On Axioms of Frobenius Like Structure in the Theory of Arrangements 3

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2. Axioms of an almost Frobenius structure, [D2] An almost Frobenius structure of charge d 6= 1 on a manifold M is a structure of a Frobenius algebra on the tangent spaces TzM = (∗z, ( , )z), where z ∈ M. It satisfies the following axioms. Axiom 1. The metric ( , )z is flat. Axiom 2. In the flat coordinates z1,...,zn for the metric, the structure constants of the multiplication

∂ ∂ k ∂ (2.1) ∗z = Ni,j(z) ∂zi ∂zj ∂zk can be represented in the form 3 k k,l ∂ L (2.2) Ni,j(z)= G (z) ∂zl∂zi∂zj for some function L(z). Here (Gi,j) is the matrix inverse to the matrix (G := ( ∂ , ∂ )). i,j ∂zi ∂zj Axiom 3. The function L(z) satisfies the homogeneity equation n ∂L 1 (2.3) zj (z) = 2L(z)+ Gi,jzizj. ∂zj 1 − d Xj=1 Xi,j Axiom 4. The Euler vector field n 1 − d ∂ (2.4) E = zj 2 ∂zj Xj=1 (1−d)2 is the unit of the Frobenius algebra. Notice that (E, E)= 4 i,j Gi,jzizj. Axiom 5. There exists a vector field P n ∂ (2.5) e = ej(z) , ∂zj Xj=1 an invertible element of the Frobenius algebra TzM for every z ∈ M, such that the operator g(z) 7→ (eg)(z) acts by the shift κ 7→ κ − 1 on the space of solutions of the system of equations 2 n ∂ g l ∂g (2.6) = κ Ni,j , i, j = 1,...,n. ∂zi∂zj ∂zl Xl=1 The solutions of that system are called twisted periods, [D2]. 2.1. Example in Section 5.2 of [D2]. Let n (2.7) F (z ,...,z )= (z − z )2 log(z − z ) 1 n 4 i j i j Xi

n ∂i − ∂j (2.8) ∂i ∗z ∂j = − for i 6= j, ∂i ∗z ∂i = − ∂i ∗z ∂j. 2 zi − zj Xj6=i The unit element of the algebra is the Euler vector ﬁeld n 1 (2.9) E = z ∂ . n j j Xj=1 ON AXIOMS OF FROBENIUS LIKE STRUCTURE IN THE THEORY OF ARRANGEMENTS 3

The operator of multiplication in the algebra by the element n ∂ equals zero. Factorizing over j=1 ∂zj this direction one obtains an almost Frobenius structure on {z + · · · + z = 0}−∪ {z = z }. 1P n i

3. Axioms of a Frobenius like structure j j=1,...,k 3.1. Objects. Let an n × k-matrix (bi )i=1,...,n be given with n > k. For i1,...,ik ∈ [1,...,n], denote

(3.1) d = det(bj ) . i1,...,ik il j,l=1,...,k σ We have dσi1,...,σik = (−1) di1,...,ik for σ ∈ Sk. We assume that di1,...,ik 6= 0 for all distinct i1,...,ik.

Remark. The numbers (di1,...,ik ) satisfy the Pl¨ucker relations. For example, for arbitrary j1,..., jk+1, i2, ..., ik ∈ [1,...,n], we have

k+1 m+1 (3.2) (−1) d c dj ,i2,...,i = 0. j1,...,jm,...,jk+1 m k m=1 X n−1 We assume that an k -dimensional complex vector space V is given with a non-degenerate symmetric bilinear form S( , ). We assume that for every k-element unordered subset {i1,...,ik} ⊂ [1,...,n] a nonzero vector

Pi1,...,ik ∈ V is given such that the vectors {Pi1,...,ik } generate V as a vector space and the only linear relations among them are n

(3.3) dj,i2,...,ik Pj,i2,...,ik = 0, ∀i2,...,ik ∈ [1,...,n]. Xj=1

Lemma 3.1. For any j ∈ [1,...,n], the vectors {Pi1,...,ik | {i1,...,ik}⊂ [1,...,n] − {j}} form a basis of V .

We assume that a (potential) function L(z1,...,zn) is given.

We assume that nonzero vectors p1(z1,...,zn),...,pn(z1,...,zn) ∈ V depending on z1,...,zn are given such that for any i2,...,ik ∈ [1,...,n], we have n

(3.4) dj,i2,...,ik pj(z1,...,zn) = 0. Xj=1 3.2. Axioms. 4 ALEXANDER VARCHENKO

3.2.1. Invariance axiom. We assume that for any i2,...,ik ∈ [1,...,n], we have n ∂L (3.5) dj,i2,...,ik = 0. ∂zj Xj=1 Given z = (z1,...,zn), for j = 1,...,n, deﬁne a linear operator pj(z)∗z : V → V by the formula ∂ ∂2kL (3.6) S(pj(z) ∗z Pi1,...,ik , Pj1,...,jk )= (z). ∂zj ∂zi1 ...∂zik ∂zj1 ...∂zjk

Lemma 3.2. The operators pj(z)∗z are well-deﬁned.

Proof. We need to check that if we substitute in (3.6) a relation of the form (3.3) instead of Pi1,...,ik or

Pj1,...,jk or if we substitute in (3.6) a relation of the form (3.4) instead of pj(z), then the right-hand side of (3.6) is zero, but that follows from assumption (3.5).

Lemma 3.3. The operators pj(z)∗z are symmetric, S(pj(z) ∗z v, w) = S(v,pj(z) ∗z w) for any v, w ∈ V .

Choose a basis {Pα | α ∈ I} of V among the vectors {Pi1,...,ik }. Here each α is an unordered k-element n−1 ∂k ∂k subset {i1,...,ik} of [1,...,n] and |I| = . Denote k := . Then (3.6) can be written as k ∂ zα ∂zi1 ...∂zik ∂ ∂2kL (3.7) S(pj ∗z Pα, Pβ)= k k (z), α, β ∈I. ∂zj ∂ zα∂ zβ α,β Denote Sα,β = S(Pα, Pβ). Let (S )α,β∈I be the matrix inverse to the matrix (Sα,β)α,β∈I . Introduce γ γ the matrix (Mj,α)α,γ∈I of the operator pj(z)∗z by the formula pj(z) ∗z Pα = γ∈I Mj,α(z)Pγ . Then 2k+1 γ ∂ L β,γ P (3.8) Mj,α = k k S . ∂zj ∂ zα∂ zβ βX∈I 3.2.2. Unit element axiom. We assume that for some a ∈ C×, we have n 1 (3.9) z p (z)∗ = Id ∈ End(V ), a j j z Xj=1 n γ γ that is, j=1 zjMj,α(z) = a δα.

3.2.3. CommutativityP axiom. We assume that the operators p1(z)∗z,...,pn(z)∗z commute. In other words, β γ β γ (3.10) [Mi,αMj,β − Mj,αMi,β] = 0. Xβ This is a quadratic relation between the (2k + 1)-st derivatives of the potential.

3.2.4. Relation between pj(z) and Pi1,...,ik axiom. We assume that we have

(3.11) pi1 (z) ∗z ···∗z pik (z)= Pi1,...,ik , for every z and all unordered k-element subsets {i1,...,ik}⊂ [1,...,n]. α More precisely, let pj(z) = α∈I Cj (z)Pα, j = 1,...,n, be the expansion of the vector pj(z) with respect to the basis {P } . Then equation (3.11) is an equation on the coeﬃcients Cα (z) and α α∈I ik β P Mim,α(z), where α, β ∈I, m = i1,...,ik−1. For example, if Pi1,...,ik is one of the basis vectors (Pα)α∈I , then (3.12) Cαk (z)M αk−1 (z)M αk−2 (z) ...M α1 (z)= δα1 . ik ik−1,αk ik−2,αk−1 i1,α2 i1,...,ik α2,...,αXk∈I ON AXIOMS OF FROBENIUS LIKE STRUCTURE IN THE THEORY OF ARRANGEMENTS 5

Remark. If k = 1, then this axiom says that pj = Pj for j ∈ [1,...,n]. In this case the commutativity axiom follows from (3.6). Deﬁne the multiplication on V by the formula

(3.13) Pi1,...,ik ∗z Pj1,...,jk = pi1 (z) ∗z pi2 (z) ∗z ···∗z pik (z) ∗z Pj1,...,jk . The multiplication is well-deﬁned due to relations (3.3), (3.4), (3.5). Lemma 3.4. The multiplication on V is commutative.

Proof. Indeed, if the subsets {i1,...,ik} and {j1,...,jk} have a nonempty intersection, say ik = jk, then

Pi1,...,ik−1,ik ∗z Pj1,...,jk−1,ik = (pi1 ∗z ···∗z pik−1 ) ∗z pik ∗z (pj1 ∗z ···∗z pjk−1 ) ∗z pik =

= (pj1 ∗z ···∗z pjk−1 ) ∗z pik ∗z (pi1 ∗z ···∗z pik−1 ) ∗z pik = Pj1,...,jk−1,ik ∗z Pi1,...,ik−1,ik .

If the subsets {i1,...,ik} and {j1,...,jk} do not intersect, then

Pi1,...,ik ∗z Pj1,...,jk = pj1 ∗z ···∗z pjk−1 ∗z pi1 ∗z (pi2 ···∗z pik−1 ∗z pik ∗z pjk )=

= pj1 ∗z ···∗z pjk−1 ∗z pi1 ∗z Pi2,...,ik−1,ik,jk = pj1 ∗z ···∗z pjk−1 ∗z pi1 ∗z Pi2,...,ik−1,jk,ik =

= Pj1,...,jk ∗z Pi1,...,ik . γ For α, β ∈I, let Pα ∗z Pβ = γ∈I Nα,β(z)Pγ . If α = {i1,...,ik} then (3.14) N γ (z)= P M βk (z)M βk−1 (z) ...M β2 (z)M γ (z). α,β ik,β ik−1,βk i2,β3 i1,β2 β2,...,βXk∈I This is a polynomial of degree k in the (2k + 1)-st derivatives of the potential. 3.2.5. Associativity axiom. We assume that the multiplication on V is associative, that is, γ β γ β (3.15) [Nα2,α3 Nα1,γ − Nα1,α2 Nγ,α3 ] = 0. Xγ∈I Theorem 3.5. For any z, the axioms 3.2.1, 3.2.2, 3.2.3, 3.2.4, 3.2.5 define on the vector space V the 1 n structure ∗z of a commutative associative algebra with unit element 1z = a j=1 zjpj(z). The algebra is Frobenius, S(u ∗ v, w) = S(u, v ∗ w), for all u, v, w ∈ V . The elements p (z),...,p (z) generate z z P 1 n (V, ∗z) as an algebra. 3.2.6. Homogeneity axiom. We assume that n ∂ a2k+1 zj L = 2kL + S(1z, 1z), ∂zj (2k)! Xj=1 where a ∈ C× is the same number as in the unit element axiom. 3.2.7. Lifting axiom. For κ ∈ C×, define on the trivial bundle V × Cn → Cn a connection ∇κ by the formula: κ ∂ (3.16) ∇j = − κpj(z)∗z, j = 1,...,n. ∂zj C× κ κ κ Lemma 3.6. For any κ ∈ , the connection ∇ is flat, that is, [∇i , ∇j ] = 0 for all i, j.

Proof. The ﬂatness for any κ is equivalent to the commutativity of the operators pi(z)∗z, pj(z)∗z and the identity ∂ (p (z)∗ )= ∂ (p (z)∗ ). The commutativity is our axiom 3.2.3 and the identity follows ∂zj i z ∂zi j z from formula (3.8). 6 ALEXANDER VARCHENKO

κ A ﬂat section s(z1,...,zn) ∈ V of the connection ∇ is a solution of the system of equations ∂s (3.17) − κpj(z) ∗z s = 0, j = 1,...,n. ∂zj We say that a ﬂat section is liftable if it has the form ∂kg (3.18) s = d2 P ∂z ...∂z i1,...,ik i1,...,ik 6 6 i1 ik 1 i1

3.2.8. Diﬀerential operator e axiom. We assume that there exists a diﬀerential operator ∂k (3.19) e = e (z) i1,...,ik ∂z ...∂z 6 6 i1 ik 1 i1

(3.20) e˜ = ei1,...,ik (z)Pi1,...,ik ∈ V 6 6 1 i1

3.3. Remark. The axioms of an almost Frobenius structure in Section 2 are similar to the axioms of a Frobenius like structure in Section 3 for k = 1. The role of the tangent bundle T M and the vectors ∂ in Section 2 is played by the trivial bundle V × Cn → Cn and the vectors p (z) ∈ V as well as the ∂zj j vectors ∂ ∈ T Cn in Section 3. ∂zj

4. An example of a Frobenius like structure

4.1. Arrangements of hyperplanes. Consider a family of arrangements C(z) = {Hi(z)}i=1,...,n of n k aﬃne hyperplanes in C . The arrangements of the family depend on parameters z = (z1,...,zn). When the parameters change, each hyperplane moves parallelly to itself. More precisely, let t = (t1,...,tk) be k coordinates on C . For i = 1,...,n, the hyperplane Hi(z) is deﬁned by the equation k j j C fi(t, z) := bi tj + zi = 0, where bi ∈ are given . Xj=1 When zi changes the hyperplane Hi(z) moves parallelly to itself. We assume that every k hyperplanes intersect transversally, that is, for every distinct i1,...,ik, we assume that d := detk (bj ) 6= 0. i1,...,ik l,j=1 il

For every k + 1 distinct indices i1,...,ik+1, the corresponding hyperplanes have nonempty intersection if and only if k+1 l−1 (4.1) fi1,...,i (z) := (−1) d b zi = 0. k+1 i1,...,il,...,ik+1 l Xl=1 ON AXIOMS OF FROBENIUS LIKE STRUCTURE IN THE THEORY OF ARRANGEMENTS 7

6 6 Cn For any 1 i1 < · · · < ik+1 n denote by Hi1...,ik+1 the hyperplane in deﬁned by the equation Cn fi1,...,ik+1 (z) = 0. The union ∆ = ∪16i1<···

C× n 4.2. Master function. Fix complex numbers a1, . . . , an ∈ such that |a| := j=1 aj 6= 0. The master function is the function P n Φ(t, z)= aj log fj(t, z). Xj=1 For fixed z, the master function is defined on the complement to our arrangement C(z) as a multivalued holomorphic function, whose derivatives are rational functions. In particular, ∂Φ = ai . For fixed z, ∂zi fi define the critical set Ck n ∂Φ Cz := t ∈ −∪j=1Hj(z) (t, z) = 0, j = 1,...,k ∂tj and the algebra of functions on the critical set C Ck n ∂Φ Az := ( −∪j=1Hj(z)) (t, z), j = 1,...,k , ∂tj . C Ck n Ck n where ( −∪j=1Hj(z)) is the algebra of regular functions on −∪j=1Hj(z). It is known that n−1 dim Az = k . For example this follows from Theorem 2.4 and Lemmas 4.1, 4.2 in [V3]. Denote by ∗ the operation of multiplication in A . z z The Grothendieck bilinear form on Az is the form 1 gh dt ∧···∧ dt (g, h) := 1 k for g, h ∈ A . z k k ∂Φ z (2πi) Γ Z j=1 ∂tj Here Γ = {t | | ∂Φ | = ǫ, j = 1,...,k}, where ǫQ > 0 is small. The cycle Γ is oriented by the condition ∂tj d arg ∂Φ ∧···∧ d arg ∂Φ > 0, The pair A , ( , ) is a Frobenius algebra. ∂t1 ∂tk z z 4.3. Algebra Az. For i = 1,...,n, denote

∂Φ ai (4.2) pi(z)= (t, z) = ∈ Az, i = 1,...,n. ∂zi fi(t, z) h i h i For any i1,...,ik ∈ [1,...,n], denote

Pi1,...,ik := pi1 (z) ∗z ···∗z pik (z) ∈ Az and wi1,...,ik := di1,...,ik pi1 (z) ∗z ···∗z pik (z) ∈ Az. Theorem 4.1.

(i) The elements p1(z),...,pn(z) generate Az as an algebra and for any i2,...,ik we have n

(4.3) dj,i2,...,ik pj(z) = 0. Xj=1 (ii) The element n 1 (4.4) 1 := z p (z) z |a| j j Xj=1 is the unit element of the algebra Az. 8 ALEXANDER VARCHENKO

(iii) The set of all elements wi1,...,ik generates Az as a vector space. The only linear relations between these elements are σ (4.5) wi1,...,ik = (−1) wσi1,...,σik , ∀σ ∈ Sk, n

wj,i2,...,ik = 0, ∀ i2,...,ik, Xj=1 in particular, the linear relations between these elements do not depend on z. (iv) The Grothendieck form has the following matrix elements:

(4.6) (wi1,...,ik , wj1,...,jk )z = 0, if |{i1,...,ik} ∩ {j1,...,jk}| < k − 1, k+1 (−1)k+1 (w , w ) = a if i ,...,i are distinct, i1,...,ik i1,...,ik−1,ik+1 z |a| il 1 k+1 Yl=1 k (−1)k (w , w ) = a a if i ,...,i are distinct i1,...,ik i1,...,ik z |a| il m 1 k Yl=1 m/∈{Xi1,...,ik} Proof. The statements of the theorem are the statements of Lemmas 3.4, 6.2, 6.3, 6.6, 6.7 in [V4] transformed with the help of the isomorphism α of Theorem 2.7 in [V4], see also Corollary 6.16 in [V4] and Theorems 2.12 and 2.16 in [V5]. Z This theorem shows that the elements wi1,...,ik deﬁne on Az a module structure over independent of z. With respect to this Z-structure the Grothendieck form is constant.

Remark. The presence of this Z-structure in Az reﬂects the presence of the Z-structure in the Orlik- Solomon algebra of the arrangement C(z), see [V5, OS].

Lemma 4.2. Multiplication in Az is deﬁned by the formulas k+1 di2,...,ik+1 ℓ+1 (4.7) pi1 (z) ∗z wi2,...,i = (−1) ai w b , k+1 ℓ i1,...,iℓ,...,ik+1 fi1,i2,...,ik+1 (z) Xℓ=1 if i1 ∈/ {i2,...,ik+1},

(4.8) pi1 (z) ∗z wi1,i2,...,ik = − pi1 (z) ∗z wm,i2,...,ik , m/∈{Xi1,...,ik} where fi1,i2,...,ik+1 (z) is deﬁned in (4.1). Proof. This is Lemma 6.8 in [V4].

Theorem 4.3. For any i0 ∈ [1,...,n], the unit element 1z ∈ Az is given by the formula k 1 (fi0,i1,...,ik (z)) (4.9) 1z = wi1,...,i . k k m k |a| 16 6 (−1) d c i1<···

(4.10) Q(z) := (1z, 1z)z. The function (−1)kQ(z) is called in [V4] the potential of ﬁrst kind. Theorem 3.11 in [V4] says that for any i1,...,ik, j1,...,jk ∈ [1,...,n], we have |a|2k ∂2kQ (4.11) (pi1 (z) ∗z ···∗z pik (z),pj1 (z) ∗z ···∗z pjk (z))z = . (2k)! ∂zi1 ...∂zik ∂zj1 ...∂zjk ON AXIOMS OF FROBENIUS LIKE STRUCTURE IN THE THEORY OF ARRANGEMENTS 9

Theorem 4.4. We have k+1 1 a (4.12) Q(z)= im (f (z))2k. |a|2k+1 2 i1,...,ik+1 6 6  d c  1 i1<···X

Theorem 4.5 ([V4]). For any i0, i1,...,ik, j1,...,jk ∈ [1,...,n], we have ∂2k+1L (4.14) (pi0 (z) ∗z pi1 (z) ∗z ···∗z pik (z),pj1 (z) ∗z ···∗z pjk (z))z = . ∂zi0 ∂zi1 ...∂zik ∂zj1 ...∂zjk This is Theorem 6.20 in [V4].

Lemma 4.6. For any i2,...,ik ∈ [1,...,n], we have n ∂L (4.15) dj,i2,...,ik = 0. ∂zj Xj=1 Proof. The lemma follows from the Pl¨ucker relations (3.2). Lemma 4.7. We have n ∂L |a|2k+1 (4.16) zj = 2kL + (1z, 1z)z. ∂zj (2k)! Xj=1

Proof. The lemma follows from the fact that each fi1,...,ik+1 is a homogeneous polynomial in z1,...,zn of degree one and from Theorem 4.4.

4.5. Bundle of algebras Az. The family of algebras Az form a vector bundle over the space of parameters z ∈ Cn − ∆. The Z-module structures of ﬁbers canonically trivialize the vector bundle n π : ⊔z∈Cn−∆Az → C − ∆. Let s(z) ∈ Az be a section of that bundle,

(4.17) s(z)= si1,...,ik (z)wi1,...,ik , i1X,...,ik where si1,...,ik (z) are scalar functions. Deﬁne the combinatorial connection on the bundle of algebras by the rule

∂s ∂si1,...,ik (4.18) := wi1,...,ik , ∀ j. ∂zj ∂zj i1X,...,ik 10 ALEXANDER VARCHENKO

This derivative does not depend on the presentation (4.17). The combinatorial connection is ﬂat. The combinatorial connection is compatible with the Grothendieck bilinear form, that is ∂ ∂ ∂ (s , s ) = s , s + s , s ∂z 1 2 z ∂z 1 2 1 ∂z 2 j j z j z for any j and sections s1,s2. For κ ∈ C×, introduce a connection ∇κ on the bundle of algebras π by the formula

κ ∂ ∇j := − κpj(z)∗z , j = 1,...,n. ∂zj This family of connections is a deformation of the combinatorial connection. Theorem 4.8. For any κ ∈ C×, the connection ∇κ is ﬂat, κ κ [∇j , ∇m] = 0, ∀ j, m. Proof. This statement is a corollary of Theorems 5.1 and 5.9 in [V3].

4.6. Flat sections. Given κ ∈ C×, the ﬂat sections

(4.19) s(z)= si1,...,ik (z)wi1,...,ik ∈ Az 6 6 1 i1

Theorem 4.9. Assume that κ is such that κ|a| ∈/ Z60 and κaj ∈/ Z60 for j = 1,...,n, then for every flat section s(z), there is a locally flat section of the homology bundle γ(z) ∈ Hk(U(C(x)), such that the coefficients si1,...,ik (z) in (4.19) are given by the following formula:

κΦ(t,z) 6 6 (4.21) si1,...,ik (z)= e d log fi1 ∧···∧ d log fik , ∀ 1 i1 < · · · < ik n. Zγ(z) Proof. This is Lemma 5.7 in [V3]. The information on κ see in Theorem 1.1 in [V1].

Theorem 4.9 identifies the connection ∇κ with the Gauss-Manin connection on the homology bundle. The flat sections are labeled by the locally constant cycles γ(z) in the complement to our arrangement. The space of flat sections is identified with the degree k homology group of the complement to the arrangement C(z). The monodromy of the flat sections is the monodromy of that homology group.

4.7. Special case. Assume that aj = 1 for all j ∈ [1,...,n]. Then |a| = n. Denote

−k κΦ(t,z) (4.22) g(z1,...,zn, κ) = κ e dt1 ∧···∧ dtk = Zγ(z) κ n −k = κ fj(t, z) dt1 ∧···∧ dtk. γ(z)   Z jY=1   ON AXIOMS OF FROBENIUS LIKE STRUCTURE IN THE THEORY OF ARRANGEMENTS 11

Lemma 4.10. If aj = 1 for all j ∈ [1,...,n], then the ﬂat section of Theorem 4.9 can be written in the form ∂kg(z, κ) (4.23) s(z)= d2 P , ∂z ...∂z i1,...,ik i1,...,ik 6 6 i1 ik 1 i1

Theorem 4.12. Assume that aj = 1 for all j ∈ [1,...,n], then there exists a diﬀerential operator ∂k (4.24) e = e (z) i1,...,ik ∂z ...∂z 6 6 i1 ik 1 i1

dt1 ∧···∧ dtk (4.27) ei1,...,i = di1,...,i Res . . . Res . k k fi1 (t,z)=0 fik (t,z)=0 n j=1 fj(t, z) ! Conversely, if we choose such coeﬃcients, then formula (4.26) holds.Q Clearly, the elemente ˜ = n 1 j=1 fj is invertible in A . z Q C× i 4.8. Summary. Fix a1, . . . , an ∈ , such that |a|= 6 1. Then the matrix (bj), algebra Az, bilinear form ( , )z, vectors pj(z), vectors Pi1,...,ik , element 1z, function L(z1,...,zn) of Sections 4.1-4.4 satisfy axioms 3.2.1, 3.2.2, 3.2.3, 3.2.4, 3.2.5, 3.2.6. If in addition we assume that aj = 1 for all j ∈ [1,...,n], then these objects also satisfy axioms 3.2.7 and 3.2.8. 12 ALEXANDER VARCHENKO

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