of Klein Surfaces ⋆

Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider singular curves of the plane; here we recover, in a different way, a result proved by Fronsdal and make it more precise. On the other hand, we are interested in Klein surfaces. The use of a complex suggested by Kontsevich and the help of Groebner bases allow us to solve the problem.


Deformation quantization
Given a mechanical system (M, F(M )), where M is a Poisson manifold and F(M ) the algebra of regular functions on M , it is important to be able to quantize it, in order to obtain more precise results than through classical mechanics. An available method is deformation quantization, which consists of constructing a star-product on the algebra of formal power series F(M ) [[ ]]. The first approach for this construction is the computation of Hochschild cohomology of F(M ).
We consider such a mechanical system given by a Poisson manifold M , endowed with a Poisson bracket {·, ·}. In classical mechanics, we study the (commutative) algebra F(M ) of regular functions (i.e., for example, C ∞ , holomorphic or polynomial) on M , that is to say the observables of the classical system. But quantum mechanics, where the physical system is described by a (non commutative) algebra of operators on a Hilbert space, gives more correct results than its classical analogue. Hence the importance to get a quantum description of the classical system (M, F(M )), such an operation is called a quantization.
One option is geometric quantization, which allows us to construct in an explicit way a Hilbert space and an algebra of operators on this space (see the book [10] on the Virasoro group and algebra for a nice introduction to geometric quantization). This very interesting method presents the drawback of being seldom applicable.
That is why other methods, such as asymptotic quantization and deformation quantization, have been introduced. The latter, described in 1978 by F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer in [5], is a good alternative: instead of constructing an ⋆ This paper is a contribution to the Special Issue on Deformation Quantization. The full collection is available at http://www.emis.de/journals/SIGMA/Deformation Quantization.html is the Weyl algebra, for every finite subgroup G of Sp 2n C. It is an interesting and classical question to compare the Hochschild homology and cohomology of A n (C) G with the Poisson homology and cohomology of the ring of invariants C[x, y] G , which is a quotient algebra of the form C[z] / f 1 , . . . , f m .
C. Fronsdal studies in [8] Hochschild homology and cohomology in two particular cases: the case where n = 1 and m = 1, and the case where n = 2 and m = 1. Besides, the appendix of this article gives another way to calculate the Hochschild cohomology in the more general case of complete intersections.
In this paper, we propose to calculate the Hochschild homology and cohomology in two particularly important cases.
• The case of singular curves of the plane, with polynomials f 1 which are weighted homogeneous polynomials with a singularity of modality zero: these polynomials correspond to the normal forms of weighted homogeneous functions of two variables and of modality zero, given in the classification of weighted homogeneous functions of [3] (this case already held C. Fronsdal's attention).
• The case of Klein surfaces X Γ which are the quotients C 2 / Γ, where Γ is a finite subgroup of SL 2 C (this case corresponds to n = 3 and m = 1). The latter have been the subject of many works; their link with the finite subgroups of SL 2 C, with the Platonic polyhedra, and with McKay correspondence explains this large interest. Moreover, the preprojective algebras, to which [6] is devoted, constitute a family of deformations of the Klein surfaces, parametrized by the group which is associated to them: this fact justifies once again the calculation of their cohomology.
The main result of the article is given by two propositions: , and for all j ≥ 3, For explicit computations, we shall make use of, and develop a method suggested by M. Kontsevich in the appendix of [8].
We will first study the case of singular curves of the plane in Section 3: we will use this method to recover the result that C. Fronsdal proved by direct calculations. Then we will refine it by determining the dimensions of the cohomology and homology spaces by means of multivariate division and Groebner bases.
Next, in Section 4, we will consider the case of Klein surfaces X Γ . For j ∈ N, we denote by HH j the Hochschild cohomology space in degree j of X Γ . We will first prove that HH 0 identifies with the space of polynomial functions on the singular surface X Γ . We will then prove that HH 1 and HH 2 are infinite-dimensional. We will also determine, for j greater or equal to 3, the dimension of HH j , by showing that it is equal to the Milnor number of the surface X Γ . Finally, we will compute the Hochschild homology spaces.
In Section 1.3 we begin by recalling important classical results about deformations.

Hochschild homology and cohomology and deformations of algebras
Consider an associative C-algebra, denoted by A. The Hochschild cohomological complex of A is where the space C p (A) of p-cochains is defined by C p (A) = 0 for p ∈ −N * , C 0 (A) = A, and for We may write it in terms of the Gerstenhaber bracket 1 [·, ·] G and of the product µ of A, as follows Then we define the Hochschild cohomology of A as the cohomology of the Hochschild cohomological complex associated to A, i.e. HH 0 (A) := Ker d (0) and for p ∈ N * , HH p (A) := Ker d (p) / Im d (p−1) .
We This means that there exists a sequence of bilinear maps m j from A × A to A of which the first term m 0 is the product of A and such that We say that (A[[ ]], m) is a deformation of the algebra (A, µ). We say that the deformation is of order p if the previous formulae are satisfied (only) for n ≤ p. The Hochschild cohomology plays an important role in the study of deformations of the algebra A, by helping us to classify them. In fact, if π ∈ C 2 (A), we may construct a first order deformation m of A such that m 1 = π if and only if π ∈ Ker d (2) . Moreover, two first order 1 Recall that for F ∈ C p (A) and H ∈ C q (A), the Gerstenhaber product is the element F • H ∈ C p+q−1 (A) defined by F • H(a1, . . . , ap+q−1) = P p−1 i=0 (−1) i(q+1) F (a1, . . . , ai, H(ai+1, . . . , ai+q), ai+q+1, . . . , ap+q−1), and the Gerstenhaber bracket is [9], and [4, page 38].
deformations are equivalent 2 if and only if their difference is an element of Im d (1) . So the set of equivalence classes of first order deformations is in bijection with HH 2 (A).
If m = p j=0 m j j , m j ∈ C 2 (A) is a deformation of order p, then we may extend m to a deformation of order p + 1 if and only if there exists m p+1 such that According to the graded Jacobi identity for [·, ·] G , the last sum belongs to Ker d (3) . So HH 3 (A) contains the obstructions to extend a deformation of order p to a deformation of order p + 1.
The Hochschild homological complex of A is where the space of p-chains is given by C p (A) = 0 for p ∈ −N * , C 0 (A) = A, and for p ∈ N * , We define the Hochschild homology of A as the homology of the Hochschild homological complex associated to A, i.e. HH 0 (A) := A / Im d 1 and for p ∈ N * , HH p (A) := Ker d p / Im d p+1 .

Presentation of the Koszul complex
We recall in this section some results about the Koszul complex used below (see the appendix of [8]).

Kontsevich theorem and notations
As in Section 1.2, we consider R = C[z] and (f 1 , . . . , f m ) ∈ R m , and we denote by A the quotient R / f 1 , . . . , f m . We assume that we have a complete intersection, i.e. the dimension of the set of solutions of the system {f 1 (z) = · · · = f m (z) = 0} is n − m. We consider the differential graded algebra 2 Two deformations m = P p j=0 mj j , mj ∈ C 2 (A) and m ′ = P p j=0 m ′ j j , m ′ j ∈ C 2 (A) are called equivalent if there exists a sequence of linear maps ϕj from A to A of which the first term ϕ0 is the identity of A and such that where η i := ∂ ∂z i is an odd variable (i.e. the η i 's anticommute), and b j an even variable (i.e. the b j 's commute).
T is endowed with the differential and the Hodge grading, defined by deg( We may now state the main theorem which will allow us to calculate the Hochschild cohomology: Theorem 1 (Kontsevich). Under the previous assumptions, the Hochschild cohomology of A is isomorphic to the cohomology of the complex ( T , d e T ) associated with the differential graded algebra T .
Remark 1. Theorem 1 may be seen as a generalization of the Hochschild-Kostant-Rosenberg theorem to the case of non-smooth spaces.
There is no element of negative degree. So the complex is as follows For each degree p, we choose a basis B p of T (p). For example for p = 0, . . . , 3, we may take Below we shall make use of the explicit matrices Mat Bp,B p+1 (d .
There is an analogous of Theorem 1 for the Hochschild homology. We consider the complex where ξ i is an odd variable and a j an even variable. Ω is endowed with the differential and the Hodge grading, defined by deg Theorem 2 (Kontsevich). Under the previous assumptions, the Hochschild homology of A is isomorphic to the cohomology of the complex ( Ω, d e Ω ) . . .
For each degree p, we will choose a basis V p of Ω(p) and we will make use of the explicit According to Theorem 2, we have, for p ∈ N, For each ideal J of C[z], we denote by J A the image of J by the canonical projection Similarly if (g 1 , . . . , g r ) ∈ A r we denote by g 1 , . . . , g r A the ideal of A generated by (g 1 , . . . , g r ).
In particular, g does not divide 0 in C[z] / J if and only if Ann J (g) = J. Finally, we denote by ∇g the gradient of a polynomial g ∈ C[z].
From now on, we consider the case m = 1 and set f := f 1 . Moreover, we use the notation ∂ j for the partial derivative with respect to z j .

Particular case where n = 1 and m = 1
In the case where n = 1 and m = 1, according to what we have seen, we have for p ∈ N, We deduce and Similarly, for p ∈ N * , and for p ∈ N, and for p ∈ N * , See [12] for a similar calculation. Theorem 3 (Classification of weighted homogeneous functions, [3]). The weighted homogeneous functions of two variables and of modality zero reduce, up to equivalence, to the following list of normal forms The singularities of types A k , D k , E 6 , E 7 , E 8 are called simple singularities. In the two following sections, we will study the Hochschild cohomology of C[z] / f , where f is one of the normal forms of the preceding table.

Description of the cohomology spaces
With the help of Theorem 1 we calculate the Hochschild cohomology of A : We begin by making cochains and differentials explicit, by using the notations of Section 2.1.
The various spaces of the complex are given by i.e., for an arbitrary p ∈ N * , and for an arbitrary p ∈ N, As in [8], we denote by ∂ ∂η k the partial derivative with respect to the variable η k , for k ∈ {1, 2}.
The matrices of d e T are therefore given by We deduce a simpler expression for the cohomology spaces We recover a result of [8] (here, we use the notations of [8] It remains to determine these spaces more explicitly. This will be done in the two following sections.

Explicit calculations in the particular case where f has separate variables
In this section, we consider the polynomial f = a 1 z k 1 +a 2 z l 2 , with k ≥ 2, l ≥ 2, and (a 1 , a 2 )∈(C * ) 2 . The partial derivatives of f are ∂ 1 f = ka 1 z k−1 1 and ∂ 2 f = la 2 z l−1 2 . We already have Besides, as f is weighted homogeneous, Euler's formula gives 1

F. Butin
But ∂ 1 f and f are relatively prime, just as ∂ 2 f and f are, hence if g ∈ A satisfies g∂ 1 f = 0 mod f , then g ∈ f , i.e. g is zero in A. So, We now determine the set First we have So the only monomials which are not in this ideal are the elements z i ]. Every polynomial P ∈ C[z] may be written in the form with α, β ∈ C[z] and a ij ∈ C. Therefore, the polynomials P ∈ C[z] such that P ∂ 2 f ∈ f, ∂ 1 f are the elements So we have calculated Ann f,∂ 1 f (∂ 2 f ). Let g = g 1 g 2 ∈ A 2 satisfy the equation Then we have It follows that From the equality z 2 ∂ 2 f = lf − l k z 1 ∂ 1 f , one deduces Then we verify that the elements g 1 and g 2 obtained in this way are indeed solutions of equation (2). Finally, we have We immediately deduce the cohomology spaces of odd degree: where the direct sum results from the following argument: if we have And by a Euclidian division in (C[z 2 ]) [z 1 ], we may write β = f q + r, where the z 1 -degree of r is smaller or equal to k − 1. So the z 1 -degree of v is also smaller or equal to k − 1, thus v ∈ f implies β = 0 and a ij = 0.
Remark 3. We obtain in particular the cohomology for the cases where f = z k+1 These cases correspond respectively to the weighted homogeneous functions of types A k , E 6 and E 8 given in Theorem 3.
The table below summarizes the results we have just obtained for the three particular cases The cases where f = z 2 1 z 2 + z k−1 2 and f = z 3 1 + z 1 z 3 2 , i.e. respectively D k and E 7 , will be studied in the next section.

Explicit calculations for D k and E 7
To study these particular cases, we use the following result about Groebner bases (Theorem 4). First, recall the definition of a Groebner basis. For g ∈ C[z], we denote by lt(g) its leading term (for the lexicographic order). Given a non-trivial ideal J of C[z], a Groebner basis of J is a finite subset G J of J \ {0} such that for all f ∈ J \ {0}, there exists g ∈ G J such that lt(g) divides lt(f ). See [15] for more details.
Definition 1. Let J be a non-trivial ideal of C[z] and let G J := [g 1 , . . . , g r ] be a Groebner basis of J. We call set of the G J -standard terms, the set of all monomials of C[z] that are not divisible by any of lt(g 1 ), . . . , lt(g r ).
Theorem 4 (Macaulay). The set of the G J -standard terms forms a basis of the quotient vector space C[z] / J.

Case of
Here we have A Groebner basis of the ideal f, ∂ 2 f is So the set of the standard terms is We We look for the normal form of the element q modulo the ideal f, But the equation with (α, β) ∈ C[z] 2 and a, b j ∈ C. Hence And with the equalities, we obtain i.e., On the other hand, a Groebner basis of ], thus Let us summarize (by using, for the direct sum, the same argument as in Section 3.2):

Homology
The study is the same as the one of the Hochschild cohomology: to get the Hochschild homology is equivalent to compute the cohomology of the complex ( Ω, d e Ω ) described in Section 2.1. We have Ω(0) = A, Ω(−2p) = Aa p 1 ⊕ Aa p−1 1 ξ 1 ξ 2 for p ∈ N * , and Ω(−2p − 1) = Aa p 1 ξ 1 ⊕ Aa p 1 ξ 2 for p ∈ N. This defines the bases V p . The differential is d e Ω = (ξ 1 ∂ 1 f + ξ 2 ∂ 2 f ) ∂ ∂a 1 . So we obtain, for p ∈ N * , the matrices The cohomology spaces read as For p ∈ N * , For p ∈ N * , From now on, we assume that f has separate variables, or f is of type D k or E 7 . Then we have {g ∈ A / g∂ 1 f = g∂ 2 f = 0} = {0}, and according to Euler's formula, for p ∈ N, ∇f . For the computation of {g ∈ A 2 / det(∇f, g) = 0} and {g∇f / g∈A} , we proceed with Groebner bases as in Section 3.3. For example, we do it for f = z 2 1 z 2 + z k−1 2 (i.e. type D k ). Let g ∈ A 2 be such that det(∇f, g) = 0. Then g 2 ∂ 1 f = 0 mod f, ∂ 2 f , i.e., according to Section 3.4.1, with (α, β) ∈ C[z] 2 and a, b j ∈ C. Hence With the equalities, we obtain i.e., We have {g∇f / g ∈ A} ⊂ {g ∈ A 2 / det(∇f, g) = 0}, thus Since A 2 / {g ∈ A 2 / det(∇f, g) = 0} ≃ {det(∇f, g) / g ∈ A 2 }, and since the map We collect in the following table the results for the Hochschild homology in the various cases 4 Case n = 3, m = 1. Klein surfaces

Klein surfaces
Given a finite group G acting on C n , we associate to it, according to Erlangen program of Klein, the quotient space C n /G, i.e. the space whose points are the orbits under the action F. Butin of G; it is an algebraic variety, and the polynomial functions on this variety are the polynomial functions on C n which are G-invariant. In the case of SL 2 C, invariant theory allows us to associate a polynomial to any finite subgroup, as explained in Proposition 4. Thus, to every finite subgroup of SL 2 C is associated the zero set of this polynomial; it is an algebraic variety, called a Klein surface. In this section we recall some results about these surfaces. See the references [17] and [7] for more details.
Proposition 3. Every finite subgroup of SL 2 C is conjugate to one of the following groups: • E 6 (tetrahedral), |E 6 | = 24; We call Klein surface the algebraic hyper-surface defined by {z ∈ C 3 / f (z) = 0}.

Theorem 5 (Pichereau). Consider the Poisson bracket defined on
where i is the contraction of a multiderivation by a differential form. Denote by HP * f (resp. HP f * ) the Poisson cohomology (resp. homology) for this bracket. Under the previous assumptions, the Poisson cohomology HP * f and the Poisson homology HP f * of (C[z 1 , z 1 , z 3 ] / f , {·, ·} f ) are given by The algebra C[x, y] is a Poisson algebra for the standard symplectic bracket {·, ·} std . As G is a subgroup of the symplectic group Sp 2 C (since Sp 2 C = SL 2 C), the invariant algebra C[x, y] G is a Poisson subalgebra of C[x, y]. The following proposition allows us to deduce, from Theorem 5, the Poisson cohomology and homology of C[x, y] G for the standard symplectic bracket.

Proposition 5. With the choice made in the preceding table for the polynomial f , the isomorphism of associative algebras
In the sequel, we will calculate the Hochschild cohomology of C[z 1 , z 1 , z 3 ]/ f , and we will immediately deduce the Hochschild cohomology of C[x, y] G , with the help of the isomorphism π. Note that the fact that π preserves the Poisson structures has no incidence on the computation of the Hochschild cohomology. Therefore, so as to simplify the calculations, we may replace the polynomial f by a simpler one, given in the following table Indeed, the linear maps defined by are isomorphisms of associative algebras.

Description of the cohomology spaces
We consider now the case A := C[z 1 , z 2 , z 3 ], / f and we want to calculate the Hochschild cohomology of A. We use the notations of Section 2.1, but we change the ordering of the basis: we shall take (η 1 η 2 , η 2 η 3 , η 3 η 1 ) instead of (η 1 η 2 , η 1 η 3 , η 2 η 3 ). The different spaces of the complex are now given by i.e., for an arbitrary p ∈ N * , We have The matrices of d e T are therefore given by We deduce For p ≥ 2, For p ∈ N * , The following section will allow us to make those various spaces more explicit.

Explicit calculations in the particular case where f has separate variables
In this section, we consider the polynomial f = a 1 z i 1 + a 2 z j 2 + a 3 z k 3 , with 2 ≤ i ≤ j ≤ k and a j ∈ C * . Its partial derivatives are and

We already have
Moreover, as f is weighted homogeneous, Euler's formula gives So we have the inclusion f ⊂ ∂ 1 f, ∂ 2 f, ∂ 3 f , thus Finally, as ∂ 1 f and f are relatively prime, if g ∈ A verifies g∂ 1 f = 0 mod f , then g ∈ f , i.e. g is zero in A. Now we determine the set First we have Thus the only monomials which are not in this ideal are the elements z p So every polynomial P ∈ C[z] may be written in the form The polynomials P ∈ C[z] such that P ∂ 3 f ∈ f, ∂ 1 f, ∂ 2 f are therefore the following ones So we have calculated Ann f,∂ 1 f,∂ 2 f (∂ 3 f ). The equation g · ∇f = 0 mod f leads to g 3 ∈ Ann f,∂ 1 f,∂ 2 f (∂ 3 f ), i.e.
And A 3 / {g ∈ A 3 / g · ∇f = 0} ≃ {g · ∇f / g ∈ A 3 }. Since the map In the following table we collect the results for the Hochschild homology in the various cases