Potentials of a Frobenius like structure
aa r X i v : . [ m a t h . AG ] D ec POTENTIALS OF A FROBENIUS LIKE STRUCTURE
CLAUS HERTLING AND ALEXANDER VARCHENKO
Abstract.
This paper proves the existence of potentials of the first andsecond kind of a Frobenius like structure in a frame which encompassesfamilies of arrangements.The frame uses the notion of matroids. For the proof of the existenceof the potentials, a power series ansatz is made. The proof that it worksrequires that certain decompositions of tuples of coordinate vector fieldsare related by certain elementary transformations. This is shown with anontrivial result on matroid partition. Introduction and main results
A Frobenius manifold comes equipped locally with a potential. If one givesa definition which does not mention this potential explicitly, one neverthelessobtains it immediately by the following elementary fact: Let z i be the coordi-nates on C n and ∂ i = ∂∂z i be the coordinate vector fields. Let M be a convexopen subset of C n and T M be the holomorphic tangent bundle of M . Let A : T M → O M be a symmetric map such that also ∂ i A ( ∂ j , ∂ k , ∂ l ) is symmetricin i, j, k, l . Then a potential F ∈ O M with ∂ i ∂ j ∂ k F = A ( ∂ i , ∂ j , ∂ k ) exists. OnFrobenius manifolds see [D, M].This paper is devoted to a nontrivial generalization of this fact. The gener-alization turns up in the theory of families of arrangements as in [V2, ch. 3].The geometry there looks at first view similar to the geometry of Frobeniusmanifolds, but at second view, it is quite different.At first view, one finds in both cases data ( M, K, ∇ K , C, S, ζ ) with the follow-ing properties. M is an open subset of C n (with coordinates z i and coordinatevector fields ∂ i = ∂∂z i ). K → M is a holomorphic vector bundle with a flatholomorphic connection ∇ K . C is a Higgs field, i.e. an O M -linear map C : O ( K ) → Ω M ⊗ O ( K ) (1.1)such that all the endomorphisms C X : K → K, X ∈ T M , commute: C X C Y = C Y C X . And C and ∇ K satisfy the integrability condition ∇ K∂ i C ∂ j = ∇ K∂ j C ∂ j for all i, j ∈ { , ..., n } (1.2) Mathematics Subject Classification.
Key words and phrases.
Potentials, matroids, Frobenius like structure, family ofarrangements.This work was supported by the DFG grant He2287/4-1 (SISYPH), the second authorwas supported in part by NSF grant DMS-1362924 and Simons Foundation grant (which is equivalent to ∇ K ( C ) = 0, see remark 4.1). S is a ∇ K -flat sym-metric nondegenerate and Higgs field invariant pairing. ζ is a global nowherevanishing section of K .At second view, one sees the differences. In the case of a Frobenius manifold, M is the Frobenius manifold, rk K = n , and (much stronger) C • ζ : T M →O ( K ) is an isomorphism and all the sections C ∂ i ζ are ∇ K -flat. One obtainsan identification of T M with K and of the coordinate vector fields ∂ i with theflat sections C ∂ i ζ .In the case of a family of arrangements, rk K ≥ n , and the ∇ K -flat sectionsin K have the following much more surprising form. Define J := { , ..., n } . Afamily of arrangements in C k with k < n as in [V2, ch. 3] comes equipped withvectors ( v i ) i ∈ J in M (1 × k, C ) = { row vectors of length k with values in C } such that h v , .., v n i = M (1 × k, C ). A subset { i , ..., i k } ⊂ J is called maximalindependent if v i , ..., v i k is a basis of M (1 × k, C ). The sections C ∂ i ...C ∂ ik ζ in K for such subsets { i , ..., i k } are ∇ K -flat.The purpose of this paper is to show that also in this situation a potential ex-ists which resembles the potential of a Frobenius manifold. This is nontrivial.The proof combines the integrability condition (1.2) with intricate combina-torial considerations which are due to the complicated form of the ∇ K -flatsections.Theorem 1.2 is the main result. Definition 1.1 gives the frame and the usednotions. The frame is in two mild aspects more general than the data above inthe case of arrangements. First, S is more general, and second, the maximalindependent subsets { i , ..., i k } ⊂ J are maximal independent with respect toan arbitrary matroid ( J, F ) of rank k . See definition 2.1 for the notion of amatroid. Definition 1.1. (a) A
Frobenius like structure of order ( n, k, m ) ∈ Z > with n ≥ k is a tuple ( M, K, ∇ K , C, S, ζ , ( J, F )) with the following prop-erties.
M, K, ∇ K , C, ζ and J are as above. S is a ∇ K -flat m -linear form S : O ( K ) m → O M , which is Higgs field invariant, i.e. S ( C X s , s , ..., s m ) = S ( s , C X s , ..., s m ) = ... = S ( s , s , ..., C X s m ) (1.3)for s , s , ..., s m ∈ O ( K ) and X ∈ T M . ( J, F ) is a matroid with rank r ( J ) = k .For any maximal independent subset { i , ..., i k } ⊂ J the section C ∂ i ...C ∂ ik ζ is ∇ K -flat.(b) Some notations: For any subset I = { i , ..., i k } ⊂ J , the differentialoperator ∂ I := ∂ i ...∂ i k and the endomorphism C I := C ∂ i ...C ∂ ik : O ( K ) →O ( K ) are well defined (they do not depend on the chosen order of the elements i , ..., i k ).(c) In the situation of (a), a potential of the first kind is a function Q ∈ O M with ∂ I ...∂ I m Q = S ( C I ζ , ..., C I m ζ ) (1.4) OTENTIALS OF A FROBENIUS LIKE STRUCTURE 3 for any m maximal independent subsets I , ..., I m ⊂ J . A potential of thesecond kind is a function L ∈ O M with ∂ i ∂ I ...∂ I m L = S ( C ∂ i C I ζ , ..., C I m ζ ) (1.5)for any m maximal independent subsets I , ..., I m ⊂ J and any i ∈ J . Theorem 1.2.
Let ( M, K, ∇ K , C, S, ζ , ( J, F )) be a Frobenius like structureof some order ( n, k, m ) ∈ Z > . Then locally (i.e. near any z ∈ M ⊂ C n )potentials of the first and second kind exist. Notice that by formulas (1.4) and (1.5) the potential of the first kind de-termines the matrix elements of the m -linear form S on the flat sections C ∂ i ...C ∂ ik ζ and the potential of the second kind determines the matrix ele-ments of the Higgs operators C ∂ i acting on the flat sections C ∂ i ...C ∂ ik ζ . Thusall information on the m -linear form and the Higgs operators is packed intothe two potential functions.At the end of the paper, several remarks discuss the case of arrangementsand the relation to Frobenius manifolds. In the case of arrangements, one hasan ( n, k, n, , L above generalizes the potential of aFrobenius manifold. For generic arrangements, a global explicit constructionof the potentials Q and L had been given in [V3]. Recently this was generalizedin [PV] to all families of arrangements as in [V2, ch. 3].Section 2 cites a nontrivial result of J. Edmonds [E, 4. Theorem] on matroidpartition and adds some considerations. Section 3 applies an implication of itto a combinatorial situation which in turn is needed in the proof of the maintheorem 1.2 in section 4. Section 4 concludes with some remarks.We thank a referee of an earlier version [HV] of this paper for pointing us tothe result on matroid partition. This led to the present version of the paperwhich uses matroids. The second author thanks MPI in Bonn for hospitalityduring his visit in 2015-2016.2. Matroid partition
Definition 2.1. (E.g. [E]) A matroid ( E, F ) is a finite set E together witha nonempty family F ⊂ P ( E ) of subsets of E , called independent sets , suchthat the following holds.(i) Every subset of an independent set is independent.(ii) For every subset A ⊂ E , all maximal independent subsets of A havethe same cardinality, called the rank r ( A ) of A .For example, if V is a vector space and ( v e ) e ∈ E is a tuple of elements whichgenerates V , one obtains a matroid where a subset B ⊂ E is independent ifand only if the tuple ( v b ) b ∈ B is a linearly independent tuple of vectors. In thecase of a family of arrangements, such a matroid will be used.The following result on matroid partition was proved by J. Edmonds [E]. C. HERTLING AND A. VARCHENKO
Theorem 2.2. [E, 4. Theorem] . Let ( E, F i ) , i = 1 , ..., m , be matroids whichare defined on the same set E . Let r i ( A ) be the rank of A ⊂ E relative to ( E, F i ) . The following two conditions are equivalent.( α ) The set E can be partitioned into a family { I i } i =1 ,...,m of sets I i ∈ F i .( β ) Any set A ⊂ E satisfies | A | ≤ m X i =1 r i ( A ) . (2.1)The implication ( α ) ⇒ ( β ) is immediate: Suppose that { I i } i =1 ,...,m is apartition of E with I i ∈ F i . Then for any A ⊂ EA = ˙ [ mi =1 A ∩ I i , | A | = m X i =1 | A ∩ I i | ≤ m X i =1 r i ( A ) . But the implication ( β ) ⇒ ( α ) is nontrivial. The proof in [E] is an involvedinductive algorithm.We are interested in the more special situation in theorem 2.6. Before, twolemmata are needed. Definition 2.3. [E] (a) A minimal dependent set of elements of a matroid iscalled a circuit .(b) For any number l ∈ Z ≥ and any finite set E with | E | ≥ l , the set F ( l,E ) := { I ⊂ E | | I | ≤ l } defines obviously a matroid ( E, F ( l,E ) ), the uniformmatroid of rank l . Lemma 2.4. [E, Lemma 2]
The union of any independent set I and anyelement e of a matroid contains at most one circuit of the matroid. Lemma 2.5.
Let ( E, F ) be a matroid. Let A , A ⊂ E be subsets. For i = 1 , ,let I i ⊂ A i be a maximal independent subset of A i . Suppose that I ∪ I is anindependent set. Then I ∪ I is a maximal independent subset of A ∪ A , and I ∩ I is a maximal independent subset of A ∩ A . Proof:
Suppose that for some element b ∈ ( A ∪ A ) − ( I ∪ I ) the union I ∪ I ∪ { b } is independent. Then for some i ∈ { , } , b ∈ A i . But I i ∪ { b } is a larger independent subset of A i than I i , a contradiction. This proves that I ∪ I is a maximal independent subset of A ∪ A .Suppose that for some element b ∈ ( A ∩ A ) − ( I ∩ I ) the union ( I ∩ I ) ∪{ b } is independent. If b ∈ I i then b / ∈ I j where { i, j } = { , } . Then I j ∪ { b } is anindependent subset of A j , a contradiction to the maximality of I j . Therefore b / ∈ I ∪ I . Thus for i = 1 ,
2, the set I i ∪ { b } ⊂ A i is dependent as it is largerthan I i . Therefore it contains a circuit C i ⊂ I i ∪{ b } . Obviously C i ∩ ( I i − I j ) = ∅ where { i, j } = { , } . Thus C = C . Both are circuits in ( I ∪ I ) ∪ { b } , acontradiction to lemma 2.4. This proves that I ∩ I is a maximal independentsubset of A ∩ A . (cid:3) Theorem 2.6.
Let ( E, F i ) , i = 1 , ..., m , be matroids which are defined on thesame set E and which satisfy together ( α ) and ( β ) in theorem 2.2. Suppose OTENTIALS OF A FROBENIUS LIKE STRUCTURE 5 that F m = F ( l,E ) for some l ∈ Z ≥ with l ≤ | E | . Suppose that the set G := { A ⊂ E | | A | = l + m − X i =1 r i ( A ) } (2.2) contains the set E .(a) Then this set G is closed under the operations union and intersection ofsets. Especially, it contains a set called A min ⊂ E which is the unique minimalelement of G with respect to the partial order given by inclusion. Of course A min = ∅ if and only if l ≥ .(b) Now suppose l ≥ . Then A min = A par where A par is the set A par := { b ∈ E | ∃ a partition { I i } i =1 ,...,m of E (2.3)such that I i ∈ F i and b ∈ I m } . Proof: (a) Choose a partition { I i } i =1 ,...,m of E with I i ∈ F i . For any subset A ⊂ E , it induces a partition A = ˙ S mi =1 A ∩ I i of A into subsets ( A ∩ I i ) ∈ F i .If A ∈ G , then by (2.2) each set A ∩ I i is a maximal independent subset of A with respect to the matroid ( E, F i ). As | A | ≥ l , especially | A ∩ I m | = l . As E itself is in G , | I m | = l , and thus A ∩ I m = I m for any set A ∈ G .Let A , A ∈ G . For any i = 1 , ..., m , lemma 2.5 applies to the maximalindependent sets A ∩ I i and A ∩ I i of A respectively A relative to the matroid( E, F i ), because also ( A ∪ A ) ∩ I i ∈ F i . Therefore ( A ∪ A ) ∩ I i is a maximalindependent subset of A ∪ A relative to ( E, F i ), and ( A ∩ A ) ∩ I i is a maximalindependent subset of A ∩ A relative to ( E, F i ). Also, I m = A ∩ I m = A ∩ I m shows I m = ( A ∪ A ) ∩ I m = ( A ∩ A ) ∩ I m . Now A ∪ A ∈ G and A ∩ A ∈ G are obvious. Therefore G is closed underthe operations union and intersection of sets.(b) A par ⊂ A min : Fix an arbitrary element b ∈ A par . Choose a partition { I i } i =1 ,...,m of E with I i ∈ F i and b ∈ I m . Recall A min ∩ I m = I m . Thus b ∈ A min . A min ⊂ A par : Fix an arbitrary element b ∈ A min . Define e E := E − { b } . Anyset A ⊂ e E does not contain A min , because b ∈ A min . Therefore any set A ⊂ e E satisfies A / ∈ G and | A | ≤ − l + m − X i =1 r i ( A ) . (2.4)Consider the matroids ( e E, e F i ), where e F i := { I ∈ F i | b / ∈ I } for i ∈ { , ..., m − } and e F m := F ( l − , e E ) . For i ∈ { , ..., m − } the rank of A ⊂ e E relative to ( e E, e F i )is equal to the rank r i ( A ) of A relative to ( E, F i ).By (2.4) and theorem 2.2, a partition { e I i } i =1 ,...,m of e E with e I i ∈ e F i exists.Now the sets I i := e I i for i = 1 , ..., m −
1, and I m := e I m ∪ { b } form a partitionof E with I i ∈ F i . This shows b ∈ A par . (cid:3) C. HERTLING AND A. VARCHENKO An equivalence between index systems
In this section we fix three positive integers n, k, m ∈ Z > with n ≥ k and amatroid ( J, F ) with underlying set J = { , ..., n } , rank function r : P ( J ) → Z ≥ and rank r ( J ) = k . Notations 3.1.
As usual Z J := { maps : J → Z } and Z J ≥ := { maps : J → Z ≥ } . The set Z J is an additive group, the set Z J ≥ is an additive monoid.For j ∈ J denote by [ j ] ∈ Z J ≥ the map with [ j ]( j ) = 1 and [ j ]( i ) = 0 forany i = j . Then any map T ∈ Z J can be written as T = P nj =1 T ( j ) · [ j ]. For T ∈ Z J denote | T | := P nj =1 T ( j ) ∈ Z . The support of T ∈ Z J is supp T := { j ∈ J | T ( j ) = 0 } . The map d H : Z J × Z J → Z ≥ , ( T , T ) X j ∈ J | T ( j ) − T ( j ) | (3.1)is a metric on Z J . On Z J one has the partial ordering ≤ with S ≤ T ⇐⇒ S ( j ) ≤ T ( j ) ∀ j ∈ J. (3.2)Any map T ∈ Z J ≥ with | T | = t ∈ Z ≥ is called a system of elements of J orsimply a system or a t -system . If S and T are systems with S ≤ T , then S isa subsystem of T . Definition 3.2.
Here l ∈ Z ≥ . Here all systems are systems of elements of J .(a) A system T ∈ Z J ≥ is a base if supp T ∈ F and | T | = k (so the supportsupp T is a maximal independent subset of J and all T ( a ) ∈ {
0; 1 } ).(b) A strong decomposition of an ( mk + l )-system T is a decomposition T = T (1) + ... + T ( m +1) into m k -systems T (1) , ..., T ( m ) and one l -system T ( m +1) such that T (1) , ..., T ( m ) are bases (and T ( m +1) is an arbitrary l -system; e.g. if l = 0 then T ( m +1) = 0 automatically).(c) An ( mk + l )-system is strong if it admits a strong decomposition.(d) A good decomposition of an N -system T with N ≥ mk + 1 is adecomposition T = T + T into two systems such that T is a strong( mk + 1)-system of elements of J .(e) Two good decompositions T + T = T and S + S = T of an N -system T with N ≥ mk + 1 are locally related , notation: ( S , S ) ∼ loc ( T , T ), if there are strong decompositions S (1)2 + ... + S ( m +1)2 = S of S and T (1)2 + ... + T ( m +1)2 = T of T with S ( j )2 = T ( j )2 for 1 ≤ j ≤ m .Of course, ∼ loc is a reflexive and symmetric relation.(f) Two good decompositions T + T = T and S + S = T of an N -system T with N ≥ mk + 1 are equivalent , notation: ( S , S ) ∼ ( T , T ), if there is a sequence σ , σ , ..., σ r for some r ∈ Z ≥ of gooddecompositions of T such that σ = ( S , S ), σ r = ( T , T ) and σ j ∼ loc σ j +1 for j = 1 , ..., r −
1. Of course, ∼ is an equivalence relation.The main result of this section is the following theorem 3.3. Theorem 3.3.
Let T ∈ Z J ≥ be an N -system for some N ≥ mk + 1 which hasgood decompositions. Then all its good decompositions are equivalent. OTENTIALS OF A FROBENIUS LIKE STRUCTURE 7
The theorem will be proved after the proofs of corollary 3.4 and lemma 3.5.Corollary 3.4 is a corollary of theorem 2.6.
Corollary 3.4.
Fix a strong ( mk + l ) -system T ∈ Z J ≥ with l ∈ Z ≥ . Then forany B ⊂ J X j ∈ B T ( j ) ≤ l + m · r ( B ) . (3.3) The set G ( T ) := { B ⊂ supp T | X j ∈ B T ( j ) = l + m · r ( B ) } (3.4) contains supp T and is closed under the operations union and intersection ofsets. Especially, it contains a set called A min ( T ) ⊂ supp T which is the uniqueminimal element with respect to inclusion. In the case l ≥ , define the set A dec ( T ) := { b ∈ J | ∃ a strong decomposition (3.5) T = T (1) + ... + T ( m +1) with b ∈ supp T ( m +1) } . Then A min ( T ) = A dec ( T ) . Proof:
We will construct from T certain lifts of the matroids ( J, F ) and(
J, F ( l,J ) ) to matroids on the set E := { , , ..., mk + l } and go with them intotheorem 2.6. Choose a map f : E → J with | f − ( j ) | = T ( j ). Define the sets F = ... = F m := { A ⊂ E | f | A : A → J injective, f ( A ) ∈ F } ⊂ P ( E ) ,F m +1 := F ( l,E ) ⊂ P ( E ) . Then (
E, F i ) for i ∈ { , ..., m + 1 } is a matroid. Together they satisfy ( α ) intheorem 2.2 (with m + 1 instead of m ) because T is a strong ( mk + l )-system.We go into theorem 2.6 with m + 1 instead of m .That T is a strong ( mk + l )-system, gives also E ∈ G and (3.3).Therefore the set A min in theorem 2.6 is well defined. The set A par is welldefined, anyway. One sees easily r ( A ) = ... = r m ( A ) = r ( f ( A )) for A ⊂ E,G = { f − ( B ) | B ∈ G ( T ) } . Therefore G ( T ) contains supp T and is closed under the operations union andintersection of sets. Now one sees also easily A min = f − ( A min ( T )) , A par = f − ( A dec ( T )) , and thus A min ( T ) = A dec ( T ). (cid:3) Lemma 3.5.
Let S and T ∈ Z J ≥ be two strong ( mk + 1) -systems. At leastone of the following two alternatives holds. ( α ) T has a strong decomposition T = T (1) + ... + T ( m +1) with T ( m +1) = [ i ] for some i ∈ supp T with T ( i ) > S ( i ) . ( β ) For any strong decomposition S = S (1) + ... + S ( m +1) a strong de-composition T = T (1) + ... + T ( m +1) with T ( m +1) = S ( m +1) exists. C. HERTLING AND A. VARCHENKO
Proof:
Suppose that ( α ) does not hold. Then for any i ∈ A dec ( T ) S ( i ) ≥ T ( i ). Especially X i ∈ A dec ( T ) S ( i ) ≥ X i ∈ A dec ( T ) T ( i ) = 1 + m · r ( A dec ( T )) . The equality uses A dec ( T ) = A min ( T ) ∈ G ( T ). Now (3.3) for S instead of T shows that ≥ can be replaced by =. Therefore A dec ( T ) ∈ G ( S ). Any elementof G ( S ) contains A min ( S ). This and the equality A dec ( S ) = A min ( S ) give A dec ( S ) = A min ( S ) ⊂ A dec ( T ) . Thus ( β ) holds. (cid:3) Proof of theorem 3.3:
Let ( S , S ) and ( T , T ) be two different gooddecompositions of an N -system T of elements of J (with N ≥ mk + 1). Then S and T are strong ( mk + 1)-systems of elements of J . At least one of thetwo alternatives ( α ) and ( β ) in lemma 3.5 holds for S and T . First case, ( α ) holds: Let T = T (1)2 + ... + T ( m +1)2 be a strong decompositionwith T ( m +1)2 = [ i ] for some i ∈ supp T with T ( i ) > S ( i ). Then a j ∈ supp T with T ( j ) > S ( j ) and T ( j ) < S ( j ) exists. The decomposition T = R + R with R = T − [ j ] + [ i ] , R = T + [ j ] − [ i ] (3.6)is a good decomposition of T because T (1)2 + ... + T ( m )2 + [ j ] is a strong de-composition of R . The good decompositions ( R , R ) and ( T , T ) are locallyrelated, ( R , R ) ∼ loc ( T , T ), and thus equivalent,( R , R ) ∼ ( T , T ) . (3.7)Furthermore, d H ( R , S ) = d H ( T , S ) − . (3.8) Second case, ( β ) holds: Let T = T (1)2 + ... + T ( m +1)2 and S = S (1)2 + ... + S ( m +1)2 be strong decompositions of T and S with T ( m +1)2 = S ( m +1)2 = [ a ] forsome a ∈ supp T . Two elements b, c ∈ supp T with T ( b ) > S ( b ), T ( b ) S ( c ) exist. Consider the decompositions of T and S , T = R + R with R = T − [ b ] + [ a ] , R = T + [ b ] − [ a ] , (3.9) S = Q + Q with Q = S − [ c ] + [ a ] , Q = S + [ c ] − [ a ] . (3.10)They are good decompositions because R has the strong decomposition R = T (1) + ... + T ( m ) + [ b ] and Q has the strong decomposition Q = S (1) + ... + S ( m ) + [ c ]. The local relations( R , R ) ∼ loc ( T , T ) and ( Q , Q ) ∼ loc ( S , S )and the equivalences( R , R ) ∼ ( T , T ) and ( Q , Q ) ∼ ( S , S ) (3.11)hold. Furthermore d H ( R , Q ) = d H ( T , S ) − . (3.12) OTENTIALS OF A FROBENIUS LIKE STRUCTURE 9
The properties (3.7), (3.8), (3.11) and (3.12) show that in both cases theequivalence classes of ( S , S ) and ( T , T ) contain good decompositions whosesecond members are closer to one another with respect to the metric d H than T and S . This shows that ( S , S ) and ( T , T ) are in one equivalence class. (cid:3) Potentials of the first and second kind
The main part of this section is devoted to the proof of theorem 1.2. At theend some remarks on the relation to families of arrangements and Frobeniusmanifolds are made.
Remark 4.1.
Here a coordinate free formulation of the integrability condition(1.2) will be given. For M, ∇ K and C as in the introduction, ∇ K ( C ) ∈ Ω M ⊗O (End( K )) is the 2-form on M with values in End( K ) such that for X, Y ∈ T M ∇ K ( C )( X, Y ) = ∇ KX ( C Y ) − ∇ KY ( C X ) − C [ X,Y ] . (4.1)Now (1.2) is equivalent to ∇ K ( C ) = 0 Proof of theorem 1.2:
Let (
M, K, ∇ K , C, S, ζ , ( J, F )) be a Frobenius likestructure of some order ( n, k, m ) ∈ Z > .We need some notations. If T ∈ Z J ≥ is a system of elements of J , then( z − x ) T := Y i ∈ J ( z i − x i ) T ( i ) for any x ∈ C n ,T ! := Y i ∈ J T ( i )! , ∂ T := Y i ∈ J ∂ T ( i ) z i , C T := Y i ∈ J C T ( i ) ∂ zi . Thus, if S and T are systems of elements of J , then ∂ T ( z − x ) S = (cid:26) T S, S !( S − T )! · ( z − x ) S − T if T ≤ S, (4.2)for any x ∈ C n .The existence of a (not just local, but even global) potential Q of the firstkind is trivial. The function Q := X T with ( ∗ ) T ! · S ( C T ζ , ζ , ..., ζ ) · z T ( m times ζ ) , (4.3)( ∗ ) : T ∈ Z J ≥ is a strong mk -system (definition 3.1(c)) . works. It is a homogeneous polynomial of degree mk and contains only mono-mials which are relevant for (1.2). In fact, one can add to this Q an arbitrarylinear combination of the monomials z T for the mk -systems T which are notstrong, so which are not relevant for (1.2).The existence of a potential L of the second kind is not trivial. Let some x ∈ M be given. We make the power series ansatz L := X T ∈ Z J ≥ a T · ( z − x ) T , (4.4) where the coefficients a T have to be determined. If T satisfies | T | ≤ mk or ifit satisfies | T | ≥ mk + 1, but does not admit a good decomposition (definition3.1 (d)), then the conditions (1.3) are empty for a T ( z − x ) T because of (4.2),so then a T can be chosen arbitrarily, e.g. a T := 0 works.Now consider T with | T | ≥ mk +1 which admits good decompositions. Theneach good decomposition T = T + T gives via (1.3) a candidate a T ( T , T ) := 1 T ! · ( ∂ T S ( C T ζ , ζ , ..., ζ )) ( x ) , (4.5)for the coefficient a T of ( z − x ) T in L . We have to show that the candidates a T ( T , T ) for all good decompositions ( T , T ) of T coincide.Suppose that two good decompositions ( T , T ) and ( S , S ) are locally re-lated, ( T , T ) ∼ loc ( S , S ) (definition 3.1 (e)), but not equal. Then there arestrong decompositions T = T (1)2 + ... + T ( m )2 + [ a ] and S = T (1)2 + ... + T ( m )2 + [ b ]with a = b , and thus also T − [ b ] = S − [ a ] ∈ Z J ≥ holds. Because any T ( j )2 , j ∈ { , ..., m } , is independent, C T ( j )2 ζ is ∇ K -flat. This and (4.3) give ∂ z b S ( C T ζ , ζ , ..., ζ )= ∂ z b S ( C ∂ za C T (1)2 ζ , C T (2)2 ζ , ..., C T ( m )2 ζ )= S ( ∇ K∂ zb ( C ∂ za ) C T (1)2 ζ , C T (2)2 ζ , ..., C T ( m )2 ζ )= S ( ∇ K∂ za ( C ∂ zb ) C T (1)2 ζ , C T (2)2 ζ , ..., C T ( m )2 ζ )= ∂ z a S ( C ∂ zb C T (1)2 ζ , C T (2)2 ζ , ..., C T ( m )2 ζ )= ∂ z a S ( C S ζ , ζ , ..., ζ ) . (4.6)This implies a T ( T , T ) = a T ( S , S ) , (4.7)so the locally related good decompositions ( T , T ) and ( S , S ) give the samecandidate for a T . Thus all equivalent (definition 3.1 (f)) good decompositionsgive the same candidate for a T . By theorem 3.3, all good decompositions of T are equivalent. Therefore they all give the same candidate for a T . Thus apotential L of the second kind exists as a formal power series as in (4.4).It is in fact a convergent power series because of the following. There arefinitely many strong ( mk + 1)-systems T . Each determines the coefficients a T for all T ≥ T . We put a T := 0 for T which do not admit good decompositions.The part of L in (4.4) which is determined by some strong ( mk + 1)-system T is a convergent power series. Thus L is the union of finitely many overlappingconvergent power series. It is easy to see that it is itself convergent. Thisfinishes the proof of theorem 1.2. (cid:3) Remark 4.2.
In [V2, ch. 3] families of arrangements are considered which giverise to Frobenius like structures (
M, K, ∇ K , C, S, ζ , ( J, F )) of order ( n, k, k and n with k < n and with a matrix B := ( b ji ) i =1 ,..,n ; j =1 ,..,k ∈ M ( n × k, C ) with rank B = k . Define J := { , ..., n } .Here the matroid ( J, F ) is the vector matroid (also called linear matroid ) of the
OTENTIALS OF A FROBENIUS LIKE STRUCTURE 11 tuple ( v i ) i ∈ J of row vectors v i := ( b ji ) j =1 ,...k of the matrix B . More precisely,a subset A ⊂ J is independent, if the tuple ( v i ) i ∈ A is a linearly independentsystem of vectors.Consider C n × C k with the coordinates ( z, t ) = ( z , ..., z n , t , ..., t k ) and withthe projection π : C n × C k → C n . Define the functions g i := k X j =1 b ji · t j , f i := g i + z i for i ∈ J (4.8)on C n × C k .We obtain on C n × C k the arrangement C = { H i } i ∈ J , where H i is the zero setof f i . Let U ( C ) := C n × C k − S i ∈ J H i be the complement. For every x ∈ C n , thearrangement C restricts to an arrangement C ( x ) on π − ( x ) ∼ = C k . For almostall x ∈ C n the arrangement C ( x ) is essential (definition in [V2]) with normalcrossings. The subset ∆ ⊂ C n where this does not hold, is a hypersurface andis called the discriminant , see [V2, 3.2]. Define M := C n − ∆.A set I = { i , ..., i k } ⊂ J is maximal independent, i.e. ( v i , ..., v i k ) is a basisof M (1 × k, C ), if and only if for some (or equivalently for any) x ∈ C n thehyperplanes H i ( x ) , ..., H i k ( x ) are transversal.Let a = ( a , ..., a n ) ∈ ( C ∗ ) n be a system of weights such that for any x ∈ M the weighted arrangement ( C ( x ) , a ) is unbalanced : See [V2] for the definitionof unbalanced , e.g. a ∈ R n> is unbalanced, also a generic system of weights isunbalanced. The master function of the weighted arrangement ( C , a ) isΦ a ( z, t ) := X i ∈ J a i log f i . (4.9)Several deep facts are related to this master function. We use some of themin the following. See [V2] for references.For z ∈ M all critical points of Φ a are isolated, and the sum µ of theirMilnor numbers is independent of the unbalanced weight a and the parameter z ∈ M . The bundle K := [ z ∈ M K z with K z := O ( U ( C ) ∩ π − ( z )) / (cid:18) ∂ Φ a ∂t j | j = 1 , ..., k (cid:19) (4.10)over M is a vector bundle of µ -dimensional algebras.It comes equipped with the section ζ of unit elements ζ ( z ) ∈ K z , a Higgsfield C , a combinatorial connection ∇ K and a pairing S . The Higgs field C : O ( K ) → Ω M ⊗ O ( K ) is defined with the help of the period mapΨ : T M → K, ∂ z i (cid:20) ∂ Φ a ∂z i (cid:21) = (cid:20) a i f i (cid:21) =: p i (4.11)by C ∂ zi ( h ) := p i · h for h ∈ K z . (4.12)Because of 0 = (cid:20) ∂ Φ a ∂t j (cid:21) = n X i =1 b ji p i , (4.13) the Higgs field vanishes on the vector fields X j := P ni =1 b ji ∂ i , j ∈ { , ..., k } , C X j = 0 for j ∈ { , ..., k } . (4.14)In fact the whole geometry of the family of arrangements is invariant withrespect to the flows of these vector fields.The sections det( b ji ) i ∈ I,j =1 ,...,k · C I ζ for all maximal independent sets I = { i , ..., i k } ⊂ J generate the bundle K , and they satisfy only relations withconstant coefficients in Z . The combinatorial connection ∇ K is the uniqueflat connection such that the sections C I ζ for I ⊂ J maximal independent are ∇ K -flat. The sections det( b ji ) i ∈ I,j =1 ,...,k · C I ζ for I ⊂ J maximal independentgenerate a ∇ K -flat Z -lattice structure on K .The pairing S comes from the Grothendieck residue with respect to thevolume form dt ∧ ... ∧ dt k Q kj =1 ∂ Φ a ∂t j . (4.15)It is symmetric, nondegenerate, ∇ K -flat, multiplication invariant and Higgsfield invariant.The existence of potentials of the first and second kind for families of ar-rangements was conjectured in [V1]. If all the k × k minors of the matrix B = ( b ji ) are nonzero, the potentials were constructed in [V1], cf. [V3]. In[PV] this was generalized to all cases in this remark 4.2. The potentials aregiven by explicit formulas in terms of the linear functions defining the hyper-planes in C n composing the discriminant. Remarks 4.3. (i) The situation in remark 4.2 is in several aspects richer thana Frobenius like structure of type ( n, k, m ). The bundle K is a bundle ofalgebras. The sections C I ζ for maximal independent sets I ⊂ J generate thebundle. The sections det( b ji ) i ∈ I,j =1 ,...,k · C I ζ generate a flat Z -lattice structurein K . The Higgs field vanishes on the vector fields X , ..., X k . The m -linearform S is a pairing ( m = 2) and is nondegenerate. We will not discuss the Z -lattice structure, but we will discuss some logical relations between the otherenrichments and some implications of them.(ii) Let ( M, K, ∇ K , C, S, ζ , V, ( v , ..., v n )) be a Frobenius like structure oforder ( n, k, m ). Suppose that it satisfies the generation condition (GC) The sections C I ζ for maximal independent sets I ⊂ J (4.16)generate the bundle K. Let µ be the rank of K . Then for any x ∈ M , the endomorphisms C X , X ∈ T x M , generate a µ -dimensional commutative subalgebra A z ⊂ End( K x ). Andany endomorphism which commutes with them is contained in this subalgebra.This gives a rank µ bundle A of commutative algebras. And the map A → K, B Bζ , (4.17)is an isomorphism of vector bundles and induces a commutative and associativemultiplication on K x for any x ∈ M , with unit field ζ ( x ). Therefore the special OTENTIALS OF A FROBENIUS LIKE STRUCTURE 13 section ζ and the generation condition (GC), which exist and hold in remark4.2, give the multiplication on the bundle K there.(iii) In the situation in (ii) with the condition (GC), the m -linear form ismultiplication invariant because it is Higgs field invariant. The condition (GC)implies also that it is symmetric: S ( C I ζ , C I ζ , ..., C I m ζ ) = S ( C I σ (1) ζ , C I σ (2) ζ , ..., C I σ ( m ) ζ )for any maximal independent sets I , ..., I m and any permutation σ ∈ S m .(iv) The following special case gives rise to Frobenius manifolds withoutEuler fields. Consider a Frobenius like structure ( M, K, ∇ K , C, S, ζ , ( J, F )) oforder ( n, ,
2) with nondegenerate pairing S , ∇ K -flat section ζ , the uniformmatroid ( J, F ) = (
J, F (1 ,J ) ) and the condition that the map C • ζ : T M → K is an isomorphism. Then the sections C ∂ i ζ generate the bundle K and are ∇ K -flat. Here M becomes a Frobenius manifold (without Euler field) whoseflat structure is the naive flat structure of C n ⊃ M . The potential L is thepotential of the Frobenius manifold. References [D] B. Dubrovin,
Geometry of 2D topological field theories , Integrable Systems and Quan-tum Groups, ed. M. Francaviglia and S. Greco. Springer lecture notes in mathematics,1620, 120–348.[E] J. Edmonds,
Matroid partition,
Mathematics of the Decision Sciences: Part 1, ed.G.B. Dantzig and A.F. Veinott. AMS, 1968, 335–345. Reprinted in: 50 years of integerprogramming 1958–2008, ed. M. J¨unger et al. Springer, 2010, 207–217.[HV] C. Hertling, A. Varchenko: Potentials of a Frobenius type structure and m bases of avector space. arXiv:1608.08423, 18 pages.[M] Y.I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces , AmericanMathematical Society Colloquium Publications, vol. 47, AMS, Providence, RI, 1999.[PV] A. Prudhom, A. Varchenko,
Potentials of a family of arrangements of hyperplanes andelementary subarrangements , arXiv:1611.03944, 24 pages.[V1] A. Varchenko,
Arrangements and Frobenius like structures , Annales de la faculte dessciences de Toulouse Ser. 6, 24 no. 1 (2015), p. 133–204.[V2] A. Varchenko,
Critical set of the master function and characteristic variety of the as-sociated Gauss-Manin differential equations , Arnold Math. J. (2015), no. 3, 253–282, DOI 10.1007/s40598-015-0020-8 .[V3] A. Varchenko,
On axioms of Frobenius like structure in the theory of ar-rangements , arXiv:1601.02208, Journal of Integrable Systems (2016) 00, 1–15, doi:10.1093/integr/xyw007 . Lehrstuhl f¨ur Mathematik VI, Universit¨at Mannheim, A5,6, 68131Mannheim, Germany
E-mail address : [email protected] Department of Mathematics, University of North Carolina at Chapel Hill,Chapel Hill, NC 27599-3250, USA
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