Table Of ContentON AXIOMS OF FROBENIUS LIKE STRUCTURE IN
THE THEORY OF ARRANGEMENTS
ALEXANDERVARCHENKO⋆
Department of Mathematics, University of North Carolina at Chapel Hill
6
Chapel Hill, NC 27599-3250, USA
1
0
2
l Abstract. AFrobeniusmanifold isamanifold with aflat metricandaFrobeniusalgebra structureon
u
J tangentspacesatpointsofthemanifold suchthatthestructureconstantsofmultiplication aregivenby
third derivatives of a potential function on themanifold with respect to flat coordinates.
4
In this paper we present a modification of that notion coming from the theory of arrangements of
] hyperplanes. Namely, given natural numbers n > k, we have a flat n-dimensional manifold and a
G vectorspaceV withanondegeneratesymmetricbilinearformandanalgebrastructureonV,depending
A on points of the manifold, such that the structure constants of multiplication are given by 2k +1-st
derivatives of a potential function on the manifold with respect to flat coordinates. We call such a
.
h structurea Frobenius like structure. Sucha structurearises whenonehas afamily ofarrangements of n
t affinehyperplanesin Ck dependingon parameters sothat thehyperplanesmoveparallely tothemselves
a
m whentheparameterschange. InthatcaseaFrobeniuslikestructurearisesonthebaseCn ofthefamily.
[
3 1. Introduction
v
8 The theory of Frobenius manifolds has multiple connections with other branches of mathematics,
0
such as quantum cohomology, singularity theory, the theory of integrable systems, see, for example,
2
2 [D1, D2, M]. The notion of a Frobenius manifold was introduced by B.Dubrovin in [D1], see also
0 [D2, M, FV, St], where numerous variants of this notion are discussed. In all alterations, a Frobenius
.
1 manifold is a manifold with a flat metric and a Frobenius algebra structure on tangent spaces at points
0 of the manifold such that the structure constants of multiplication are given by third derivatives of a
6
potential function on the manifold with respect to flat coordinates.
1
: In this paper we present a modification of that notion coming from the theory of arrangements
v
of hyperplanes. Namely, given natural numbers n > k, we have a flat n-dimensional manifold and a
i
X vector space V withanondegenerate symmetricbilinear formand analgebra structureon V, depending
r on points of the manifold, such that the structure constants of multiplication are given by 2k +1-st
a
derivatives of a potential function on the manifold with respect to flat coordinates. We call such a
structure a Frobenius like structure. Such a structure arises when one has a family of arrangements of n
affine hyperplanes in Ck dependingon parameters so that the hyperplanes move parallely to themselves
when the parameters change. In that case a Frobenius like structure arises on the base Cn of the family.
On such families see, for example, [MS, SV].
In Section 2, we give Dubrovin’s definition of almost Frobenius structure. In Section 3, we give
the definition of Frobenius like structure motivated by Dubrovin’s definition and results of [V4]. The
definition consists of eight axioms: 3.2.1, 3.2.2, 3.2.3, 3.2.4, 3.2.5, 3.2.6, 3.2.7, 3.2.8. In Section 4, we
consider a family of arrangements and construct a Frobenius like structure on its base.
The paper had been written while the author visited the MPI in Bonn in 2015-2016. The author
thanks the MPI for hospitality, Yu.I.Manin for interest in this work, P.Dunin-Barkowski, B.Dubrovin,
and C. Hertling for useful discussions.
⋆E-mail: [email protected], supported in part by NSFgrant DMS-1362924 and Simons Foundation grant #336826.
2 ALEXANDERVARCHENKO
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 T M = (∗ ,(, ) ), where z ∈ M. It satisfies the following axioms.
z z z
Axiom 1. The metric (, ) is flat.
z
Axiom 2. In the flat coordinates z ,...,z for the metric, the structure constants of the multiplication
1 n
∂ ∂ ∂
(2.1) ∗ = Nk (z)
∂z z ∂z i,j ∂z
i j k
can be represented in the form
∂3L
(2.2) Nk (z) = Gk,l (z)
i,j ∂z ∂z ∂z
l i j
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) z (z) = 2L(z)+ G z z .
j i,j i j
∂z 1−d
j
j=1 i,j
X X
Axiom 4. The Euler vector field
n
1−d ∂
(2.4) E = z
j
2 ∂z
j
j=1
X
(1−d)2
is the unit of the Frobenius algebra. Notice that (E,E) = G z z .
4 i,j i,j i j
Axiom 5. There exists a vector field
P
n
∂
(2.5) e = e (z) ,
j
∂z
j
j=1
X
an invertible element of the Frobenius algebra T M for every z ∈ M, such that the operator g(z) 7→
z
(eg)(z) acts by the shift κ 7→ κ−1 on the space of solutions of the system of equations
∂2g n ∂g
(2.6) = κ Nl , i,j = 1,...,n.
∂z ∂z i,j∂z
i j l
l=1
X
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 )2log(z −z )
1 n i j i j
4
i<j
X
and let the metric be ( ∂ , ∂ ) = δ . The third derivatives of F(z) give the multiplication law of
∂zi ∂zj i,j
tangent vectors ∂ := ∂ :
i ∂zi
n∂ −∂
i j
(2.8) ∂ ∗ ∂ = − for i6= j, ∂ ∗ ∂ = − ∂ ∗ ∂ .
i z j i z i i z j
2 z −z
i j
j6=i
X
The unit element of the algebra is the Euler vector field
n
1
(2.9) E = z ∂ .
j j
n
j=1
X
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<j i j
The vector field e is
n n
1
(2.10) e= − ∂ , f(t):= (t−z ).
f′(z ) j i
j
j=1 i=1
X Y
Equations (2.6) take the form
κ
(2.11) ∂ ∂ g = − (∂ g−∂ g), i6= j
i j i j
z −z
i j
with solutions
n
(2.12) g(z ,...,z ) = (t−z )κdt.
1 n i
Z i=1
Y
3. Axioms of a Frobenius like structure
3.1. Objects. Let an n×k-matrix (bj)j=1,...,k be given with n > k. For i ,...,i ∈ [1,...,n], denote
i i=1,...,n 1 k
j
(3.1) d = det(b ) .
i1,...,ik il j,l=1,...,k
We have d = (−1)σd for σ ∈ S . We assume that d 6= 0 for all distinct i ,...,i .
σi1,...,σik i1,...,ik k i1,...,ik 1 k
Remark. The numbers (d ) satisfy the Plu¨cker relations. For example, for arbitrary j , ..., j ,
i1,...,ik 1 k+1
i , ..., i ∈ [1,...,n], we have
2 k
k+1
(3.2) (−1)m+1dj1,...,jcm,...,jk+1djm,i2,...,ik = 0.
m=1
X
We assume that an n−1 -dimensional complex vector space V is given with a non-degenerate sym-
k
metric bilinear form S(, ).
(cid:0) (cid:1)
We assume that for every k-element unordered subset {i ,...,i } ⊂ [1,...,n] a nonzero vector
1 k
P ∈ V is given such that the vectors {P } generate V as a vector space and the only linear
i1,...,ik i1,...,ik
relations among them are
n
(3.3) d P = 0, ∀i ,...,i ∈ [1,...,n].
j,i2,...,ik j,i2,...,ik 2 k
j=1
X
Lemma 3.1. For any j ∈ [1,...,n], the vectors {P | {i ,...,i } ⊂ [1,...,n]−{j}} form a basis
i1,...,ik 1 k
of V. (cid:3)
We assume that a (potential) function L(z ,...,z ) is given.
1 n
We assume that nonzero vectors p (z ,...,z ),...,p (z ,...,z ) ∈ V depending on z ,...,z are
1 1 n n 1 n 1 n
given such that for any i ,...,i ∈ [1,...,n], we have
2 k
n
(3.4) d p (z ,...,z )= 0.
j,i2,...,ik j 1 n
j=1
X
3.2. Axioms.
4 ALEXANDERVARCHENKO
3.2.1. Invariance axiom. We assume that for any i ,...,i ∈[1,...,n], we have
2 k
n
∂L
(3.5) d = 0.
j,i2,...,ik∂z
j
j=1
X
Given z = (z ,...,z ), for j = 1,...,n, define a linear operator p (z)∗ :V → V by the formula
1 n j z
∂ ∂2kL
(3.6) S(p (z)∗ P ,P ) = (z).
j z i1,...,ik j1,...,jk ∂z ∂z ...∂z ∂z ...∂z
j i1 ik j1 jk
Lemma 3.2. The operators p (z)∗ are well-defined.
j z
Proof. We need to check that if we substitute in (3.6) a relation of the form (3.3) instead of P or
i1,...,ik
P or if we substitute in (3.6) a relation of the form (3.4) instead of p (z), then the right-hand side
j1,...,jk j
of (3.6) is zero, but that follows from assumption (3.5). (cid:3)
Lemma 3.3. The operators p (z)∗ are symmetric, S(p (z)∗ v,w) = S(v,p (z)∗ w) for any v,w ∈
j z j z j z
V. (cid:3)
Chooseabasis{P |α ∈ I}ofV amongthevectors{P }. Hereeachαisanunorderedk-element
α i1,...,ik
subset {i ,...,i } of [1,...,n] and |I|= n−1 . Denote ∂k := ∂k . Then (3.6) can be written as
1 k k ∂kzα ∂zi1...∂zik
(cid:0) ∂(cid:1) ∂2kL
(3.7) S(p ∗ P ,P )= (z), α,β ∈ I.
j z α β ∂z ∂kz ∂kz
j α β
Denote S = S(P ,P ). Let (Sα,β) be the matrix inverse to the matrix (S ) . Introduce
α,β α β α,β∈I α,β α,β∈I
γ γ
the matrix (M ) of the operator p (z)∗ by the formula p (z)∗ P = M (z)P . Then
j,α α,γ∈I j z j z α γ∈I j,α γ
(3.8) Mγ = ∂2k+1L Sβ,γ. P
j,α ∂z ∂kz ∂kz
j α β
β∈I
X
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),
j j z
a
j=1
X
that is, n z Mγ (z) = aδγ.
j=1 j j,α α
3.2.3. CPommutativity axiom. We assume that the operators p (z)∗ ,...,p (z)∗ commute. In other
1 z n z
words,
β γ β γ
(3.10) [M M −M M ]= 0.
i,α j,β j,α i,β
β
X
This is a quadratic relation between the (2k+1)-st derivatives of the potential.
3.2.4. Relation between p (z) and P axiom. We assume that we have
j i1,...,ik
(3.11) p (z)∗ ···∗ p (z) = P ,
i1 z z ik i1,...,ik
for every z and all unordered k-element subsets {i ,...,i } ⊂ [1,...,n].
1 k
More precisely, let p (z) = Cα(z)P , j = 1,...,n, be the expansion of the vector p (z) with
j α∈I j α j
respect to the basis {P } . Then equation (3.11) is an equation on the coefficients Cα(z) and
α α∈IP ik
β
M (z), where α,β ∈ I, m = i ,...,i . For example, if P is one of the basis vectors (P ) ,
im,α 1 k−1 i1,...,ik α α∈I
then
(3.12) Cαk(z)Mαk−1 (z)Mαk−2 (z)...Mα1 (z) = δα1 .
ik ik−1,αk ik−2,αk−1 i1,α2 i1,...,ik
α2,.X..,αk∈I
ON AXIOMS OF FROBENIUS LIKE STRUCTURE IN THE THEORY OF ARRANGEMENTS 5
Remark. If k = 1, then this axiom says that p = P for j ∈ [1,...,n]. In this case the commutativity
j j
axiom follows from (3.6).
Define the multiplication on V by the formula
(3.13) P ∗ P = p (z)∗ p (z)∗ ···∗ p (z)∗ P .
i1,...,ik z j1,...,jk i1 z i2 z z ik z j1,...,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 {i ,...,i } and {j ,...,j } have a nonempty intersection, say i = j ,
1 k 1 k k k
then
P ∗ P = (p ∗ ···∗ p )∗ p ∗ (p ∗ ···∗ p )∗ p =
i1,...,ik−1,ik z j1,...,jk−1,ik i1 z z ik−1 z ik z j1 z z jk−1 z ik
= (p ∗ ···∗ p )∗ p ∗ (p ∗ ···∗ p )∗ p = P ∗ P .
j1 z z jk−1 z ik z i1 z z ik−1 z ik j1,...,jk−1,ik z i1,...,ik−1,ik
If the subsets {i ,...,i } and {j ,...,j } do not intersect, then
1 k 1 k
P ∗ P =p ∗ ···∗ p ∗ p ∗ (p ···∗ p ∗ p ∗ p )=
i1,...,ik z j1,...,jk j1 z z jk−1 z i1 z i2 z ik−1 z ik z jk
= p ∗ ···∗ p ∗ p ∗ P = p ∗ ···∗ p ∗ p ∗ P =
j1 z z jk−1 z i1 z i2,...,ik−1,ik,jk j1 z z jk−1 z i1 z i2,...,ik−1,jk,ik
= P ∗ P .
j1,...,jk z i1,...,ik
(cid:3)
γ
For α,β ∈ I, let P ∗ P = N (z)P . If α= {i ,...,i } then
α z β γ∈I α,β γ 1 k
(3.14) Nγ (z) = P Mβk (z)Mβk−1 (z)...Mβ2 (z)Mγ (z).
α,β ik,β ik−1,βk i2,β3 i1,β2
β2,.X..,βk∈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,
(3.15) [Nγ Nβ −Nγ Nβ ] = 0.
α2,α3 α1,γ α1,α2 γ,α3
γ∈I
X
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
structure ∗ of a commutative associative algebra with unit element 1 = 1 n z p (z). The algebra
z z a j=1 j j
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 1 n
P
(V,∗ ) as an algebra. (cid:3)
z
3.2.6. Homogeneity axiom. We assume that
n ∂ a2k+1
z L = 2kL + S(1 ,1 ),
j z z
∂z (2k)!
j
j=1
X
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:
∂
(3.16) ∇κ = − κp (z)∗ , j = 1,...,n.
j ∂z j z
j
Lemma 3.6. For any κ∈ C×, the connection ∇κ is flat, that is, [∇κ,∇κ]= 0 for all i,j.
i j
Proof. The flatness for any κ is equivalent to the commutativity of the operators p (z)∗ , p (z)∗ and
i z j z
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). (cid:3)
6 ALEXANDERVARCHENKO
A flat section s(z ,...,z ) ∈ V of the connection ∇κ is a solution of the system of equations
1 n
∂s
(3.17) − κp (z)∗ s = 0, j = 1,...,n.
j z
∂z
j
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
16i1<X···<ik6n i1 ik
for a scalar function g(z ,...,z ). If a section is liftable, the function g(z) is not unique (an arbitrary
1 n
polynomial of degree less than k can be added). Such a function g will be called a twisted period.
We assume that for generic κ all flat sections of ∇κ are liftable and the set of all exceptional κ is
contained in a finite union of arithmetic progressions in C.
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
16i1<X···<ik6n i1 ik
such that the operator g(z) 7→ (eg)(z) acts as the shift κ7→ κ−1 on the space of solutions of equations
(3.17) and (3.18). We also require that the element
(3.20) e˜= e (z)P ∈ V
i1,...,ik i1,...,ik
16i1<X···<ik6n
is invertible for every z.
Notice that the operator e annihilates all polynomials in z of degree less than k.
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 TM 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 ∂ ∈ TCn in Section 3.
∂zj
4. An example of a Frobenius like structure
4.1. Arrangements of hyperplanes. Consider a family of arrangements C(z) = {H (z)} of n
i i=1,...,n
affinehyperplanes in Ck. Thearrangements of the family dependon parameters z =(z ,...,z ). When
1 n
the parameters change, each hyperplane moves parallelly to itself. More precisely, let t = (t ,...,t ) be
1 k
coordinates on Ck. For i = 1,...,n, the hyperplane H (z) is defined by the equation
i
k
f (t,z) := bjt +z = 0, where bj ∈ C are given.
i i j i i
j=1
X
When z changes the hyperplane H (z) moves parallelly to itself.
i i
We assume that every k hyperplanes intersect transversally, that is, for every distinct i ,...,i , we
1 k
assume that
d := detk (bj) 6= 0.
i1,...,ik l,j=1 il
For every k+1 distinct indices i ,...,i , the corresponding hyperplanes have nonempty intersection
1 k+1
if and only if
k+1
(4.1) fi1,...,ik+1(z) := (−1)l−1di1,...,ibl,...,ik+1zil = 0.
l=1
X
ON AXIOMS OF FROBENIUS LIKE STRUCTURE IN THE THEORY OF ARRANGEMENTS 7
For any 1 6 i < ··· < i 6 n denote by H the hyperplane in Cn defined by the equation
1 k+1 i1...,ik+1
fi1,...,ik+1(z) = 0. The union ∆ = ∪16i1<···<ik+16nHi1,...,ik+1 ⊂ Cn is called the discriminant. We will
consider only z ∈ Cn−∆. In that case the arrangement C(z) = {H (z)} has normal crossings.
i i=1,...,n
4.2. Master function. Fix complex numbers a ,...,a ∈ C× such that |a| := n a 6= 0. The
1 n j=1 j
master function is the function
P
n
Φ(t,z) = a logf (t,z).
j j
j=1
X
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
∂Φ
C := t ∈Ck −∪n H (z) (t,z) = 0, j = 1,...,k
z j=1 j ∂t
(cid:26) (cid:12) j (cid:27)
and the algebra of functions on the critical set (cid:12)
(cid:12)
∂Φ
A := C(Ck −∪n H (z)) (t,z), j = 1,...,k ,
z j=1 j ∂t
(cid:28) j (cid:29)
.
where C(Ck − ∪n H (z)) is the algebra of regular functions on Ck − ∪n H (z). It is known that
j=1 j j=1 j
dimA = n−1 . For example this follows from Theorem 2.4 and Lemmas 4.1, 4.2 in [V3].
z k
Denote by ∗ the operation of multiplication in A .
z z
(cid:0) (cid:1)
The Grothendieck bilinear form on A is the form
z
1 gh dt ∧···∧dt
1 k
(g,h) := for g,h ∈ A .
z (2πi)k k ∂Φ z
ZΓ 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
darg ∂Φ ∧···∧darg ∂Φ > 0, The pair A ,(, ) is a Frobenius algebra.
∂t1 ∂tk z z
(cid:0) (cid:1)
4.3. Algebra A . For i= 1,...,n, denote
z
∂Φ a
i
(4.2) p (z) = (t,z) = ∈ A , i= 1,...,n.
i z
∂z f (t,z)
i i
h i h i
For any i ,...,i ∈ [1,...,n], denote
1 k
P := p (z)∗ ···∗ p (z) ∈ A and w := d p (z)∗ ···∗ p (z) ∈A .
i1,...,ik i1 z z ik z i1,...,ik i1,...,ik i1 z z ik z
Theorem 4.1.
(i) The elements p (z),...,p (z) generate A as an algebra and for any i ,...,i we have
1 n z 2 k
n
(4.3) d p (z) = 0.
j,i2,...,ik j
j=1
X
(ii) The element
n
1
(4.4) 1 := z p (z)
z j j
|a|
j=1
X
is the unit element of the algebra A .
z
8 ALEXANDERVARCHENKO
(iii) The set of all elements w generates A as a vector space. The only linear relations between
i1,...,ik z
these elements are
(4.5) w = (−1)σw , ∀σ ∈ S ,
i1,...,ik σi1,...,σik k
n
w = 0, ∀ i ,...,i ,
j,i2,...,ik 2 k
j=1
X
in particular, the linear relations between these elements do not depend on z.
(iv) The Grothendieck form has the following matrix elements:
(4.6) (w ,w ) = 0, if |{i ,...,i }∩{j ,...,j }| <k−1,
i1,...,ik j1,...,jk z 1 k 1 k
(−1)k+1 k+1
(w ,w ) = a if i ,...,i are distinct,
i1,...,ik i1,...,ik−1,ik+1 z |a| il 1 k+1
l=1
Y
(−1)k 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]. (cid:3)
This theorem shows that the elements w define on A a module structure over Z independent
i1,...,ik z
of z. With respect to this Z-structure the Grothendieck form is constant.
Remark. The presence of this Z-structure in A reflects the presence of the Z-structure in the Orlik-
z
Solomon algebra of the arrangement C(z), see [V5, OS].
Lemma 4.2. Multiplication in A is defined by the formulas
z
k+1
d
(4.7) pi1(z)∗z wi2,...,ik+1 = fi1,ii22,.,....,.,iikk++11(z) ℓ=1(−1)ℓ+1aiℓwi1,...,ibℓ,...,ik+1,
X
if i ∈/ {i ,...,i },
1 2 k+1
(4.8) p (z)∗ w = − p (z)∗ w ,
i1 z i1,i2,...,ik i1 z m,i2,...,ik
m∈/{Xi1,...,ik}
where f (z) is defined in (4.1).
i1,i2,...,ik+1
Proof. This is Lemma 6.8 in [V4]. (cid:3)
Theorem 4.3. For any i ∈ [1,...,n], the unit element 1 ∈ A is given by the formula
0 z z
1 (f (z))k
(4.9) 1 = i0,i1,...,ik w .
z |a|k 16i1<X···<ik6n km=0(−1)mdi0,i1,...,icm,...,ik i1,...,ik
i0∈/{i1,...,ik}
Q
Proof. This is Theorem 6.12 in [V4]. (cid:3)
4.4. Potential functions. Denote
(4.10) Q(z) := (1 ,1 ) .
z z z
The function (−1)kQ(z) is called in [V4] the potential of first kind. Theorem 3.11 in [V4] says that for
any i ,...,i ,j ,...,j ∈ [1,...,n], we have
1 k 1 k
|a|2k ∂2kQ
(4.11) (p (z)∗ ···∗ p (z),p (z)∗ ···∗ p (z)) = .
i1 z z ik j1 z z jk z (2k)! ∂z ...∂z ∂z ...∂z
i1 ik j1 jk
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 d2 i1,...,ik+1
16i1<·X··<ik+16n mY=1 i1,...,icm,...,ik+1
Cf. formulas (4.22) and (5.58) in [V4].
Proof. Denote by Q∨ the right-hand side of (4.12). Both Q(z) and Q∨(z) are homogeneous polynomials
of degree 2k in z ,...,z . The2k-th derivatives of Q(z) are given by formulas (4.11) and(4.5). Adirect
1 n
verification shows that the 2k-th derivatives of Q∨(z) also are given by the same formulas (4.5). Hence
Q(z) = Q∨(z). (cid:3)
Introduce the potential function
(4.13)
k+1
1 a
L(z ,...,z ) = im (f (z))2klog(f (z)).
1 n (2k)! d2 i1,...,ik+1 i1,...,ik+1
16i1<·X··<ik+16n mY=1 i1,...,icm,...,ik+1
In [V4] this function was called the potential of second kind.
Theorem 4.5 ([V4]). For any i ,i ,...,i ,j ,...,j ∈ [1,...,n], we have
0 1 k 1 k
∂2k+1L
(4.14) (p (z)∗ p (z)∗ ···∗ p (z),p (z)∗ ···∗ p (z)) = .
i0 z i1 z z ik j1 z z jk z ∂z ∂z ...∂z ∂z ...∂z
i0 i1 ik j1 jk
This is Theorem 6.20 in [V4].
Lemma 4.6. For any i ,...,i ∈ [1,...,n], we have
2 k
n
∂L
(4.15) d = 0.
j,i2,...,ik∂z
j
j=1
X
Proof. The lemma follows from the Plu¨cker relations (3.2). (cid:3)
Lemma 4.7. We have
n ∂L |a|2k+1
(4.16) z = 2kL + (1 ,1 ) .
j z z z
∂z (2k)!
j
j=1
X
Proof. The lemma follows from the fact that each f is a homogeneous polynomial in z ,...,z
i1,...,ik+1 1 n
of degree one and from Theorem 4.4. (cid:3)
4.5. Bundle of algebras A . The family of algebras A form a vector bundle over the space of
z z
parameters z ∈ Cn − ∆. The Z-module structures of fibers canonically trivialize the vector bundle
π : ⊔z∈Cn−∆Az → Cn−∆.
Let s(z)∈ A be a section of that bundle,
z
(4.17) s(z) = s (z)w ,
i1,...,ik i1,...,ik
i1X,...,ik
where s (z) are scalar functions. Define the combinatorial connection on the bundle of algebras by
i1,...,ik
the rule
∂s ∂s
(4.18) := i1,...,ikw , ∀ j.
∂z ∂z i1,...,ik
j j
i1X,...,ik
10 ALEXANDERVARCHENKO
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 (cid:18) j (cid:19)z (cid:18) j (cid:19)z
for any j and sections s ,s .
1 2
For κ∈ C×, introduce a connection ∇κ on the bundle of algebras π by the formula
∂
∇κ := − κp (z)∗ , j = 1,...,n.
j ∂z j z
j
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 ] = 0, ∀ j,m.
j m
Proof. This statement is a corollary of Theorems 5.1 and 5.9 in [V3]. (cid:3)
4.6. Flat sections. Given κ∈ C×, the flat sections
(4.19) s(z) = s (z)w ∈ A
i1,...,ik i1,...,ik z
16i1<X···<ik6n
of the connection ∇κ are given by the following construction.
The function eκΦ(t,z) = ( ni=1fi(t,z)ai)κ defines a rank one local system Lκ on U(C(z)) := Ck −
∪n H (z) the complement to the arrangement C(z), whose horizontal sections over open subsets of
i=1 i
Q
U(z) are uni-valued branches of eκΦ(t,z) multiplied by complex numbers, see, for example, [V2]. The
vector bundle
(4.20) ⊔x∈Cn−∆ Hk(U(C(x)),Lκ|U(C(x))) → Cn−∆,
called the homology bundle, has the canonical flat Gauss-Manin connection.
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) ∈H (U(C(x)), such that the
k
coefficients s (z) in (4.19) are given by the following formula:
i1,...,ik
(4.21) s (z) = eκΦ(t,z)dlogf ∧···∧dlogf , ∀ 1 6 i < ··· < i 6 n.
i1,...,ik i1 ik 1 k
Zγ(z)
Proof. This is Lemma 5.7 in [V3]. The information on κ see in Theorem 1.1 in [V1]. (cid:3)
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 a = 1 for all j ∈ [1,...,n]. Then |a| = n. Denote
j
(4.22) g(z ,...,z ,κ) = κ−k eκΦ(t,z)dt ∧···∧dt =
1 n 1 k
Zγ(z)
κ
n
= κ−k f (t,z) dt ∧···∧dt .
j 1 k
Zγ(z) j=1
Y
