > -1 since (X, ∆) is klt. Using (⋆) it easily implies

(1 - 1 m 1 )(D 1 • E 1 ) < 1 and (1 - 1 m 2 )(D 2 • E n ) < 1. Because (1 -1 m ) ≥ 1 2 we get D 1 • E 1 = D 2 • E n = 1
and we are thus in the first case. Assume now that n = 1 and notice that then D i meets E 1 for i = 1, 2. We then have

(e 1 -2) + (1 - 1 m 1 )(D 1 • E 1 ) + (1 - 1 m 2 )(D 2 • E 1 ) = βe 2 1
for some β < 1 (by the klt condition). The solutions are easily determined (i) either

E 1 • D 1 = E 1 • D 2 = 1 and m 1 , m 2 arbitrary, (ii) or (up to permutation) E 1 • D 1 = m 1 = 2 and E 1 • D 2 = 1
, m 2 arbitrary. This last case (which is not normal crossings) is excluded by two blow-ups which make the total transform of ∆ a normal crossings divisor. We indeed get the following extended dual graph (in which the E ′ i , i = 1, 2, 3 appear as white circles in the order E ′ 1 , E ′ 3 , E ′ 2 , while the left (resp. upper) black dot is the strict transform of D 1 (resp. D 2 ):

v f f f v Figure 2. A non-klt case.
We also have: e ′ 1 = e 1 + 2, e ′ 3 = 1 and e ′ 2 = 2. A direct resolution of the linear system (⋆) shows that this pair is not klt.

Assume next that our dual graph contains a fork (of white vertices). We know that K X ′ ≡ 0 in this case (e i = 2, ∀i). Let D 1 , E 1 , . . . , E n-2 , E n-1 , E n be a shortest subgraph connecting the component D 1 to the fork E n-2 , E n-1 , E n , with end points the last two vertex. The sequence is so labelled that D 1 meets E 1 only, and that E 1 , . . . , E n-2 is a chain. Consider the curve

B := -(1 - 1 m 1 )(D 1 • E 1 ) n-2 ℓ=1 E ℓ + 1 2 (E n-1 + E n ) .
A computation similar to the preceding one (but simpler since now K X ′ ≡ 0) shows that A • E j = B • E j , ∀j. Thus A = B and we are in the second case. The only case left is no fork and only one component in ∆ ′ . This is the last diagram (with transversal intersection by the klt and integral conditions, as above) since the degenerate case where n = 2 in the preceding case would lead to the existence of a fork by connectedness of E.

Remark A.1. In the first case, the computation gives additionally: (e 1 -1)m 1 = (e n -1)m 2 and also e i = 2, ∀i = 1, n. The arguments of [BHPVdV04, III (5.1)] show that the singularity of X is cyclic of type A N,q with N = (n -1)(e 1 -1)(e n -1) + (e 1 -1) + (e r -1) and q = (n -1)(e 1 -1) + 1 with N q = (e n -1) + e1-1

(n-1)(e1-1)+1 . Lemma A.3. Let (X, ∆) be a two-dimensional germ of pair as above. This germ is klt if and only if has a finite local fundamental group. [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]Prop. 4.18]. Otherwise, if the singularity is of type A n , an easy computation shows (as in [Lam86, Ch. IV, §13-14]) that the local fundamental group has order nm 1 m 2 . If the singularity is of type D n , we reduce it to the preceding case by a double cover.

Proof: If ∆ = 0, it is [
To deal with the smooth case, we need preliminary observations (A.4, A.1 and A.6 below) on orbifold étale covers.

A.2. Orbifold étale covers. This first lemma ensure that orbifold étale covers of klt pairs will remain klt.

Lemma A.4 (Prop. 5.20 [START_REF] Kollár | Birational geometry of algebraic varieties[END_REF]). Let g : (X ′ , ∆ ′ ) → (X, ∆) be a finite map between normal germs of varieties. Assume that K X ′ + ∆ ′ and K X + ∆ are Q-Cartier 9 , and that K X ′ + ∆ ′ = g * (K X + ∆). Then K X ′ + ∆ ′ is klt if and only if K X + ∆ is klt.

We come back in dimension 2 with integral pairs. Definition A.1. We say that a finite map g : (X ′ , ∆ ′ ) → (X, ∆) of degree d is orbifold étale if (i) the map g ramifies only above the support of

∆ = k (1 -1 m k )D k , (ii) g has order of ramification r k dividing m k along D k , (iii) ∆ ′ = g * ( k (1 -r k m k )D k ). Remark A.2.
In this preceding case the local fundamental group of (X ′ , ∆ ′ ) has index d in the local fundamental group of (X, ∆). Moreover from the ramification formula we get:

K ′ X + ∆ ′ = g * (K X + r k=1 (1 - 1 r k )D k ) + r k=1 (1 - r k m k )g * ( D k r k ) = g * (K X + ∆) since (1 -1 r k ) + (1 -r k m k ) 1 r k = 1 -1 m k .
It also follows that (X, ∆) is klt if and only if so is (X ′ , ∆ ′ ). We shall apply this remark only when r 1 = m 1 , and r k = 1 for r ≥ 2. In this case ∆ ′ = g * ( k=r k=2 (1 -1 m k ).D k ) and the component D 1 'disappears'. 9 In dimension 2 the Q-Cartier assumption is superfluous.

Example A.1. We now always assume that X is a smooth germ. 1. Assume that D 1 is a cusp of equation x p = y q (p, q coprime) and of multiplicity m 1 which we write symbolically as ∆ 1 = (p, q; m 1 ). Then X ′ is the singularity of equation z m1 = y q -x p . Thus if (X, ∆) is klt we have 1 p + 1 q + 1 m1 > 1. And so (p, q, m 1 ) is up to order either (2, 2, m) or (2, 3, m), m = 3, 4, 5. Moreover if the support of ∆ is reducible (if r ≥ 2) then X ′ must be an A n singularity and the support of ∆ ′ must have at most two components. Thus (p, q; m 1 ) is, up to order, (2, 2; m) and r ≤ 3. If D 1 is not a priori assumed to be a cusp, but just be given by a parametrisation x(t) = t q , y(t) = t p with p, q coprime, then since the cover of degree m 1 ramified exactly over D 1 will be a klt singularity, and so will have the usual normal form in suitable coordinates. We thus see a posteriori that D 1 was indeed a cusp in suitable coordinates.

2. Assume that the support of ∆ has two smooth components D 1 , D 2 tangent at order p ≥ 2 at the origin. They have (in this order), in suitable coordinates, equations: y = 0 and y = x p . Consider the map: g : C 2 → X given by: g(u, v) = (x, y) := (u, v m1 ). Thus using the above remark A.2, we see that making this (orbifold étale) cover g we are lead to the case (again klt) where

∆ ′ = g * ( k=r k=2 (1 - 1 m k )D k ), with D ′ 2 having equation v m1 = u p .
p = m 1 = d or p = d = m 1 or m 1 = d = p.
A.3. The smooth case. We consider next the remaining case in which the germ X is smooth. Write ∆ = r k=1 (1 -1 m k )D k . For k = 1 . . . r, let t k ≥ 1 be the multiplicity of the germ D k at the origin. Thus D k is smooth if and only if t k = 1. We say that an irreducible germ of curve in X ≃ C 2 has a (p, q)-cusp at the origin if its equation in suitable coordinates is y q -x p = 0 (with p and q coprime); in this case the multiplicity is inf(p, q). Lemma A.5. If (X, ∆) is klt, the only possibilities for the data r, t k , m k are the following ones.

(1) ∀k,

t k = 1. Then r ≤ 3. If r = 3 then 1 m1 + 1 m2 + 1 m3 > 1 ( i.e. ei- ther (m 1 , m 2 , m 3 ) = (2, 2, m 3 ) with m 3 ≥ 2 arbitrary or (m 1 , m 2 , m 3 ) = (2, 3, m 3 ) with 2 ≤ m 3 ≤ 5).
(2) t k = 2 for some k. Then either r = 1 and D k has a (2, q)-cusp at the origin with 1 2 + 1 q + 1 m k > 1; or r = 2 (and assume k = 1). There are two subcases: (a) D 1 has a (2, q)-cusp with m 1 = 2 and D 2 is smooth and has intersection multiplicity 2 with D 1 ; m 2 and q (odd) are arbitrary. (b) D 1 has a (2, 3)-cusp with m 1 = 2, D 2 (smooth) has intersection multiplicity 3 with D 1 and m 2 = 2. (3) t k = 3 for some k. Then r = 1 and D 1 is a (3, 5)-cusp or (3, 4)-cusp with m 1 = 2.

Proof: Let us make a blow-up f : X 1 → X at the origin with exceptional divisor E 1 and denote K

1 := K X1 and K = K X . Let ∆ 1 be the strict transform of ∆ in X 1 . Then K 1 + ∆ 1 = f * (K + ∆) + cE 1 where c = 1 -k t k (1 -1 m k ). The klt condition implies c > -1 that is k t k (1 -1 m k ) < 2. Since (1 -1 m k ) ≥ 1, this implies k t k < 4.
Thus r ≤ 3 and we get the following list of possible values: 1. if t k ≥ 3 for some k then r = 1 and t 1 = 3. 2. if t k = 2 for some k then r = 1, 2 and (t 1 , t 2 ) = (2, 1) if r = 2.

3. if ∀k, t k = 1 then r = 1, 2 or 3. We now examine the distribution of possible multiplicities.

If t 1 ≥ 2 and r = 1 the étale orbifold cover (see A.1 above) of degree m 1 ramified along the (p, q)-cusp D 1 leads to the singularity z m1 = y q -x p with the zero orbifold divisor, which is klt if and only if the claimed inequality 1 p + 1 q + 1 m1 > 1 holds. If t 1 = 2, 3, we thus get the cases described in cases 2 and 3 of lemma A.5. We are left with the case t 1 = 2, t 2 = 1, r = 2. Thus D 1 is a (2, q)-cusp of multiplicity m 1 and D 2 smooth of multiplicity m 2 with q odd. We distinguish two cases according to whether the intersection multiplicity of D 1 and D 2 is 2 or more.

In the first case, the orbifold étale cover of (X, ∆) ramified to order m 2 along D 2 leads to the orbifold divisor ∆ ′ supported on the locus of equation x 2 = y m2.q and with multiplicity m 1 . From (X ′ , ∆ ′ ) still being klt we derive 1 2 + 1 qm2 + 1 m1 > 1 and it yields m 1 = 2 since qm 2 ≥ 6.

If the intersection multiplicity of D 1 and D 2 is 3 or more, the orbifold étale cover of (X, ∆) ramified to order m 2 along D 2 leads to the orbifold divisor ∆ ′ supported on the locus of equation x 2m2 = y q , and with multiplicity m 1 (still klt). If q and m 2 are coprime we get the inequality 1 q + 1 2m2 + 1 m1 > 1 whose the only possible solution is m 1 = m 2 = 2 and q = 3.

We now show that q and m 2 are coprime, which will complete the proof. Otherwise let d > 1 be their gcd. Then ∆ ′ consists of d components of multiplicity m 1 and equations x t = εy s with t = q d and s = 2m2 d . These components need to be smooth since d > 1. Thus either t = 1 or s = 1. Since q is odd it is not divisible by 2m 2 and so q = d divides m 2 . Because ∆ ′ is supported by d = q ≤ 3 smooth components, we finally get d = q = 3 components having pairwise tangency of order 2m2 q ≥ 2 at the origin. This contradicts lemma A.6 below. Lemma A.6. Let (X, ∆) be a two-dimensional smooth germ, the support of ∆ consisting of s ≥ 2 smooth germs having two-by-two contact order t ≥ 1 at the origin and each of multiplicity m k (k = 1, . . . , r). Then (X, ∆) is klt if and only if

k (1 - 1 m k ) < 1 + 1 t .
The solutions (t, r, m 1 ≤ m 2 ≤ • • • ≤ m r ) are: ⋆ t = 1, r = 2 and (m 1 , m 2 ), ⋆ t = 1, r = 3 and (2, 2, m 3 ) or (2, 3, 3 ≤ m 3 ≤ 5), ⋆ t = 2, r = 2 and (2, m 2 ), (3, m 2 ≤ 5), ⋆ t = 3, r = 2 and (2, m 2 ≤ 5), ⋆ t ≥ 4, r = 2 and (2, 2). Proof: We get a log-resolution of this pair in performing t successive suitable point blow-ups with successive exceptional divisors E 1 , . . . , E t . If b : X ′ → X is the resulting composition one checks that in this process

K X ′ = b * (K X ) + t h=1 hE h while b * (∆) = ∆ ′ + ( k (1 - 1 m k ))( t h=1 hE h )
where ∆ ′ is the strict transform of ∆. We thus see that the initial pair is klt if and only if the coefficient of E t in K X ′ + ∆ ′ -b * (K X + ∆) is strictly greater than -1, i.e. if

k (1 - 1 m k ) < 1 + 1 t .
The solutions are then easily determined. (2, q) m 2 r

(2, 3) (1 -1 m k )D k , the D k being smooth at the origin. We know that r ≤ 3. When r = 1 any m 1 leads to the klt situation. When r = 2 the possible situations are described in the lemma A.6. We thus assume that r = 3 and that not all three components are normal crossings.

2 2 i 1 mi > 1 1 m + 1 p + 1 q > 1 1 m + 1 n + 1 rp > 1 q odd
We shall then show that (after reordering) D 1 and D 2 are tangent to order t ≥ 2 while D 3 is transversal to them and that 1 m1 + 1 m2 + 1 tm3 > 1. So that one has either m 1 = m 2 = 2, m 3 and t arbitrary or 2 = t = m 1 = m 3 and m 2 = 3.

Assume indeed that D 1 and D 2 have contact order t ≥ 2. We first remark that D 3 is transversal to them. Assume not, then after possible reordering of the components, we may assume that the order of contact of D 3 with D 1 and D 2 is at least t. In this situation, the proof of the above lemma A.6 still applies to show that 3 k=1 (1 -1 m k ) < 1 + 1 t ≤ 1 + 1 2 , which is impossible since m k ≥ 2, ∀k. The conclusion now simply follows from considering the orbifold étale cover ramified to order m 3 along D 3 , which replaces ∆ by ∆ ′ , supported by two smooth components of multiplicities m 1 and m 2 and having contact order t ′ = tm 3 . This implies by lemma A.6 that 1 m1 + 1 m2 + 1 tm3 > 1 since the inequality k (1 -1 m k ) < 1 + 1 t ′ is equivalent to k 1 m k + 1 t ′ > 1 when r = 2. The solutions are the classical ones. A.4. The local fundamental groups. Proposition A.2. Let (X, ∆) be a two-dimensional germ of integral pair. If this germ is klt it has a finite local fundamental group.

Proof: If X is singular this is lemma A.3. When X is smooth, this is a direct consequence of the construction of suitable orbifold étale covers as explained above.

Proposition A. 1 .

 1 The following list exhaust the classification of integral pairs which are klt with a smooth ambient surface. We use the notation of[START_REF] Uludağ | Orbifolds and their uniformization, Arithmetic and geometry around hypergeometric functions[END_REF].

Figure 3 .

 3 Figure 3. Germs of klt pairs with smooth base (and conditions on the coefficients)

And even, more naturally, to special orbifolds.

Or Y ∼ = P 1 and J X has one or two multiple fibres. This case is easily treated similarly.


Modris Tenisons
email: modris.tenisons@google.com

PhD, Senior Dainis Zeps

ORNAMENTAL SIGN LANGUAGE IN THE FIRST ORDER TRACERY BELTS

Keywords: binary coding, binary matrices, ornaments, asymmetry, sign coding, sieve in ornamental pattern, first order complexity, hemiolia principle

We consider ornamental sign language of first order where principles of sieve displacement, of asymmetric building blocks as base of ornament symmetry, color exchangeability and side equivalence principles work. The generic aspects of sieve and genesis of ornamental pattern and ornament sign in it are discussed. The hemiolia principle for ornamental genesis is introduced. The discoverer of most of these principles were artist Modris Tenisons [4, 5, 6, 7 (refs. 23, 24), 8 (ref. 65)]. Here we apply systematical research using simplest mathematical arguments.

We come to conclusions that mathematical argument in arising ornament is of much more significance than simply symmetries in it as in image. We are after to inquire how ornament arises from global aspects intertwinned with these local. We raise argument of sign's origin from code rather from image, and its eventual impact on research of ornamental patterns, and on research of human prehension of sign and its connection with consciousness.

Introduction

Ornaments and ornamental patterns are part of both historical and cultural richness of different nations. What message they convey? Can such question be justified scientifically? Either, has such question legible meaning what concerns exact sciences? In order to answer such and similar questions we must study them. In this article we study mathematically one type of ornament pattern found, e.g., in ornamental belts of Baltic coutries: for that reason we introduce notion first order complexity ornamental sign language.

Latvian national ornamental tracery belts, or, simpler, ornamental belts are sufficiently rich area to study [START_REF] Tenisons | Ornaments and Symmetry: Language of Signs of Ornamental Tracery (interview with Modris Tenisons)[END_REF][START_REF] Vytautas | Research into the Baltic sashes symbol system: from the scientific interpetation to the authentic traditional folk typology. Baltiškų Juostų simbolių sistemos paieškos: Nuo mokslinės intepretacijos iki autentiškos tradicinės liaudiškos tipologijos[END_REF][START_REF] Vytautas | Ornament of Lithuanian traditional pick-up woven sashes: typology and semantics[END_REF][START_REF] Vint | The belt of Lielvarde. On hypothesis of Tõnis Vint[END_REF]]. The best known example is belt of Lielvarde (little town in Latvia) [START_REF] Tenisons | Ornaments and Symmetry: Language of Signs of Ornamental Tracery (interview with Modris Tenisons)[END_REF][START_REF] Vytautas | Ornament of Lithuanian traditional pick-up woven sashes: typology and semantics[END_REF][START_REF] Vint | The belt of Lielvarde. On hypothesis of Tõnis Vint[END_REF], that has attracted enormous interest of different researchers, maybe less from side of mathematicians. Possible reason for that may be the fact that this type of belts has very complicate ornamental tracery. Nevertheless there are some sufficiently rich patterns of belts that are much simpler, e.g., belt of Nica (small rural district in Latvia) (see 1.pict.) [3]. Just this type of belts we are going to study in this article.

In this article we systematize the many years experience of Modris Tenisons in the area of the research of Latvian ornamental belts [4, 5, 6, 7 (refs. 23, 24), 8 (ref. 65)]. Next to experience of Latvian artist and ornamentalistic researcher Modris Tenisons, we come accross with researchers from other Baltic countries, namely, Lithuanian researcher Vytautas Tumėnas [START_REF] Vytautas | Contemporary folk art in the centre and in the province: changes of values. Tautodailė centre ir periferijoje: vetybių perkainojimas[END_REF][START_REF] Vytautas | Research into the Baltic sashes symbol system: from the scientific interpetation to the authentic traditional folk typology. Baltiškų Juostų simbolių sistemos paieškos: Nuo mokslinės intepretacijos iki autentiškos tradicinės liaudiškos tipologijos[END_REF][START_REF] Vytautas | Ornament of Lithuanian traditional pick-up woven sashes: typology and semantics[END_REF] and Estonian researcher Tõnis Vint [START_REF] Vint | The belt of Lielvarde. On hypothesis of Tõnis Vint[END_REF].

The principle of removal of sieve in the first order belt and the code of the ornamental belt

Analyzing different kinds of ornamental belts Modris Tenisons came to persuasion that there may be distinguished one type of belts that may be called first order belts. The ornament in these belts may be divided into two parts, scilicet, the part of the sieve, and the part of the code. We research in this article just this type of belts called first order belts. The language of ornamental signs used in theses belts we call first order complexity ornamental sign language.

First order belts are characterized by property that there can be removed sieve in them leaving code alone. The principle of sieve removal or separation and coding in this sieve has been described in the article of Modris Tenisons and Armands Strazds [3]. The principle has been fixed also by patent of Modris Tenisons [START_REF] Tenisons | Ornaments and Symmetry: Language of Signs of Ornamental Tracery (interview with Modris Tenisons)[END_REF].

If ornamental tracery in belt such as Nica belt is considered as two colored squared pattern, where checks in it has two colors, then there is the part that doesn't change, and the changing part. The permanent part is called sieve, and the changing partcode part. The sieve consists from two dual parts, where each is lattice of cross elements, that is called cross lattice. Each cross element consists from five checks, naturally forming sign of cross. See two dual cross lattices in pict. 3 in the middle, where cross elements of cross lattices are correspondingly of red and grey colors. The changing part, that we call code part, consists from checks,

where each stands for one code unit. Ornamental tracery is built filling (coloring) filling places of codes with color of one or other cross lattices. More precisely, assuming code to be binary code consisting from zeros and units, choosing by coding for zeros one color of cross lattice and for units other color of code lattice we get ornament as image of ornamental code.

On the spot we get first fundamental property, videlicet, there arise eventually two traceries from one fixed ornamental code, interchanging color either of code or cross elements. We have as if four choices here but further we would see that opposite coloring of all ornamental pattern didn't give other ornament. For coding it would mean that opposite code, i.e., interchanging units and zeros, doesn't give other ornamental pattern for human prehension, though mathematically does. In the pict. 3 we see ornamental pattern with six code places in one row. We would say that this is the belt of breadth six as is Nica belt. We consider in this article only ornamental traceries and belts of breadth six, which is in way minimal nontrivial belt but sufficiently complex and problematic as represented by Nica belt.

In the example in pict. 3. ornamental code by fixing only half due to symmetry might be 011,010,100, 010, 011, 011,010,011,010,100 or in octal number system, 3242332324.

Picture 3. Pattern of ornamental trasery on right that is coded in the sieve that stands on the left. In the middle picture sieve is divided into two dual parts, cross lattices, where each is colored distinctly, red and grey resp. Places for coding are clearly seen as check elements, respectively in white.

Hemiolia principle -one and a half principle in ornament genesis

We may try to specify sieve and part of code by some generic aspect with some inquiry where from it could arise. Let us suggest in background of sieve be tracery of checkerboard with checks, and code being as if displacement of this background with shrinking of scale by at the same time, namely, the shifted squares are checks. By this approach we have as if two grounds for ornament to be enacted, scilicet, non changing background with checks and changing by code ground with checks. To see this in picture 3 in middle, imagine grey (or red) places in be squares where corners are overlapped or partly covered with places of codes, squares, where white places have shrinked by scale . In this play of enacting ornamental tracery some places respond as if belonging to one background part, say, red, but some to othersay, white.

We see that here ratio 3/2 plays a decisive role. The ratio is significant in some mystical teachings.

We call this principle of arising ornament from volume parts of background and volume parts of code as if interplaying between themselves that we describe here hemiolia principle. In Greek   one and a half.

Sieve displacement

The notion of sieve displacement arises by observing that displacing sieve by three checks or, what is the same, displacement of code by one row gives other ornament pattern that is equivalent of change of colors of cross lattices. Thus, if we add empty code line in the ornament coding, we get quite new ornamental pattern that what we were to obtain by sieve displacement.

Mathematically the things we consider here are almost trivial, but they effect on human imagination not in the least sense in trivial manner. The consideration in this direction see below. Moreover, let us bear in mind that ornamental belts' weavers who are directly connected with these belts, because just they are who have invented them, don't know these simple facts as clearly as they stand in mathematical setting and nevertheless they are able to obtain miracles in literal sense like in case of Nica belt, not even to speak about belt of Lielvarde.

Picture 4. Two ornament patterns from the same ornamental code. We may say that one is obtained by other using sieve displacement. We can see that we can't fix the difference by "mathematically non equipped eye" as some simple trick. Neither did most prolific masters as weavers of the most wonderful belts, who only "knew" the rule by some "instrumental" sense that they couldn't formulate directly. Picture 5. The belt of Nica in two variations. On the right sieve displacement is performed by adding empty code row at the beginning giving another pattern of Nica belt tracery. Visual effect is such as if we had quite another sample of Nica belt.

In pict. 4 we see two ornament patterns from the same ornamental code. We may say that one is obtained by other using the sieve displacement. We can see that we can't fix the difference by "mathematically non equipped eye" as some simple trick. Neither did most prolific masters as weavers of the most wonderful belts, who only "knew" the rule by some "instrumental" sense that couldn't formulate directly.

The sieve removal and sieve displacement Modris Tenisons discovered in the late seventies of previous century. The discovery is fixed in patent as intellectual game [START_REF] Tenisons | Intellectual game, patent of design pattern[END_REF].

Creation of ornamental tracery from asymmetric elements

Modris Tenisons has developed idea that first order belt code should be built from asymmetric elements, i.e., that in the base of symmetric ornamental tracery stands asymmetry. In order to illustrate this idea, let us characterize type of ornamental belt that is built from asymmetric elements that are called seeds of chaos.

Let us consider elementtwo-colored matrix where four checks are painted out asymmetrically with respect to middle row, middle column and both diagonals. How many such elements exist that are not equal with respect to named already symmetric transformations? It turns out thatten, see pict. 6. This simple but not trivial mathematical fact we formulate as theorem of Modris Tenisons.

Theorem 1 (Modris Tenisons

). There are exactly binary matrices with four units that are without automorphisms and not isomorphic with respect to reflections versus middle row, middle column and both diagonals.

Proof. Theorem may be checked by direct enumeration. It is important to notice that all mentioned reflections are considered with respect both asymmetry (non automorphisms) and non isomorphism. See in picture 6 all these 10 possible matrices where units are designated by red checks and zeros by white checks. Picture 6. Ten asymmetric elements named by M. Tenisons seeds of chaos. Each of them characterize equivalence class of eight elementsmatrices which are asymmetric versus vertical and horizontal and diagonal reflections. Together there are 80 such elements because their asymmetry doesn't allow the number of elements to "break down" due to symmetries. To compare with, symmetric such elements are in equivalence classes. Factorization as simple as in case of asymmetry is "broken down" by two extra symmetric elements that are symmetric versus all allowed symmetries, i.e., doesn't divide .

The next theorem is very relevant for the building of ornamental traceries in the first order belts.

Theorem 2 There are exactly binary matrices with units that are without automorphisms with respect to reflections versus middle row and middle column and both diagonals.

Proof This theorem differ from previous that condition of non isomorphism is removed.

Because elements are asymmetric in the mentioned sense they factorize in 10 classes of equivalence where each class has elements. Altogether we get matrices.

Let us try to get this number 80 without reference to previous theorem. In total we have binary matrices with units. Let us subtract the symmetric ones. We have symmetric matrices with respect to middle row 12, accordingly, with no unit in middle row -3, and with two unites -9. This number should be multiplied by 4. Two matrices were with all units in all corners, and no unit in corners. These came in the count with repetition because of excessive symmetry, thus they we should re-subtract. Thus, we get . ■

Let us try to build ornamental tracery from these asymmetric elements we call seeds of chaos that are in total factorized in clases of equivalence. Let us first try to count how many ornamental signs we may build in the most simple ornamental tracery from these asymmetric elements.

Definition 1

Let us call first level ornamental sign code or, simpler, ornamental sign code or sign code element that is built from 4 asymmetric elements which are symmetric either with respect to vertical and horizontal reflection, or with respect to rotation of element, all versus center of element, (see. pict. 7).

Definition 2 Let us say that two sign codes are equivalent if they are elements reflection of first and second row. Definition 2 introduces simple principle of right and left side symmetry in the belt, namely, belt is readable from both sides equally. This principle has more deep methodological meaning in the making distinction between human prehension and mathematical, see below.

Let us consider allowed sign codes by definition 1 (see in pict. 7). Pict. 7. Illustration of definition 1. Let us put in place of symbol "R" whatever asymmetric element with whatever orientation. On left, element show how code of sign may be built using reflections of element. On the right in two pictures code of sign is built using two possible rotations of element. Two depicted cases by definition 2 are dealt as equivalent. This equivalence doesn't affect first way of building of code of sign because code in this case is already symmetric with respect to vertical reflection.

Theorem 3 (Modris Tenisons) There are nonequivalent codes of signs.

Proof Let us show that only different codes of signs can be built with allowed operations by definitions 1 and 2.

Let us consider the way to build code of sign using definition 1 where asymmetric element should be symmetric with respect to vertical and horizontal reflections, (see pict. 7 left side). In place of asymmetric element we can place elements according theorem 2, which all should give different codes of signs. We get codes of sign.

Let us consider second way of how to build code of sign by rotation (see pict. 7 middle and right side). There similarly we get codes of sign, but half of these codes would be symmetric with respect to vertical reflection and according definition 2 should be excluded as being equivalent. In the result we get only non equivalent codes of signs. Together we get codes of sign as stated by theorem.

Other ways of building code of sign we do not have. Thus, theorem is proved. ■

The doubling of ornamental signs by displacement of sieve

We already considered sieve displacement that gives another ornamental tracery. As we saw sieve consists from two dual lattices of cross elements that may be merged into one by displacement by three checks. Let one lattice be one color and another in second color: putting code in color one we get one ornamental tracery, and other color would give other.

The may be attained by displacement of sieve, that could be performed, say, by entering empty line of code. But this consideration gives us way to build from one code of sign two different ornamental signs. This gives us right for next theorem.

Theorem 4 (Modris Tenisons). 120 non equivalent codes of sign give 240 distinct ornament signs using sieve displacement.

Proof If signs are divided into two classes with respect to sieve configuration, then no sign from one class may be equal with sign from other class, because the sieve itself plays decisive role: sieve in one class is as if displaced versus other class. This argument is sufficient for theorem to be proved. See pict. 9 for illustration. ■ Pict. 8. Sieve displacement with respect to the same code. Sign "Fish" or "zivtiņa".

Pict. 9. Changing colors in ornamental tracery we get another one. However this change does not effect to the same extent than by sieve displacement: we may discern previous ornamental tracery by our imagination.

Actually the ornamental tracery is invariant with respect to color change whatsoever and to color interchange ditto.

Pictures 9 and 10 show that sieve displacement and color interchange work in different way in ornament building.

Actually we could introduce one more feasibility to double number of codes of signs, scilicet, in place of asymmetric code allowing to be its dual element, namely, asymmetric element with five units. But this would lead to some disbalance of colors or brightness of colors in ornamental tracery, scilicet, 5-unit elements would give more colored tracery that 4-unit elements and coming them alternately would give place to disbalance of brightness of ornament. Thus, here we see where pure mathematical argument may come in conflict with some artistic principle and human prehension: mathematics would say that we may have sign codes whereas artistic principles would force us to refrain only to sign codes.

The principle of the opening of field information

Modris Tenisons came to building ornament signs in little different way, namely, using principle of the opening of field of information. Let us try to formalize his approach, using systematic argument.

Let us first consider a little different proof to theorem 3. Let us consider pict.8 on the left where we use reflection of elements in four diagonal corners. Let us do this in two directions, vertical and horizontal getting in this way ways to build code of sign by use of reflection. Similarly let us do rotation of element according schema in pict.8 on right: we do rotation in each corner getting codes of signs. Except, two pictures are to be used to escape overlap. Together we get schemata that for asymmetric elements should give different codes of signs. In this way we didn't use even definition 2. Why? It turns out that performance of rotation only in one direction is equivalent with use of definition 2 that introduces equivalence of left and right sides of ornamental tracery. Allowing rotation in both directions would result in getting repetition of codes that definition 2 excludes. Pict.9. Example of building code of sign. On left one asymmetric element gives four codes of sign, correspondingly in each corners of asymmetric element in the center. In picture in center and on right from the same asymmetric element we get codes of sign by twos in two pictures because in one picture codes would overlap. Clearly is seen principle of double overlap by rotation symmetry.

Let us try to follow idea of Modris Tenisons about opening of the field of information. If in place of asymmetric elements we had symmetric, then the number of different elements would be less because every symmetry tends to reduce corresponding number at least by . Some analogy may be the fact that elements with five zeros are classes of equivalence that do not factorize as simple as in case of asymmetry, namely, number of all matrices is . Two exceedingly symmetric matrices, with units in corners and no unit in corner, breaks down simplicity of count, say, .

Let us try to reason how Modris Tenisons does it in his workshops of ornament creation (see [START_REF] Lounesto | Clifford Algebras and Spinors[END_REF]). Let us take asymmetric element and open it in all direction with reflections, filling corner squares too as in pict. 9. Going on with such "opening" we wouldn't get new information because of repetitions. If in place of asymmetric element we had taken something symmetric we didn't get even this amount of information, but we had already repetitions. Speaking informally, information for us is that that do not repeat. But, let us imagine in place of what we see are dimensions in hyperspace: repetitions there would mean no information already directly. In place of information here we might use termaccess to information, and quantitydegrees of freedom, and that would give us full justification for use of these notions in way Modris Tenisons do.

Let us consider how many unrepeatable information we might get. We have equivalence classes of asymmetric elements and elements. Informally we might say that theorem 3 counts number of information or degrees of freedom in the generation of code of first level.

Pict. 9.

field where each of them is matrix for building code of sign where with slash is depicted imaginable asymmetric Such slashes in the language of building of signs may denote their generic element that artists are used to. On right examples of such sign buildings: the sign of Māra ("Māras zīme") that arises from union of two opposite, masculine and feminine, elements; "eternity"; "fish sign"; "crayfish sign" or "cancer". See other examples in [START_REF] Vytautas | Contemporary folk art in the centre and in the province: changes of values. Tautodailė centre ir periferijoje: vetybių perkainojimas[END_REF][START_REF] Vytautas | Research into the Baltic sashes symbol system: from the scientific interpetation to the authentic traditional folk typology. Baltiškų Juostų simbolių sistemos paieškos: Nuo mokslinės intepretacijos iki autentiškos tradicinės liaudiškos tipologijos[END_REF][START_REF] Vytautas | Ornament of Lithuanian traditional pick-up woven sashes: typology and semantics[END_REF]. All these signs except last may be placed in this field of code. The slash works as if variable that may have assigned possible values. For example, we may form samples of "Māras zīme", etc.

Sign language alphabet

Let us consider simple idea how from sieve displacement and ornament coding we may come to sign alphabet in first order (complexity) belts. Two following rows in code in case when ornament code width is, say, six may be divided into two following rows of code as binary field that may be divided into binary fields. Each such field may be considered as letter in 16 sign alphabet. For binary codes it would be too trivial step without much use. In case these binary fields "work" together with sieve, they make 16 signs for ornamnetal sign language alphabet. Actually we get two such possible alphabets using sieve displacement. Every belt of width 6 may be cut into such square elements standing for ornamental sign alphabet letters because we should get invariantly only 16 such alphabet letters by simple binary code argument.

Pict. 9. Illustration how alphabet arises from simple binary code. On left example of two binary columns from two rowsbinary matrix . Putting this code into sieve we receive one alphabet letter of sign alphabet, as in pict. in center and right where places of code are marked. Every belt of width 6 may be cut into such square elements standing for ornamental sign alphabet letters because we should get invariantly only 16 such alphabet letters by simple binary code argument.

Modris Tenisons patent [START_REF] Tenisons | Intellectual game, patent of design pattern[END_REF] actually is for this sign alphabet that in indirect way contains ideas of sieve displacement and ornamental coding.

Pict. 10. 16 letters of ornamental sign language alphabet. Would this alphabet of Modris Tenisons sufficient to code belt of Nica? ... in sense any first order ornamental pattern may be composed by these squares as puzzle? Mathematically it is trivially if we only notice that this is full binary code in sieve. By sieve displacement we get another such alphabet that is different from this one.

The belt of Nica and its investigation

Modris Tenisons has forwarded idea that the belt of Nica is built from asymmetric elements too similarly as we described higher [START_REF] Tenisons | Ornaments and Symmetry: Language of Signs of Ornamental Tracery (interview with Modris Tenisons)[END_REF]. But it turns out not to be so simple. If we directly search after asymmetric elements in the belt of Nica, we do not get them sufficiently many to get complete cover of all the belt. We come to question how the code of belt of Nica is built if we want to stick around idea of Tenisons about asymmetry as basis of symmetry in ornamental tracery of belt?

We may now ask. How weaver may come to belt that is rich as belt of Nica? One interpretation is sufficiently simple. Together with experience by weaving instrumental experience heaps up. Weavers better remember "instrumental" experience than, say, visual, namely, they remember technological and instrumental information how they do anything by weaving rather in some other way. In much simpler case, exempli gratia, knitter remembers combinations of stitches rather than anything else. The same but in more complex setting applies for weaver of belts. The belt of Lielvarde shows that this assumed instrumental experience that might be incredibly complex may produce real wonders. Widely used allusion to information from Cosmos in this contex [START_REF] Lounesto | Clifford Algebras and Spinors[END_REF] might have some ground though not subject to direct scientific argument up to now. But what to say with respect to belt of Nica when we want to discover asymmetry in its ornamental tracery? Here we may be forced to attack this question from another side and ask: don't we have to deal here not so much with local code but volume or distributive code? To find distributive function or pieces of holograms if any in the structure of code might be much harder task than code's investigation locally. If so, just distributive function would be real code of belt of Nice, in case we were successful in its discovery. At present first author is working in this direction too.

"Belt of Nica teaches us": the aesthetics in belts of first order

First order ornamental belts are convenient object of investigations because we may speak about some level of aesthetics that may be characterizes sufficiently simply by quantitative argument. Firstly, belt is readable equivalently 1) changing colors to opposite; 2) changing sides to opposite (with correction of right side turn argument). Secondly, belts are coded in asymmetric coding. Thirdly, ornamental pattern has balanced color ratio, i.e., already seed of chaos incarnates relation 4:5, and sieve consists from two dual lattices of both colors. Fourth, signs in ornamental patter are inseparable, i.e., sign transforms to other sign, signs overlapping form signs, and so on.

The interchange of colors in ornamental tracery of belts says that belt is readable in both ways equivalently, leaving space for some aesthetic or artistic difference in reference to human psyche. But there is way to read belt in both ways simultaneously, namely, when we see as if belt in both ways. Then we may start to see belt as if border between two colors, namely, we do not see "black on white" or "white on black" but sign as border between both colors. After all that is what we are after: belt is the union of two elements, masculine and feminine. In Latvian tradition this says that belt is the sign of Māra, "Māras zīme". Christian tradition may say "sign of Christ". What concerns partial equivalence of sides of belt, left and right sides in belt are equivalent in sense they are readable in both ways, but right side remains prevalent over left side as right side (hand) is prevalent over left side (hand) in human nature settings: we discern right and left side as homines sapientes, but nature above us might have some indifference against this distinction. This same message we see in mathematics and physics. Lorentz transforms and right and left glove non-interchangeability are guided by the same signature of quaternion [START_REF] Tenisons | Intellectual game, patent of design pattern[END_REF]. Double cover of by don't care of prevalence of right over left, but makes space for such prevalence. What we have in case of belts is two dimension case is mostly convenient three dimensional case. It is nice to see that 2-dimensional case leaves over these characteristic two points "left" and "right" of unit circle in complex plane that would be actual for thee-dimensional and higher cases. That we see in ornamental belts incarnated in the principle of partial equivalence of sides of belt.

Modris

Pict.11. Piece of ornamental tracery got from one seed of chaos.

12. The belt produced from 10 asymmetric elements, using random number generator to choose different seeds of chaos, according principles explained in this article. How about aesthetics in this example? We don't have that simplicity that in case of Nica belt. Do we lack some more simple aesthetic principle here?

Genesis of ornamental pattern: sign from code or vice versa

In this article we explain genesis of ornamental sign more from side of code than from side we perceive it as image. Moreover, we may conclude that ornamental code may creep in indirectly in other way too, e.g., via instrumental experience of weaver. Now we may question on more general level asking: how ornamental pattern arise? Via code or as an image? This question is not so redundant, because researchers of ornamental patterns use so much effort to explain where from one or other pattern could arise basically using "image approach" [START_REF] Milašius | The Mathematical Basis of Ornamentation of Patterned Woven Fabrics. FIBRES & TEXTILES in Eastern Europe[END_REF][START_REF] Vytautas | Contemporary folk art in the centre and in the province: changes of values. Tautodailė centre ir periferijoje: vetybių perkainojimas[END_REF][START_REF] Vytautas | Research into the Baltic sashes symbol system: from the scientific interpetation to the authentic traditional folk typology. Baltiškų Juostų simbolių sistemos paieškos: Nuo mokslinės intepretacijos iki autentiškos tradicinės liaudiškos tipologijos[END_REF][START_REF] Vytautas | Ornament of Lithuanian traditional pick-up woven sashes: typology and semantics[END_REF], because "code approach" should require some understanding how code generates sign. If code can generate an ornamental pattern as if from itselft, the picture or this type or reserch changes cardinally. New aspect at least comes before researchersinstrumental experience as source of producer of ornamental patterns.

Belts of higher order and their eventual investigation

We have already mentioned belt of Lielvarde, that is much more complex than, say, belt of Nica. In order to research belt of Lielvarde it isn't sufficient to remove sieve and apply asymmetry on some simple level. Belt of Lielvarde is build as if on several levels, sieve if discernable, is used on several levels, scilicet, at least two, and sieve may have more complex structure than simple pair of dual cross lattices. Nevertheless, Lielvarde belt is produced by human beings, though very skilful masters. How to attack investigation of these belts with exact methods similar we try to use in this article? Our answer is simple enough: we must first research first level belts, then maybe nearest samples that step outside this first level. Moreover, we step as if in new typology of genesis of belts, scilicet, based on arising sign from code rather than as image. To support this approach we must apply new mathematical methods along with these already used in researches as in [START_REF] Milašius | The Mathematical Basis of Ornamentation of Patterned Woven Fabrics. FIBRES & TEXTILES in Eastern Europe[END_REF]. Modris Tenisons attributes Lielvardes belt to fifth or even higher level of complexity, but he has no exact means to say what specifically; all this is as if in field of artistic estimation, not in reach of exact scientific criteria. All this shows necessity to do researches and typology of ornamental patterns in gradual way, from simpler to more complex. Making research in field as complex as of Lielvardes belt researchers face question: is this only problem of artists or, say, researches of Living Ethics? Many speak about information from Cosmos. What is this? Are these things subject to research or only area of religion or mystics? Exact sciences tend to say that they do not want to have anything shared with Living Ethics? But they may have common area of investigations, scilicet, belt of Lielvarde. Now when question has become even more compex we may have for this a new justification: we may ask -Cosmos provide us with visual or instrumental information? See [START_REF] Zeps | Our Ability to Research Comes Before Understanding of What We Research[END_REF].

Does there exist first order complexity in nature?

The research of belts is not only recreative entertainment that doesn't have anything in connection with research in nature or mathematics. Question about what is primarycode or imagemight be more actual than we assume not only in ornamentalistic but in nature too.

Ornamentalistic plays important role in different areas of science, arts, where these areas may intertwine. We may use language of artists and try to unite with precise language of mathematics similarly as we tried to do in this article. Ornamentalistic would be the branch where without this synthesis were hard to get along. But here may come other areas into touch too, say, physics, biology, psychology. The last is mentioned in very interesting aspect in the movie "The belt of Lielvarde. On hypothesis of Tõnis Vint". Let us take look into message of Tõnis Vint (Estonia) (transl. by D.Z.): "Through times and lands signs unite us and tell. For example, these Mexican temple walls hide about world three layers of information: everyday objective, facts about events, and phenomena in nature description. For this last epistemology gifted were only some chosen people. Several notions for one symbol were present also in Chinese tables of I-Cin : heavenvirility, earthfemininity, and parallel notions: heaven creation father, earthsubmission mother, water, moon, danger, fire, light, son, and other meanings.

To son, moon, fire and water our ancients attributed magical signification. In this way geometric ornaments already long ago were used for practical reasons.

American Indians lived in precisely created circles, that afterwards were called medical circles, because, how archeological excavations indicated, in these settlements living Indians of Siu tribe didn't know illnesses. According legend, each child for this tribe had sage or philosophical father specially chosen. Taking away child into wild nature, having explored his character and biophysical features, Indian sage draw individual symbol for boy or girl that served as singular person's code or card. Using these cards youth got acquainted, families united in friendly contacts, and all lived in great friendly family in psychological concordance, and illnesses for harmonic balanced people went past. Using this circle there was possibility to influence human's psyche more directly too."

This was said before year 1980. What we have now? Have we solved problem that was not problem for Indians of Siu tribe? I don't think so? Why? The main problem is about ourselves who consider such evidences as legends that are not much worth for direct research. But why after finding Rosetta Stone there came investigators, and Champollion himself, to decipher the script? The script of Nica doesn't look like script that could be deciphered or deserves to be deciphered? We here maid attempt to show that maybe we heavily err.

Picture 1 .

 1 Belt of Nica [3] in two settings. The ornamental tracery should be read linearly from left upper corner to the right lower corner. See in appendix binary code of this belt. Length of code 751 rows. Image of the belt here created by computer program. In lower part belt of Nica with opposite coloring. The one of main principles in the first order ornamental tracery is that both colors are as if in equilibrium what concerns amount of elments in it both globaly and locally.

Pict. 10 .

 10 The code of the belt of Nica and its ornamental tracery.

  Tenisons as artist uses rather different way of expression, saying, what we encounter as aesthetics in Nica belt we must attribute to what belt of Nica teaches us [4]. Thus we must say "Belt of Nica teaches us." "Belt of Nica teaches us -In all should prevail simplicy. Belt of Nica teaches us -Nothing excessive. ... Colors should be in equilibrium. ... Belt should be read between colors ... And so on."

Acknowledgements

We thank professor Matti Pitkänen (Finland) for helpuful discussions and corrections.