Rank 2 bundles with meromorphic connections with poles of Poincaré rank 1
aa r X i v : . [ m a t h . AG ] S e p RANK 2 BUNDLES WITHMEROMORPHIC CONNECTIONS WITHPOLES OF POINCAR ´E RANK 1
CLAUS HERTLING
Abstract.
Holomorphic vector bundles on C × M , M a complexmanifold, with meromorphic connections with poles of Poincar´erank 1 along { } × M arise naturally in algebraic geometry. Theyare called ( T E )-structures here. This paper takes an abstract pointof view. It gives a complete classification of all (
T E )-structures ofrank 2 over germs (
M, t ) of manifolds. In the case of M a point,they separate into four types. Those of three types have universalunfoldings, those of the fourth type (the logarithmic type) not.The classification of unfoldings of ( T E )-structures of the fourthtype is rich and interesting. The paper finds and lists also all(
T E )-structures which are basic in the following sense: Togetherthey induce all rank 2 (
T E )-structures, and each of them is notinduced by any other (
T E )-structure in the list. Their base spaces M turn out to be 2-dimensional F-manifolds with Euler fields. Thepaper gives also for each such F-manifold a classification of all rank2 ( T E )-structures over it. Also this classification is surprisinglyrich. The backbone of the paper are normal forms. But also themonodromy and the geometry of the induced Higgs fields and ofthe bases spaces are important.
Contents
1. Introduction 22. The two-dimensional F-manifolds and their Euler fields 83. (
T E )-structures in general 103. 1. Definitions 103. 2. (
T E )-structures with trace free pole part 133. 3. (
T E )-structures over F-manifolds with Euler fields 163. 4. Birkhoff normal form 19
Date : September 29, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Meromorphic connections, isomonodromic deforma-tions, (
T E )-structures.This work was funded by the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) – 242588615.
3. 5. Regular singular (
T E )-structures 203. 6. Marked (
T E )-structures and moduli spaces for them 234. Rank 2 (
T E )-structures over a point 274. 1. Separation into 4 cases 274. 2. The case (Sem) 284. 3. Joint considerations on the cases (Bra), (Reg) and (Log) 304. 4. The case (Bra) 324. 5. The case (Reg) with tr U = 0 394. 6. The case (Log) with tr U = 0 425. Rank 2 ( T E )-structures over germs of regular F-manifolds 446. 1-parameter unfoldings of logarithmic (
T E )-structures overa point 486. 1. Numerical invariants for such (
T E )-structures 486. 2. 1-parameter unfoldings with trace free pole part oflogarithmic pure (
T LE )-structures over a point 546. 3. Generically regular singular (
T E )-structures over ( C , t = 0 and notsemisimple monodromy 637. Marked regular singular rank 2 ( T E )-structures 688. Unfoldings of rank 2 (
T E )-structures of type (Log) over apoint 778. 1. Classification results 788. 2. (
T E )-structures over given F-manifolds with Euler fields 858. 3. Proof of Theorem 8.5 899. A family of rank 3 (
T E )-structures with a functionalparameter 94References 971.
Introduction
A holomorphic vector bundle H on C × M , M a complex manifold,with a meromorphic connection ∇ with a pole of Poincar´e rank 1 along { } × M and no pole elsewhere, is called a ( T E )-structure. The aim ofthis paper is the local classification of all rank 2 (
T E )-structures, overarbitrary germs (
M, t ) of manifolds.Before we talk about the results, we will put these structures intoa context, motivate their definition, mention their occurence in alge-braic geometry, and formulate interesting problems. The rank 2 caseis the first interesting case and already very rich. In many aspects itis probably typical for arbitrary rank, in some not. And it is certainlythe only case where such a thorough classification is feasible. ANK 2 (
T E )-STRUCTURES 3
The pole of Poincar´e rank 1 along { } × M of the pair ( H, ∇ ) meansthe following. Let t = ( t , ..., t n ) be holomorphic coordinates on M with coordinate vector fields ∂ , ..., ∂ n , and let z be the standard co-ordinate on C . Then ∇ ∂ z σ for a holomorphic section σ ∈ O ( H ) of H is in z − O ( H ), and ∇ ∂ j σ is in z − O ( H ). The pole of order twoalong ∂ z is the first case beyond the easy and tame case of a pole oforder 1, i.e. a logarithmic pole. The pole of order 1 along ∂ i gives agood variation property, a generalization of Griffiths transversality forvariations of Hodge structures. It is the most natural constraint for anisomonodromic family of bundles on C with poles of order 2 at 0. So, apole of Poincar´e rank 1 is in some sense the first case beyond the caseof connections with logarithmic poles.In algebraic geometry, such connections arise naturally. A distin-guished case is the Fourier-Laplace transformation (with respect to thecoordinate z ) of the Gauss-Manin connection of a family of holomor-phic functions with isolated singularities [He03, ch. 8][Sa02, VII]. Thepaper [He03] defines ( T ERP )-structures, which are (
T E )-structureswith additional real structure and pairing and which generalize varia-tions of Hodge structures. Also the notion (
T EZP )-structure makessense, which is a (
T E )-structures with a flat Z -lattice bundle on C ∗ × M and a certain pairing. A family of holomorphic functions with iso-lated singularities (and some topological well-behavedness) gives riseto a ( T EZP )-structure over the base space of the family [He02, ch.11.4][He03, ch. 8]In [He02] and other papers of the author, a Torelli problem is con-sidered. We formulate it here as the following question: Does the(
T EZP )-structure of a holomorphic function germ with an isolatedsingularity determine the (
T EZP )-structure of the universal unfoldingof the function germ? The first one is a (
T E )-structure over a point t .The second one is a ( T E )-structure over a germ (
M, t ) of a manifold M . It it an unfolding of the first ( T E )-structure with a primitive Higgsfield . The base space M is an F-manifold with Euler field .We explain these notions. A second (
T E )-structure over a mani-fold M is an unfolding of a first ( T E )-structure over a submanifold of M if the restriction of the second ( T E )-structure to the submanifoldis isomorphic to the first (
T E )-structure. If ϕ : M (2) → M (1) is amorphism and if ( H, ∇ ) is a ( T E )-structure over M (1) , then the pullback ϕ ∗ ( H, ∇ ) is a ( T E )-structure over M (2) . An unfolding of a ( T E )-structure is universal if it induces any unfolding via a unique map ϕ (see Definition 3.15 (b)+(c) for details). C. HERTLING
If ( H → C × M, ∇ ) is a ( T E )-structure, then define the vector bundle K := H | { }× M on M and the Higgs field C := [ z ∇ ] ∈ Ω ( M, End( K ))on K . The endomorpisms C X = [ z ∇ X ] : O ( K ) → O ( K ) for X ∈ T M commute. And they commute with the endomorphism U := [ z ∇ ∂ z ] : O ( K ) → O ( K ) (see Definition 3.8 and Lemma 3.12). The Higgs field C is primitive if on each sufficiently small subset U ⊂ K a section ζ U exists such that the map T U → O ( K ) , X C X ζ U , is an isomorphism(see Definition 3.13).An F-manifold with
Euler field is a complex manifold M togetherwith a holomorphic commutative and associative multiplication ◦ on T M which comes equipped with the integrability condition (2.1), witha unit field e ∈ T M (with ◦ e = id) and an Euler field E ∈ T M withLie E ( ◦ ) = ◦ (see [HM99] or Definition 2.1). A ( T E )-structure over M with primitive Higgs field induces on the base manifold M the structureof an F-manifold with Euler field (see Theorem 3.14 for details).A result of Malgrange [Ma86] (cited in Theorem 3.16 (c)) says thata ( T E )-structure over a point t has a universal unfolding if the endo-morphism U : K → K (here K is a vector space) is regular, i.e. it hasonly one Jordan block for each eigenvalue. Theorem 3.16 (b) gives ageneralization from [HM04]. A special case of this generalization saysthat a ( T E )-structure with primitive Higgs field over a germ (
M, t )is its own universal unfolding (see Theorem 3.16 (a)). A supplementfrom [DH17] says that then the base space is a regular F-manifold (seeDefinition 2.4 and Theorem 2.5).Malgrange’s result makes life easy if one starts with a (
T E )-structureover a point whose endomorphism U is regular. But if one starts witha ( T E )-structure over a point such that U is not regular, then in gen-eral it has no universal unfolding, and the study of all its unfoldingsbecomes very interesting. The second half of this paper (the sections6 – 8) studies this situation in rank 2. The Torelli problem for a holo-morphic function germ with an isolated singularity is similar: The en-domorphism U of its ( T EZP )-structure is never regular (except if thefunction has an A -singularity), but I hope that the ( T EZP )-structuredetermines nevertheless somehow the specific unfolding with primitiveHiggs field, which comes from the universal unfolding of the originalfunction germ.Now sufficient background is given. We describe the contents of thispaper.The short section 2 recalls the classification of the 2-dimensionalgerms of F-manifolds with Euler fields (Theorem 2.2 from [He02] andTheorem 2.3 from [DH20-3]). It treats also regular F-manifolds (Defi-nition 2.4 and Theorem 2.5 from [DH17]).
ANK 2 (
T E )-STRUCTURES 5
Section 3 recalls many general facts on (
T E )-structures: their defini-tion, their presentation by matrices, formal (
T E )-structures, unfoldingsand universal unfoldings of (
T E )-structures, Malgrange’s result andthe generalization in [HM04], (
T E )-structures over F-manifolds, (
T E )-structures with primitive Higgs fields, regular singular (
T E )-structuresand elementary sections, Birkhoff normal form for (
T E )-structures (notall have one, Theorem 3.20 cites existence results of Plemely and ofBolibroukh and Kostov). Not written before, but elementary is a cor-respondence between (
T E )-structures with trace free endomorphism U and arbitrary ( T E )-structures (the Lemmata 3.9, 3.10 and 3.11).New is the notion of a marked (
T E )-structure. It is needed for theconstruction of moduli spaces. Theorem 3.28 (which builds on resultsin [HS10]) constructs such moduli spaces, but only in the case of regularsingular (
T E )-structures. And it starts with a good family of regularsingular (
T E )-structures. There are two open problems. It is notclear how to generalize this notion of a good family beyond the caseof regular singular (
T E )-structures. And we hope, but did not provefor rank ≥
3, that any regular singular (
T E )-structure (over M withdim M ≥
1) is a good family of regular singular (
T E )-structures. Forrank 2 this is true, it follows from Theorem 8.5.Section 4 gives the classification of rank 2 (
T E )-structures over apoint t . There are 4 types, which we call (Sem), (Bra), (Reg) and (Log)(for semisimple, branched, regular singular and logarithmic ). In thetype (Sem) U has two different eigenvalues, in the type (Log) U ∈ C · id,in the types (Bra) and (Reg) U has a 2 × U is trace free, a ( T E )-structure of type (Log) has a logarithmicpole, a (
T E )-structure of type (Reg) has a regular singular, but notlogarithmic pole, and the pull back of a (
T E )-structure of type (Bra)by a branched cover of C of order 4 has a meromorphic connectionwith semisimple pole of order 3 (see Lemma 4.8). The semisimplecase (Sem) is not central in this paper. Therefore we do not discussit in detail and do not introduce Stokes structures. For the othertypes (Bra), (Reg) and (Log), section 4 discusses normal forms andtheir parameters. All ( T E )-structures of type (Bra) have nice Birkhoffnormal forms (Theorem 4.9), but not all of type (Reg) (Theorem 4.15and the Remarks 4.17) and type (Log) (Theorem 4.18 and the Remarks4.20). The types (Reg) and (Log) become transparent by the use ofelementary sections.A (
T E )-structure of type (Sem) or (Bra) or (Reg) over a point t sat-isfies the hypothesis of Malgrange’s result, namely, the endomorphism U : K → K is regular. Therefore it has a universal unfolding, and anyunfolding of it is induced by this universal unfolding. Here life is easy. C. HERTLING
Section 5 discusses this. Also because of this fact, the semisimple caseis not central in this paper.The sections 6 – 8 are devoted to the study of (
T E )-structures over agerm (
M, t ) such that the restriction to t is a ( T E )-structure of type(Log). Then the set of points over which the (
T E )-structure restrictsto one of type (Log) is either a hypersurface or the whole of M . In thefirst case, it restricts to a fixed generic type (Sem) or (Bra) or (Reg)over points not in the hypersurface. In the second case, the generictype is (Log).Section 6 starts this study. It considers the cases with trace free U and dim M = 1. It has three parts. In the first part, invariants of such1-parameter families are studied. In a surprisingly direct way, con-straints on the difference of the leading exponents (defined in Theorem4.18) of the logarithmic ( T E )-structure over t are found, and the mon-odromy in the generic cases (Sem) and (Reg) turns out to be semisimple(Theorem 6.2). By Plemely’s result (and our direct calculations), thesecases come equipped with Birkhoff normal forms. Theorem 6.3 in thesecond part classifies all ( T E )-structures over (
M, t ) with trace free U , dim M = 1, logarithmic restriction to t and Birkhoff normal form.Theorem 6.7 in the third part classifies all generically regular singular( T E )-structures over (
M, t ) with dim M = 1, logarithmic restrictionto t , and whose monodromy has a 2 × T E )-structures from Theorem 3.28 explicit in the rank 2 cases. It buildson the classification results for the types (Reg) and (Log) in section4. The long Theorem 7.4 describes the moduli spaces and offers 5figures in order to make this more transparent. The moduli spaceshave countably many topological components, and each componentconsists of an infinite chain of projective spaces which are either theprojective line P or the Hirzebruch surface F or e F (which is obtainedby blowing down in F the unique ( − T E )-structuresover a 2-dimensional germ (
M, t ) such that the restriction to t hasa logarithmic pole, such that the Higgs field is generically primitive,and such that the induced structure of an F-manifold with Euler fieldextends to all of M . Theorem 8.1 (d) offers explicit normal forms. ANK 2 (
T E )-STRUCTURES 7
Corollary 8.3 starts with any logarithmic rank 2 (
T E )-structure over apoint t and lists the ( T E )-structures in Theorem 8.1 (d) which unfoldit.Theorem 8.5 is the most fundamental result of section 8. Table (8.12)in it is a sublist of the (
T E )-structures in Theorem 8.1 (d). Theorem8.5 states that any unfolding of a rank 2 (
T E )-structure of type (Log)over a point is induced by one (
T E )-structure in table (8.12). In thegeneric cases (Reg) and (Bra) these are precisely those in Theorem8.1 (d) with primitive Higgs field, but in the generic cases (Sem) and(Bra) table (8.12) contains many (
T E )-structures with only generi-cally primitive Higgs field. All the (
T E )-structures in table (8.12)are universal unfoldings of themselves, also those with only genericallyprimitive Higgs field. Almost all logarithmic (
T E )-structures over apoint have several unfoldings which do not induce one another. Onlythe logarithmic (
T E )-structures over a point whose monodromy has a2 × T E )-structures exist over each of them. It turns out that the nilpotentF-manifold N with the Euler field E = t ∂ + t r (1 + c t r − ) ∂ for r ≥ T E )-structure over itif c = 0, and it has no ( T E )-structure with primitive Higgs field overit if c = 0 or r ≥
3. But most 2-dimensional F-manifolds with Eulerfields have one or countably many families of (
T E )-structures with 1or 2 parameters over them.The third part of section 8 is the proof of Theorem 8.5.In many aspects, the (
T E )-structures of rank 2 are probably typicalalso for higher rank. But section 9 makes one phenomenon explicitwhich arises only in rank ≥
3. Section 9 presents a family of rank 3(
T E )-structures with primitive Higgs fields over a fixed 3-dimensionalglobally irreducible F-manifold with nowhere regular Euler field, suchthat the family has a functional parameter . The example is essentiallydue to M. Saito, it is a Fourier-Laplace transformation of the mainexample in a preliminary version of [SaM17] (though he considers onlythe bundle and connection over a 2-dimensional submanifold of theF-manifold).This paper has some overlap with [DH20-2] and [DH20-3]. In[DH20-2, ch. 8] (
T E )-structures over the 2-dimensional F-manifolds I ( m ) were studied. They are of generic type (Sem). In [DH20-3]( T E )-structures over the 2-dimensional F-manifold N (with all possi-ble Euler fields) were studied. They are of generic types (Bra), (Reg) or C. HERTLING (Log). But in [DH20-2] and [DH20-3] the focus was on (
T E )-structureswith primitive Higgs fields. Those with generically primitive, but notprimitive Higgs fields were not considered. And the approach to theclassification was very different. It relied on the formal classification ofrank 2 ( T )-structures in [DH20-1]. The approach here is independentof these three papers.I would like to thank Liana David for a lot of joint work on ( T E )-structures.2.
The two-dimensional F-manifolds and their Eulerfields F -manifolds were first defined in [HM99]. Their basic propertieswere developed in [He02]. An overview on them and on more recentresults is given in [DH20-2]. Definition 2.1. (a) An
F-manifold ( M, ◦ , e ) (without Euler field) is acomplex manifold M with a holomorphic commutative and associativemultiplication ◦ on the holomorphic tangent bundle T M and with aglobal holomorphic vector field e ∈ T M with e ◦ = id ( e is called a unitfield ), which satisfies the following integrability condition:Lie X ◦ Y ( ◦ ) = X ◦ Lie Y ( ◦ ) + Y ◦ Lie X ( ◦ ) for X, Y ∈ T M . (2.1)(b) Given an F-manifold ( M, ◦ , e ), an Euler field on it is a global vectorfield E ∈ T M with Lie E ( ◦ ) = ◦ .In this paper we are mainly interested in the 2-dimensional F-manifolds and their Euler fields. They were classified in [He02]. Theorem 2.2. [He02, Theorem 4.7]
In dimension 2, (up to isomor-phism) the germs of F-manifolds fall into three types:(a) The semisimple germ. It is called A , and it can be given asfollows. ( M,
0) = ( C ,
0) with coordinates u = ( u , u ) and e k = ∂∂u k ,e = e + e , e j ◦ e k = δ jk · e j . (2.2) Any Euler field takes the shape E = ( u + c ) e + ( u + c ) e for some c , c ∈ C . (2.3) (b) Irreducible germs, which (i.e. some holomorphic representativesof them) are at generic points semisimple. They form a series I ( m ) , ANK 2 (
T E )-STRUCTURES 9 m ∈ Z ≥ . The germ of type I ( m ) can be given as follows. ( M,
0) = ( C ,
0) with coordinates t = ( t , t ) and ∂ k := ∂∂t k ,e = ∂ , ∂ ◦ ∂ = t m − e. (2.4) Any Euler field takes the shape E = ( t + c ) ∂ + 2 m t ∂ for some c ∈ C . (2.5) (c) An irreducible germ, such that the multiplication is everywhereirreducible. It is called N , and it can be given as follows. ( M,
0) = ( C ,
0) with coordinates t = ( t , t ) and ∂ k := ∂∂t k ,e = ∂ , ∂ ◦ ∂ = 0 . (2.6) Any Euler field takes the shape E = ( t + c ) ∂ + g ( t ) ∂ for some c ∈ C (2.7)and some function g ( t ) ∈ C { t } . The family of Euler fields in (2.7) on N can be reduced by coordinatechanges which respect the multiplication of N to a family with twocontinuous parameters and one discrete parameter. This classificationis proved in [DH20-3]. It is recalled in Theorem 2.3. The group Aut( N )of automorphisms of the germ N of an F-manifold is the group ofcoordinate changes of ( C ,
0) which respect the multiplication of N .It is Aut( N ) = { ( t , t ) ( t , λ ( t )) | λ ∈ C { t } (2.8)with λ ′ (0) = 0 and λ (0) = 0 . } Theorem 2.3.
Any Euler field on the germ N of an F-manifold canbe brought by a coordinate change in Aut( N ) to a unique one in thefollowing family of Euler fields. E = ( t + c ) ∂ + ∂ , (2.9) E = ( t + c ) ∂ , (2.10) E = ( t + c ) ∂ + c t ∂ , (2.11) E = ( t + c ) ∂ + t r (1 + c t r − ) ∂ , (2.12) where c , c ∈ C , c ∈ C ∗ and r ∈ Z ≥ . The group
Aut( N , E ) ofcoordinate changes of ( C , which respect the multiplication of N and this Euler field is Aut( N , E ) = { ( t , t ) ( t , γ ( t ) t ) | γ as in (2.14) } , (2.13) case (2.9) (2.10) (2.11) (2.12) γ ∈ { } C { t } ∗ C ∗ { e πil/ ( r − | l ∈ Z } (2.14)A special class of F-manifolds, the regular F-manifolds, is related toa result of Malgrange on universal unfoldings of ( T E )-structures, seethe Remarks 3.17.
Definition 2.4. [DH17, Definition 1.2] A regular F-manifold is an F-manifold ( M, ◦ , e ) with Euler field E such that at each t ∈ M theendomorphism E ◦ | t : T t M → T t M is a regular endomorphism, i.e. ithas for each eigenvalue only one Jordan block. Theorem 2.5. [DH17, Theorem 1.3 ii)]
For each regular endomor-phism of a finite dimensional C -vector space, there is a unique (up tounique isomorphism) germ ( M, t ) of a regular F-manifold such that E ◦ | t is isomorphic to this endomorphism. (For a normal form of thisgerm of an F-manifold, see [DH17, Theorem 1.3 i)] ). Remark 2.6.
In dimension 2, this theorem is an easy consequenceof the Theorems 2.2 and 2.3. The germs of regular 2-dimensional F-manifolds are as follows:(i) The germ A in Theorem 2.2 (a) with any Euler field E =( u + c ) e + ( u + c ) e as in (2.3) with c , c ∈ C , c = c .(ii) The germ N in Theorem 2.2 (c) with any Euler field E =( t + c ) ∂ + ∂ as in (2.9) with c ∈ C .3. ( T E ) -structures in general
3. 1.
Definitions.
A (
T E )-structure is a holomorphic vector bundleon C × M , M a complex manifold, with a meromorphic connection ∇ with a pole of Poincar´e rank 1 along { } × M and no pole elsewhere.Here we consider them together with the weaker notion of ( T )-structureand the more rigid notions of a ( T L )-structure and a (
T LE )-structure.The structures had been considered before in [HM04], and they arerelated to structures in [Sa02, VII] and in [Sa05].
Definition 3.1. (a) Definition of a (T)-structure ( H → C × M, ∇ ): H → C × M is a holomorphic vector bundle. ∇ is a map ∇ : O ( H ) → z − O C × M · Ω M ⊗ O ( H ) , (3.1)which satisfies the Leibniz rule, ∇ X ( a · s ) = X ( a ) · s + a · ∇ X s for X ∈ T M , a ∈ O C × M , s ∈ O ( H ) , and which is flat (with respect to X ∈ T M , not with respect to ∂ z ), ∇ X ∇ Y − ∇ Y ∇ X = ∇ [ X,Y ] for X, Y ∈ T M . ANK 2 (
T E )-STRUCTURES 11
Equivalent: For any z ∈ C ∗ , the restriction of ∇ to H | { z }× M is a flatholomorphic connection.(b) Definition of a (TE)-structure ( H → C × M, ∇ ): H → C × M isa holomorphic vector bundle. ∇ is a flat connection on H | C ∗ × M with apole of Poincar´e rank 1 along { } × M , so it is a map ∇ : O ( H ) → (cid:0) z − O C × M · Ω M + z − O C × M · d z (cid:1) ⊗ O ( H ) (3.2)which satisfies the Leibniz rule and is flat.(c) Definition of a (TL)-structure ( H → P × M, ∇ ): H → P × M is a holomorphic vector bundle. ∇ is a map ∇ : O ( H ) → (cid:0) z − O P × M + O P × M (cid:1) · Ω M ⊗ O ( H ) , (3.3)such that for any z ∈ P − { } , the restriction of ∇ to H | { z }× M is a flatconnection. It is called pure if for any t ∈ M the restriction H | P ×{ t } isa trivial holomorphic bundle on P .(d) Definition of a (TLE)-structure ( H → P × M, ∇ ): It is simulta-neously a ( T E )-structure and a (
T L )-structure, where the connection ∇ has a logarithmic pole along {∞} × M . The ( T LE )-structure iscalled pure if the (
T L )-structure is pure.
Remarks 3.2.
Here we write the data in Definition 3.1 (a)–(b) and thecompatibility conditions between them in terms of matrices. Considera (
T E )-structure ( H → C × M, ∇ ) of rank rk H = r ∈ N . We will fixthe notations for a trivialization of the bundle H | U × M for some smallneighborhood U ⊂ C of 0. Trivialization means the choice of a basis v = ( v , ..., v r ) of the bundle H | U × M . Also, we choose local coordinates t = ( t , ..., t n ) with coordinate vector fields ∂ i = ∂/∂t i on M . We write ∇ v = v · Ω withΩ = r X i =1 z − · A i ( z, t )d t i + z − B ( z, t )d z, (3.4) A i ( z, t ) = X k ≥ A ( k ) i z k ∈ M r × r ( O U × M ) , (3.5) B ( z, t ) = X k ≥ B ( k ) z k ∈ M r × r ( O U × M ) , (3.6)with A ( k ) i , B ( k ) ∈ M r × r ( O M ), but this dependence on t ∈ M is usuallynot written explicity. The flatness 0 = dΩ + Ω ∧ Ω of the connection ∇ says for i, j ∈ { , ..., n } with i = j z∂ i A j − z∂ j A i + [ A i , A j ] , (3.7)0 = z∂ i B − z ∂ z A i + zA i + [ A i , B ] . (3.8) These equations split into the parts for the different powers z k for k ≥ A ( − i = B ( − = 0),0 = ∂ i A ( k − j − ∂ j A ( k − i + k X l =0 [ A ( l ) i , A ( k − l ) j ] , (3.9)0 = ∂ i B ( k − − ( k − A ( k − i + k X l =0 [ A ( l ) i , B ( k − l ) ] . (3.10)In the case of a ( T )-structure, B and all equations except (3.4) whichcontain B are dropped.Consider a second ( T E )-structure ( e H → C × M, e ∇ ) of rank r over M , where all data except M are written with a tilde. Let v and e v betrivializations. A holomorphic isomorphism from the first to the second( T E )-structure maps v · T to e v , where T = T ( z, t ) = P k ≥ T ( k ) z k ∈ M r × r ( O ( C , × M ) with T ( k ) ∈ M r × r ( O M ) and T (0) invertible satisfies v · Ω · T + v · d T = ∇ ( v · T ) = v · T · e Ω . (3.11)(3.11) says more explicitly0 = z∂ i T + A i · T − T · e A i , (3.12)0 = z ∂ z T + B · T − T · e B. (3.13)These equations split into the parts for the different powers z k for k ≥ T ( − := 0):0 = ∂ i T ( k − + k X l =0 ( A ( l ) i · T ( k − l ) − T ( k − l ) · e A ( l ) i ) , (3.14)0 = ( k − T ( k − + k X l =0 ( B ( l ) · T ( k − l ) − T ( k − l ) · e B ( l ) ) . (3.15)The isomorphism here fixes the base manifold M . Such isomorphismsare called gauge isomorphisms . A general isomorphism is a compositionof a gauge isomorphism and a coordinate change on M (a coordinatechange induces an isomorphism of ( T E )-structures, see Lemma 3.6).
Remark 3.3.
In this paper we care mainly about (
T E )-structures overthe 2-dimensional germs of F-manifolds with Euler fields. For each ofthem except ( N , E = ( t + c ) ∂ ), the group of coordinate changes of( M,
0) = ( C ,
0) which respect the multiplication and E is quite small,see Theorem 2.3. Therefore in this paper, we care mainly about gauge isomorphisms of the ( T E )-structures over these F-manifolds with Eulerfields.
ANK 2 (
T E )-STRUCTURES 13
Definition 3.4.
Let M be a complex manifold.(a) The sheaf O M [[ z ]] on M is defined by O M [[ z ]]( U ) := O M ( U )[[ z ]]for an open subset U ⊂ M (with O M ( U ) and O M [[ z ]]( U ) the sections of O M and O M [[ z ]] on U ). Observe that the germ ( O M [[ z ]]) t for t ∈ M consists of formal power series P k ≥ f k z k whose coefficients f k ∈ O M,t have a common convergence domain. In the case of ( M, t ) = ( C n , O C n [[ z ]] =: C { t, z ]].(b) A formal ( T ) -structure over M is a free O M [[ z ]]-module O ( H )of some finite rank r ∈ N together with a map ∇ as in (3.1) where O C × M is replaced by O M [[ z ]] which satisfies properties analogous to ∇ in Definition 3.1 (a), i.e. the Leibniz rule for X ∈ T M , a ∈ O M [[ z ]] , s ∈O ( H ) and the flatness condition for X, Y ∈ T M .A formal ( T E ) -structure is defined analogously: In Definition 3.1 (b)one has to replace O C × M by O M [[ z ]]. Remarks 3.5. (a) The formulas in the Remarks 3.2 hold also for for-mal ( T )-structures and formal ( T E )-structures if one replaces O C × M , O U × M and O ( C , × M by O M [[ z ]].The following lemma is obvious. Lemma 3.6.
Let ( H → C × M, ∇ ) be a ( T E ) -structure over M , andlet ϕ : M ′ → M be a holomorphic map between manifolds. One canpull back H and ∇ with id × ϕ : C × M ′ → C × M . We call the pullback ϕ ∗ ( H, ∇ ) . It is a ( T E ) -structure over M ′ . We say that the pullback ϕ ∗ ( H, ∇ ) is induced by the ( T E ) -structure ( H, ∇ ) via the map ϕ . Remarks 3.7. (i) We will give in Theorem 8.5 and in Corollary5.1 & Lemma 5.2 (iv) a classification of rank 2 (
T E )-structures overgerms (
M, t ) = ( C ,
0) of 2-dimensional manifolds such that any rank2 (
T E )-structure over a germ ( M ′ , s ) is obtained as the pull back ϕ ∗ ( H, ∇ ) of a rank 2 ( T E )-structure in the classification via a holo-morphic map ϕ : ( M ′ , s ) → ( M, t ).(ii) Here the behaviour of the ( T E )-structure ( H, ∇ ) over ( M, t ) =( C ,
0) with coordinates t = ( t , t ) along t is quite trivial. It is conve-nient to split it off. The next subsection does this in greater generality.3. 2. ( T E ) -structures with trace free pole part.Definition 3.8. Let ( H → C × M, ∇ ) be a ( T E )-structure. Definethe vector bundle K := H | { }× M over M . The pole part of the ( T E )-structure is the endomorphism U : K → K which is defined by U := [ z ∇ ∂ z ] : K → K. (3.16)The pole part is trace free if tr U = 0 on M . The following lemma gives formal invariants of a (
T E )-structure.
Lemma 3.9.
Let ( H → C × M, ∇ ) be a ( T E ) -structure of rank r ∈ N over a manifold M . By a formal invariant of the ( T E ) -structure, wemean an invariant of its formal isomorphism class.(a) Its pole part U , that means the pair ( K, U ) up to isomorphism, isa formal invariant of the ( T E ) -structure. Especially, the holomorphicfunctions δ (0) := det U ∈ O M and ρ (0) := r tr U ∈ O M are formalinvariants.(b) For any t ∈ M , fix an O M,t -basis v of O ( H ) (0 ,t ) , consider thematrices in (3.4) – (3.6) , consider the function ρ (1) := r tr B (1) ∈ O M,t ,and consider the functions δ ( k ) ∈ O M,t for k ∈ Z ≥ which are definedby writing det B as a power series det B = X k ≥ δ ( k ) z k . (3.17) Then the functions δ (1) and ρ (1) are independent of the choice of thebasis v . The locally for any t defined functions δ (1) and ρ (1) glue toglobal holomorphic functions δ (1) ∈ O M and ρ (1) ∈ O M . They areformal invariants. Furthermore, the function ρ (1) is constant on anycomponent of M . Proof: U , δ (0) , ρ (0) and δ (1) are formal invariants because of (3.13): e B = T − BT + z · T − ∂ z T. For ρ (1) , observe additionally e B (1) = ( T (0) ) − B (1) T (0) + [( T (0) ) − B (0) T (0) , ( T (0) ) − T (1) ] . Recall also that the trace of a commutator of matrices is 0. Therefore ρ (1) is a formal invariant.(3.10) for k = 2 implies ∂ i tr( B (1) ) = 0, so the function ρ (1) is con-stant. (cid:3) The following lemma is obvious.
Lemma 3.10.
Let ( H → C × M, ∇ ) be a ( T E ) -structure of rank r ∈ N over a manifold M .(a) Consider a holomorphic function g : M → C . The trivial linebundle H [1] = C × ( C × M ) → C × M over C × M with connection ∇ [1] := d + d( gz ) defines a ( T E ) -structure of rank 1 over M , whosesheaf of sections with connection is called E g/z .(b) ( O ( H ) , ∇ ) ⊗ E g/z for g as in (a) is a ( T E ) -structure.(c) The ( T E ) -structure ( H [2] → C × M, ∇ [2] ) with ( O ( H [2] ) , ∇ [2] ) =( O ( H ) , ∇ ) ⊗ E ρ (0) /z has trace free pole part. And, of course, ANK 2 (
T E )-STRUCTURES 15 ( O ( H ) , ∇ ) ∼ = ( O ( H [2] ) , ∇ [2] ) ⊗ E − ρ (0) /z . If v is a C { t, z } -basis of O ( H ) = O ( H [2] ) , then the matrix valued connection 1-forms Ω and Ω [2] of ∇ and ∇ [2] with respect to this basis satisfy Ω = Ω [2] − d( ρ (0) z ) · r .(d) (Definition) Consider a ( T E ) -structure ( H [3] → C × M [3] , ∇ [3] ) with trace free pole part. Consider the manifold M [4] := C × M [3] with (local) coordinates t on C and t ′ on M [3] , and the projection ϕ [4] : M [4] → M [3] , ( t , t ′ ) t ′ . Define the ( T E ) -structure ( H [4] → C × M [4] , ∇ [4] ) with ( O ( H [4] ) , ∇ [4] ) = ( ϕ [4] ) ∗ ( O ( H [3] ) , ∇ [3] ) ⊗ E t /z .(e) If the ( T E ) -structure ( H [2] , ∇ [2] ) is induced by the ( T E ) -structure ( H [3] , ∇ [3] ) via a map ϕ : M → M [3] , then the ( T E ) -structure ( H, ∇ ) is induced by the ( T E ) -structure ( H [4] , ∇ [4] ) via the map ( − ρ (0) , ϕ ) : M → M [4] = C × M [3] . Part (c) allows to go from an arbitrary (
T E )-structure to one withtrace free pole part, and to go back to the original one. Part (e)considers two (
T E )-structures as in part (c), an original one and anassociated one with trace free pole part. If the associated one is in-duced by a third (
T E )-structure, then the original one is induced by aclosely related (
T E )-structure with one parameter more. Lemma 3.11continues Lemma 3.10.
Lemma 3.11.
Let ( H → C × ( M, t ) , ∇ ) be a ( T E ) -structure of rank r ∈ N over a germ ( M, t ) of a manifold, with coordinates t = ( t , ..., t n ) and ∂ i := ∂/∂t i . We suppose t = 0 so that O ( C × M, (0 ,t )) = C { t, z } .Recall the functions ρ (0) and ρ (1) of the ( T E ) -structure from Lemma3.9.Consider the ( T E ) -structure ( H [2] , ∇ [2] ) from Lemma 3.10 with tracefree pole part which is defined by ( O ( H [2] ) , ∇ [2] ) := ( O ( H ) , ∇ ) ⊗ E ρ (0) /z .Here H [2] = H , but ∇ [2] = ∇ + d( ρ (0) z ) · id . The matrices A i and B in (3.4) – (3.6) for the ( T E ) -structure ( H [2] , ∇ [2] ) of any C { t, z } -basis v of O ( H [2] ) satisfy A (0) i = tr B (0) = tr( B (1) − ρ (1) r ) . (3.18) The basis v can be chosen such that the matrices satisfy A i = tr( B − zρ (1) r ) . (3.19) Proof:
Any C { t, z } -basis v of O ( H [2] ) = O ( H ) satisfiestr B (0) = tr U [2] = 0 as ( H [2] , ∇ [2] ) has trace free pole part,tr A (0) i = 0 because of tr ∂ i B (0) = ∂ i tr B (0) = 0and (3.10) for k = 1 , tr( B (1) − ρ (1) r ) = 0 by Lemma 3.9 and especiallyΩ = Ω [2] − d( ρ (0) z ) · r = Ω [2] − n X i =1 z − ∂ρ (0) ∂t i · r d t i + z − ρ (0) · r d z. Start with an arbitrary basis v , consider the function g := 1 r X k ≥ − tr B ( k ) k − · z k − ∈ z C { t, z } , (3.20)consider T := e g · r , and e v := v · T . (3.13) gives e B = B + T − z ∂ z T = B + (cid:0) − X k ≥ tr B ( k ) z k (cid:1) · r r , so tr e B ( k ) = 0 for k ≥ e B (1) = B (1) , e B (0) = B (0) .Therefore now suppose tr( B − zρ (1) r ) = 0. (3.10) for k ≥ A ( l ) i = 0 for l ≥
2, because tr ∂ i B ( l ) = ∂ i tr B ( l ) = 0.Finally, we consider T = T (0) = e h · r for a suitable function h ∈ C { t } . Then e B = B , e A ( k ) i = A ( k ) i for k = 1, and e A (1) i = A (1) i + ∂ i h · r .Se we need h ∈ C { t } with ∂ i h = − r tr A (1) i . Such a function existsbecause (3.9) for k = 2 implies ∂ i tr A (1) j = ∂ j tr A (1) i . We have obtaineda basis v with tr( B − zρ (1) r ) = 0 and tr A i = 0 for all i . (cid:3)
3. 3. (
T E ) -structures over F-manifolds with Euler fields. Thepole part of a ( T )-structure (or a ( T E )-structure) over C × M along { } × M induces a Higgs bundle (together with U ). This is elementary(e.g. [He03] or [DH20-1]). Lemma 3.12.
Let ( H → C × M, ∇ ) be a ( T ) -structure. Define K := H | { }× M . Then C := [ z ∇ ] ∈ Ω ( M, End( K )) , more explicitly C X [ a ] := [ z ∇ X a ] for X ∈ T M , a ∈ O ( H ) , (3.21) is a Higgs field, i.e. the endomorphisms C X , C Y : K → K for X, Y ∈T M commute.If ( H → C × M ) is a ( T E ) -structure, then its pole part U : K → K commutes with all endomorphisms C X , X ∈ T M , short: [ C, U ] = 0 . ANK 2 (
T E )-STRUCTURES 17
Definition 3.13.
The Higgs field of a ( T )-structure or a ( T E )-structure ( H → C × M, ∇ ) is primitive if there is an open cover V of M and for any U ∈ V a section ζ U ∈ O ( K | U ) (called a local primi-tive section ) with the property that the map T U ∋ X → C X ζ U ∈ O ( K )is an isomorphism.The Theorems 3.14 and 3.16 show in two ways that primitivity ofa Higgs field is a good condition. Theorem 3.14 was first proved in[HHP10, Theorem 3.3] (but see also [DH20-1, Lemma 10]). Theorem 3.14. A ( T ) -structure ( H → C × M, ∇ ) with primitive Higgsfield induces a multiplication ◦ on T M which makes M an F -manifold.A ( T E ) -structure ( H → C × M, ∇ ) with primitive Higgs field inducesin addition a vector field E on M , which, together with ◦ , makes M an F -manifold with Euler field. The multiplication ◦ , unit field e andEuler field E (the latter in the case of a ( T E ) -structure), are definedby C X ◦ Y = C X C Y , C e = Id , C E = −U , (3.22) where C is the Higgs field defined by ∇ , and U is defined in (3.16) . Definition 3.15 recalls the notions of an unfolding and of a universalunfolding of a (
T E )-structure over a germ of a manifold from [HM04,Definition 2.3]. It turns out that any (
T E )-structure over a germ ofa manifold with primitive Higgs field is a universal unfolding of itself.But we will see in Theorem 8.5 also (
T E )-structures which are universalunfoldings of themselves, but where the Higgs bundle is only genericallyprimitive. Still in the examples which we consider, the base manifoldis an F-manifold with Euler field globally.Malgrange [Ma86] proved that a (
T E )-structure over a point t hasa universal unfolding with primitive Higgs field if the endomorphism U : K t → K t is regular , i.e. it has for each eigenvalue only one Jordanblock. A generalization was given by Hertling and Manin [HM04, The-orem 2.5]. Theorem 3.16 cites in part (b) the generalization. Part (a)is the special case of a ( T E )-structure with primitive Higgs field. Part(c) is the special case of a (
T E )-structure over a point, Malgrange’sresult.
Definition 3.15.
Let ( H → C × ( M, t ) , ∇ ) be a ( T E )-structure overa germ (
M, t ) of a manifold.(a) An unfolding of it is a ( T E )-structure ( H [1] → C × ( M × C l , ( t , , ∇ [1] ) over a germ ( M × C l , ( t , l ∈ Z ≥ ) together with a fixed isomorphism i [1] : ( H → C × ( M, t ) , ∇ ) (3.23) → ( H [1] → C × ( M × C l , ( t , , ∇ [1] ) | C × ( M ×{ } , ( t , . (b) One unfolding ( H [1] → C × ( M × C l , ( t , , ∇ [1] , i [1] ) induces asecond unfolding ( H [2] → C × ( M × C l , ( t , , ∇ [2] , i [2] ) if there are aholomorphic map germ ϕ : ( M × C l , ( t , → ( M × C l , ( t , , (3.24)which is the identity on M ×{ } , and an isomorphism j from the secondunfolding to the pullback of the first unfolding by ϕ such that i [1] = j | C × ( M ×{ } , ( t , ◦ i [2] . (3.25)(Then j is uniquely determined by ϕ and (3.25).)(c) An unfolding is universal if it induces any unfolding via a uniquemap ϕ .By definition of a universal unfolding in part (c), a ( T E )-structurehas (up to canonical isomorphism) at most one universal unfolding,because any two universal unfoldings induce one another by uniquemaps.
Theorem 3.16. (a) [HM04, Theorem 2.5] A ( T E ) -structure over agerm ( M, t ) with primitive Higgs field is a universal unfolding of itself.(b) [HM04, Theorem 2.5] Let ( H → C × ( M, t ) , ∇ ) be a ( T E ) -structure over a germ ( M, t ) of a manifold. Let ( K → ( M, t ) , C ) bethe induced Higgs bundle over ( M, t ) . Suppose that a vector ζ t ∈ K t with the following properties exists.(IC) (Injectivity condition) The map C • ζ t : T t M → K t isinjective.(GC) (Generation condition) ζ t and its images under iterationof the maps U | t : K t → K t and C X : K t → K t for X ∈ T t M generate K t .Then a universal unfolding of the ( T E ) -structure over a germ ( M × C l , ( t , ( l ∈ N suitable) exists. It is unique up to isomorphism. ItsHiggs field is primitive.(c) [Ma86] A ( T E ) -structure over a point t has a universal unfoldingwith primitive Higgs field if the endomorphism [ z ∇ ∂ z ] = U : K t → K t is regular, i.e. it has for each eigenvalue only one Jordan block. In thatcase, the germ of the F-manifold with Euler field which underlies theuniversal unfolding, is by definition (Definition 2.4) regular. ANK 2 (
T E )-STRUCTURES 19
Remarks 3.17. (i) A germ ((
M, t ) , ◦ , e, E ) of a regular F-manifold isuniquely determined by the regular endomorphism E ◦ | t : T t → T t (Theorem 2.5).(ii) Consider the germ ( M,
0) = ( C ,
0) of a 2-dimensional F-manifoldwith Euler field E in Theorem 2.2. It is regular if and only if E ◦ | t =0 / ∈{ λ id | λ ∈ C } . In the semisimple case (Theorem 2.2 (a)) this holds ifand only if c = c . In the cases I ( m ) ( m ≥
3) it does not hold. Inthe case of N with E = t ∂ + g ( t ) ∂ it holds if and only if g (0) = 0.See also Remark 2.6.(iii) Theorem 3.16 (c) implies that a ( T E )-structure with primitiveHiggs field over a germ (
M, t ) of a regular F-manifold with Euler fieldis determined up to gauge isomorphism by the restriction of the ( T E )-structure to t .(iv) Lemma 3.6, Definition 3.8, Lemma 3.9, Lemma 3.10, Lemma3.11, Lemma 3.12, Definition 3.13, Theorem 3.14 and Definition 3.15hold or make sense also for formal ( T )-structures or ( T E )-structures.But the proof of Theorem 3.16 used in an essential way holomorphic ( T E )-structures. We do not know whether Theorem 3.16 holds also forformal (
T E )-structures.3. 4.
Birkhoff normal form.Definition 3.18.
Let ( H → C × M, ∇ ) be a ( T E )-structure over amanifold M with coordinates t = ( t , ..., t n ). A Birkhoff normal form consists of a basis v of H and associated matrices A , ..., A n , B as in(3.4) such that A ( k )1 = ... = A ( k ) n = 0 for k ≥ , B ( k ) = 0 for k ≥ , ∂ i B (1) = 0 . (3.26) Remarks 3.19. (i) Such a basis defines an extension of the (
T E )-structure to a pure (
T LE )-structure. Then it is a basis of the (
T LE )-structure whose restriction to {∞} × M is flat with respect to theresidual connection (that is just the restriction of the connection ∇ ofthe underlying ( T L )-structure to H | {∞}× M ). Then the conditions (3.9)and (3.10) boil down to0 = [ A (0) i , A (0) j ] , ∂ i A (0) j = ∂ j A (0) i , (3.27)0 = [ A (0) i , B (0) ] , ∂ i B (0) + A (0) i + [ A (0) i , B (1) ] , ∂ i B (1) . (3.28)Such a basis is relevant for the construction of Frobenius manifolds (seee.g. [DH20-2]).(ii) Vice versa, if the ( T E )-structure has an extension to a pure(
T LE )-structure, then a basis v of the ( T LE )-structure exists whoserestriction to {∞} × M is flat with respect to the residual connection. Then this basis v and the associated matrices form a Birkhoff normalform.(iii) A Birkhoff normal form does not always exist. But if a Birkhoffnormal form of the restriction of a ( T E )-structure over M to a point t ∈ M exists, it extends to a Birkhoff normal form of the ( T E )-structure over the germ (
M, t ) [Sa02, VI Theorem 2.1] (or [DH20-2,Theorem 5.1 (c)]).(iv) The problem whether a ( T E )-structure over a point has an ex-tension to a pure (
T LE )-structure is a special case of the
Birkhoffproblem , which itself is a special case of the Riemann-Hilbert-Birkhoffproblem. The book [AB94] and chapter IV in [Sa02] are devoted tothese problems and results on them.Here the following two results on the Birkhoff problem will be useful.But in fact, we will use part (a) only in the case of a (
T E )-structureover a point t with a logarithmic pole at z = 0, in which case it istrivial. Theorem 3.20.
Let ( H → C × { t } , ∇ ) be a ( T E ) -structure over apoint t .(a) (Plemely, [Sa02, IV Corollary 2.6 (1)] ) If the monodromy issemisimple, the ( T E ) -structure has an extension to a pure ( T LE ) -structure.(b) (Bolibroukh and Kostov, [Sa02, IV Corollary 2.6 (3)] ) The germ O ( H ) ⊗ C { z } C { z } [ z − ] is a C { z } [ z − ] -vector space of dimension r =rk H ∈ N on which ∇ acts.If no C { z } [ z − ] sub vector space of dimension in { , ..., r − } existswhich is ∇ -invariant, then the ( T E ) -structure has an extension to apure ( T LE ) -structure.
3. 5.
Regular singular ( T E ) -structures. A (
T E )-structure over apoint t is regular singular if all its holomorphic sections have moderategrowth near 0. A good tool to treat this situation are special sectionsof moderate growth, the elementary sections . Definition 3.21 explainsthem and other basic notations. We work with a simply connectedmanifold M , so that the only monodromy is the monodromy alongclosed paths in the punctured z -plane going around 0. One importantcase is the case of a germ ( M, t ) of a manifold. The most importantcase is the case of a point, M = { t } . Definition 3.21.
Let ( H → C × M, ∇ ) be a ( T E )-structure of rank r = rk H ∈ N over a simply connected manifold M . We associate thefollowing data to it. ANK 2 (
T E )-STRUCTURES 21 (a) H ′ := H | C ∗ × M is the flat bundle on C ∗ × M . H ∞ denotesthe C -vector space (of dimension r ) of global flat multivalued sec-tions on H ′ . Let M mon be the monodromy on it with semisimple part M mons , unipotent part M monu , nilpotent part N mon := log M monu so that M monu = e N mon , and with eigenvalues in the finite set Eig( M mon ) ⊂ C .For λ ∈ C , let H ∞ λ := ker( M mons − λ id : H ∞ → H ∞ ) be the generalizedeigenspace in H ∞ of the monodromy with eigenvalue λ . It is not { } if and only if λ ∈ Eig( M mon ).(b) For α ∈ C , define the finite dimensional C -vector space C α ofthe following global sections of H ′ , C α := { σ ∈ O ( H ′ )( C ∗ ) | ( ∇ z∂ z − α id) r ( σ ) = 0 , ∇ ∂ i ( σ ) = 0 } (3.29)(where t = ( t , ..., t n ) are local coordinate and ∂ i are the coordinatevector fields). Observe z k · C α = C α + k for k ∈ Z . For each α the map s ( ., α ) : H ∞ e − πiα → C α , (3.30) A s ( A, α ) := z α · e − logz · N mon / πi A (log z ) , is an isomorphism. So, C α = { } if and only if e − πiα ∈ Eig( M mon ).The sections s ( A, α ) are called elementary sections .(c) A holomorphic section σ of H ′ | ( U −{ } ) × U for U ⊂ C a neigh-borhood of 0 ∈ C and U ⊂ M open in M can be written uniquely asan (in general infinite) sum of elementary sections es ( σ, α ) ∈ O U · C α with coefficients in O U , σ = X α : e − πiα ∈ Eig( M Mon ) es ( σ, α ) . (3.31)In order to see this, choose numbers α j ∈ C and elementary sections s j ∈ C α j for j ∈ { , ..., r } such that s , ..., s r form a global basis of H ′ .Then σ = r X j =1 a j s j with (3.32) a j = a j ( z, t ) = ∞ X k = −∞ a kj ( t ) z k ∈ O ( U −{ } ) × U . (3.33)Here (3.33) is the expansion of a j as a Laurent series in z with holo-morphic coefficients a kj ∈ O U in t . Then es ( σ, α )( z, t ) = X j : α − α j ∈ Z a α − α j ,j ( t ) z α − α j s j . (3.34)(d) A holomorphic section σ as in (c) has moderate growth if a bound b ∈ R with es ( σ, α ) = 0 for all α with Re( α ) < b exists. The sheaf V > −∞ on C × M of all sections of moderate growth is V > −∞ := M α : − < Re( α ) ≤ O C × M [ z − ] · C α . (3.35)The Kashiwara-Malgrange V -filtration is given by the locally free sub-sheaves for r ∈ R , V r := M α : Re( α ) ∈ [ r,r +1[ O C × M · C α . (3.36) Definition 3.22. (a) A (
T E )-structure ( H → C × M, ∇ ) over a simplyconnected manifold M is regular singular if O ( H ) ⊂ V > −∞ , so if all itsholomorphic sections have moderate growth near 0.(b) A ( T E )-structure ( H → C × M, ∇ ) over a simply connectedmanifold M is logarithmic if it has a basis v whose connection 1-formΩ has a logarithmic pole along { } × M (then this holds for any basis).In the notations of (3.4)–(3.6) that means A (0) i = B (0) = 0. Then therestriction of ∇ to K := H | { }× M is well-defined. It is called the residualconnection ∇ res . And then the residue endomorphism is Res = [ ∇ z∂ z ] : K → K . Theorem 3.23. (Well known, e.g. [He02, Theorem 7.10 and Theorem8.7] ) Let ( H → C × M, ∇ ) with H | C ∗ × M = H ′ be a logarithmic ( T E ) -structure over a simply connected manifold.(a) The bundle H has a global basis which consists of elemen-tary sections s j ∈ C α j , j ∈ { , ..., rk H } , for some α j ∈ C . Es-pecially, ( O ( H ) , ∇ ) = ϕ ∗ t ( O ( H | C ×{ t } ) , ∇ ) for any t ∈ M , where ϕ t : M → { t } is the projection. So it is just the pull back of a loga-rithmic ( T E ) -structure over a point. Especially, it is a regular singular ( T E ) -structure.(b) The residual connection ∇ res is flat. In the notations (3.4) – (3.6) ,its connection 1-form is P ni =1 A (1) i d t i . The residue endomorphism Res is ∇ res -flat. In the notations (3.4) – (3.6) , it is given by B (1) .(c) The endomorphism e − πi Res : K → K has the same eigenval-ues as the monodromy M mon , but it might have a simpler Jordan blockstructure. If no eigenvalues of Res differ by a nonzero integer (non-resonance condition) then e − πi Res has the same Jordan block structureas the monodromy M mon . Remarks 3.24. (i) Part (a) of Theorem 3.23 implies that a logarithmic(
T E )-structure over a simply connected manifold M is the pull back ϕ ∗ (( H, ∇ ) | C ×{ t } ) of its restriction to t for any t ∈ M .(ii) In the case of a regular singular ( T E )-structure over a simplyconnected manifold M , one can choose elementary sections s j ∈ C α j , ANK 2 (
T E )-STRUCTURES 23 j ∈ { , ..., rk H } , for some α j ∈ C , such that they form a basis of H ∗ and such the extension to { } × M which they define, is a logarithmic( T E )-structure. Then the base change from any local basis of H to thebasis ( s , ..., s rk H ) of this new ( T E )-structure is meromorphic, so thetwo (
T E )-structures give the same meromorphic bundle. This obser-vation fits to the usual definition of meromorphic bundle with regularsingular pole.(iii) The property of a section to have moderate growth, is invariantunder pull back. Therefore also the property of a (
T E )-structure to beregular singular is invariant under pull back.3. 6.
Marked ( T E ) -structures and moduli spaces for them. Itis easy to give a (
T E )-structure ( H → C × M, ∇ ) with nontrivial Higgsfield and which is thus not the pull back of the ( T E )-structure over apoint, such that nevertheless the (
T E )-structures over all points t ∈ M are isomorphic as abstract ( T E )-structures. Examples are given in Re-mark 7.1 (ii). The existence of such (
T E )-structures obstructs theconstruction of nice Hausdorff moduli spaces for (
T E )-structures up toisomorphism. The notion of a marked ( T E )-structure hopefully reme-dies this. But in the moment, we have only results in the regular singu-lar cases. Definition 3.25 gives the notion of a marked ( T E )-structure.Definition 3.26 defines good families of marked regular singular (
T E )-structures. Definition 3.28 defines a functor for such families. Theorem3.29 states that this functor is represented by a complex space. It buildson results in [HS10, ch. 7] Several remarks discuss what is missing inthe other cases and what more we have in the regular singular rank 2case, thanks to the Theorems 6.3, 6.7 and 8.5.
Definition 3.25. (a) A reference pair ( H ref, ∞ , M ref ) consists of a fi-nite dimensional (reference) C -vector space H ref, ∞ together with anautomorphism M ref of it.(b) Let M be a simply connected manifold. A marking on a ( T E )-structure ( H → C × M, ∇ ) is an isomorphism ψ : ( H ∞ , M mon ) → ( H ref, ∞ , M ref ). Here H ∞ is (as in Definition 3.21) the space of globalflat multivalued sections on the flat bundle H ′ := H | C ∗ × M , and M mon is its monodromy. ( H ref, ∞ , M ref ) is a reference pair. The isomorphism ψ of pairs means an isomorphism ψ : H ∞ → H ref, ∞ with ψ ◦ M mon = M ref ◦ ψ . A marked ( T E )-structure is a (
T E )-structure with a marking.(c) An isomorphism between two marked (
T E )-structures(( H (1) , ∇ (1) ) , ψ (1) ) and (( H (2) , ∇ (2) ) , ψ (2) ) over the same base space M (1) = M (2) and with the same reference pair ( H ref, ∞ , M ref ) is a gaugeisomorphism ϕ between the unmarked ( T E )-structures such that the induced isomorphism ϕ ∞ : H (1) , ∞ → H (2) , ∞ is compatible with themarking, ψ (2) ◦ ϕ ∞ = ψ (1) . (3.37)(d) Set ( H ref, ∞ ,M ref ) denotes the set of marked ( T E )-structuresover a point with the same reference pair ( H ref, ∞ , M ref ). AndSet ( H ref, ∞ ,M ref ) ,reg ⊂ Set ( H ref, ∞ ,M ref ) denotes the subset of marked reg-ular singular ( T E )-structures over a point with the same reference pair( H ref, ∞ , M ref ).We hope that Set ( H ref, ∞ ,M ref ) carries for any reference pair( H ref, ∞ , M ref ) a natural structure as a complex space. Theorem 3.29says that this holds for Set ( H ref, ∞ ,M ref ) ,reg and that this space representsa functor of good families of marked regular singular ( T E )-structures.Definition 3.26 gives a notion of a family of marked ( T E ) -structures andthe notion of a good family of marked regular singular ( T E ) -structures . Definition 3.26.
Let X be a complex space. Let t be an abstractpoint and ϕ : X → { t } be the projection. Let ( H ref, ∞ , M ref ) be areference pair. Let ( H ref, ∗ , ∇ ref ) be a flat bundle on C ∗ × { t } withmonodromy M ref and whose space of global flat multivalued sectionsis identified with H ref, ∞ . Let i : C ∗ × X ֒ → C × X be the inclusion.(a) A family of marked ( T E ) -structures over X is a pair ( H, ψ ) withthe following properties.(i) H is a holomorphic vector bundle on C × X , i.e. the linearspace associated to a locally free sheaf O ( H ) of O C × X -modules.Denote H ′ := H | C ∗ × X .(ii) ψ is an isomorphism ψ : H ′ → ϕ ∗ H ref, ∗ such that the re-striction of the induced flat connection on H ′ to C ∗ × { x } forany x ∈ X makes H | C ×{ x } into a ( T E )-structure over the point x , i.e. the connection has a pole of order ≤ H | C ×{ x } .(b) Consider a family ( H, ψ ) of marked regular singular (
T E )-structures over X . The marking ψ induces for each x ∈ X canonicalisomorphisms ψ : H ∞ ( x ) → H ref, ∞ , (3.38) ψ : C α ( x ) → C ref,α ( α ∈ C with e − πiα ∈ Eig( M ref )) ,ψ : V r ( x ) → V ref,r ( r ∈ R ) , where H ∞ ( x ) , C α ( x ) , V r ( x ) and C ref,α , V ref,r are defined for the ( T E )-structure over x respectively for ( H ref, ∗ , ∇ ) as in Definition 3.21. ANK 2 (
T E )-STRUCTURES 25
The family (
H, ψ ) is called good if some r ∈ R and some N ∈ N existwhich satisfy O ( H | C ×{ x } ) ⊃ V r ( x ) for any x ∈ X, (3.39)dim C O ( H | C ×{ x } ) /V r ( x ) = N for any x ∈ X. (3.40) Remarks 3.27. (i) The notion of a family of marked (
T E )-structuresis too weak. For example, it contains the following pathological familyof logarithmic (
T E )-structures of rank 1 over X := C (with coordinate t ) and with trivial monodromy. Write s ∈ C for a generating flatsection. Define H by O ( H ) = O C × X · ( t + z l ) s for some l ∈ N . (3.41)The marked ( T E )-structures over all points t ∈ C ∗ ⊂ X = C are iso-morphic and even equal, the one over t = 0 is different. The dimension O ( H | C ×{ t } ) /V l ( t ) is equal to l for t ∈ C ∗ and equal to 0 for t = 0.Therefore this family is not good in the sense of Definition 3.26 (b).Also, z ∇ ∂ z ( t + z l ) s = lz l s is not a section in O ( H ), although for eachfixed t ∈ X , the restriction to C × { t } is a section in O ( H | C ×{ t } ).(ii) Theorem 3.29 gives evidence that the notion of a good family ofmarked regular singular ( T E )-structures is useful. But it is not clear apriori whether any regular singular (
T E )-structure ( H → C × M, ∇ )over a simply connected manifold M is a good family of marked regularsingular ( T E )-structures over X = M . A marking can be imposed as M is simply connected. But the condition (3.40) is not clear a priori.Theorem 8.5 will show this for regular singular rank 2 ( T E )-structures.It builds on the Theorems 6.3 and 6.7 which show this for regularsingular rank 2 (
T E )-structures over M = C .(iii) For not regular singular ( T E )-structures, we do not see an easyreplacement of condition (3.40). Is the condition z ∇ ∂ z O ( H ) ⊂ O ( H )useful? Definition 3.28.
Fix a reference pair ( H ref, ∞ , M ref ).(a) Define the functor M ( H ref, ∞ ,M ref ) , reg from the category of complexspaces to the category of sets by M ( H ref, ∞ ,M ref ) , reg ( X ) (3.42):= { ( H, ψ ) | ( H, ψ ) is a good family of markedregular singular (
T E )-structures over X } , and, for any morphism f : Y → X of complex spaces and any element( H, ψ ) of M ( H ref, ∞ ,M ref ) , reg ( X ), define M ( H ref, ∞ ,M ref ) , reg ( f )( H, ψ ) := f ∗ ( H, ψ ) (b) Choose r ∈ R and N ∈ N . Define the functor M ( H ref, ∞ ,M ref ) ,r,N from the category of complex spaces to the category of sets by M ( H ref, ∞ ,M ref ) ,r,N ( X ) (3.43):= { ( H, ψ ) | ( H, ψ ) is a good family of markedregular singular (
T E )-structures over X which satisfies (3.39) and (3.40) for the given r and N } . and, for any morphism f : Y → X of complex spaces and any element( H, ψ ) of M ( H ref, ∞ ,M ref ) ,r,N ( X ), define M ( H ref, ∞ ,M ref ) ,r,N ( f )( H, ψ ) := f ∗ ( H, ψ ) Theorem 3.29. (a) The functors M ( H ref, ∞ ,M ref ) , reg and M ( H ref, ∞ ,M ref ) ,r,N are represented by complex spaces, which arecalled M ( H ref, ∞ ,M ref ) , reg and M ( H ref, ∞ ,M ref ) ,r,N . In the case of M ( H ref, ∞ ,M ref ) ,r,N , the complex space has even the structure of a projec-tive algebraic variety. As sets M ( H ref, ∞ ,M ref ) ,reg = Set ( H ref, ∞ ,M ref ) ,reg . Proof:
The proof for M ( H ref, ∞ ,M ref ) ,r,N can be copied from the proofof Theorem 7.3 in [HS10]. Here it is relevant that r and N with (3.39)and (3.40) imply the existence of an r ∈ R with r < r and V r ( x ) ⊃ O ( H | C ×{ x } ) for any x ∈ X. (3.44)In [HS10], ( T ERP )-structures are considered. (3.39) and (3.44) aredemanded there. (3.40) is not demanded there explicitly, but it followsfrom the properties of the pairing there, and this is used in Lemma7.2 in [HS10]. The additional conditions of (
T ERP )-structures are notessential for the arguments in the proof of Lemma 7.2 and Theorem7.3 in [HS10]. Therefore these proofs apply also here and give thestatements for M ( H ref, ∞ ,M ref ) ,r,N .Let us call ( r, N ) ∈ R × N and ( e r, e N ) ∈ R × N compatible if n ∈ Z with ( e r, e N ) = ( r + n, N + n · dim H ref, ∞ ) exists. In thecase n > M ( H ref, ∞ ,M ref ) , e r, e N is a union of M ( H ref, ∞ ,M ref ) ,r,N andadditional irreducible components. Thus for fixed ( r, N ) the union S n ∈ N M ( H ref, ∞ ,M ref ) ,r + n,N + n · dim H ref, ∞ is a complex space with in gen-eral countably many irreducible (and compact) components. And M ( H ref, ∞ ,M ref ) , reg is the union of these unions for all possible ( r, N )(as Eig( M mon ) is finite, in each interval of length 1, only finitely many r are relevant). (cid:3) Remarks 3.30. (i) For each reference pair ( H ref, ∞ , M ref ) withdim H ref, ∞ = 2, the representing complex space M ( H ref, ∞ ,M ref ) , reg for ANK 2 (
T E )-STRUCTURES 27 the functor M ( H ref, ∞ ,M ref ) , reg is given in Theorem 7.4. There the topo-logical components are unions S n ∈ N M ( H ref, ∞ ,M ref ) ,r + n,N + n · dim H ref, ∞ and have countably many irreducible components which are either iso-morphic to P or to the Hirzebruch surface F or to the variety e F obtained by blowing down the ( − F . And M ( H ref, ∞ ,M ref ) , reg is a union of countably many copies of one topological component.(ii) Corollary 7.3 says that any marked rank 2 regular singular ( T E )-structure ( H → C × M, ∇ , ψ ) with reference pair ( H ref, ∞ , M ref ) is agood family of marked regular singular ( T E )-structures. Therefore andbecause of Theorem 3.29, such a (
T E )-structure is induced by a mor-phism ϕ : M → M ( H ref, ∞ ,M ref ) , reg . This is crucial for the usefulness ofthe space M ( H ref, ∞ ,M ref ) , reg . We hope that Corollary 7.3 and this impli-cation are also true for higher rank regular singular ( T E )-structures.4.
Rank 2 ( T E ) -structures over a point Here we will classify the rank 2 (
T E )-structures over a point.4. 1.
Separation into 4 cases.
They separate naturally into 4 cases.
Definition 4.1.
Let ( H → C , ∇ ) be a rank 2 ( T E )-structure over apoint t = 0. Its formal invariants δ (0) , ρ (0) , δ (1) , ρ (1) from Lemma 3.9are complex numbers. The eigenvalues of −U are called u , u ∈ C .They are given by ( x − u )( x − u ) = x + 2 ρ (0) x + δ (0) . We separatefour cases:(Sem) U has two different eigenvalues − u and − u ∈ C , i.e.0 = δ (0) − ( ρ (0) ) .(Bra) U has only one eigenvalue (which is then ρ (0) ) and one2 × δ (1) − ρ (0) ρ (1) = 0.(Reg) U has only one eigenvalue (which is then ρ (0) ) and one2 × δ (1) − ρ (0) ρ (1) = 0.(Log) U = ρ (0) · id.Here (Sem) stands for semisimple , (Bra) for branched , (Reg) for regularsingular and (Log) for logarithmic .First we will treat the semisimple case (Sem). Then the cases (Bra),(Reg) and (Log) will be considered together. Lemma 4.8 will justifythe names (Bra) and (Reg). Finally, the three cases (Bra), (Reg) and(Log) will be treated one after the other. The following lemma givessome first information. Its proof is straightforward. Lemma 4.2.
Let ( H → C , ∇ ) be a rank 2 ( T E ) -structure over a point.Denote by ( e H → C , e ∇ ) the ( T E ) -structure with trace free pole part with ( O ( e H ) , e ∇ ) = ( O ( H ) , ∇ ) ⊗ E ρ (0) /z from Lemma 3.10 (b) (called ( H [2] → C , ∇ [2] ) there), and denote its invariants from Lemma 3.9 by e U , e δ (0) , e ρ (0) , e δ (1) , e ρ (1) . Then e U = U − ρ (0) id , (4.1) e δ (0) = δ (0) − ( ρ (0) ) , e ρ (0) = 0 , e δ (1) = δ (1) − ρ (0) ρ (1) , e ρ (1) = ρ (1) . ( e H → C , e ∇ ) is of the same type (Sem) or (Bra) or (Reg) or (Nil)as ( H → C , ∇ ) . The following table characterizes of which type the ( T E ) -structures ( H → C , ∇ ) and ( e H → C , e ∇ ) are. (Sem) (Bra) (Reg) (Log) e δ (0) = 0 e δ (0) = 0 , e δ (1) = 0 e δ (0) = e δ (1) = 0 , e U 6 = 0 e U = 0 (4.2) Especially, e U = 0 implies e δ (0) = e δ (1) = 0 .
4. 2.
The case (Sem).
A (
T E )-structure over a point with a semisim-ple endomorphism U with pairwise different eigenvalues is formallyisomorphic to a socalled elementary model , and its holomorphic iso-morphism class is determined by its Stokes structure. These two factsare well known. A good reference is [Sa02, II 5. and 6.]. The olderreference [Ma83a] considers only the underlying meromorphic bundle,so ( O ( H ) ⊗ C { z } C { z } [ z − ] , ∇ ).In order to formulate the result for rank 2 ( T E )-structures moreprecisely, we need some notation.
Definition 4.3.
Choose numbers u , u , α , α ∈ C . Consider the flatbundle H ′ → C ∗ with flat connection ∇ and a basis f = ( f , f ) ofglobal flat multivalued sections f and f with the monodromy f ( z · e πi ) = f ( z ) (cid:18) e − πiα e − πiα (cid:19) . (4.3)The new basis v = ( v , v ) which is defined by v ( z ) = f ( z ) (cid:18) e u /z z α e u /z z α (cid:19) (4.4)(for some choice of log( z )) is univalued. It defines a ( T E )-structurewith z ∇ ∂ z v = v · B and B = (cid:18) − u + zα − u + zα (cid:19) (4.5)This ( T E )-structure is called an elementary model . The numbers α and α are called the regular singular exponents . The formal in-variants δ (0) , ρ (0) , δ (1) , ρ (1) ∈ C of the ( T E )-structure and the tuple
ANK 2 (
T E )-STRUCTURES 29 ( u , u , α , α ) (up to joint exchange of the indices 1 and 2) are equiv-alent because of δ (0) − ( ρ (0) ) = −
14 ( u − u ) , ρ (0) = − u + u , (4.6) δ (1) − ρ (0) ρ (1) = u − u α − α ) , ρ (1) = α + α . (4.7)Therefore also the tuple ( u , u , α , α ) (up to joint exchange of theindices 1 and 2) is a formal invariant of the ( T E )-structure.
Theorem 4.4. (a) Any rank 2 ( T E ) -structure over a point with endo-morphism U with two different eigenvalues is formally isomorphic to aunique elementary model in Definition 4.3. Here − u and − u are theeigenvalues of U .(b) The ( T E ) -structure in (a) is up to holomorphic isomorphism de-termined by the numbers u , u , α , α and two more numbers s , s ∈ C , the Stokes parameters. It is holomorphically isomorphic to theelementary model to which it is formally isomorphic if and only if s = s = 0 .(c) Any such tuple ( u , u , α , α , s , s ) ∈ ( C − { ( u , u ) | u ∈ C } ) × C determines such a ( T E ) -structure. A second tu-ple ( e u , e u , e α , e α , e s , e s ) = ( u , u , α , α , s , s ) determines an iso-morphic ( T E ) -structure if and only if ( e u , e u , e α , e α , e s , e s ) =( u , u , α , α , s , s ) . Part (a) follows for example from [Sa02, II Theorem 5.7] togetherwith [Sa02, II Remark 5.8] (Theorem 5.7 considers only the underlyingmeromorphic bundle; Remark 5.8 takes care of the holomorphic bun-dle). For the parts (b) and (c), one needs to deal in detail with theStokes structure. We will not do it here, as the semisimple case is notcentral in this paper. We refer to [Sa02, II 5. and 6.] or to [HS11].
Remarks 4.5. (i) Malgrange’s unfolding result, Theorem 3.16 (c), ap-plies to these (
T E )-structures. Such a (
T E )-structure has a uniqueuniversal unfolding. The parameters ( α , α , s , s ) are constant, theparameters ( u , u ) are local coordinates on the base space. The basespace is an F-manifold of type A with Euler field E = u e + u e .See Remark 5.3 (iii).(ii) We do not offer normal forms for the ( T E )-structures in Theorem4.4 for three reasons: (1) As said in (i), the (
T E )-structures aboveunfold uniquely to (
T E )-structures over germs of F-manifolds. In thatsense they are easy to deal with. (2) It looks difficult to write downnormal forms. (3) Normal forms should be considered together with the
Stokes parameters, and the corresponding Riemann-Hilbert map fromthe space of monodromy data ( α , α , s , s ) to a space of parametersfor normal forms should be studied. This is a nontrivial project, whichdoes not fit into the main aims of this paper.4. 3. Joint considerations on the cases (Bra), (Reg) and (Log).Notations 4.6. (i) We shall use the following matrices, C := , C := (cid:18) (cid:19) , D := (cid:18) − (cid:19) , E := (cid:18) (cid:19) , (4.8)and the relations between them, C = 0 , D = C , E = 0 , (4.9) C D = C = − DC , [ C , D ] = 2 C , (4.10) C E = 12 ( C − D ) , EC = 12 ( C + D ) , [ C , E ] = − D, (4.11) DE = E = − ED, [ D, E ] = 2 E. (4.12)Consider a ( T E )-structure ( H → C , ∇ ) over a point with U of type(Bra), (Reg) or (Log). Then U has only one eigenvalue, which is ρ (0) ∈ C . We can and will restrict to C { z } -bases v of O ( H ) such that thematrix B ∈ M × ( C { z } ) with z ∇ ∂ z v = v · B has the shape B = b C + b C + zb D + zb E with b , b , b , b ∈ C { z } . (4.13)Write as in Remark 3.2 B = P k ≥ B ( k ) z k with B ( k ) ∈ M × ( C ), andwrite b j = X k ≥ b ( k ) j z k with b ( k ) j ∈ C for j ∈ { , } , (4.14) zb j = X k ≥ b ( k ) j z k with b ( k ) j ∈ C for j ∈ { , } . (4.15)Then the formal invariants δ (0) , ρ (0) , δ (1) and ρ (1) of Lemma 3.9 are givenby ρ (0) = b (0)1 , ρ (1) = b (1)1 , (4.16) δ (0) − ( ρ (0) ) = 0 , δ (1) − ρ (0) ρ (1) = − b (0)2 b (1)4 . (4.17)We are in the case (Bra) if b (0)2 = 0 and b (1)4 = 0, in the case (Reg) if b (0)2 = 0 and b (1)4 = 0, and in the case (Log) if b (0)2 = 0. ANK 2 (
T E )-STRUCTURES 31
Consider T ∈ GL ( C { z } ) and the new basis e v = v · T and its matrix e B = P k ≥ e B ( k ) z k with z ∇ z∂ z e v = e v · e B . Write T = τ C + τ C + τ D + τ E with (4.18) τ j = X k ≥ τ ( k ) j z k , τ ( k ) j ∈ C . Then e B is determined by (3.13), which is0 = z ∂ z T + B · T − T · e B (4.19)= C (cid:16) z ∂ z τ + ( b − e b ) τ + z ( b − e b ) τ z ( b − e b ) τ + ( b − e b ) τ (cid:17) + C (cid:16) z ∂ z τ + ( b − e b ) τ + ( b − e b ) τ + z ( − b − e b ) τ + ( b + e b ) τ (cid:17) + D ( z ∂ z τ + z ( b − e b ) τ + z ( b + e b ) τ b − e b ) τ + ( − b − e b ) τ (cid:17) + E ( z ∂ z τ + z ( b − e b ) τ + z ( − b − e b ) τ + ( b − e b ) τ + z ( b + e b ) τ (cid:17) . We will use this quite often in order to construct or compare normalforms. The following immediate corollary of the proof of Lemma 3.11provides a reduction of b . Corollary 4.7.
The base change matrix T = e g · C with g as in (3.20) leads to e b j with e b = b (0)1 + zb (1)1 = ρ (0) + zρ (1) , e b = b , e b = b , e b = b , (4.20)From now on we will work in this section only with bases v with b = ρ (0) + zρ (1) . This is justified by Corollary 4.7.Furthermore, we will consider from now on in this section mainly( T E )-structures with trace free pole part (Definition 3.8, ρ (0) = tr U = 0). See the Lemmata 3.10 and 3.11 for the relation to thegeneral case.The next lemma separates the cases (Bra) and (Reg). Lemma 4.8.
Consider a ( T E ) -structure over a point with U of type(Bra) or type (Reg) and with trace free pole part (so U is nilpotent butnot 0).The ( T E ) -structure is regular singular if and only if it is of type(Reg). If it is of type (Bra), then the pullback of O ( H ) ⊗ C { z } C { z } [ z − ] by the map C → C , x x = z, is the space of germs at 0 of sectionsof a meromorphic bundle on C with a meromorphic connection withan order 3 pole at 0 with semisimple pole part with eigenvalues κ and κ = − κ with − κ = δ (1) ∈ C ∗ . Thus κ is a formal invariant of the ( T E ) -structure of type (Bra). Proof:
Consider a C { z } -basis v of O ( H ) such that its matrix B isas in (4.13) and such that b = zρ (1) . This is possible by Corollary 4.7and the assumption ρ (0) = 0. As U is nilpotent, but not 0, b (0)2 = 0.Now δ (1) = − b (0)2 b (1)4 , so δ (1) = 0 ⇐⇒ b (1)4 = 0.Consider the case b (1)4 = 0, and consider the pullback of the ( T E )-structure by the map C → C , x x = z . Then d zz = 4 d xx and z∂ z = x∂ x and ∇ x∂ x v = v · X k ≥ B ( k ) x k − , (4.21) ∇ x∂ x ( v · x D ) = ( v · x D )4 (cid:16) x − X k ≥ ( b ( k )2 C + b ( k +1)4 E ) x k (4.22)+ ρ (1) C + (cid:0)
14 + X k ≥ b ( k +1)3 x k (cid:1) D (cid:17) . One sees a pole of order 3 with matrix 4( b (0)2 C + b (1)4 E ) of the polepart. It is tracefree and has the eigenvalues κ and κ = − κ with κ = 4 b (0)2 b (1)4 ∈ C ∗ . This shows the claims in the case b (1)4 = 0.Consider the case b (1)4 = 0, and consider the pullback of the ( T E )-structure by the map C → C , x x = z . Then d zz = 2 d xx and z∂ z = x∂ x and ∇ x∂ x v = v · X k ≥ B ( k ) x k − , (4.23) ∇ x∂ x ( v · x D ) = ( v · x D )2 (cid:16) ρ (1) C + 12 D (4.24)+ X k ≥ ( b ( k )2 C + b ( k +2)4 E + b ( k +1)3 D ) x k (cid:17) . One sees a logarithmic pole. Therefore also the sections v and v havemoderate growth, and the ( T E )-structure is regular singular. (cid:3)
4. 4.
The case (Bra).
The following theorem gives complete con-trol on the (
T E )-structures over a point of the type (Bra). HereEig( M mon ) ⊂ C is the set of eigenvalues of the monodromy of sucha ( T E )-structure (it has 1 or 2 elements).
Theorem 4.9. (a) Consider a ( T E ) -structure over a point of the type(Bra). The formal invariants ρ (0) , ρ (1) and δ (1) ∈ C from Lemma 3.9and the set Eig( M mon ) are invariants of its isomorphism class. To-gether they form a complete set of invariants. That means, the isomor-phism class of the ( T E ) -structure is determined by these invariants. ANK 2 (
T E )-STRUCTURES 33 (b) Any such ( T E ) -structure has a C { z } -basis v of O ( H ) such thatits matrix is in Birkhoff normal form, and more precisely, the matrix B has the shape B = ( ρ (0) + zρ (1) ) C + b (0)2 C + zb (1)3 D + zb (1)4 E (4.25) where b (0)2 , b (1)4 ∈ C ∗ and b (1)3 ∈ C satisfy − b (0)2 b (1)4 = δ (1) − ρ (0) ρ (1) and Eig( M mon ) = { e − πi ( ρ (1) ± b (1)3 ) } . Remarks 4.10. (i) Because of part (a), two Birkhoff normal forms asin (4.25) with data ( ρ (0) , ρ (1) , b (0)2 , b (1)3 , b (1)4 ) and ( e ρ (0) , e ρ (1) , e b (0)2 , e b (1)3 , e b (1)4 )give isomorphic ( T E )-structures if and only if e ρ (0) = ρ (0) , e ρ (1) = ρ (1) , e b (0)2 e b (1)4 = b (0)2 b (1)4 and e b (1)3 ∈ {± b (1)3 + k | k ∈ Z } . But the pure ( T LE )-structures which they define, are isomorphic only if additionally e b (1)3 ∈{± b (1)3 } .(ii) We could restrict to Birkhoff normal forms with b (0)2 = 1 or with b (1)4 = 1. But in view of the ( T E )-structures in the 4th case in Theorem6.3 we prefer not to do that.
Proof of Theorem 4.9:
The proof has 3 steps.
Step 1:
We will show that the hypothesis in Theorem 3.20 (b) onthe existence of a Birkhoff normal form is satisfied.Suppose on the contrary that the C { z } [ z − ]-vector space O ( H ) ⊗ C { z } C { z } [ z − ] contains a 1-dimensional ∇ -invariant subspace.Then that subspace is generated by a section σ = 0 with z ∇ ∂ z σ = h ( z ) σ for some function h ∈ C { z } [ z − ].Let v be a C { z } -basis of O ( H ) with matrix B as in (4.13) with b = ρ (0) + zρ (1) . Then b (0)2 b (1)4 = 0 holds because of (4.17), and becauseof it σ = v or σ = v is impossible. Therefore we can choose σ = gv + v = v · (cid:18) g (cid:19) with g ∈ C { z } [ z − ] − { } . Then h · v · (cid:18) g (cid:19) = h · σ = ∇ z∂ z σ = ∇ z∂ z ( v · (cid:18) g (cid:19) ) (4.26)= v · (cid:16) z − B (cid:18) g (cid:19) + (cid:18) z∂ z g (cid:19)(cid:17) = v · (cid:18) z − ( ρ (0) + zρ (1) ) g + b g + b + z∂ z gz − ( ρ (0) + zρ (1) ) + z − b g − b (cid:19) . The first line minus g times the second line gives the equation0 = − z − b g + 2 b g + b + z∂ z g. (4.27) The meromorphic function g = 0 has a degree deg z g ∈ Z . But (4.27)leads because of b (0)2 = 0 and b (1)4 = 0 with any possible degree deg z g to a contradiction. Therefore the hypothesis of Theorem 3.20 (b) issatisfied. Therefore Theorem 3.20 (b) can be applied. Any ( T E )-structure over a point in the case (Bra) admits a Birkhoff normal form.
Step 2:
Analysis of the Birkhoff normal forms. The matrix B of aBirkhoff normal form can be chosen with b = ρ (0) + zρ (1) because ofCorollary 4.7. Then it has the shape B = ( ρ (0) + zρ (1) ) C + ( b (0)2 + zb (1)2 ) C + zb (1)3 D + zb (1)4 E (4.28)with b (0)2 = 0 and b (1)4 = 0.Consider the new basis e v = v · T and its matrix e B where T = C + τ (0)2 C for some τ (0)2 ∈ C . (4.29)(4.19) gives0 = ( b − e b ) + z ( b (1)4 − e b ) τ (0)2 , b − e b ) + ( b − e b ) τ (0)2 + z ( − b (1)3 − e b ) τ (0)2 , b (1)3 − e b ) + ( b (1)4 + e b ) τ (0)2 , b (1)4 − e b ) , so e b = e b (1)4 = b (1)4 , e b = b , e b = e b (1)3 = b (1)3 + b (1)4 τ (0)2 , e b (0)2 = b (0)2 , e b (1)2 = b (1)2 − b (1)3 τ (0)2 − b (1)4 ( τ (0)2 ) . (4.30) τ (0)2 can be chosen such that e b (1)2 = 0. Then the Birkhoff normal form e B has the shape in (4.25).Suppose now that B has this shape, so b = b (0)2 . The choice τ (0)2 := − b (1)3 /b (1)4 in (4.29) leads to e b = b , e b = b , e b = b and e b = − b . (4.31)Consider the new basis e v = v · T and its matrix e B where T = C + τ (0)3 D for some τ (0)3 ∈ C − {± } . (4.32) ANK 2 (
T E )-STRUCTURES 35 (4.19) gives 0 = ( b − e b ) + z ( b (1)3 − e b ) τ (0)3 , b (0)2 − e b ) + ( b (0)2 + e b ) τ (0)3 , z ( b (1)3 − e b ) + ( b − e b ) τ (0)3 , b (1)4 − e b ) + ( − b (1)4 − e b ) τ (0)3 . so e b = b , e b = b (1)3 , e b = b (0)2 τ (0)3 − τ (0)3 , e b = b (1)4 − τ (0)3 τ (0)3 . (4.33)So, in a Birkhoff normal form in (4.25), one can change b (0)2 and b (1)4 arbitrarily with constant product b (0)2 b (1)4 and without changing b = ρ (0) + zρ (1) and b (1)3 .Consider the new basis e v = v · T and its matrix e B where T = (1 + zτ (1)1 ) C + τ (0)2 C + zτ (1)3 D + zτ (1)4 E (4.34)for some τ (1)1 , τ (0)2 , τ (1)3 , τ (1)4 ∈ C . We are searching for coefficients τ (1)1 , τ (0)2 , τ (1)3 , τ (1)4 ∈ C such that e b = b , e b = b , e b = b , e b = b + ε with ε = ± . (4.35)Under these constraints, (4.19) gives0 = τ (1)1 − ετ (1)3 , − b (1)3 − ε ) τ (0)2 + 2 b (0)2 τ (1)3 , zτ (1)3 − ε (1 + zτ (1)1 ) + b (1)4 τ (0)2 − b (0)2 τ (1)4 , τ (1)4 − b (1)4 τ (1)3 + (2 b (1)3 + ε ) τ (1)4 . With τ (1)1 = ετ (1)3 , these equations boil down to the inhomogeneouslinear system of equations ε = − b (1)3 − ε b (0)2 b (1)4 − b (0)2 − b (1)4 b (1)3 + ε + 1 τ (0)2 τ (1)3 τ (1)4 . (4.36)The determinant of the 3 × − b (0)2 b (1)4 = 0. Therefore thesystem (4.36) has a unique solution ( τ (0)2 , τ (1)3 , τ (1)4 ) t . Thus a new basis e v = v · T with (4.35) exists. Iterating this construction, one finds that one can change the matrix B in (4.25) by a holomorphic base change to a matrix e B with e b = b , e b = b , e b = b , e b = b + k (4.37)for any k ∈ Z .Putting together (4.30), (4.31), (4.33) and (4.37), one sees that twoBirkhoff normal forms as in (4.25) with data ( ρ (0) , ρ (1) , b (0)2 , b (1)3 , b (1)4 ) and( e ρ (0) , e ρ (1) , e b (0)2 , e b (1)3 , e b (1)4 ) give isomorphic ( T E )-structures if e ρ (0) = ρ (0) , e ρ (1) = ρ (1) , e b (0)2 e b (1)4 = b (0)2 b (1)4 and e b (1)3 ∈ {± b (1)3 + k | k ∈ Z } . This shows if in Remark 4.10 (i). Step 3:
Discussion of the invariants. By Lemma 3.9, ρ (0) , ρ (1) and δ (1) are even formal invariants of the ( T E )-structure. The setEig( M mon ) is obviously an invariant of the isomorphism class of the( T E )-structure.The Birkhoff normal form in (4.25) gives a pure (
T LE )-structurewith a logarithmic pole at ∞ . From its pole part at ∞ and Theorem3.23 (c) one reads offEig( M mon ) = { e − πi ( ρ (1) ± b (1)3 ) } . (4.38)As ρ (1) is an invariant of the ( T E )-structure, also the set {± b (1)3 + k | k ∈ Z } is an invariant of the ( T E )-structure.Together with Step 2, this shows only if in Remark 4.10 (i) and allstatements in Theorem 4.9. (cid:3)
Corollary 4.11.
The monodromy of a ( T E ) -structure over a pointof the type (Bra) has a × Jordan block if its eigenvalues coincide(equivalently, if b (1)3 ∈ Z for some (or any) Birkhoff normal form inTheorem 4.9 (b)). Proof:
Consider a (
T E )-structure over a point of the type (Bra)such that the eigenvalues of its monodromy coincide. Then for anyBirkhoff normal form in Theorem 4.9 (b) b (1)3 ∈ Z , and one can choosea Birkhoff normal form with b (1)3 ∈ { , − } . The induced pure ( T LE )-structure has at ∞ a logarithmic pole, and its residue endomorphism[ ∇ e z∂ e z ], where e z = z − , is given by the matrix − ( ρ (1) C + b (1)3 D + b (1)4 E ).In the case b (1)3 = 0, the nonresonance condition in Theorem 3.23 (c)is satisfied, so Theorem 3.23 (c) can be applied. Because of b (1)4 = 0,the monodromy has a 2 × b (1)3 = − , the meromorphic base change e v := v · (cid:18) z
00 1 (cid:19)
ANK 2 (
T E )-STRUCTURES 37 gives the new connection matrix e B = ( ρ (0) + z ( ρ (1) + 12 )) C + zb (0)2 C + b (1)4 E. Again, the pole at ∞ is logarithmic. Now the nonresonance conditionin Theorem 3.23 (c) is satisfied. Because of b (0)2 = 0, the monodromyhas a 2 × (cid:3) For (
T E )-structures of the type (Bra), formal isomorphism is coarserthan holomorphic isomorphism.
Lemma 4.12.
Consider a ( T E ) -structure over a point of the type(Bra). By Lemma 3.9, the numbers ρ (0) , ρ (1) and δ (1) are formal in-variants of the ( T E ) -structure.(a) The set Eig( M mon ) and the equivalent set {± b (1)3 + k | k ∈ Z } areholomorphic invariants, but not formal invariants.(b) The ( T E ) -structure with Birkhoff normal form in (4.25) is for-mally isomorphic to the ( T E ) -structure with Birkhoff normal form in (4.25) with the same values ρ (0) , ρ (1) , b (0)2 and b (1)4 , but with an arbitrary e b (1)3 . Proof:
Part (a) follows from part (b). For the proof of part (b), wehave to find T ∈ GL ( C [[ z ]]) such that T , B in (4.25) and e B = ( ρ (0) + zρ (1) ) C + b (0)2 C + z e b (1)3 D + zb (1)4 E with e b (1)3 ∈ C arbitrary satisfy (4.19). Here (4.19) says0 = z∂ z τ + ( b (1)3 − e b (1)3 ) τ , (4.39)0 = z ∂ z τ + z ( − b (1)3 − e b (1)3 ) τ + 2 b (0)2 τ , z ∂ z τ + z ( b (1)3 − e b (1)3 ) τ + zb (1)4 τ − b (0)2 τ , z∂ z τ − b (1)4 τ + ( b (1)3 + e b (1)3 ) τ . This is equivalent to0 = τ (0)3 = τ (0)4 , (4.40)0 = kτ ( k )1 + ( b (1)3 − e b (1)3 ) τ ( k )3 for k ≥ , k − − b (1)3 − e b (1)3 ) τ ( k − + 2 b (0)2 τ ( k )3 for k ≥ , k − τ ( k − + ( b (1)3 − e b (1)3 ) τ ( k − + b (1)4 τ ( k − − b (0)2 τ ( k )4 for k ≥ , − b (1)4 τ ( k )3 + ( k + b (1)3 + e b (1)3 ) τ ( k )4 for k ≥ . This is equivalent to τ (0)3 = τ (0)4 = 0 , (4.41) τ ( k )1 = − k ( b (1)3 − e b (1)3 ) τ ( k )3 for k ≥ , b (0)2 τ ( k )3 = ( b (1)3 + e b (1)3 + 1 − k ) τ ( k − for k ≥ ,b (0)2 τ (1)4 = b (1)4 τ (0)2 + ( b (1)3 − e b (1)3 ) τ (0)1 ,b (0)2 τ ( k )4 = b (1)4 τ ( k − + ( k − − k − b (1)3 − e b (1)3 ) )(2 b (0)2 ) − ( b (1)3 + e b (1)3 + 2 − k ) τ ( k − for k ≥ , b (1)4 τ (0)2 + (1 + b (1)3 + e b (1)3 )( b (1)3 − e b (1)3 ) τ (0)1 , b (1)4 (2 k + 1) τ ( k )2 + ( k + 1 + b (1)3 + e b (1)3 )( k + − k ( b (1)3 − e b (1)3 ) )(2 b (0)2 ) − ( b (1)3 + e b (1)3 + 1 − k ) τ ( k − for k ≥ . One can choose τ (0)1 ∈ C ∗ freely. Then the equations (4.41) have uniquesolutions τ − τ (0)1 , τ , τ , τ ∈ C [[ z ]]. Therefore T ∈ GL ( C [[ z ]]) existssuch that T , B as in (4.25) and e B as above satisfy (4.19). This showspart (b). (cid:3) Remarks 4.13. (i) Because of Lemma 4.12, the set Eig( M mon ) takeshere the role of the Stokes structure: It distinguishes the holomorphicisomorphism classes within one formal isomorphism class.(ii) The following ( T E )-structures have trivial Stokes structure. Theproof of Lemma 4.8 leads to these (
T E ) structures. They are the (
T E )-structures with Eig( M mon ) = { λ , λ } with λ = − λ , respectively with b (1)3 ∈ ( ± + Z ) for any Birkhoff normal form in (4.25).Choose a number α (1) ∈ C . Consider the rank 2 bundle H ′ → C ∗ with flat connection ∇ and flat multivalued basis f = ( f , f ) withmonodromy given by f ( ze πi ) = f ( z ) · ie − πiα (1) · ( C + E ) . (4.42)The eigenvalues are ± ie − πiα (1) . Choose numbers t ∈ C and t ∈ C ∗ .The following basis of H ′ is univalued. v := f · e t /z z α (1) (cid:18) z − / e t z − / z / e t z − / z − / e − t z − / − z / e − t z − / (cid:19) . (4.43) ANK 2 (
T E )-STRUCTURES 39
The matrix B with z ∇ ∂ z v = v · B is B = ( − t + zα (1) ) C − t C − z D − z t E. (4.44)So here ρ (1) = α (1) , ρ (0) = − t , δ (1) − ρ (0) ρ (1) = − t .(iii) Part (ii) generalizes to a ( T E )-structure over M = C withcoordinates t = ( t , t ). Consider the rank 2 bundle H ′ → C ∗ × M withflat connection and flat multivalued basis f = ( f , f ) with monodromygiven by f ( ze πi , t ) = f ( z, t ) · ie − πiα (1) · ( C + E ) . (4.45)The basis v in (4.43) is univalued. The matrices A , A and B in itsconnection 1-form Ω as in (3.4)–(3.6) are given by (4.44) and A = C , A = C + zE. (4.46)The restriction to a point t ∈ C × C ∗ is a ( T E )-structure of type (Bra)with trivial Stokes structure. The restriction to a point t ∈ C × { } isa ( T E )-structure of type (Log).4. 5.
The case (Reg) with tr U = 0 . The (
T E )-structures over apoint of the type (Reg) with tr U = 0 are the regular singular ( T E )-structures over a point which are not logarithmic. They can be easilyclassified using elementary sections. Theorem 4.15 splits them intothree cases (one in part (a), two in part (b): α = α and α − α ∈ N ). Notations 4.14.
Start with a (
T E )-structure ( H → C , ∇ ) of rank2 over a point. Recall the notions from Definition 3.21: H ′ := H | C ∗ , M mon , M mons , M monu , N mon , Eig( M mon ), H ∞ , H ∞ λ , C α for α ∈ C with e − πiα ∈ Eig( M mon ), s ( A, α ) ∈ C α for A ∈ H ∞ e − πiα , es ( σ, α ) ∈ C α for σ a holomorphic section on H | U −{ } for U ⊂ C a neighborhoodof 0. Now the eigenvalues of M mon are called λ and λ ( λ = λ isallowed). The sheaf V > −∞ simplifies here to a C { z } [ z − ]-vector spaceof dimension 2, V > −∞ := (cid:26) C { z } [ z − ] · C α ⊕ C { z } [ z − ] · C α if λ = λ , C { z } [ z − ] · C α if λ = λ , (4.47)where α , α ∈ C with e − πiα j = λ j . V > −∞ is the space of sections ofmoderate growth. Theorem 4.15.
Consider a regular singular, but not logarithmic, rank2 ( T E ) -structure ( H → C , ∇ ) over a point. Associate to it the data inthe notations 4.14.(a) The case N mon = 0 : There exist unique numbers α , α with e − πiα j = λ j and α = α and the following properties: There exist elementary sections s ∈ C α − { } and s ∈ C α − { } and a number t ∈ C ∗ such that O ( H ) = C { z } ( s + t s ) ⊕ C { z } ( zs ) (4.48)= C { z } ( s + t − s ) ⊕ C { z } ( zs ) . (4.49) The isomorphism class of the ( T E ) -structure is uniquely determined bythe information N mon = 0 and the set { α , α } . The numbers α and α are called leading exponents .(b) The case N mon = 0 (thus λ = λ ): There exist unique numbers α , α with e − πiα j = λ and α − α ∈ Z ≥ and the following properties:Choose any elementary section s ∈ C α − ker( z ∇ ∂ z − α : C α → C α ) .The elementary section s ∈ C α with ( z ∇ ∂ z − α )( s ) = z α − α s . (4.50) is a generator of ker( z ∇ ∂ z − α : C α → C α ) . Then O ( H ) = C { z } ( s + t s ) ⊕ C { z } ( zs ) (4.51) for some t ∈ C . If α > α then t is in C ∗ and is independent of thechoice of s . If α = α , then one can replace s by s [ new ]1 := s + t s ,and then t [ new ]2 = 0 . The isomorphism class of the ( T E ) -structure isuniquely determined by the information N mon = 0 and the pair ( α , α ) if α = α and the triple ( α , α , t ) if α > α . The numbers α and α are called leading exponents . Proof:
First, (a) and (b) are considered together. Let β , β ∈ C bethe unique numbers with e − πiβ j = λ j and − < Re( β j ) ≤
0. Chooseelementary sections e s ∈ C β and e s ∈ C β which form a global basisof H ′ . In the case N mon = 0 (then β = β ) choose them such that e s / ∈ ker( z ∇ ∂ z − β : C β → C β ) and e s ∈ ker( z ∇ ∂ z − β : C β → C β ).Let σ [1]1 , σ [1]2 ∈ O ( H ) be a C { z } -basis of O ( H ) . Write( σ [1]1 , σ [1]2 ) = ( e s , e s ) (cid:18) b b b b (cid:19) with b ij ∈ C { z } [ z − ] . (4.52)Recall that the degree deg z g of a Laurent series g = P j ∈ Z g ( j ) z j ∈ C { z } [ z − ] is the minimal j with g ( j ) = 0 if g = 0, and deg z ∞ .In the case N mon = 0 and λ = λ (then β = β ), we supposemin(deg z b , deg z b ) ≤ min(deg z b , deg z b ). If it does not hold apriori, we can exchange e s and e s .In any case, we suppose deg z b ≤ deg z b . If it does not hold apriori, we can exchange σ [1]1 and σ [1]2 . ANK 2 (
T E )-STRUCTURES 41
Again in the case N mon = 0 and λ = λ , we suppose deg z b < deg z b . If it does not hold a priori, we can replace e s by a certainlinear combination of e s and e s .Now e b := z − deg z b b ∈ C { z } ∗ is a unit. Consider α := β +deg z b and s := z deg z b e s ∈ C α and the new basis ( σ [2]1 , σ [2]2 ) of O ( H ) with( σ [2]1 , σ [2]2 ) := ( σ [1]1 , σ [1]2 ) (cid:18)e b − − b − b (cid:19) = ( s , e s ) (cid:18) e b − b b − b − b b (cid:19) . (4.53)Consider m := deg z ( b − b − b b ) ∈ Z (+ ∞ is impossible) and α := β +( m −
1) and s := z m − e s ∈ C α . Write z − m +1 e b − b = c + c with c ∈ C [ z − ] and c ∈ z C { z } . We can replace σ [2]2 by σ [3]2 := zs and σ [2]1 = s + ( c + c ) s by σ [3]1 = s + c s .(a) Consider the case N mon = 0. If λ = λ then deg z b ≥ deg z b +1 and thus( c + c ) s = e b − b e s ∈ C { z } · z deg z b · C β (4.54) ⊂ C { z } · z deg z b +1 · C β = C { z } · C α +1 . In any case (whether λ = λ or λ = λ ), we must have c = 0. Elsethe ( T E )-structure is logarithmic. As the pole has precisely order 2, c is a constant = 0 (if λ = λ , here we need (4.54)), which is now called t . This implies m − z b . In the case N mon = 0 and λ = λ we have β = β and α − α = m − − deg z b = deg z b − deg z b > , (4.55)so especially α = α .(b) Consider the case N mon = 0. Then s is generator of ker( z ∇ ∂ z − α : C α → C α ), and we can rescale it such that (4.50) holds. Firstconsider the case c = 0. As the pole has precisely order 2, we musthave α = α . Then (4.51) holds with t = 0. Now consider the case c = 0. Then σ [3] = ( σ [3]1 , σ [3]2 ) satisfies z ∇ ∂ z σ [3] = σ [3] (cid:18) α z − ( z∂ z − α + α )( c ) + z α − α − α + 1 . (cid:19) (4.56)First case, α − α ∈ Z < : The coefficient of z α − α − in z − ( z∂ z − α + α )( c )+ z α − α − is 1. Therefore the pole order is >
2, a contradiction.Second case, α ≥ α : As the pole has precisely order 2, c is aconstant = 0, which is now called t . Then (4.51) holds, and t ∈ C ∗ . In the case α − α ∈ N , t is obviously independent of the choice of s . (cid:3) Corollary 4.16 is an immediate consequence of Theorem 4.15.
Corollary 4.16.
The set of regular singular, but not logarithmic, rank2 ( T E ) -structures over a point is in bijection with the set { (0 , { α , α } ) | α , α ∈ C , α = α }∪ { (1 , α , α ) | α = α ∈ C }∪ { (1 , α , α , t ) | α , α ∈ C , α − α ∈ N , t ∈ C ∗ } . The first set parametrizes the cases with N mon = 0 , the second andthird set parametrize the cases with N mon = 0 . Theorem 4.15 describesthe corresponding ( T E ) -structures. Remarks 4.17.
The connection matrices for the special bases in The-orem 4.15 can be written down easily.The basis in (4.48): ∇ z∂ z ( s + t s , zs ) = ( s + t s , zs ) (cid:18) α z − ( α − α ) t α + 1 (cid:19) . (4.57)The basis in (4.49) with e t := t − : ∇ z∂ z ( s + e t s , zs ) = ( s + e t s , zs ) (cid:18) α z − ( α − α ) e t α + 1 (cid:19) . (4.58)The basis in (4.51) with (4.50): ∇ z∂ z ( s + t s , zs ) = ( s + t s , zs ) · (4.59) (cid:18) α z − ( α − α ) t + z α − α − α + 1 (cid:19) . Finally, in the case N mon = 0 and t ∈ C ∗ , we consider with e t := t − also the basis ( s + e t s , zs ). Again (4.50) is assumed: ∇ z∂ z ( s + e t s , zs ) = ( s + e t s , zs ) · (4.60) (cid:18) α + z α − α e t z α − α +1 z − ( α − α ) e t − z α − α − e t α + 1 − z α − α e t (cid:19) .
4. 6.
The case (Log) with tr U = 0 . The (
T E )-structures over apoint of the type (Log) with tr U = 0 are the logarithmic ( T E )-structures over a point. Just as the regular singular (
T E )-structures,they can easily be classified using elementary sections. Theorem 4.18splits them into two cases. We use again the notations 4.14.
ANK 2 (
T E )-STRUCTURES 43
Theorem 4.18.
Consider a logarithmic rank 2 ( T E ) -structure ( H → C , ∇ ) over a point. Associate to it the data in the notations 4.14.(a) The case N mon = 0 : There exist unique numbers α , α with e − πiα j = λ j and the following property: There exist elementary sec-tions s ∈ C α − { } and s ∈ C α − { } such that O ( H ) = C { z } s ⊕ C { z } s . (4.61) The isomorphism class of the ( T E ) -structure is uniquely determined bythe information N mon = 0 and the set { α , α } . The numbers α and α are called leading exponents .(b) The case N mon = 0 (thus λ = λ ): There exist unique numbers α , α with e − πiα j = λ and α − α ∈ Z ≥ and the following properties:Choose any elementary section s ∈ C α − ker( ∇ z∂ z − α : C α → C α ) .The elementary section s ∈ C α with ( z ∇ ∂ z − α )( s ) = z α − α s . (4.62) is a generator of ker( z ∇ ∂ z − α : C α → C α ) . Then O ( H ) = C { z } s ⊕ C { z } s . (4.63) The isomorphism class of the ( T E ) -structure is uniquely determined bythe information N mon = 0 and the set { α , α } . The numbers α and α are called leading exponents . Proof:
First, (a) and (b) are considered together. By Theorem3.23 (a), O ( H ) is generated by two elementary sections s ∈ C α and s ∈ C α for some numbers α and α . The numbers α and α are theeigenvalues of the residue endomorphism. So, they are unique. Thisfinishes already the proof of part (a).(b) Consider the case N mon = 0. We can renumber s and s ifnecessary, so that afterwards α − α ∈ Z ≥ . If α = α , then O ( H ) = C { z } C α , and s and s can be changed so that s ∈ C α − ker( ∇ z∂ z − α ) and s ∈ ker( ∇ z∂ z − α : C α → C α ) − { } satisfy (4.62). Thennothing more has to be shown. Consider the case α − α ∈ N . If s ∈ C α − ker( ∇ z∂ z − α ), then ( ∇ z∂ z − α )( s ) is not in O ( H ) , and thus thepole is not logarithmic, a contradiction. Therefore s ∈ ker( ∇ z∂ z − α : C α → C α ). Then necessarily s ∈ C α − ker( ∇ z∂ z − α : C α → C α ).We can rescale s so that (4.62) holds. Nothing more has to be shown. (cid:3) Corollary 4.19 is an immediate consequence of Theorem 4.18.
Corollary 4.19.
The set of logarithmic rank 2 ( T E ) -structures over apoint is in bijection with the set { (0 , { α , α } ) | α , α ∈ C , }∪ { (1 , α , α ) | α , α ∈ C , α − α ∈ Z ≥ } . The first set parametrizes the cases with N mon = 0 , the second setparametrizes the cases with N mon = 0 . Theorem 4.18 describes thecorresponding ( T E ) -structures. Remarks 4.20. (i) The connection matrices for the special bases inTheorem 4.18 can be written down easily.The basis in (4.61): ∇ z∂ z ( s , s ) = ( s , s ) (cid:18) α α (cid:19) . (4.64)The basis in (4.63): ∇ z∂ z ( s , s ) = ( s , s ) (cid:18) α z α − α α (cid:19) . (4.65)The basis ( s , s ) gives a Birkhoff normal form in the cases N mon = 0and in the cases ( N mon = 0 & α = α ). In the cases ( N mon =0 & α − α ∈ N ), a Birkhoff normal form does not exist.5. Rank 2 ( T E ) -structures over germs of regularF-manifolds This section discusses unfoldings of (
T E )-structures over a point t of type (Sem) or (Bra) or (Reg). Here Malgrange’s unfolding resultTheorem 3.16 (c) applies. It provides a universal unfolding for the( T E )-structure over t . Any unfolding is induced by the universalunfolding. The universal unfoldings turn out to be precisely the ( T E )-structures with primitive Higgs fields over germs of regular F-manifolds.The sections 6 and 8 discuss unfoldings of (
T E )-structures over apoint of type (Log). Section 8 treats arbitrary such unfoldings. Section6 prepares this. It treats 1-parameter unfoldings with trace free poleparts of logarithmic (
T E )-structures over a point.If one starts with a (
T E )-structure with primitive Higgs field overa germ (
M, t ) of a regular F-manifold, then the endomorphism U | t : K t → K t is regular.Vice versa, if one starts with a ( T E )-structure over a point t witha regular endomorphism U : K t → K t , then it unfolds uniquely toa ( T E )-structure with primitive Higgs field over a germ of a regularF-manifold by Malgrange’s result Theorem 3.16 (c). The germ of the
ANK 2 (
T E )-STRUCTURES 45 regular F-manifold is uniquely determined by the isomorphism classof U : K t → K t (i.e. its Jordan block structure). And the ( T E )-structure is uniquely determined by its restriction to t .The following statement on the rank 2 cases is an immediate con-sequence of Malgrange’s unfolding result Theorem 3.16 (c), the classi-fication of germs of regular 2-dimensional F-manifolds in Remark 2.6(building on the Theorems 2.2 and 2.3, see also Remark 3.17 (ii)) andthe classification of the rank 2 ( T E )-structures into the cases (Sem),(Bra), (Reg) and (Log) in Definition 4.3.
Corollary 5.1. (a) For any rank 2 ( T E ) -structure over a point t except those of type (Log), the endomorphism U : K t → K t is regular.The ( T E ) -structure has a unique universal unfolding. This unfoldinghas a primitive Higgs field. Its base space is a germ ( M, t ) = ( C , of an F-manifold with Euler field and is as follows: Type F-manifold Euler field(Sem) A P i =1 ( u i + c i ) e i with c = c (Bra) or (Reg) N t ∂ + g ( t ) ∂ with g (0) = 0 (5.1) In the case of (Bra) or (Reg), a coordinate change brings E to the form t ∂ + ∂ .(b) Any unfolding of a rank 2 ( T E ) -structure over t with regularendomorphism U : K t → K t is induced by the universal unfolding in(a). Because of the existence and uniqueness of the universal unfolding,it is not really necessary to give it explicitly. On the other hand, inrank 2, it is easy to give it explicitly. The following lemma offers oneway.
Lemma 5.2.
Let ( H → C , ∇ ) by a ( T E ) -structure over a point withmonodromy M mon of some rank r ∈ N . It has an unfolding whichis a ( T E ) -structure ( H ( unf ) → C × M, ∇ ) , where M = C × C ∗ withcoordinates t = ( t , t ) (on C ⊃ M ), with the following properties.(a) The monodromy around t = 0 is ( M mon ) − .(b) The original ( T E ) -structure is isomorphic to the one over t =(0 , .(c) If v is a C { z } -basis of O ( H ) with z ∇ ∂ z v = v B , then H ( unf ) has over ( C , × M a basis v such that the matrices A , A and B in (3.4) – (3.6) are as follows. A = C , (5.2) A = − B ( zt ) , (5.3) B = − t C + t B ( zt ) = − t A − t A . (5.4) (d) If U | t is regular and rank H = 2 , then the Higgs field of the ( T E ) -structure H ( unf ) is everywhere primitive. Therefore then M isan F-manifold with Euler field. The Euler field is E = t ∂ + t ∂ .(e) If U | t is regular and rank H = 2 , the ( T E ) -structure over thegerm ( M, t ) is the universal unfolding of the one over t . Proof:
Let f = ( f , ..., f r ) be a flat multivalued basis of H ′ := H | C ∗ . Let M mat ∈ GL r ( C ) be the matrix of its monodromy, so f ( ze πi ) = f · M mat . Let v = ( v , ..., v r ) be a C { z } -basis of O ( H ) .Let B ∈ GL r ( C { z } ) be the matrix with z ∇ ∂ z v = v B . Considerthe matrix Ψ( z, t ) with multivalued entries with v = f · Ψ( z ) . (5.5)Then Ψ( ze πi ) = ( M mat ) − · Ψ( z ) , (5.6)Ψ − ∂ z Ψ = z − B ( z ) . (5.7)Embed the flat bundle H ′ := H | C ∗ as the bundle over t = (0 ,
1) intoa flat bundle H ( mf ) ′ → C ∗ × M with monodromy M mon around z = 0and monodromy ( M mon ) − around t = 0. The flat multivalued basis f of H ′ extends to a flat multivalued basis f of H ( mf ) ′ with f ( ze πi , t ) = f ( z, t ) M mat , (5.8) f ( z, t , t e πi ) = f ( z, t )( M mat ) − . (5.9)The tuple of sections v with v = f · e t /z Ψ( z/t ) (5.10)is univalued, it is a basis of H ( mf ) ′ in a neighbourhood of { } × M , andit has the connection matrices in (5.2)–(5.4): The calculations for A ANK 2 (
T E )-STRUCTURES 47 and B are ∇ ∂ v = f · e t /z · ( − z/t )( ∂ z Ψ)( z/t )= f · e t /z · ( − z/t )Ψ( z/t )( z/t ) − B ( z/t )= v · ( − /z ) B ( z/t ) , ∇ ∂ z v = f · e t /z · (( − t /z )Ψ( z/t ) + (1 /t )( ∂ z Ψ)( z/t ))= f · e t /z · (( − t /z )Ψ( z/t ) + (1 /t )Ψ( z/t )( z/t ) − B ( z/t ))= v · (( − t /z ) C + ( t /z ) B ( z/t )) . Therefore v defines a ( T E )-structure, which we call ( H ( unf ) → C × M, ∇ ). It unfolds the one over t = (0 , H → C , ∇ ).It rests to show (d) and (e). Suppose rank H = 2. Then U | t isregular if and only if ( B ) (0) / ∈ C · C . Then also A (0)2 ( t ) = − ( B ) (0) / ∈ C · C , so then the Higgs field of the ( T E )-structure H ( unf ) is everywhereprimitive. Because of B (0) = − t A (0)1 − t A (0)2 , the Euler field is E = t ∂ + t ∂ . (e) follows from (d) and Malgrange’s result Theorem 3.16(c). (cid:3) Remarks 5.3. (i) In the cases (Reg) we will see the universal un-foldings again in section 7, in the Remarks 7.2. In a first step in theRemarks 7.1, the value t in the normal form in the Remarks 4.17 isturned into a parameter in P . The Remarks 7.2 add another parame-ter t in C . Then the Higgs field becomes primitive and the base space C × P becomes a 2-dimensional F-manifold with Euler field. For each t ∈ C × C ∗ , the ( T E )-structure over t is of type (Reg), and the ( T E )-structure over the germ (
M, t ) is a universal unfolding of the one over t .(ii) In the cases (Bra), the following formulas give a universal un-folding over ( C ,
0) of any (
T E )-structure of type (Bra) over the point0 (see Theorem 4.9 for their classification), such that the Euler field is E = ( t + c ) ∂ + ∂ . Here ρ (1) ∈ C , b (0)3 ∈ C , b (0)2 , b (1)4 ∈ C ∗ . A = C , (5.11) A = − b (0)2 C − z ( 12 + b (1)3 ) D − zb (1)4 e t E, (5.12) B = ( − t − c ) C + b (0)2 C + z ( ρ (1) C + b (1)3 D + b (1)4 e t E )= ( − t − c ) A − A + zρ (1) C − z D. (5.13) (iii) In the cases (Sem), a ( T E )-structure over a point extendsuniquely to a (
T E )-structure over the universal covering M of themanifold { ( u , u ) ∈ C , | u = u } , see [Ma83b] [Sa02, III Theorem2.10]. For each t ∈ M the ( T E )-structure over t is of type (Sem),and the ( T E )-structure over the germ (
M, t ) is the universal unfoldingof the ( T E )-structure over t .6. ( T E ) -structures over a point This section classifies unfoldings over (
M, t ) = ( C ,
0) with trace freepole part of logarithmic (
T E )-structures over the point t .It is a preparation for section 8, which treats arbitrary unfoldings of( T E )-structures of type (Log) over a point.Subsection 6. 1: An unfolding with trace free pole part over(
M, t ) = ( C ,
0) of a logarithmic rank 2 (
T E )-structure over t willbe considered. Invariants of it will be defined. Theorem 6.2 gives con-straints on these invariants and shows that the monodromy is semisim-ple if the generic type is (Sem) or (Bra).By Theorem 3.20 (a) (which is trivial in our case because of the log-arithmic pole at z = 0 of the ( T E )-structure over t ) and Remark 3.19(iii), the ( T E )-structure has a Birkhoff normal form, i.e. an extensionto a pure (
T LE )-structure, if its monodromy is semisimple.Subsection 6. 2: All pure (
T LE )-structures over (
M, t ) = ( C , t are clas-sified in Theorem 6.3. These comprise all with semisimple monodromyand thus all with generic types (Sem) or (Bra).Subsection 6. 3: All ( T E )-structures over (
M, t ) = ( C ,
0) with tracefree pole part and with logarithmic restriction over t whose mon-odromies have a 2 × M, t ) = ( C ,
0) of logarithmic rank 2 (
T E )-structuresover t .6. 1. Numerical invariants for such ( T E ) -structures. The nextdefinition gives some numerical invariants for such (
T E )-structures.Recall the invariants δ (0) and δ (1) in Lemma 3.9. Definition 6.1.
Let ( H → C × ( M, t ) , ∇ ) be a ( T E )-structure withtrace free pole part over (
M, t ) = ( C ,
0) (with coordinate t ) whose ANK 2 (
T E )-STRUCTURES 49 restriction over t = 0 is logarithmic. Let M ⊂ C be a neighborhoodof 0 on which the ( T E )-structure is defined. On M − { } it has a fixedtype, (Sem) or (Bra) or (Reg) or (Log), which is called the generic type of the ( T E )-structure. Lemma 4.2 characterizes the generic type interms of (non)vanishing of δ (0) , δ (1) ∈ t C { t } and U :(Sem) (Bra) (Reg) (Log) δ (0) = 0 δ (0) = 0 , δ (1) = 0 δ (0) = δ (1) = 0 , U 6 = 0 U = 0For the generic types (Sem), (Bra) and (Reg), define k ∈ N by k := max( k ∈ N | U ( O ( H ) ) ⊂ t k O ( H ) . (6.1)For the generic types (Sem) and (Bra) define k ∈ Z by k := (cid:26) deg t δ (0) − k for the generic type (Sem) , deg t δ (1) − k for the generic type (Bra) . (6.2)The following theorem gives for the generic type (Bra) and part ofthe generic type (Sem) restrictions on the eigenvalues of the residueendomorphism of the logarithmic pole at z = 0 of the ( T E )-structureover t = 0. And it shows that the monodromy is semisimple if thegeneric type is (Sem) or (Bra). Theorem 6.2.
Let ( H → C × ( M, t ) , ∇ ) be a rank 2 ( T E ) -structurewith trace free pole part over ( M, t ) = ( C , whose restriction over t = 0 is logarithmic. Recall the invariant ρ (1) ∈ C from Lemma 3.9(b), and recall the invariants k ∈ N and k ∈ Z from Definition 6.1 ifthe generic type is (Sem) or (Bra).(a) Suppose that the generic type is (Sem).(i) Then k ≥ k .(ii) If k > k then the eigenvalues of the residue endomorphism ofthe logarithmic pole at z = 0 of the ( T E ) -structure over t are ρ (1) ± k − k k + k ) . Their difference is smaller than 1. Especially, the eigenvaluesof the monodromy are different, and the monodromy is semisimple.(iii) Also if k = k , the monodromy is semisimple.(b) Suppose that the generic type is (Bra).(i) Then k ∈ N .(ii) The eigenvalues of the residue endomorphism of the logarithmicpole at z = 0 of the ( T E ) -structure over t are ρ (1) ± − k k + k ) . Their dif-ference is smaller than 1. Especially, the eigenvalues of the monodromyare different, and the monodromy is semisimple. Proof:
By Lemma 3.11, a C { t, z } -basis v of the germ O ( H ) (0 , can be chosen such that the matrices A and B ∈ M × ( C { t, z } ) with z ∇ ∂ t v = vA and z ∇ ∂ z v = vB satisfy (3.19), 0 = tr A = tr( B − zρ (1) C ), or, more explicitly, A = a C + a D + a E with a , a , a ∈ C { t, z } , (6.3) B = zρ (1) C + b C + b D + b E with b , b , b ∈ C { t, z } . (6.4)Write a j = P k ≥ a ( k ) j z k and a ( k ) j = P l ≥ a ( k ) j,l t l ∈ C { t } , and analogouslyfor b j . Condition (3.8) says here0 = z∂ t B − z ∂ z A + zA + [ A, B ]= C (cid:2) z∂ t b + za (0)2 − X k ≥ ( k − a ( k )2 z k +1 + 2 a b − a b (cid:3) (6.5)+ D (cid:2) z∂ t b + za (0)3 − X k ≥ ( k − a ( k )3 z k +1 − a b + a b (cid:3) (6.6)+ E (cid:2) z∂ t b + za (0)4 − X k ≥ ( k − a ( k )4 z k +1 − a b + 2 a b (cid:3) . (6.7)(a) Suppose that the generic type is (Sem).(i) By definition of k and k , k = min(deg t b (0)2 , deg t b (0)3 , deg t b (0)4 ) , (6.8) k + k = deg t (( b (0)3 ) + b (0)2 b (0)4 ) ≥ k , (6.9)thus k ≥ k .(ii) Suppose k > k . By a linear change of the basis v , we canarrange that k = deg t b (0)2 . The base change matrix T = C + b (0)3 /b (0)2 · E ∈ GL ( C { t } ) gives the new basis e v = v · T with matrix e B (0) = T − B (0) T = b (0)2 C + ( b (0)4 + ( b (0)3 ) /b (0)2 ) E. We can make a coordinate change in t such that afterwards b (0)2 b (0)4 + ( b (0)3 ) = γ t k + k for an arbitrarily chosen γ ∈ C ∗ . Then a diagonal base change leads toa basis which is again called v with matrices which are again called A and B with b (0)3 = 0 , b (0)2 = γt k , b (0)4 = γt k . (6.10) ANK 2 (
T E )-STRUCTURES 51
Now the vanishing of the coefficients in C { t } of C · z , C · z , D · z , D · z and E · z in (6.5)–(6.7) tells the following. C · z : a (0)3 = 0 .C · z : 0 = k γt k − + a (0)2 (1 + 2 b (1)3 ) − a (1)3 γt k , so deg t a (0)2 = k − , k γ + a (0)2 ,k − (1 + 2 b (1)3 , ) .D · z : a (0)2 γt k = a (0)4 γt k , so a (0)4 = a (0)2 t k − k , so deg a (0)4 = k − , and a (0)4 ,k − = a (0)2 ,k − .D · z : a (0)2 b (1)4 + a (1)2 γt k = a (0)4 b (1)2 + a (1)4 γt k , so b (1)4 , = 0 (here k > k is used) .E · z : 0 = k γt k − + a (0)4 (1 − b (1)3 ) + 2 a (1)3 γt k , so 0 = k γ + a (0)4 ,k − (1 − b (1)3 , ) . This shows b (1)4 , = 0 , b (1)3 , = k − k k + k ) ∈ ( − , ∩ Q . (6.11)With respect to the basis v | (0 , of K (0 , , the matrix of the residueendomorphism of the logarithmic pole at z = 0 of the ( T E )-structureover t = 0 is B (1) (0) = ρ (1) C + b (1)3 , D + b (1)2 , C . It is semisimple with the eigenvalues ρ (1) ± b (1)3 , , whose difference issmaller than 1. The monodromy is semisimple with the two differenteigenvalues exp( − πi ( ρ (1) ± b (1)3 , )).(iii) Suppose k = k . As in the proof of (ii), we can make a co-ordinate change in t and then obtain a C { t, z } -basis e v of O ( H ) (0 , with e b (0)3 = 0 , e b (0)2 = e b (0)4 = γt k for an arbitrarily chosen γ ∈ C ∗ . Now the constant base change matrix T = (cid:18) − (cid:19) gives the basis v = e v · T with b (0)2 = b (0)4 = 0 , b (0)3 = γt k . (6.12) The vanishing of the coefficients in C { t } of C · z , E · z , D · z , C · z and E · z in (6.5)–(6.7) tells the following. C · z : a (0)2 = 0 .E · z : a (0)4 = 0 .D · z : 0 = k γt k − + a (0)3 , so a (0)3 = − k γt k − .C · z : b (1)2 = b (0)3 a (0)3 a (1)2 = − k · t · a (1)2 , so b (1)2 , = 0 .E · z : b (1)4 = b (0)3 a (0)3 a (1)4 = − k · t · a (1)4 , so b (1)4 , = 0 . With respect to the basis v | (0 , of K (0 , , the matrix of the residueendomorphism of the logarithmic pole at z = 0 of the ( T E )-structureover t = 0 is B (1) (0) = ρ (1) C + b (1)3 , D. It is diagonal with the eigenvalues ρ (1) ± b (1)3 , . Therefore the monodromyhas the eigenvalues exp( − πi ( ρ (1) ± b (1)3 , )).If b (1)3 , ∈ C − ( Z − { } ), the eigenvalues of the residue endomorphismdo not differ by a nonzero integer. Because of Theorem 3.23 (c), thenthe monodromy is semisimple.We will show that the monodromy is also in the cases b (1)3 , ∈ Z − { } semisimple, by reducing these cases to the case b (1)3 , = 0.Suppose b (1)3 , ∈ N . The case b (1)3 , ∈ Z < can be reduced to this caseby exchanging v and v . We will construct a new ( T E )-structure over(
M, t ) = ( C ,
0) with the same monodromy and again with trace freepole part and of generic type (Sem) with logarithmic restriction over t , but where B (1) (0) is replaced by e B (1) (0) = ( ρ (1) + 12 ) + ( b (1)3 , −
12 ) D. (6.13)Applying this sufficiently often, we arrive at the case b (1)3 , = 0, whichhas semisimple monodromy. ANK 2 (
T E )-STRUCTURES 53
The basis e v := v · (cid:18) z (cid:19) of H ′ := H | C ∗ × ( M,t ) in a neighborhood of(0 ,
0) defines a new (
T E )-structure over ( M,
0) because of z ∇ ∂ t e v = e v (cid:0) z − a C + a D + za E (cid:1) and a (0)2 = 0 , (6.14) z ∇ ∂ z e v = e v (cid:0) z ( ρ (1) + 12 ) C + z − b C + ( b − z
12 ) D + zb E (cid:1) (6.15)and b (0)2 = 0 . Of course, it has the same monodromy. The restriction over t = 0 hasa logarithmic pole at z = 0 because b (1)2 = − k ta (1)2 and b (0)3 = γt k with k ∈ N . Its generic type is still (Sem). Its numbers e k and e k satisfy e k + e k = deg t det e U = deg t ( b (0)3 ) = 2 k . The assumption e k < e k would lead together with part (ii) to two different eigenvalues of themonodromy, a contradiction. Therefore e k = e k = k . Thus we are inthe same situation as before, with b (1)3 , diminuished by .(b) Suppose that the generic type is (Bra).(i) and (ii) U is nilpotent, but not 0. We can choose a C { t, z } -basis v of O ( H ) (0 , such that B (0) = b (0)2 C , so b (0)3 = b (0)4 = 0 . (6.16)Then δ (1) = − b (0)2 b (1)4 . Here deg t b (0)2 = k and deg t δ (1) = k + k , so k = deg t b (1)4 ≥
0. We can make a coordinate change in t such thatafterwards b (0)2 b (1)4 = γ t k + k for an arbitrarily chosen γ ∈ C ∗ . Then a diagonal base change leads toa basis which is again called v with matrices which are again called A and B with b (0)2 = γt k , b (0)3 = b (0)4 = 0 , b (1)4 = γt k . (6.17) The vanishing of the coefficients in C { t } of C · z , D · z , C · z , D · z and E · z in (6.5)–(6.7) tells the following. C · z : a (0)3 = 0 .D · z : a (0)4 = 0 .C · z : 0 = k γt k − + a (0)2 (1 + 2 b (1)3 ) − a (1)3 γt k , so deg t a (0)2 = k − , k γ + a (0)2 ,k − (1 + 2 b (1)3 , ) .D · z : a (0)2 γt k = a (1)4 γt k , so t k = a (1)4 t k a (0)2 , so k = 1 + deg a (1)4 ≥ , and a (1)4 = a (0)2 t k − k .E · z : 0 = k γt k − + 2 a (1)3 γt k − a (1)4 b (1)3 , so 0 = k γ − a (0)2 ,k − b (1)3 , . This shows k ≥ , b (1)4 , = 0 , b (1)3 , = − k k + k ) ∈ ( − , ∩ Q . (6.18)With respect to the basis v | (0 , of K (0 , , the matrix of the residueendomorphism of the logarithmic pole at z = 0 of the ( T E )-structureover t = 0 is B (1) (0) = ρ (1) C + b (1)3 , D + b (1)2 , C . It is semisimple with the eigenvalues ρ (1) ± b (1)3 , , whose difference issmaller than 1. The monodromy is semisimple with the two differenteigenvalues exp( − πi ( ρ (1) ± b (1)3 , )). (cid:3)
6. 2. ( T LE ) -structures over a point. Such unfoldings arethemselves pure (
T LE )-structures over ( C , t = 0 have a logarithmic pole at z = 0. Theorem 6.3classifies such pure ( T LE )-structures. The underlying (
T E )-structureswere subject of Definition 6.1 and Theorem 6.2. They gave their generictype and invariants ( k , k ) ∈ N (for the generic types (Sem) and(Bra)) and k ∈ N (for the generic type (Reg)). Theorem 6.3 will givean invariant k ∈ N also for the generic type (Log) with Higgs field = 0. Lemma 3.9 (b) gave the invariant ρ (1) ∈ C . The coordinate on C is again called t . ANK 2 (
T E )-STRUCTURES 55
Theorem 6.3.
Any pure rank 2 ( T LE ) -structure over ( M, t ) = ( C , with trace free pole part and with logarithmic restriction over t has af-ter a suitable coordinate change in t a unique Birkhoff normal formin the following list. Here the Birkhoff normal form consists of twomatrices A and B which are associated to a global basis v of H whoserestriction to {∞} × ( M, t ) is flat with respect to the residual connec-tion along {∞} × ( M, t ) , via z ∇ ∂ t v = vA and z ∇ ∂ z v = vB . Thematrices have the shape A = a (0)2 C + a (0)3 D + a (0)4 E, (6.19) B = zρ (1) C − γtA + zb (1)2 C + zb (1)3 D (6.20) with a (0)2 , a (0)3 , a (0)4 ∈ C [ t ] , ρ (1) , γ ∈ C , b (1)2 , b (1)3 ∈ C (so here zb (1)4 E does not turn up, resp. b (1)4 = 0 ). The left column of the following listgives the generic type of the underlying ( T E ) -structure and, dependingon the type, the invariant k ∈ N or the invariants k , k ∈ N fromDefinition 6.1 of the underlying ( T E ) -structure. The invariant ρ (1) ∈ C is arbitrary and is not listed in the table. ζ ∈ C , α ∈ R ≥ ∪ H , α ∈ C − {− } , k ∈ N and k ∈ N are invariants in some cases. Inthe first 6 cases, a (0) i is determined by b (0) i = − γta (0) i . generic type& invariants γ b (0)2 b (0)3 b (0)4 b (1)2 b (1)3 (Sem) k − k > k + k t k t k k − k k + k ) k − k ∈ N k + k t k ζ t ( k + k ) / (1 − ζ ) t k k − k k + k ) k = k k t k α (Bra) , k , k k + k t k t k + k − t k +2 k − k k + k ) (Reg) , k α k t k α (Reg) , k k t k γ a (0)2 a (0)3 a (0)4 b (1)2 b (1)3 (Log) 0 k t k − − (Log) 0 0 0 0 0 α (Log) 0 0 0 0 1 0Before the proof, several remarks on these Birkhoff normal forms aremade. The proof is given after the Remarks 6.6. Remarks 6.4. (i) The matrix B (0) = zB (1) (0) is the matrix of thelogarithmic pole at z = 0 of the restriction over t = 0 of the ( T E )-structure. In all cases except the 6th case and the 9th case, it is z ( ρ (1) C + b (1)3 D ), so it is diagonal. In these cases the monodromyis semisimple with eigenvalues exp( − πi ( ρ (1) ± b (1)3 )). In the 6th caseand the 9th case, this matrix is z ( ρ (1) C + C ). Then the matrix andthe monodromy have a 2 × − πiρ (1) ). In all cases, the leading exponents (definedin Theorem 4.18) of the logarithmic ( T E )-structure over t are called α & α , and they are α / = ρ (1) ± b (1)3 , i.e. α + α ρ (1) , α − α = 2 b (1)3 . (6.21)The 6th and 9th cases turn up again in Theorem 6.7. See the Remarks6.8 (iv)–(vi).(ii) In the generic types (Sem), the critical values satisfy u = − u because the pole part is trace free, − u + u = ρ (0) = 0. They and theregular singular exponents α & α can be calculated with the formulas(4.6) and (4.7): δ (0) = − b (0)2 b (0)4 − ( b (0)3 ) = − t k + k , (6.22) u / = ± r
14 ( u − u ) = ± p − δ (0) = ± t ( k + k ) / , (6.23) α + α ρ (1) , (6.24) α − α = u − δ (1) = k − k > k − k k + k ζ gen. type (Sem) with k − k ∈ N , − α gen. type (Sem) with k = k . (6.25)If k = k then { α , α } = { α , α } , but if k > k then { α , α } 6 = { α , α } , except if ζ ∈ {± } .(iii) In the generic type (Bra), ρ (1) ∈ C is arbitrary, b (1)3 = − k k + k ) ,and δ (1) varies as follows, δ (1) = k k + k t k + k . (6.26)(iv) In the 5th, 7th and 8th cases in Theorem 6.3, the monodromyis semisimple and the ( T E )-structure is regular singular. Associateto it the data in Definition 3.18: H ′ := H | C × ( M,t ) , M mon , N mon ,Eig( M mon ) = { λ , λ } , H ∞ , C α for α ∈ C with e − πiα j ∈ { λ , λ } .The leading exponents of the logarithmic ( T E )-structure over t arecalled α & α as in (i). The leading exponents of the ( T E )-structureover t ∈ C − { } are now called α & α . Possibly after renumbering ANK 2 (
T E )-STRUCTURES 57 λ & λ and α & α and α & α , we have e − πiα j = e − πiα j = λ j andthe relations in the following table,in Theorem 6.3 α α α α ρ (1) + α ρ (1) − α α α − ρ (1) − ρ (1) + α α ρ (1) + α ρ (1) − α α α (6.27)And there exist sections s j ∈ C α j − { } with O ( H ) = C { t, z } ( s + −
11 + α t k s ) ⊕ C { t, z } ( zs ) in the 5th case,(6.28) O ( H ) = C { t, z } ( s + t k z − s ) ⊕ C { t, z } s in the 7th case, (6.29) O ( H ) = C { t, z } s ⊕ C { t, z } s in the 8th case. (6.30)One confirms (6.28) and (6.30) immediately by calculating the matrices A and B with z ∇ ∂ t v = vA and z ∇ ∂ z v = vB for v the basis in (6.28)–( ?? ).(v) Theorem 6.7 contains for the 6th and 9th cases in Theorem 6.3a description similar to part (iv). See the Remarks 6.8 (iv)–(vi). Remarks 6.5.
These remarks study the behaviour of the (
T E )-structures in Theorem 6.3 under pull back via maps ϕ : ( C , → ( C , ϕ with ϕ ( s ) = s n for some n ∈ N are again normal forms inTheorem 6.3.(i) A general observation: Let ( H → C × ( M, t ) , ∇ ) be a ( T E )-structure over (
M, t ) = ( C ,
0) of rank r ∈ N . Let v be C { t, z } -basis of O ( H ) with z ∇ ∂ t v = vA and z ∇ ∂ z v = vB and A, B ∈ M r × r ( C { t, z } ).Choose n ∈ N and consider a map ϕ : ( C , → ( C , s ϕ ( s ) = t .Then the pull back ( T E )-structure ϕ ∗ ( H, ∇ ) has the basis e v ( z, s ) = v ( z, ϕ ( s )). The matrices e A, e B ∈ M r × r ( C { s, z } ) with z ∇ ∂ s e v = e v e A , z ∇ ∂ z e v = e v e B are e A = ∂ s ϕ ( s ) · A ( z, ϕ ( s )) , e B = B ( z, ϕ ( s )) . (6.31)(ii) These formulas (6.31) show for the 1st to 7th cases in the list inTheorem 6.3 the following: The pull back via ϕ : ( C , → ( C ,
0) with ϕ ( s ) = s n for some n ∈ N of such a ( T E )-structure with invariants( k , k ) or k is a ( T E )-structure in the same case where the invariants( k , k ) or k are replaced by ( e k , e k ) = ( nk , nk ) or e k = nk , andwhere all other invariants coincide with the old invariants. (iii) The following table says which of the ( T E )-structures in the 1stto 7th cases in the list in Theorem 6.3 are not induced by other such(
T E )-structures.generic type & invariants not induced if(
Sem ) : k − k > k , k ) = 1( Sem ) : k − k ∈ N , ζ = 0 gcd( k , k ) = 1( Sem ) : k − k ∈ N , ζ = 0 gcd( k , k + k ) = 1( Sem ) : k = k k = k = 1( Bra ) gcd( k , k ) = 1( Reg ) : N mon = 0 k = 1( Reg ) : N mon = 0 k = 1( Log ) k = 1 (6.32)(iv) In the 8th and 9th cases, the ( T E )-structure is induced by itsrestriction over t via the map ϕ : ( M, t ) → { t } , so it is constantalong M .(v) The formulas (6.28) and (6.29) confirm part (ii) for the 5th and7th cases in Theorem 6.3. Formula (6.30) confirms part (iv) in the8th case in Theorem 6.3. Analogous statements to part (ii) and part(iv) hold for the cases in Theorem 6.7. They follow from the formulas(6.49), (6.50) and (6.51) there, which are analogous to (6.28), (6.29)and (6.30). See the Remarks 6.8 (ii) and (iii). Remarks 6.6.
In the 2nd and 4th cases in the list in Theorem 6.3,another C { t, z } -basis e v of O ( H ) with nice matrices e A and e B is e v = v · T with (6.33) T = C + a (0)3 a (0)2 E = (cid:26) C + ζ t ( k − k ) / E in the 2nd case, C + t k E in the 4th case.In the 2nd case e A = − γ − ( t k − C + t k − E ) + z k − k ζ t ( k − k − / E, (6.34) e B = zρ (1) C − γt e A + zb (1)3 D. In the 4th case e A = − γ − t k − C + zk t k − E, (6.35) e B = zρ (1) C − γt e A + zb (1)3 D. These matrices are not in Birkhoff normal form. The basis e v is stilla global basis of the pure ( T LE )-structure, but the sections e v j | {∞}× M are not flat with respect to the residual connection along {∞} × M . ANK 2 (
T E )-STRUCTURES 59
Proof of Theorem 6.3:
Consider any pure (
T LE )-structure over(
M, t ) = ( C ,
0) with trace free pole part and with logarithmic re-striction to t . Choose a global basis v of H whose restriction to {∞} × ( M, t ) is flat with respect to the residual connection along {∞} × ( M, t ). Its matrices A and B with z ∇ ∂ t v = vA and z ∇ ∂ z v = vB have because of (3.18) (in Lemma 3.11) the shape (6.19) and B = zρ (1) C + ( b (0)2 + zb (1)2 ) C + ( b (0)3 + zb (1)3 ) D + ( b (0)4 + zb (1)4 ) E (6.36)with a (0) j ∈ C { t } , b (0) j ∈ t C { t } , b (1) j ∈ C . They satisfy the relations(3.28) (and, equivalently, (6.5)–(6.7)), so, more explicitly, a (0) i b (0) j = a (0) j b (0) i for ( i, j ) ∈ { (2 , , (2 , , (3 , } , (6.37) − ∂ t b (0)2 − ∂ t b (0)3 − ∂ t b (0)4 = b (1)3 − b (1)2 − b (1)4 b (1)2 b (1)4 − b (1)3 a (0)2 a (0)3 a (0)4 . (6.38)First we consider the cases when all a (0) j are 0. Then also all b (0) j are 0, because of b (0) j ∈ t C { t } and because of the differential equations(6.38). Then B = zB (1) , and it is clear that this matrix can be broughtto the form B = zρ (1) C + zα D or B = zρ (1) C + zC by a constantbase change. α ∈ C can be replaced by − α , so α ∈ R ≥ ∪ H isunique. This gives the last two cases in the list. There the generic typeis (Log).For the rest of the proof, we consider the cases when at least one a (0) j is not 0. Then (6.37) says( b (0)2 , b (0)3 , b (0)4 ) = b (0) j a (0) j · ( a (0)2 , a (0)3 , a (0)4 ) , so B (0) = b (0) j a (0) j · A (0) . (6.39)If b (0) j = 0 then b (0)2 = b (0)3 = b (0)4 = 0, and the generic type is (Log). If b (0) j = 0, then the generic type is (Sem) or (Bra) or (Reg). Consider fora moment the cases when the residue endomorphism of the logarithmicpole at z = 0 of the ( T E )-structure over t is semisimple. By Theorem6.1, these cases include the generic types (Sem) and (Bra). Then alinear base change gives b (1)2 = b (1)4 = 0, so that the 3 × e β j := deg t b (0) j ∈ N . A coordinatechange in t leads to b (0) j = b (0) j, e β j · t e β j . The differential equation in (6.38)leads to a (0) j = a (0) j, e β j − · t e β j − , and to b (0) j /a (0) j = − γt for some γ ∈ C ∗ . Define β = (1 + 2 b (1)3 ) /γ, β = 1 /γ, β = (1 − b (1)3 ) /γ. (6.40)Now (6.39) and the differential equations in (6.38) show e β j = β j and b (0)2 = 0 or ( β ∈ N and b (0)2 = b (0)2 ,β · t β ) = 0 ,b (0)3 = 0 or ( β ∈ N and b (0)3 = b (0)3 ,β · t β ) = 0 , (6.41) b (0)4 = 0 or ( β ∈ N and b (0)4 = b (0)4 ,β · t β ) = 0 . Now we discuss the generic types (Sem), (Bra), (Reg) and (Log)separately.
Generic type (Sem):
By Theorem 6.2, we can choose the basis v such that b (1)2 = b (1)4 = 0. In the cases k > k , by Theorem 6.2, b (1)3 isup to the sign unique, and we can choose it to be b (1)3 = k − k k + k ) ∈ ( − , ∩ Q (possibly by exchanging v and v ). In the cases k = k we write α := b (1)3 ∈ C . We can change its sign and get a unique α ∈ R ≥ ∪ H .We make a suitable coordinate change in t and obtain b (0)2 , b (0)3 , b (0)4 asin (6.41). The relations (6.8) and (6.9) still hold. (6.9) implies( b (0)2 b (0)4 = 0 , β + β = k + k ) or ( b (0)3 = 0 , β = k + k )(or both). In both cases (6.40) gives γ = 2 k + k . (6.42)Thus( β , β , β ) = (cid:26) ( k , k + k , k ) if k > k , ( k (1 + 2 α ) , k , k (1 − α ) if k = k . (6.43)In the cases k > k , we have β < β < β . Then (6.41) and therelation (6.8) imply b (0)2 = 0, so b (0)2 ,β = 0. The nonvanishing δ (0) = 0implies b (0)2 b (0)4 + ( b (0)3 ) = 0.In the case k − k > t and adiagonal base change allow to reduce the triple ( b (0)2 ,β , b (0)3 ,β , b (0)4 ,β ) ∈ C to a triple (1 , ζ , − ζ ) with ζ ∈ C unique.In the case k − k > β / ∈ N , so b (0)3 = 0, and a linearcoordinate change in t and a diagonal base change allow to reduce thepair ( b (0)2 ,β , b (0)4 ,β ) ∈ ( C ∗ ) to the pair (1 , ANK 2 (
T E )-STRUCTURES 61
In the cases k = k and α = 0, (6.8) and (6.43) imply b (0)2 = b (0)4 =0. Then a linear coordinate change in t allows to reduce b (0)3 ,β to thevalue 1.In the cases k = k and α = 0, as in the proof of Theorem 6.2(a)(iii), a base change with constant coefficients leads to b (0)2 = b (0)4 = 0.Then a linear coordinate change in t allows to reduce b (0)3 ,β to the value1. In all cases of generic type (Sem), we obtain the normal forms inthe list in Theorem 6.3. Generic type (Bra):
By Theorem 6.2, we can choose the basis v such that b (1)2 = b (1)4 = 0, and b (1)3 is up to the sign unique. We canchoose it to be b (1)3 = − k k + k ) ∈ ( − , ∩ Q (possibly by exchanging v and v ). We make a suitable coordinatechange in t and obtain b (0)2 , b (0)3 , b (0)4 as in (6.41). The nonvanishing δ (1) = 0 and deg t δ (1) = k + k say0 = δ (1) = − b (1)3 b (0)3 , so b (0)3 = 0 and1 /γ = β = deg b (0)3 = deg δ (1) = k + k , γ = 1 k + k , (6.44)( β , β , β ) = ( k , k + k , k + 2 k ) . (6.45)The relation (6.8) still holds, and it implies b (0)2 = 0. The vanishing δ (0) = 0 says b (0)2 ,β b (0)4 ,β + ( b (0)3 ,β ) = 0. Together with b (0)2 ,β = 0 and b (0)3 ,β =0 it implies b (0)4 ,β = 0. A linear coordinate change in t and a diagonalbase change allow to reduce the triple ( b (0)2 ,β , b (0)3 ,β , b (0)4 ,β ) ∈ ( C ∗ ) to thetriple (1 , , − Generic type (Reg):
First we consider the case when the residueendomorphism of the logarithmic pole at z = 0 of the ( T E )-structureover t is semisimple. Then a linear base change gives b (1)2 = b (1)4 = 0.And a suitable coordinate change in t gives b (0)2 , b (0)3 , b (0)4 as in (6.41).First consider the case b (1)3 = 0. Then the vanishing 0 = δ (1) = − b (1)3 b (0)3 says b (0)3 = 0. Now the vanishing 0 = δ (0) = − b (0)2 b (0)4 saysthat either b (0)2 = 0 or b (0)4 = 0. Both together cannot be 0 as thegeneric type is (Reg) and not (Log). After possibly exchanging v and v , we suppose b (0)2 = 0, b (0)4 = 0. Now k = β . Write α := 2 b (1)3 ∈ C . By (6.40), γ = 1 + α k . (6.46)A diagonal base change allows to reduce b (0)2 ,β to 1.Now consider the case b (1)3 = 0. Then β = β = β = 1 /γ , and thisis equal to k , as β j ∈ N for at least one j . Write α := b (1)3 = 0. Then(6.46) still holds. By a base change with constant coefficients, we canobtain b (0)2 = t k and b (0)3 = 0. The vanishing 0 = δ (0) = − b (0)2 b (0)4 tells b (0)4 = 0. For α = 0 as well as for α = 0, we obtain the normal formin the 5th case in the list in Theorem 6.3.Finally consider the case when the residue endomorphism of thelogarithmic pole at z = 0 of the ( T E )-structure over t has a 2 × b (1)3 = b (1)4 = 0 and b (1)2 = 1. We will lead the assumption b (0)4 = 0 as well asthe assumption b (0)4 = 0 , b (0)3 = 0 to a contradiction.Assume b (0)4 = 0. Denote β := deg t b (0)4 ∈ N . A coordinate change in t leads to b (0)4 = − β t β . The differential equation in (6.38) for b (0)4 gives a (0)4 = t β − . Now (6.39) gives b (0)3 = − β ta (0)3 . The differential equationin (6.38) for b (0)3 becomes ∂ t ( ta (0)3 ) = β a (0)3 + β t β − . This equation has no solution in C { t } , a contradiction.Assume b (0)4 = 0 , b (0)3 = 0. The same arguments as for the case b (0)4 = 0 give a contradiction if we replace ( b (0)4 , a (0)4 , b (0)3 , a (0)3 ) by( b (0)3 , a (0)3 , b (0)2 , a (0)2 ).Therefore b (0)4 = 0 , b (0)3 = 0, b (0)2 = 0. Now k = deg t b (0)2 . A coordi-nate change in t leads to b (0)2 = t k . The differential equations (6.38)gives a (0)4 = a (0)3 = 0, a (0)2 = − k t k − . We obtain the normal form inthe 6th case in the list in Theorem 6.3. Generic type (Log):
Now b (0)2 = b (0)3 = b (0)4 = 0. The cases whenall a (0) i = 0, were considered above. We assume now a (0) j = 0 for some j ∈ { , , } . The equations (6.38) become a homogeneous system oflinear equations with a nontrivial solution. Therefore the determinantof the 3 × − b (1)3 ) − b (1)2 b (1)4 . Itsvanishing tells det( B (1) − zρ (1) C ) = − . As tr( B (1) − zρ (1) C ) = 0, this ANK 2 (
T E )-STRUCTURES 63 matrix has the eigenvalues ± . Therefore a linear base change gives b (1)2 = b (1)4 = 0 , b (1)3 = − . Now the system of equations (6.38) gives a (0)3 = a (0)4 = 0, whereas a (0)2 is arbitrary in C { t }−{ } . Denote k := 1+deg t a (0)2 ∈ N . A coordinatechange in t leads to a (0)2 = k t k − . We obtain the normal form in theseventh case in the list in Theorem 6.3. (cid:3)
6. 3.
Generically regular singular ( T E ) -structures over ( C , with logarithmic restriction over t = 0 and not semisimplemonodromy. The only 1-parameter unfoldings with trace free polepart of logarithmic (
T E )-structures over a point, which are not coveredby Theorem 6.3, have generic type (Reg) or (Log) and not semisimplemonodromy. This follows from Theorem 6.2 and Theorem 3.20 (a).These (
T E )-structures are classified in Theorem 6.7. Some of them arein the 6th or 9th case in Theorem 6.3, but most are not.
Theorem 6.7.
Consider a rank 2 ( T E ) -structure ( H → C × ( M, t ) , ∇ ) over ( M, t ) = ( C , which is generically regular singular (so of generictype (Reg) or (Log)), which has trace free pole part, whose restrictionover t is logarithmic, and whose monodromy has a × Jordan block.Associate to it the data in Definition 3.18: H ′ := H | C × ( M,t ) , M mon , N mon , Eig( M mon ) = { λ } , H ∞ , C α for α ∈ C with e − πiα = λ .The leading exponents of the ( T E ) -structures over t = 0 come fromTheorem 4.15 (b) if the generic type is (Reg) and from Theorem 4.18(b) if the generic type is (Log). In both cases the leading exponents areindependent of t and are still called α & α . Recall α − α ∈ Z ≥ .The leading exponents of the logarithmic ( T E ) -structure over t = 0 from Theorem 4.18 (b) are now called α & α . Recall α − α ∈ Z ≥ .Precisely one of the three cases (I), (II) and (III) in the followingtable holds. case (I) α = α α = α + 1 thus α − α ∈ N case (II) α = α + 1 α = α case (III) α = α α = α (6.47) Choose any section s ∈ C α − ker( ∇ z∂ z − α : C α → C α ) . It de-termines uniquely a section s ∈ ker( ∇ z∂ z − α : C α → C α ) − { } with ( ∇ z∂ z − α )( s ) = z α − α s . (6.48) Then O ( H ) (0 , = C { t, z } ( s + f s ) ⊕ C { t, z } zs (6.49)for some f ∈ t C { t } − { } in case (I), O ( H ) (0 , = C { t, z } ( s + f s ) ⊕ C { t, z } zs (6.50)for some f ∈ t C { t } − { } in case (II), O ( H ) (0 , = C { t, z } s ⊕ C { t, z } s in case (III). (6.51) The function f in the cases (6.49) and (6.50) is independent of thechoice of s , so it is an invariant of the gauge equivalence class of the ( T E ) -structure. Before the proof, some remarks are made.
Remarks 6.8. (i) (6.48) gives ∇ z∂ z (( s , s )) = ( s , s ) (cid:18) α z α − α α (cid:19) = ( s , s ) · z − B (6.52)with B = z α + α C + z α − α +1 C + z α − α D (ii) The generic type is (Log) in the case (III). This ( T E )-structureis induced by its restriction over t = 0 via the projection ϕ : ( M, t ) →{ t } . The matrices A and B for the basis v = ( s , s ) are A = 0 and B as in (6.52).(iii) The generic type is (Reg) in the cases (I) and (II). In these casesthe ( T E )-structure is induced by the special cases of (6.49) respectively(6.50) with e f = t via the map ϕ = f : ( C , → ( C , A and B for the basis v = ( s + f s , zs ) in (6.49)( ⇒ case (I), ⇒ α − α ∈ N ) are A = ∂ t f · C , B = (cid:18) zα α − α ) f + z α − α z ( α + 1) (cid:19) . (6.53)The matrices A and B for the basis v = ( s + f s , zs ) in (6.50) ( ⇒ case (II), ⇒ α − α ∈ Z ≥ ) are A = ∂ t f · C , B = (cid:18) zα + z α − α +1 f z α − α +2 ( α − α ) f − z α − α f z ( α + 1) − z α − α +1 f (cid:19) . (6.54)(v) The invariant k ∈ N from (6.1) is here k = deg t f ∈ N in thecase (6.49) and the case ((6.50) & α − α ∈ N ). It is k = 2 deg t f ∈ N in the case ((6.50) & α = α ). A suitable coordinate change in t reduces f to f = t k respectively f = t k / .(vi) The overlap of the ( T E )-structures in Theorem 6.3 and in The-orem 6.7 is as follows.
ANK 2 (
T E )-STRUCTURES 65
The case (6.49) with α = α + 1 and f = − t k is the 6th case inTheorem 6.3 with ρ (1) = α .The case (6.51) with α = α is the 9th case in Theorem 6.3 with ρ (1) = α . Proof of Theorem 6.7:
Choose any section s ∈ C α − ker( ∇ z∂ z − α : C α → C α ) −{ } . It determines uniquely a section s ∈ ker( ∇ z∂ z : C α → C α ) − { } with( ∇ z∂ z − α ) s = z α − α s . (6.55)Then O ( H | C ×{ t } ) = C { z } s | C ×{ t } ⊕ C { z } s | C ×{ t } . (6.56)Choose a C { t, z } -basis v = ( v , v ) of O ( H ) (0 , which extends this C { z } -basis of O ( H | C ×{ t } ) . It has the shape v = ( s , s ) · F with F = (cid:18) f f f f (cid:19) and f , f ∈ C { t, z } [ z − ] , f ( z,
0) = f ( z,
0) = 1 , (6.57) f , f ∈ t C { t, z } [ z − ] . We write f j = P k ≥ deg z f j f ( k ) j z k with f ( k ) j = P l ≥ f ( k ) j,l t l ∈ C { t } and f ( k ) j,l ∈ C . Also we write det F = P k ≥ deg z det F (det F ) ( k ) z k with(det F ) ( k ) ∈ C { t } .A meromorphic function g ∈ C { t, z } [ z − ] on a neighborhood U ⊂ C × M which is holomorphic and not vanishing on ( U − { } × M ) ∪{ (0 , } is in C { t, z } ∗ . This and the facts that v and ( s , s ) are basesof H | U −{ }× M for some neighborhood U ⊂ C × M of (0 ,
0) and that v | C ×{ t } = ( s , s ) imply(det F ) ∈ C { t, z } ∗ , so especially (det F ) ( k ) = 0 for k < . (6.58)Write k j := deg z f j ∈ Z ∪ {∞} ( ∞ if f j = 0). Recall (6.57). It implies f (0)1 , f (0)4 ∈ C { t } ∗ and f ( k )1 , f ( k )4 ∈ t C { t } for k = 0 and f ( k )2 , f ( k )3 ∈ t C { t } for all k . Especially k ≤ k ≤
0. We distinguish four cases.Precisely one of them holds.Case f ( I ): 0 = k ≤ k , > min( k , k ) . (6.59)Case g ( II ): 0 = k ≤ k , > min( k , k ) . (6.60)Case ] ( III ): 0 = k = k , ≤ k , ≤ k . (6.61)Case ] ( IV ): 0 > min( k , k ) , > min( k , k ) . (6.62) We will show: Case f ( I ) leads to (6.49) and case (I), case g ( II ) leads to(6.50) and case (II), case ] ( III ) leads to (6.51) and case (III), and case ] ( IV ) is impossible. Case ] ( III ) : Then F ∈ GL ( C { t, z } ) and a base change leads to thenew basis e v = ( s , s ). With( α , α , s , s ) = ( α , α , s , s ) , (6.63)this gives (6.51) and case (III). Case f ( I ) : Then f ∈ C { t, z } ∗ , and a base change leads to a newbasis v [1] = ( s , s ) · F [1] with f [1]1 = 1 , f [1]2 = 0 , f [1]4 = det F [1] ∈ C { t, z } ∗ , f [1]3 ∈ t C { t, z } [ z − ] , As k [1]4 = 0, we have k [1]3 <
0. A base change leads to a new basis v [2] = ( s , s ) · F [2] with F [2] = C + f [2]3 C , with f [2]3 ∈ tz − C { t } [ z − ] − { } . The covariant derivative z ∇ ∂ t v [2]1 = z∂ t f [2]3 · v [2]2 must be in O ( H ) . Thisshows f [2]3 ∈ tz − C { t } − { } . With( α , α , s , s , f ) = ( α , α − , s , z − s , zf [2]3 ) , (6.64)this gives (6.49) and case (II). Case g ( II ) : Then f ∈ C { t, z } ∗ , and a base change leads to a newbasis v [1] = ( s , s ) · F [1] with f [1]4 = 1 , f [1]3 = 0 , f [1]1 = det F [1] ∈ C { t, z } ∗ , f [1]2 ∈ t C { t, z } [ z − ] . As k [1]1 = 0, we have k [1]2 <
0. A base change leads to a new basis v [2] = ( s , s ) · F [2] with F [2] = C + f [2]2 E, with f [2]2 ∈ tz − C { t } [ z − ] − { } . The covariant derivative z ∇ ∂ t v [2]2 = z∂ t f [2]2 · v [2]1 must be in O ( H ) . Thisshows f [2]2 ∈ tz − C { t } − { } . With( α , α , s , s ) = ( α − , α , z − s , s ) , (6.65)this gives (6.50) and almost case (II). ”Almost” because we still haveto show α − α ∈ Z ≥ . This follows from the summand − z α − α f inthe left lower entry in the matrix B in (6.54). Case g ( IV ) : Exchange v and v if k > k or if k = k anddeg t f ( k )1 > deg t f ( k )2 . Keep the basis v if not. The new basis v [1] satisfies min( k , k ) = k [1]1 ≤ k [1]2 , and in the case k [1]1 = k [1]2 it satisfiesdeg t ( f [1]1 ) ( k [1]1 ) ≤ deg t ( f [1]2 ) ( k [1]1 ) . ANK 2 (
T E )-STRUCTURES 67
By replacing v [1]2 by a suitable element in v [1]2 + C { t, z } v [1]1 , we obtain anew basis v [2] either with f [2]2 = 0 or with k [2]1 < k [2]2 and deg t ( f [2]1 ) ( k [2]1 ) > deg t ( f [2]2 ) ( k [2]2 ) .The case f [2]2 = 0 is impossible, as then we would have f [2]1 f [2]4 =det F [2] ∈ C { t, z } ∗ , so f [2]1 ∈ C { t, z } ∗ and 0 = k [2]1 , but also k [2]1 = k [1]1 =min( k , k ) <
0. For the same reason, f [2]3 = 0 is impossible. f [2]4 = 0 is impossible as then we would have − f [2]2 f [2]3 = det F [2] ∈ C { t, z } ∗ , so f [2]2 , f [2]3 ∈ C { t, z } ∗ , 0 = k [2]2 = k [2]3 and k [2]4 = ∞ , so0 = min( k [2]3 , k [2]4 ) = min( k , k ) <
0, a contradiction.Write l := deg t ( f [2]2 ) ( k [2]2 ) ∈ N , l := deg t ( f [2]1 ) [ k [2]1 ) − l ∈ N ,l := deg t ( f [2]3 ) ( k [2]3 ) ∈ N , l := deg t ( f [2]4 ) ( k [2]4 ) ∈ N . Multiplying v [2]1 and v [2]2 by suitable units in C { t } , we obtain a basis v [3] with k [3] j = k [2] j and( f [3]1 ) ( k [3]1 ) = t l + l , ( f [3]2 ) ( k [3]2 ) = t l , ( f [3]3 ) ( k [3]3 ) = t l · u , ( f [3]4 ) ( k [3]4 ) = t l · u for some units u , u ∈ C { t } ∗ . We still have 0 > k [3]1 < k [3]2 andmin( k [3]3 , k [3]4 ) <
0. Consider z ∇ ∂ t ( v [3]1 ) = z∂ t f [3]1 · s + z∂ t f [3]3 · s ∈ O ( H ) (0 , = C { t, z } v [3]1 ⊕ C { t, z } v [3]2 . The leading nonvanishing monomial in z∂ t f [3]1 is z k [3]1 +1 t l + l − . Thisimplies k [3]2 = k [3]1 + 1 ≤
0. Therefore k [3]1 + k [3]4 < k [3]2 + k [3]3 < F [3] ) ( k ) for k < k [3]1 + k [3]4 = k [3]2 + k [3]3 < , so k [3]4 = k [3]3 + 1 ≤ , k [3]3 < , f [3]1 ) ( k [3]1 ) ( f [3]4 ) ( k [3]3 +1) − ( f [3]2 ) ( k [3]1 +1) ( f [3]3 ) ( k [3]3 ) = t l (cid:0) t l + l u − t l u (cid:1) , so l = l + l , u = u . We can write v [3] = ( s , s ) ( t l + l + zg ) z k [3]1 ( t l + zg ) z k [3]1 +1 ( t l + l u + zg ) z k [3]3 ( t l u + zg ) z k [3]3 +1 ! with some suitable g , g , g , g ∈ C { t, z } . This shows O ( H ) (0 , ∩ (cid:0) z k [3]1 +2 C { t, z } s + C { t, z } [ z − ] s (cid:1) (6.66)= z C { t, z } v [3]1 + z C { t, z } v [3]2 + C { t, z } ( zv [3]1 − t l v [3]2 ) ⊂ O ( H ) (0 , ∩ (cid:0) C { t, z } [ z − ] s + z k [3]3 +2 C { t, z } s (cid:1) . Now consider the element z ( ∇ z∂ z − ( α + k [3]1 ))( v [3]1 )= z ∂ z ( zg ) z k [3]1 s + ( t l + l + zg ) z k [3]1 +1+ α − α s + z ∂ z ( zg ) z k [3]3 s + ( t l + l u + zg )( k [3]3 + α − α − k [3]1 ) z k [3]3 +1 s . of O ( H ) (0 , . It is contained in the first line of (6.66), and thereforealso in the third line of (6.66). But this leads to a contradiction, whenwe compare the coefficient of s . Here observe k [3]3 + α − α − k [3]1 < = > ⇐⇒ k [3]1 + 1 + α − α < = > k [3]3 + 1 . This contradiction shows that case ] ( IV ) is impossible. (cid:3) Marked regular singular rank 2 ( T E ) -structures The regular singular rank 2 (
T E )-structures over points were sub-ject of the subsections 4. 5 and 4. 6, those over ( C ,
0) were subject ofTheorem 6.3 and Remark 6.4 (iv) and of Theorem 6.7 and the Remarks6.8.First we will consider in the Remarks 7.1 (i)+(ii) regular singularrank 2 (
T E )-structures over P , which arise naturally from the Theo-rems 4.15 and 4.18. The ( T E )-structures over the germs ( P ,
0) and( P , ∞ ) appeared already in Remark 6.4 (iv) and in Theorem 6.7.With the construction in Lemma 3.10 (d), each of these ( T E )-structures over P extends to a rank 2 ( T E )-structure of generic type(Reg) or (Log) over C × P with primitive Higgs field. With Theo-rem 3.14, the base manifold C × P obtains a canonical structure asF-manifold with Euler field. For each t ∈ C × C ∗ , the ( T E )-structureover the germ ( C × P , t ) is a universal unfolding of its restrictionover t . For each t ∈ C × { , ∞} , the ( T E )-structure over the germ( C × P , t ) will reappear in the Theorems 8.1, 8.5 and 8.6. See Remark7.2 (i)+(ii).Then we will observe in Corollary 7.3 that any marked regular sin-gular ( T E )-structure is a good family of marked regular singular (
T E )-structures (over points) in the sense of Definition 3.26 (b).
ANK 2 (
T E )-STRUCTURES 69
In Theorem 7.4 we will determine the moduli spaces M ( H ref, ∞ ,M ref ) , reg for marked regular singular rank 2 ( T E )-structures, which were subjectof Theorem 3.29. The parameter space P of each ( T E )-structure over P in the Remarks 7.1 (i)+(ii) embeds into one of these moduli spaces,after the choice of a marking. Also these embeddings will be describedin Theorem 7.4.Because of Corollary 7.3, any marked regular singular rank 2 ( T E )-structure over a manifold M is induced by a holomorphic map M → M ( H ref, ∞ ,M ref ) , reg , where ( H ref, ∞ , M ref ) is the reference pair used in themarking of the ( T E )-structure. Remark 7.5 says something about thehorizontal direction(s) in the moduli spaces.
Remarks 7.1. (i) Consider the manifold M (3) := P with coordinate t on C ⊂ P and coordinate e t := t − on P − { } ⊂ P .With the projection M (3) → { } , we pull back the flat bundle H ′ → C ∗ in Theorem 4.15 to a flat bundle H (3) ′ on C ∗ × M . Recall thenotations 4.14.Now we read v := ( s + t s , zs ) in (4.57) and (4.59) as a basis ofsections on H (3) ′ | C ∗ × C , and e v := ( s + e t s , zs ) in (4.58) and (4.60) asa basis of sections on H (3) ′ | C ∗ × ( P −{ } ) . One sees immediately z ∇ ∂ v = v C , z ∇ e ∂ e v = e v C . (7.1)and again (4.57) resp. (4.59) and (4.58) resp. (4.60). Therefore v and e v are in any case bases of a ( T E )-structure ( H (3) → C × M (3) , ∇ (3) ) on C × C ⊂ C × M (3) respectively C × ( P −{ } ) ⊂ C × M (3) . The restricted( T E )-structures over t ∈ C ∗ are those in Theorem 4.15. They areregular singular, but not logarithmic. Their leadings exponents α & α are independent of t ∈ C ∗ . The ( T E )-structures over t = 0 and over e t = 0 (so t = ∞ ) are logarithmic except for the case ( N mon = 0 & α = α ), in which case the one over t = 0 is regular singular, but notlogarithmic. Their leading exponents are called α & α and α ∞ & α ∞ .Then over 0 over ∞ α = α α ∞ = α + 1 α = α + 1 α ∞ = α (7.2)except that in the case ( N mon = 0 & α = α ) we have α = α , α = α . For use in Theorem 7.3, we write the base space for the ( T E )-structure over P with leading exponents α & α as M (3) , ,α ,α ∼ = P in the case N mon = 0 and as M (3) , =0 ,α ,α ∼ = P in the case N mon = 0. (ii) We extend the case N mon = 0 from Theorem 4.15 (a) to the case α = α . (4.57) and (4.58) still hold, but now the restricted ( T E )-structures over points in M (3) = P are all logarithmic, though the( T E )-structure over M (3) is not logarithmic, but only regular singular.(7.1) still holds. In this case, the leading exponents are constant andare α and α + 1 (so, not α and α = α ). Similarly to (i), the basespace is called M (3) , ,α ,log ∼ = P . Remarks 7.2. (i) The construction in Lemma 3.10 (d) extends a (
T E )-structure ( H (3) → C × M (3) , ∇ (3) ) in Remark 7.1 (i) or (ii) with M (3) = P to a ( T E )-structure ( H (4) → C × M (4) , ∇ (4) ) with M (4) = C × M (3) = C × P , via ( O ( H (4) ) , ∇ (4) ) = ( ϕ (4) ) ∗ ( O ( H (3) ) , ∇ (3) ) ⊗ E t /z ,where t is the coordinate on the first factor C in C × P , and where ϕ (4) : M (4) → M (3) , ( t , t ) t , is the projection. Define v (4) :=( ϕ (4) ) ∗ ( v in Remark 7.1 (i) or (ii)).Then the matrices A and B with z ∇ ∂ i v (4) = v (4) A i and z ∇ ∂ z v (4) = v (4) B are unchanged, and A = C , so A = C , A = C (as in (7.1)) , B is as in (4.57) or (4.59) . (7.3)The Higgs field is everywhere on M (4) primitive. By Theorem 3.14, M (4) = C × P is an F-manifold with Euler field. The unit field is ∂ ,the multiplication is given by ∂ ◦ ∂ = 0 and e ∂ ◦ e ∂ = 0. So, eachgerm ( M (4) , t ) is the germ N . The Euler field is E = t ∂ + ( α − α ) t ∂ = t ∂ + ( α − α ) e t e ∂ (7.4) (cid:26) in the case (4.57) and (4.58) and in (ii) above,and in the case (4.59) and (4.60) with α − α ∈ N ,E = t ∂ − ∂ = t ∂ + e t e ∂ (7.5)in the case (4.59) and (4.60) with α = α . (ii) For t (4) ∈ C × C ∗ ⊂ M (4) , the ( T E )-structure ( H (4) → C × ( M (4) , t (4) ) , ∇ ) is a universal unfolding of the one over t , because thatone is of type (Reg) and because the Higgs field is primitive. SeeCorollary 5.1.(iii) Let ( H → C × ( M, t ) , ∇ ) be a regular singular unfolding ofa regular singular, but not logarithmic rank 2 ( T E )-structure over t .Because of part (ii), it is induced by the ( T E )-structure ( H (4) , ∇ (4) )via a map ( M, t ) → ( M (4) , t (4) ) for some t (4) ∈ { } × C ∗ . Because itis regular singular, the image of the map is in { } × C ∗ ⊂ { } × M (3) .As there the leading exponents are constant, they are also constant onthe unfolding ( H, ∇ ). ANK 2 (
T E )-STRUCTURES 71
Theorem 7.3.
Any marked regular singular rank 2 ( T E ) -structure (seeDefinition 3.15 (b), especially, M is simply connected) is a good familyof marked regular singular ( T E ) -structures (over points) in the senseof Definition 3.26 (b). Proof:
Let (( H → C × M, ∇ ) , ψ ) be a regular singular rank 2 ( T E )-structure with a marking ψ , i.e. an isomorphism ψ from ( H ∞ , M mon )to a reference pair ( H ref, ∞ , M ref ). We have to show the conditions(3.39) and (3.40) for a good family of marked regular singular ( T E )-structures.By definition of a marking, M is simply connected, so especially itis connected. The subset M [ log ] := { t ∈ M | U | t = 0 } (7.6)= { t ∈ M | the ( T E )-structure over t is logarithmic } is a priori a subvariety (in fact, it is either ∅ or a hypersurface or equalto M ).First consider the case M [ log ] = M . Choose any point t ∈ M andany disk ∆ ⊂ M through t . The restriction of the ( T E )-structureover the germ (∆ , t ) is in the case N mon = 0 isomorphic to one inthe 7th or 8th or 9th case in Theorem 6.3. In the case N mon = 0,it is isomorphic to one in case (III) in Theorem 6.7. In either casethe leading exponents are constant on ∆, because of table (6.27) inRemark 6.4 (iv) and because of the definition of case (III) in Theorem6.7. Therefore they are constant on M . We call them α gen and α gen .Now consider the case M [ log ] $ M . For each t ∈ M − M [ log ] , the( T E )-structure over the germ (
M, t ) has constant leading exponentsbecause of Remark 7.2 (iii). Therefore the leading exponents are con-stant on M − M [ log ] . We call these generic leading exponents α gen and α gen .For t ∈ M [ log ] choose a generic small disk ∆ ⊂ M through t . Then∆ − { t } ⊂ M − M [ log ] . The restriction of the ( T E )-structure over thegerm (∆ , t ) is in the case N mon = 0 isomorphic to one in the 5th casein Theorem 6.3. In the case N mon = 0, it is isomorphic to one in case(I) or case (II) in Theorem 6.7. In either case, the leading exponents( α ( t ) , α ( t )) of the ( T E )-structure over t are either ( α gen + 1 , α gen )or ( α gen , α gen + 1), because of table (6.27) in Remark 6.4 (iv) andbecause of the definition of the cases (I) and (II) in Theorem 6.7.Remark 6.4 (iv) and Theorem 6.7 provide generators of O ( H | C × (∆ ,t ) ) (0 ,t ) which are certain linear combinations of elementarysections. The shape of these generators and the almost constancy of the leading exponents imply the two conditions, O ( H | C ×{ t } ) (0 ,t ) ⊃ V r for any t ∈ M (7.7)where r := max(Re( α gen ) + 1 , Re( α gen ) + 1) , dim C O ( H | C ×{ t } ) (0 ,t ) /V r is independent of t ∈ M, (7.8)which are the conditions (3.39) and (3.40) for a good family of markedregular singular ( T E )-structures. (cid:3)
The following theorem describes the moduli space M ( H ref, ∞ ,M ref ) , reg from Theorem 3.29 for the marked regular singular rank 2 ( T E )-structures as infinite unions of curves isomorphic to P such that thefamilies of ( T E )-structures over these curves are the (
T E )-structuresin the Remarks 7.1 (i)+(ii). Part (a) treats the cases with N mon = 0,part (b) treats the cases with N mon = 0. Recall the definitions of M (3) , ,α ,α , M (3) , =0 ,α ,α and M (3) , ,α ,log in the Remarks 7.1 (i) and(ii). Theorem 7.4.
Let ( H ref, ∞ , M ref ) be a reference pair with dim H ref, ∞ = 2 . Let Eig( M ref ) = { λ , λ } be the set of eigenvaluesof M ref . Let β , β ∈ C be the unique numbers with e − πiβ j = λ j and − < Re β j ≤ .(a) The case N mon = 0 .(i) The cases with λ = λ . Then M ( H ref, ∞ ,M ref ) ,reg = ˙ [ l ∈ Z (cid:16) [ l ∈ Z M (3) , ,β + l + l ,β − l (cid:17) . (7.9) Its topological components are the unions in brackets, so S l ∈ Z M (3) , ,β + l + l ,β − l . Each component is a chain of P ’s, thepoint ∞ of M (3) , ,α ,α is identified with the point of M (3) , ,α +1 ,α − . ✉ ✉ ✉ ✉ (cid:18) α − α + 2 (cid:19)(cid:18) α − α + 1 (cid:19)(cid:18) α α + 1 (cid:19) (cid:18) α α (cid:19) (cid:18) α + 1 α (cid:19)(cid:18) α + 1 α − (cid:19)(cid:18) α + 2 α − (cid:19) (Log) (Log) (Log) (Log)(Reg) (Reg) (Reg) ✻ ✻ ✻ ✻❄ ❄ ❄ ❄ Figure 1.
One topological component in part (a) (i)
ANK 2 (
T E )-STRUCTURES 73 (ii) The cases with λ = λ (so β = β ). Then M ( H ref, ∞ ,M ref ) ,reg = ˙ [ l ∈ Z (cid:16) [ l ∈ N F β + l + l ,β + l − l (cid:17) (7.10) ∪ ˙ [ l ∈ Z (cid:16)e F β + l +1 ,β + l ∪ [ l ∈ N F β + l +1+ l ,β + l − l (cid:17) . Here F α ,α is for all possible α , α the Hirzebruch surface F , and e F α ,α − is the surface e F , which is obtained from F by blowing downthe unique ( − -curve in F . The unions in brackets are the topologicalcomponents. They are chains of Hirzebruch surfaces. A (+2) -curve of F α ,α is identified with the ( − -curve of F α +1 ,α − (and a (+2) -curveof e F α ,α − is identified with the ( − curve in F α +1 ,α − ). The ( T E ) -structures over the points in the ( − -curves are logarithmic, and alsothe ( T E ) -structure over the singular point of e F α ,α − is logarithmic.The ( T E ) -structures over all other points of F α ,α and e F α ,α are regu-lar singular, but not logarithmic, and have leading exponents α & α .For each F α ,α , and also for e F α ,α after blowing up the singular pointto a ( − -curve, the fibers of it as a P -fiber bundle over P are isomor-phic to M (3) , ,α ,α . The ( − -curve in F α ,α − (the F with l = 1 ineach topological component in the first line of (7.10) ) is isomorphic to M (3) , ,α − ,log , and the ( T E ) -structures over its points are logarithmicwith leading exponents α , α − .(b) The cases with N mon = 0 (and thus λ = λ , β = β ). Then M ( H ref, ∞ ,M ref ) ,reg = ˙ [ l ∈ Z (cid:16) [ l ∈ N M (3) , =0 ,β + l + l ,β + l − l (cid:17) (7.11) ∪ ˙ [ l ∈ Z (cid:16) [ l ∈ N M (3) , =0 ,β + l +1+ l ,β + l − l (cid:17) . Its topological components are the unions in brackets. Each componentis a chain of P ’s, the point ∞ of M (3) , =0 ,α ,α is identified with thepoint of M (3) , =0 ,α +1 ,α − . Proof:
We consider only marked (
T E )-structures with a fixed ref-erence pair ( H ref, ∞ , M ref ). Because of the markings, we can identifyfor each such ( T E )-structure its pair ( H ∞ , M mon ) with the referencepair ( H ref, ∞ , M ref ). Thus also the spaces C α can be identified for allmarked ( T E )-structures.(a) (i) and (b) In both parts, there is no harm in fixing elementarysections s ∈ C α and s ∈ C α as in Theorem 4.15. Then Theorem4.15 lists all marked ( T E )-structures with the given reference pair. The ✉ ✉ ✉ ✉✉ ✉ ✉ ✉✉ ✉ ✉ ✉ (cid:18) α α − (cid:19)(cid:18) α α − (cid:19)(cid:18) α + 1 α − (cid:19)(cid:18) α + 1 α − (cid:19)(cid:18) α + 2 α − (cid:19)(cid:18) α + 2 α − (cid:19)(cid:18) α + 3 α − (cid:19) (Log) (Log) (Log)(Reg) (Reg) (Reg) M (3) , ,α − ,log (Log) ✻ ✻ ✻ ✻❄ ❄ ❄ ❄ F α ,α − F α +1 ,α − F α +2 ,α − Figure 2.
One topological component in part (a) (ii)Remarks 7.1 (i) just put these marked (
T E )-structures into familiesparametrized by the spaces M (3) , ,α ,α resp. M (3) , =0 ,α ,α . Most log-arithmic ( T E )-structures (which are classified in Theorem 4.18) turnup in two such families. This leads to the identification of the point ∞ in M (3) , / =0 ,α ,α with the point 0 in M (3) , / =0 ,α +1 ,α − . Only each ofthe logarithmic ( T E )-structures with N mon = 0 and leading exponents α = α turns up in only one P , in the space M (3) , =0 ,α ,α − . There itis over the point 0.(a) (ii) Here the leading exponents satisfy α − α ∈ Z − { } , andwe index them such that α − α ∈ N . We fix a basis σ , σ of C α and define σ := z α − α σ ∈ C α , σ = z α − α σ ∈ C α . Then becauseof Theorem 4.15 (a), we can write all marked regular singular, but notlogarithmic ( T E )-structures with leading exponents α & α in twocharts C × C ∗ with coordinates ( r , t ) and ( r , t ), O ( H ) = C { z } ( σ + t ( σ + r σ )) ⊕ C { z } ( z ( σ + r σ )) , (7.12) O ( H ) = C { z } ( σ + t ( σ + r σ )) ⊕ C { z } ( z ( σ + r σ )) . The charts overlap where r , r ∈ C ∗ , with r = r − , t = − t r . (7.13) ANK 2 (
T E )-STRUCTURES 75 ✉ ✉ ✉✉ ✉ ✉ ✉✉ ✉ ✉ (cid:18) α α (cid:19) (cid:18) α α − (cid:19)(cid:18) α + 1 α − (cid:19)(cid:18) α + 1 α − (cid:19)(cid:18) α + 2 α − (cid:19)(cid:18) α + 2 α − (cid:19)(cid:18) α + 3 α − (cid:19) (Log) (Log) (Log) (Log)(Reg) (Reg) (Reg) ✻ ✻ ✻❄ ❄ ❄✻❄ e F α ,α − F α +1 ,α − F α +2 ,α − Figure 3.
Another topological component in part (a) (ii) ✉ ✉ ✉ ✉ (cid:18) α α (cid:19) (cid:18) α + 1 α (cid:19)(cid:18) α + 1 α − (cid:19)(cid:18) α + 2 α − (cid:19)(cid:18) α + 2 α − (cid:19)(cid:18) α + 3 α − (cid:19) (Log) (Log) (Log)(Reg) (Reg) (Reg) ✻ ✻ ✻❄ ❄ ❄ Figure 4.
Another topological component in part (b)Compactification to t = 0 and t = ∞ (and t = 0 and t = ∞ )gives the Hirzebruch surface F = F α ,α . The curve with t = 0 (and t = 0) is the ( − T E )-structures (see Theorem 4.18) with leadingexponents α and α + 1, O ( H ) = C { z } ( σ ) ⊕ C { z } ( z ( σ + r σ )) , O ( H ) = C { z } ( σ ) ⊕ C { z } ( z ( σ + r σ )) , ✉ ✉ ✉ ✉ (cid:18) α α (cid:19) (cid:18) α α − (cid:19)(cid:18) α + 1 α − (cid:19)(cid:18) α + 1 α − (cid:19)(cid:18) α + 2 α − (cid:19)(cid:18) α + 2 α − (cid:19)(cid:18) α + 3 α − (cid:19) (Log) (Log) (Log) (Log)(Reg) (Reg) (Reg) ✻ ✻ ✻ ✻❄ ❄ ❄ ❄ Figure 5.
One topological component in part (b)The curve with t = ∞ (and t = ∞ ) is a (+2)-curve. Over this curve,we have the family of marked logarithmic ( T E )-structures with leadingexponents α + 1 and α . Therefore the (+2)-curve in F α ,α must beidentified with the ( − F α +1 ,α − .In the case α − α = 2, the ( − F α ,α is the moduli space M (3) , ,α − ,log from Remark 7.1 (i).In the case α − α = 1, the ( − F α ,α has to be blowndown, as then for t = 0 O ( H ) = C { z } ( σ ) ⊕ C { z } ( z ( σ + r σ )) = C { z } C α = V α is independent of the parameter r .The projection ( r , t ) r extends to the P -fibration of F α ,α over P . The fibers are isomorphic to M (3) , ,α ,α . Affine coordinateson these fibers are t and e t = t − or t and e t = t − . (cid:3) Remarks 7.5. (i) Consider a marked regular singular rank 2 (
T E )-structure (( H → C × M, ∇ ) , ψ ). There is a unique map ϕ : M → M ( H ref, ∞ ,M ref ) ,reg , which maps t ∈ M to the unique point in M ( H ref, ∞ ,M ref ) ,reg over which one has up to marked isomorphism thesame marked ( T E )-structure as over t . Corollary 7.3 and the fact thatthe moduli space represents the moduli functor M ( H ref, ∞ ,M ref ) ,reg , im-ply that ϕ is holomorphic.Because M is (simply) connected, the map ϕ goes to one irreduciblecomponent of the moduli space, so to one M (3) , / =0 ,α ,α ∼ = P in theparts (a)(i) and (b) in Theorem 7.4 and to one F α ,α or to e F α ,α − inpart (a)(ii).(ii) In fact, in part (a)(ii) the map ϕ goes even to a projective curvewhich is isomorphic to one curve M (3) , ,α ,α or to the curve M (3) , ,α ,log .This holds for the ( T E )-structure over any manifold, as it holds by
ANK 2 (
T E )-STRUCTURES 77
Remark 6.4 (iv) and Theorem 6.7 for the (
T E )-structures over the1-dimensional germ ( C , M (3) , ,α ,α are the (0)-curves in the P fibration of F α ,α over P (in the case of e F α +1 ,α each fiber of F α +1 ,α over P embeds also into the blown down surface e F α +1 ,α ). The curveisomorphic to M (3) , ,α ,log is the ( − F α ,α − .(iii) We have here a notion of horizontal directions which is similarto that for classifying spaces of Hodge structures. There it comes fromGriffiths transversality. Here it comes from the part of the pole ofPoincar´e rank 1, which says that the covariant derivatives ∇ ∂ j alongvector fields on the base space see only a pole of order 1.In the cases of the F α ,α with α − α ∈ N − { , } , the horizontaldirections are the tangent spaces to the fibers of the P fibration. In thecases of F α ,α − and e F α ,α − , the horizontal directions contain thesetangent spaces. But on points in the ( − F α ,α − and on thesingular point in e F α ,α − , any direction is horizontal. Remarks 7.6.
If we forget the markings of the (
T E )-structures inone moduli space M ( H ref, ∞ ,M ref ) ,reg and consider the unmarked ( T E )-structures up to isomorphism, we obtain in the cases N mon = 0 count-ably many points, one for each intersection point or intersection curveof two irreducible components, and one for each irreducible component.On the contrary, in the cases N mon = 0, the unmarked and the marked( T E )-structures almost coincide, as the choice of an elementary section s in Theorem 4.15 (b) fixes uniquely the elementary section s with(4.50). The set of unmarked ( T E )-structures up to isomorphism is stillalmost in bijection with the moduli space M ( H ref, ∞ ,M ref ) ,reg in the case N mon = 0. Only the components M (3) , =0 ,α ,α − {∞} boil down tosingle points.8. Unfoldings of rank 2 ( T E ) -structures of type (Log)over a point The sections 5 and 8 together treat all rank 2 (
T E )-structures overgerms (
M, t ) of manifolds. Section 5 treated the unfoldings of ( T E )-structures of types (Sem) or (Bra) or (Reg) over t . Section 8 will treatthe unfoldings of ( T E )-structures of type (Log) over t .It builds on section 6, which classified the unfoldings with trace freepole parts over ( M, t ) = ( C ,
0) of a logarithmic rank 2 (
T E )-structureover t and on section 7, which treated arbitrary regular singular rank2 ( T E )-structures. Here the Lemmata 3.10 and 3.11 are helpful. They allow to go from arbitrary (
T E )-structures to (
T E )-structures withtrace free pole parts and vice versa.Subsection 8. 1 gives the classification results. Subsection 8. 2 ex-tracts from them a characterization of the space of all (
T E )-structureswith generically primitive Higgs fields over a given germ of a 2-dimensional F-manifold with Euler field. Subsection 8. 3 gives theproof of Theorem 8.5.First we characterize in Theorem 8.1 the 2-parameter unfoldings ofrank 2 (
T E )-structures of type (Log) over a point such that the Higgsfield is generically primitive and induces an F-manifold structure onthe underlying germ (
M, t ) of a manifold. Theorem 8.1 is a ratherimmediate implication of Theorem 6.3 and Theorem 6.7 together withthe Lemmata 3.10 and 3.11. Part (d) gives an explicit classification.The other results in this section will all refer to this classification.Corollary 8.3 lists for any logarithmic rank 2 ( T E )-structure over apoint t all unfoldings within the set of ( T E )-structures in Theorem8.1 (a). The proof consists of inspection of the explicit classification inTheorem 8.1 (d).Theorem 8.5 is the main result of this section. It lists a finite subsetof the unfoldings in Theorem 8.1 (d) with the following property: Anyunfolding of a rank 2 (
T E )-structure of type (Log) over a point isinduced by a (
T E )-structure in this list. The (
T E )-structures in thelist turn out to be universal unfoldings of themselves.The proof of Theorem 8.5 is long. It is deferred to subsection 8. 3.The results of section 6 are crucial, especially Theorem 6.3 and Theo-rem 6.7.Finally, Theorem 8.6 lists the rank 2 (
T E )-structures over a germ(
M, t ) of a manifold such that the Higgs field is primitive (so that( M, t ) becomes a germ of an F-manifold with Euler field) and therestriction over t is of type (Log). This list turns out to be a sublistof the one in Theorem 8.5. Theorem 8.6 follows easily from Theorem8.1.Theorem 8.6 is also contained in the papers [DH20-2] and [DH20-3],the generic type (Sem) is in [DH20-2], the generic types (Bra), (Reg)and (Log) are in [DH20-3]. The proofs there are completely different.They build on the formal classification of ( T )-structures in [DH20-1].8. 1. Classification results.Theorem 8.1. (a) Consider a rank 2 ( T E ) -structure ( H → C × ( M, t ) , ∇ ) over a 2-dimensional germ ( M, t ) with restriction over t of type (Log), with generically primitive Higgs field, and such that the ANK 2 (
T E )-STRUCTURES 79 induced F-manifold structure on generic points of M extends to all of M .There is a unique rank 2 ( T E ) -structure ( H [3] → C × ( C , , ∇ [3] ) over ( C , (with coordinate t ) with trace free pole part, with nonvan-ishing Higgs field and with logarithmic restriction over t = 0 such that ( O ( H ) , ∇ ) arises from ( O ( H [3] ) , ∇ [3] ) as follows. There are coordinates t = ( t , t ) on ( M, t ) such that ( M, t ) = ( C , and a constant c ∈ C such that ( O ( H ) , ∇ ) ∼ = pr ∗ ( O ( H [3] ) , ∇ [3] ) ⊗ E ( t + c ) /z , (8.1) where pr : ( M, t ) → ( C , , ( t , t ) t (see Lemma 3.10 (a) for E ( t + c ) /z ).The ( T E ) -structure ( H, ∇ ) is of type (Log) over ( C × { } , and ofone generic type (Sem) or (Bra) or (Reg) or (Log) over ( C × C ∗ , .(b) Vice versa, if ( H [3] , ∇ [3] ) is as in (a) and c ∈ C , then the ( T E ) -structure ( O ( H ) , ∇ ) := pr ∗ ( O ( H [3] ) , ∇ [3] ) ⊗ E ( t + c ) /z over ( M, t ) =( C , satisfies the properties in (a).(c) The rank 2 ( T E ) -structures ( H [3] , ∇ [3] ) over ( C , with trace freepole part, nonvanishing Higgs field and logarithmic restriction over are classified in Theorem 6.3 and Theorem 6.7. They are in suitablecoordinates the first 7 of the 9 cases in the list in Theorem 6.3 and thecases (6.49) and (6.50) with f = k t k for some k ∈ N in Theorem6.7. (Though here the 6th case in Theorem 6.3 is part of the cases (6.49) and (6.50) in Theorem 6.7.)(d) The explicit classification of the ( T E ) -structures ( H, ∇ ) in (a)is as follows. There are coordinates ( t , t ) such that ( M, t ) = ( C , ,and there is a C { t, z } -basis v of O ( H ) whose matrices A , A , B ∈ M × ( C { t, z } ) with z ∇ ∂ i v = vA i , z ∇ ∂ z v = vB are in the followinglist of normal forms. The normal form is unique. Always A = C . (8.2) Always M is an F-manifold with Euler field in one of the normal formsin Theorem 2.2 and 2.3 (in the case (i) the product ∂ ◦ ∂ is only almostin the normal form in Theorem 2.2; in the case (iii) with α = − theEuler field is only almost in the normal form in Theorem 2.3). (i) Generic type (Sem): invariants k , k ∈ N with k ≥ k , c , ρ (1) ∈ C , ζ ∈ C if k − k ∈ N , α ∈ R ≥ ∪ H if k = k , γ := 2 k + k , (8.3) A = − γ − ( t k − C + t k − E ) if k − k > − γ − ( t k − C + ζ t ( k + k ) / − D + (1 − ζ ) t k − E )if k − k ∈ N , − γ − t k − D if k = k ,B = ( − t − c + zρ (1) ) C + ( − γt ) A + (cid:26) z k − k k + k ) D if k > k ,zα D if k = k , F-manifold I ( k + k ) (with I (2) = A ) , with ∂ ◦ ∂ = γ − t k + k · ∂ ,E = ( t + c ) ∂ + γt ∂ Euler field . (ii) Generic type (Bra): invariants k , k ∈ N , c , ρ (1) ∈ C , γ := 1 k + k , (8.4) A = − γ − ( t k − C + t k + k − D − t k +2 k − E ) B = ( − t − c + zρ (1) ) C + ( − γt ) A + z − k k + k ) D, F-manifold N , with ∂ ◦ ∂ = 0 ,E = ( t + c ) ∂ + γt ∂ Euler field . (iii) Generic type (Reg): invariants c , ρ (1) ∈ C , α ∈ C − {− } if N mon = 0 , α ∈ Z if N mon = 0 , k ∈ N if N mon = 0 , e k ∈ N if ANK 2 (
T E )-STRUCTURES 81 N mon = 0 (with k = e k if α = − , and k = 2 e k if α = − ), γ := 1 + α k , (8.5) A = ( − γ − t k − C if N mon = 0 , e k t e k − C if N mon = 0 ,B = ( − t − c + zρ (1) ) C + ( − γt ) A + z α D + N mon = 0 ,z α +1 C if N mon = 0 , α ∈ Z ≥ , − z − α − t e k C + z − α t e k D + z − α +1 E if N mon = 0 , α ∈ Z < , F-manifold N , with ∂ ◦ ∂ = 0 ,E = ( ( t + c ) ∂ + γt ∂ if α = − , ( t + c ) ∂ + e k t e k +12 ∂ if α = − . ) Euler field . (iv) Generic type (Log): invariants k ∈ N , c , ρ (1) ∈ C , A = k t k − C , (8.6) B = ( − t − c + zρ (1) ) C − z D F-manifold N , with ∂ ◦ ∂ = 0 ,E = ( t + c ) ∂ Euler field . Theorem 8.1 is proved after Remark 8.2.
Remark 8.2.
The other normal forms in Remark 6.6 for the generictype (Sem) with k − k ∈ N and for the generic type (Bra) give thefollowing other normal forms. In both cases, the formulas for A = C , γ , B , the F-manifold and E are unchanged, only the matrix A changes.For the generic type (Sem) with k − k ∈ N , A becomes A = − γ − ( t k − C + t k − E ) + z k − k ζ t ( k − k − / E. (8.7)For the generic type (Bra), A becomes A = − γ − t k − C + zk t k − E. (8.8) Proof of Theorem 8.1:
We prove the parts of Theorem 8.1 in theorder (c), (d), (b), (a).(c) Consider a rank 2 (
T E )-structure ( H [3] → C × ( C , , ∇ [3] ) (withcoordinate t on ( C , restriction over t = 0. If it admits an extension to a pure ( T LE )-structure, it is contained in Theorem 6.3. If not, then it is containedin Theorem 6.7. The condition that the Higgs field is not vanishing,excludes the 8th and 9th cases in Theorem 6.3 and the case (6.51) =case (III) in Theorem 6.7, see the Remarks 6.8 (ii) and (iii).(d) Part (d) makes for such a (
T E )-structure ( H [3] , ∇ [3] ) the ( T E )-structure ( O ( H ) , ∇ ) = pr ∗ ( O ( H [3] ) , ∇ [3] ) ⊗ E ( t + c ) /z explicit. The coo-ordinate t and the matrix A in Theorem 6.3 and in Remark 6.8 (iv)become now t and A . Here the matrices in the 6th case in Theorem6.3 are not used, but the matrices in Remark 6.8 (iv). The function f in Remark 6.8 (iv) is now specialized to f = t k / if α = α ( ⇒ case (II) and (6.54)) and to f = t k if α − α ∈ N (case (I) and (6.53)or case (II) and (6.54)). The new matrix B is ( − t − c ) C plus thematrix B in Theorem 6.3 and in Remark 6.8 (iv).In the normal forms in Remark 6.8 (iv) we replaced α and α by ρ (1) and α as follows, ρ (1) := α + α + 12 , α := (cid:26) α − α − ∈ Z ≥ in (6.53) ,α − α − ∈ Z < in (6.54) . (8.9)(b) Now part (b) follows from inspection of the normal forms in part(d).(a) Consider a ( T E )-structure as in (a). Choose coordinates t =( t , t ) on ( M, t ) such that ( M, t ) = ( C ,
0) and the germ of the F-manifold is in a normal form in Theorem 2.2 (especially e = ∂ ) andthe Euler field has the form E = ( t + c ) ∂ + g ( t ) ∂ for some c ∈ C and some g ( t ) ∈ C { t } .Choose any C { t, z } -basis v of O ( H ) and consider its matrices A , A , B with z ∇ ∂ i v = vA i , z ∇ ∂ z v = vB . Now ∂ = e implies A (0)1 = C . We make a base change with the matrix T ∈ GL ( C { t, z } )which is the unique solution of the differential equation ∂ T = − (cid:16)X k ≥ A ( k )1 z k − (cid:17) T, T ( z, , t ) = C . Then the matrices e A , e A , e B of the new basis e v = vT satisfy e A = C , ∂ e A = 0 , ∂ e B = − C , (8.10)because (3.12) for i = 1 and (3.7) and (3.8) give0 = z∂ T + A T − T e A = C T − T e A = T ( C − e A ) , z∂ e A − z∂ e A + [ e A , e A ] = z∂ e A , z∂ B − z ∂ z e A + z e A + [ e A , e B ] = z ( ∂ B + C ) . ANK 2 (
T E )-STRUCTURES 83
In Lemma 3.10 (c) and Lemma 3.11 we considered the (
T E )-structure ( O ( H [2] ) , ∇ [2] ) = ( O ( H ) , ∇ ) ⊗ E ρ (0) /z with trace free polepart. Here ρ (0) = − t − c .(8.10) shows that ( H [2] , ∇ [2] ) is the pull back of its restriction( H [3] , ∇ [3] ) to ( { } × C , ⊂ ( C , O ( H ) , ∇ ) ∼ =( O ( H [2] ) , ∇ [2] ) ⊗ E − ρ (0) /z in Lemma 3.10 (c) imply (8.1). (cid:3) Corollary 8.3.
The following table gives for each logarithmic rank 2 ( T E ) -structure over a point t its unfoldings within the set of ( T E ) -structures in Theorem 8.1 (d). Here the set { α , α } ⊂ C is the set ofleading exponents in Theorem 4.18 of the logarithmic ( T E ) -structureover t . So, in the case N mon = 0 , α and α ∈ C are arbitrary.In the case N mon = 0 , they satisfy α − α ∈ Z ≥ . Two conditionsare c = 0 and ρ (1) = α + α . The other conditions and the otherinvariants (though without their definition domains) are given in thetable. All invariants in Theorem 8.1 (d) which are not mentioned here,are arbitrary here. generic type invariants N mon condition( Sem ) : k > k k , k , ζ = 0 α − α = ± k − k k + k ( Sem ) : k = k k , k , α = 0 α − α = ± α ( Bra ) k , k = 0 α − α = ± − k k + k ( Reg ) k , α = 0 α − α = ± α ( Reg ) e k , α = 0 α − α = | α | ( Log ) k = 0 α − α = ± Proof:
This follows from inspection of the cases in Theorem 8.1 (d). (cid:3)
Remark 8.4.
Beware of the following:(i) In the generic case (Sem) with k = k we have α ∈ R ≥ ∪ H .Here e α = − α is excluded, as it gives an isomorphic unfolding.(ii) In the generic cases (Reg) with α − α ∈ C − { } almost always α = α − α and e α = − α give (for the same k ∈ N respectively e k ∈ N ) two different unfoldings. The only exception is the case N mon = 0and α − α = ±
1, as then α = − k ∈ N .(iv) Unfoldings of generic type (Sem) with k > k and of generictype (Bra) exist only if α − α ∈ ( − , ∩ Q ∗ and N mon = 0. Theorem 8.5. (a) Any unfolding of a rank 2 ( T E ) -structure of type(Log) over a point is induced by one in the following subset of ( T E ) -structures in Theorem 8.1 (d). generic type & invariants condition( Sem ) : k − k > k , k ) = 1( Sem ) : k − k ∈ N , ζ = 0 gcd( k , k ) = 1( Sem ) : k − k ∈ N , ζ = 0 gcd( k , k + k ) = 1( Sem ) : k = k k = k = 1( Bra ) gcd( k , k ) = 1( Reg ) : N mon = 0 k = 1( Reg ) : N mon = 0 e k = 1( Log ) : ( N mon = 0) k = 1 (8.12) (b) The inducing ( T E ) -structure is not unique only if the orig-inal ( T E ) -structure has the form ϕ ∗ ( O ( H [5] ) , ∇ [5] ) ⊗ E − ρ (0) /z where ( H [5] , ∇ [5] ) is a logarithmic ( T E ) -structure over a point t [5] and ϕ :( M, t ) → { t [5] } is the projection, and ( H [5] , ∇ [5] ) is not one with N mon = 0 and equal leading exponents α = α . Then the original ( T E ) -structure is of type (Log) everywhere with Higgs field endomor-phisms C X ∈ O ( M,t ) · id for any X ∈ T ( M,t ) .(c) The ( T E ) -structures in the list in (a) are universal unfoldings ofthemselves. The proof of Theorem 8.5 will be given in subsection 8. 3.
Theorem 8.6.
The set of rank 2 ( T E ) -structures with primitive (notjust generically primitive) Higgs field over a germ ( M, t ) of an F-manifold and with restriction of type (Log) over t is (after the choiceof suitable coordinates) the proper subset of those in the list (8.12) inTheorem 8.5 which satisfy k = 1 respectively e k = 1 . In the cases(Reg) and (Log), it coincides with the list (8.12) . In the cases (Sem)and (Bra), it is a proper subset. Proof:
The set of rank 2 (
T E )-structures with primitive Higgs fieldover a germ (
M, t ) of an F-manifold and with restriction of type (Log)over t consists by Theorem 8.1 (a)+(d) of those ( T E )-structures inTheorem 8.1 (d) which satisfy A ( t = 0) / ∈ C · C . This holds if andonly if k = 1 respectively e k = 1 ( e k = 1 if the generic type is (Reg)and N mon = 0), and then A ( t = 0) ∈ {− γC , − γD, C } . Obviously,this is a proper subset of those in table (8.12) in the generic cases(Sem) and (Bra), and it coincides with those in table (8.12) in thegeneric cases (Reg) and (Log). (cid:3) ANK 2 (
T E )-STRUCTURES 85
8. 2. (
T E ) -structures over given F-manifolds with Euler fields.Remarks 8.7. For a given germ ((
M, t ) , ◦ , e, E ) of an F-manifold withEuler field, define B ((( M, t ) , ◦ , e, E )) := { ( T E )-structures over (
M, t ) (8.13)with generically primitive Higgs field, inducingthe given F-manifold structure with Euler field } ,B ((( M, t ) , ◦ , e, E )) := { ( T E )-structures in B (8.14)which are in table (8.12) } ,B ((( M, t ) , ◦ , e, E )) := { ( T E )-structures in B (8.15)with primitive Higgs fields } Now we can answer the questions, how big these sets are. Of-ten we write B j instead of B j ((( M, t ) , ◦ , e, E )), when the germ(( M, t ) , ◦ , e, E ) is fixed.(i) First we consider the cases when the germ (( M, t ) , ◦ , e, E ) isregular. Compare Remark 2.6 and Remark 3.17 (ii). By Malgrange’sunfolding result Theorem 3.16 (c), any ( T E )-structure over (
M, t )is the universal unfolding of its restriction over t , and it is its ownuniversal unfolding. So then B = B = B , and the classification ofthe ( T E )-structures over points in section 4 determines this space B .In the case of A with E = ( u + c ) e + ( u + c ) e with c = c ,any ( T E )-structure over t is of type (Sem). Theorem 4.4 tells thatthen B is connected and 4-dimensional. The parameters are the tworegular singular exponents and two Stokes parameters.In the case of N with E = ( t + c ) ∂ + ∂ , any ( T E )-structure over t is either of type (Bra) or of type (Reg). Then B has one componentfor type (Bra) and countably many components for type (Reg).The component for type (Bra) is connected and 3-dimensional. Theparameters are given in Theorem 4.9, they are ρ (1) , δ (1) and Eig( M mon )(here ρ (0) ( t ) = − c is fixed, and one eigenvalue and ρ (1) determine theother eigenvalue).Corollary 4.16 gives the countably many components for type (Reg).One is 1-dimensional, the others are 2-dimensional.(ii) Now we consider the cases when the germ (( M, t ) , ◦ , e, E ) isnot regular. Then E | t = c ∂ for some c ∈ C . If ( O ( H ) , ∇ ) is a( T E )-structure in B j ((( M, t ) , ◦ , e, E )), then ( O ( H ) , ∇ ) ⊗ E − c /z is a( T E )-structure in B j ((( M, t ) , ◦ , e, E − c ∂ )). Therefore we can andwill restrict to the cases with E | t = 0. Theorem 8.1 (d) gives the (
T E )-structures in B , Theorem 8.5 (a)gives the ( T E )-structures in B , and Theorem 8.6 gives the ( T E )-structures in B . For each germ (( M, t ) , ◦ , e, E ) with E | t = 0 B ⊃ B ⊃ B . (8.16)In the cases A and I ( m ), the Euler field with E | t = 0 is unique on( M, t ), therefore we do not write it down.In the case of I ( m ) with m ∈ N (this includes the case A = I (2)) B ( I ( m )) ∼ = ˙ [ ( k ,k ) ∈ N : k + k = m,k ≥ k C , (8.17) B ( I ( m )) ∼ = ˙ [ ( k ,k ) ∈ N : k + k = m,k ≥ k , gcd( k ,m/ C ,B ( I ( m )) ∼ = C , here ( k , k ) = (1 , m − . The 2 continuous parameters are the regular singular exponents of the(
T E )-structures at generic points in M .In the case of I ( m ) with m ≥ odd , B ( I ( m )) ∼ = ˙ [ ( k ,k ) ∈ N : k + k = m,k >k C , (8.18) B ( I ( m )) ∼ = ˙ [ ( k ,k ) ∈ N : k + k = m,k >k , gcd( k ,k )=1 C ,B ( I ( m )) ∼ = C , here ( k , k ) = (1 , m − . For odd m ≥
3, the regular singular exponents of the (
T E )-structuresat generic points in M coincide and give the continuous parameter.Especially, for m ∈ { , } B ( I ( m )) = B ( I ( m )) = B ( I ( m )) ∼ = (cid:26) C if m = 2 , C if m = 3 . (8.19)The F-manifold N allows by Theorem 2.3 many nonisomorphic Eu-ler fields with E | t = 0, the cases (2.10)–(2.12) with c = 0. The case (2.10), E = t ∂ : Here each ( T E )-structure has generictype (Log) and semisimple monodromy. Here B ( N , E ) ∼ = ˙ [ k ∈ N C , (8.20) B ( N , E ) = B ( N , E ) ∼ = C , here k = 1 . The continuous parameter is ρ (1) in Theorem 8.1 (d) (iv) or, equiva-lently, one of the two residue eigenvalues (which are ρ (1) ± ). The case (2.12), E = t ∂ + t r (1 + c t r − ) for some r ∈ Z ≥ andsome c ∈ C : Here each ( T E )-structure has generic type (Reg) and
ANK 2 (
T E )-STRUCTURES 87 satisfies N mon = 0. Here B ( N , E ) (cid:26) = ∅ if c ∈ C ∗ , ∼ = C if c = 0 , (8.21) B ( N , E ) = B ( N , E ) (cid:26) = ∅ if c ∈ C ∗ or r ≥ , = B ( N , E ) if c = 0 and r = 2 . So, ( N , E ) with c ∈ C ∗ does not allow ( T E )-structures over it, and( N , E ) with c = 0 and r ≥ T E )-structures over itwith primitive Higgs field. If B j ( N , E ) = ∅ then B j ( N , E ) ∼ = C andthe continuous parameter is ρ (1) in Theorem 8.1 (d) (iii). The case (2.11), E = t ∂ + c t ∂ for some c ∈ C ∗ : This is a richcase. Here we decompose B j = B j ( N , E ) as B j = B ( Reg ) , j ˙ ∪ B ( Reg ) , =0 j ˙ ∪ B ( Bra ) j , (8.22)where the first set contains ( T E )-structures of generic type (Reg) with N mon = 0, the second set contains ( T E )-structures of generic type(Reg) with N mon = 0, and the third set contains ( T E )-structures ofgeneric type (Bra). Then B ( Reg ) , ∼ = ˙ [ k ∈ N C , B ( Reg ) , = B ( Reg ) , ∼ = C , (8.23) B ( Reg ) , =01 (cid:26) = ∅ if c ∈ C ∗ − Q ∗ , ∼ = ˙ S ( k ,α ) ∈ N × Z : k c =1+ α C if c ∈ Q ∗ , (8.24) B ( Reg ) , =02 = B ( Reg ) , =03 (cid:26) = ∅ if c ∈ C − Z , ∼ = C if c ∈ Z − { } ,B ( Bra )1 = B ( Bra )2 = B ( Bra )3 = ∅ if c − ∈ C ∗ − Z ≥ , (8.25) B ( Bra )1 ∼ = ˙ S ( k ,k ) ∈ N : k + k = c − C ,B ( Bra )2 ∼ = ˙ S ( k ,k ) ∈ N : k + k = c − , gcd( k ,k )=1 C ,B ( Bra )3 ∼ = C , here ( k , k ) = (1 , c − − . if c − ∈ Z ≥ . Remarks 8.8. (i) Theorem 8.1 (d) (i) tells how many (
T E )-structuresexist over the F-manifold with Euler field I ( m ), such that the Higgsbundle is generically primitive and induces this F-manifold structure.There are [ m ] many holomorphic families from the different choices of( k , k ) ∈ N with k ≥ k and k + k = m . They have 2 parametersif m is even and 1 parameter if m is odd, compare (8.17) and (8.18).For each I ( m ), only one of these families consists of ( T E )-structureswith primitive Higgs fields.(ii) Consider m ≥
3. Write M = C for the F-manifold I ( m ) inTheorem 2.2, and M [ log ] = C × { } for the subset of points where the multiplication is not semisimple. Over these points the restricted( T E )-structures are of type (Log). We checked that there are [ m ] manyStokes structures which give ( T E )-structures on M − M [ log ] . Because of(i), all these ( T E )-structures extend holomorphically over M [ log ] , andthey give the [ m ] holomorphic families of ( T E )-structures on I ( m ) in(i).(iii) Especially remarkable is the case A = I (2). There Theorem8.1 (a)+(d)(i) implies directly that each holomorphic ( T E )-structureover A with generically primitive Higgs field has primitive Higgs fieldand is an elementary model (Definition 4.3), so it has trivial Stokesstructure.(iv) This result is related to much more general work in [CDG17]and [Sa19] on meromorphic connections over the F-manifold A n nearpoints where some of the canonical coordinates coincide. Let us restrictto the special case of a neighborhood of a point where all canonicalcoordinates coincide. This generalizes the germ at 0 of A to the germat 0 of A n .[CDG17, Theorem 1.1] and [Sa19, Theorem 3] both give the trivial-ity of the Stokes structure. Though their starting points are slightlyrestrictive. [CDG17] starts in our notation from pure ( T LE )-structureswith primitive Higgs fields. The step before in the case of A , passingfrom a ( T E )-structure over A to a pure ( T LE )-structure, is done es-sentially in our Theorem 6.2 (a)(iii). Our argument for the triviality ofthe Stokes structure is then contained in the proof of Theorem 6.3.[Sa19] starts in our notation from (
T E )-structures which are alreadyformally isomorphic to sums L ni =1 E u i /z z α i . Then it is shown that theyare also holomorphically isomorphic to such sums. In this special case,Corollary 5.7 in [DH20-2] give this implication, too.(v) In (ii) we stated that in the case of I ( m ) with m ≥
3, each(
T E )-structure on M − M [ log ] with primitive Higgs field extends holo-morphically to M . In the case of N this does not hold in general. Forexample, start with the flat rank 2 bundle H ′ → C ∗ × M where M = C (with coordinates t = ( t , t )) with semisimple monodromy with twodifferent eigenvalues λ and λ . Choose α , α ∈ C with e − πiα j = λ j .Let s j ∈ C α j be generating elementary sections. Define the new basis v = ( v , v ) = ( e t /z ( s + e − /t s ) , e t /z ( zs )) (8.26) ANK 2 (
T E )-STRUCTURES 89 on H ′ | M ′ where M ′ := M − C × { } . Then z ∇ ∂ v = v · C , (8.27) z ∇ ∂ v = v · t − e − /t C ,z ∇ ∂ z v = v · (cid:16) − t C + ( α − α ) e − /t C + z (cid:18) α α + 1 (cid:19)(cid:17) . So, we obtain a regular singular (
T E )-structure on M ′ with primitiveHiggs field. The F-manifold structure on M ′ is given by e = ∂ and ∂ ◦ ∂ = 0, so it is N , and the Euler field is E = t ∂ + ( α − α ) t ∂ .F-manifold and Euler field extend from M ′ to M , but not the ( T E )-structure.8. 3.
Proof of Theorem 8.5. (a) Let ( H → C × ( M, t ) , ∇ ) be an un-folding of a ( T E )-structure of type (Log) over t . The ( T E )-structure( H [2] → C × ( M, t ) , ∇ [2] ) in Lemma 3.10 (c) with ( O ( H [2] ) , ∇ [2] ) =( O ( H ) , ∇ ) ⊗ E − ρ (0) /z has trace free pole part. Lemma 3.10 (d) and (e)apply. Because of them, it is sufficient to prove that the ( T E )-structure( H [2] , ∇ [2] ) is induced by a ( T E )-structure ( H [3] → C × ( M [3] , t [3] ) , ∇ [3] )over ( M [3] , t [3] ) = ( C ,
0) via a map ϕ : ( M, t ) → ( M [3] , t [3] ), where the( T E )-structure ( H [3] , ∇ [3] ) is one of the ( T E )-structures in the 1st to7th cases in Theorem 6.3 or one of the (
T E )-structures in the cases(I) or (II) in Theorem 6.7 with invariants as in table (8.12). Thenthe (
T E )-structure ( H [4] , ∇ [4] ) which is constructed in Lemma 3.10 (d)from ( H [3] , ∇ [3] ) is one of the ( T E )-structures in Theorem 8.1 (d) withinvariants as in table (8.12), and it induces by Lemma 3.10 (e) the(
T E )-structure ( H, ∇ ).From now on we suppose ρ (0) = 0, so ( H, ∇ ) = ( H [2] , ∇ [2] ). Weconsider the invariants δ (0) , δ (1) ∈ O M,t and U and the four possible generic types (Sem), (Bra), (Reg) and (Log), which are defined by thefollowing table, analogously to Definition 6.1,(Sem) (Bra) (Reg) (Log) δ (0) = 0 δ (0) = 0 , δ (1) = 0 δ (0) = δ (1) = 0 , U 6 = 0 U = 0First we treat the generic types (Reg) and (Log), then the generictype (Sem) and (Bra). Generic types (Reg) and (Log):
Then the (
T E )-structure ( H, ∇ )is regular singular. We can use the results in section 7 (which built onthe Theorems 6.3 and 6.7). Choose a marking for the ( T E )-structure( H, ∇ ). Then by Remark 7.5 (i), there is a unique map ϕ : ( M, t ) → M ( H ref, ∞ ,M ref ) ,reg which maps t ∈ M to the point in the moduli spaceover which one has up to isomorphism the same marked ( T E )-structure as over t . The map ϕ is holomorphic. By Remark 7.5 (i)+(ii) it maps( M, t ) to one projective curve which is isomorphic to M (3) , ,α ,α or M (3) , =0 ,α ,α or M (3) , ,α ,log . The ( T E )-structure ( H, ∇ ) is induced bythe ( T E )-structure over this curve via the map ϕ . The point t ismapped to 0 or ∞ in the cases M (3) , ,α ,α or M (3) , =0 ,α ,α (not 0 in thecase M (3) , =0 ,α ,α ) as the ( T E )-structure over t is logarithmic. Thegerms at 0 and ∞ in M (3) , ,α ,α and M (3) , =0 ,α ,α (not 0 in the case M (3) , =0 ,α ,α ) and the germ at any point t (3)2 in M (3) , ,α ,log are containedin table (8.12). This shows Theorem 8.5 for the generic cases (Reg) and(Log). Generic types (Sem) and (Bra):
We choose a (connected and sufficiently small) representative M ofthe germ ( M, t ), and we choose on it coordinates t = ( t , ..., t m ) (with m = dim M ) with t = 0. We denote by M [ log ] the analytic hypersur-face M [ log ] := (cid:26) ( δ (0) ) − (0) if the generic type is (Sem),( δ (1) ) − (0) if the generic type is (Bra). (8.28)It contains t . Choose a disk ∆ ⊂ M through t with ∆ − { t } ⊂ M − M [ log ] . The restricted ( T E )-structure ( H, ∇ ) | C × (∆ ,t ) has thesame generic type as the ( T E )-structure ( H, ∇ ). The restricted ( T E )-structure ( H, ∇ ) | C × (∆ ,t ) is isomorphic to a ( T E )-structure in the cases1, 2, 3 or 4 in Theorem 6.3.The parameters of the restricted (
T E )-structure ( H, ∇ ) | C × (∆ ,t ) aregiven in the following table.generic type parameters( Sem ) k , k ∈ N with k ≥ k , ρ (1) ∈ C , (cid:26) ζ ∈ C if k − k ∈ N ,α ∈ R ≥ ∪ H if k = k . ( Bra ) k , k ∈ N , ρ (1) ∈ C . (8.29)There is a unique pair ( k , k ) ∈ N with ( k , k ) ∈ Q > · ( k , k ) andwith the conditions in table (8.30),generic type & invariants conditions( Sem ) : k − k > k , k ) = 1( Sem ) : k − k ∈ N , ζ = 0 gcd( k , k ) = 1( Sem ) : k − k ∈ N , ζ = 0 k − k ∈ N , gcd( k , k + k ) = 1( Sem ) : k = k k = k = 1( Bra ) gcd( k , k ) = 1 (8.30) ANK 2 (
T E )-STRUCTURES 91
In fact, it is the pair ( k , k ) ∈ N of minimal numbers which satisfies( k , k ) ∈ N · ( k , k ) (8.31)and which satisfies in the case (Sem) with k − k ∈ N and ζ = 0additionally k − k ∈ N .We denote by ( H [3] → C × ( M [3] , t [3] ) , ∇ [3] ) the ( T E )-structureover ( M [3] , t [3] ) = ( C ,
0) which has ( k , k ) instead of ( k , k ), butwhich has the same other parameters as the restricted ( T E )-structure( H, ∇ ) | C × (∆ ,t ) .We have seen in the Remarks 6.5 (ii) and (iii) that the restricted( T E )-structure ( H, ∇ ) | C × (∆ ,t ) is induced by the ( T E )-structure( H [3] , ∇ [3] ) via the branched covering ϕ ∆ : (∆ , t ) → ( M [3] , t [3] ) with ϕ ∆ ( τ ) = τ k /k . Here τ denotes that coordinate on ∆ with which( H, ∇ ) | C × (∆ ,t ) can be brought to a normal form in the cases 1, 2, 3and 4 in Theorem 6.3.It rests to extend ϕ ∆ to a map ϕ : M → M [3] such that ( H, ∇ ) isinduced by ( H [3] , ∇ [3] ) via this map ϕ . Claim:
There exists a unique holomorphic function ϕ ∈ O M with ϕ | ∆ = ϕ ∆ , (8.32) δ (0) = − ϕ k + k if the generic type is (Sem) , (8.33) δ (1) = k k + k · ϕ k + k if the generic type is (Bra) . (8.34) Proof of the Claim:
Choose any point t [1] ∈ M [ log ] and any disk∆ [1] through t [1] with ∆ [1] − { t [1] } ⊂ M − M [ log ] . In order to show theexistence of a function ϕ ∈ O M with (8.33) respectively (8.34), it issufficient to show that δ (0) | ∆ [1] respectively δ (1) | ∆ [1] has at t [1] a zero ofan order which is a multiple of k + k .The restricted ( T E )-structure ( H, ∇ ) | C × (∆ [1] ,t [1] ) has the same generictype as ( H, ∇ ) and is isomorphic to a ( T E )-structure in the cases 1, 2,3 or 4 in Theorem 6.3. Its invariants k and k are here called k [1]1 and k [1]2 , in order to distinguish them from the invariants of ( H, ∇ ) | (∆ ,t ) .We want to show ( k [1]1 , k [2]2 ) ∈ N · ( k , k ) . (8.35) We did not say much about the Stokes structure. Here we need thefollowing properties of it, if the generic type is (Sem). k = k (8.36) (1) ⇐⇒ ( H, ∇ ) | C ×{ t [2] } has trivial Stokes structure for t [2] ∈ ∆ − { t } (2) ⇐⇒ ( H, ∇ ) | C ×{ t [2] } has trivial Stokes structure for t [2] ∈ ∆ [1] − { t [1] } (3) ⇐⇒ k [1]2 = k [1]1 . (1) = ⇒ and (3) ⇐ = are obvious from the normal form in the 3rd case in The-orem 6.3. It is not hard to see that the normal forms for fixed t ∈ C ∗ in the 1st and 2nd case in Theorem 6.3 are not holomorphically iso-morphic to an elementary model in Definition 4.3 (see also Remark8.8 (ii)). This shows (1) ⇐ = and (3) = ⇒ . The equivalence (2) ⇐⇒ is a conse-quence of the invariance of the Stokes structure within isomonodromicdeformations.In the generic type (Sem) with k = k we have also k [1]2 = k [1]1 and k = k = 1, and thus especially (8.35).Now consider the cases with k > k . This comprises the generic type(Bra) and gives in the generic type (Sem) also k > k and k [1]2 > k [1]1 .So ( H, ∇ ) | C × (∆ [1] ,t [1] ) is in the 1st, 2nd or 4th case in Theorem 6.3. Thenumber b (1)3 in Theorem 6.3 is uniquely determined by the properties b (1)3 ∈ Q ∩ ] − ,
0[ and Eig( M mon ) = { exp( − πi ( ρ (1) ± b (1)3 ) } (see Remark6.4 (i) for the second property). Therefore k − k k + k ) = k − k k + k ) = b (1)3 = k [1]1 − k [1]2 k [1]1 + k [1]2 ) in the case (Sem) , (8.37) − k k + k ) = − k k + k ) = b (1)3 = − k [1]2 k [1]1 + k [1]2 ) in the case (Bra) . This implies ( k [1]1 , k [1]2 ) ∈ Q > · ( k , k ). In the cases with gcd( k , k ) = 1(8.35) follows.If gcd( k , k ) = 1, then the generic type is (Sem), k − k ∈ N , k − k ∈ N , and the invariant ζ of ( H, ∇ ) | C × (∆ ,t ) is ζ = 0. Butthen the regular singular exponents α and α of the restriction of the( T E )-structure ( H, ∇ ) over points in M − M [ log ] are invariants of the( T E )-structure ( H, ∇ ). By (6.25) and (8.37) also ζ is an invariant ofthe ( T E )-structure ( H, ∇ ). Now ζ = 0 implies k [1]2 − k [1]1 ∈ N . Again(8.35) follows. ANK 2 (
T E )-STRUCTURES 93 (6.22) and (6.26) imply that δ (0) | ∆ [1] respectively δ (1) | ∆ [1] has at t [1] a zero of an order which is a multiple of k + k . Therefore a function ϕ ∈ O M with (8.33) respectively (8.34) exists.(6.22) and (6.26) for ( H, ∇ ) | C × (∆ ,t ) tell δ (0) | ∆ = − τ k + k = − ( τ k /k ) k + k = − ( ϕ ∆ ) k + k in the case (Sem), δ (1) | ∆ = k k + k τ k + k = k k + k ( τ k /k ) k + k = k k + k ( ϕ ∆ ) k + k in the case (Bra).Therefore a function ϕ as in the claim exists and is unique. ( (cid:3) )Now compare the ( T E )-structures ( H, ∇ ) and ϕ ∗ ( H [3] , ∇ [3] ) over M .Both extend to pure ( T LE )-structures. For ϕ ∗ ( H [3] , ∇ [3] ), one usesthe pull back ϕ ∗ ( v [3] ) of the basis v [3] which gives for ( H [3] , ∇ [3] ) theBirkhoff normal form in Theorem 6.3. For ( H, ∇ ) one starts with theanalogous basis v ∆ for H | C × ∆ which gives for ( H, ∇ ) | C × ∆ the Birkhoffnormal form in Theorem 6.3. It has a unique extension v to C × M which still yields a Birkhoff normal form. Compare Remark 3.19 (ii)for this.The Remarks 6.5 (ii) and (iii) (or simply the Birkhoff normal formsin Theorem 6.3) tell that the map ( ϕ ∗ v [3] ) | C × ∆ v ∆ = v | C × ∆ is anisomorphism of pure ( T LE )-structures.Now consider a point t [2] ∈ ∆ − { t } and its image t [4] := ϕ ( t [2] ) ∈ M [3] − { t [3] } = C − { } . Over the germ ( M [3] , t [4] ), the ( T E )-structure( H [3] , ∇ [3] ) is the part with trace free pole part of a universal unfoldingof ( H [3] , ∇ [3] ) | C ×{ t [4] } . Therefore in a neighborhood U ⊂ M of t [2] , the( T E )-structure ( H, ∇ ) | C × U is induced by ( H [3] , ∇ [3] ) | C × ( M [3] ,t [4] ) via amap e ϕ : U → M [3] . We can choose it such that e ϕ | ∆ = ϕ ∆ . (8.38)(6.22) and (6.26) tell δ (0) | U = − ( e ϕ ) k + k in the case (Sem), (8.39) δ (1) | U = k k + k ( e ϕ ) k + k in the case (Bra). (8.40)(8.38)–(8.40) and the Claim imply e ϕ = ϕ | U . Therefore the matricesin Birkhoff normal form for the basis v of ( H, ∇ ) coincide on C × U with the matrices in Birkhoff normal form for the basis ϕ ∗ ( v [3] ) of ϕ ∗ ( H [3] , ∇ [3] ). As all matrices are holomorphic on C × M , they coincidepairwise on C × M . Therefore the pure ( T LE )-structure ( H, ∇ ) with basis v is isomorphic to the pure ( T LE )-structure ϕ ∗ ( H [3] , ∇ [3] ) withbasis ϕ ∗ ( v [3] ). This finishes the proof of part (a) of Theorem 8.5.(b) If the original ( T E )-structure ( H, ∇ ) has the form ϕ ∗ ( O ( H [5] ) , ∇ [5] ) ⊗ E − ρ (0) /z then the ( T E )-structure ( H [2] , ∇ [2] )with trace free pole part which was associated to ( H, ∇ ) at thebeginning of the proof of part (a), has the form ϕ ∗ ( O ( H [5] ) , ∇ [5] ).Then any ( T E )-structure ( H [3] , ∇ [3] ) over ( M [3] , t [3] ) = ( C ,
0) works,whose restriction over t [3] is the given logarithmic ( T E )-structure( H [5] , ∇ [5] ).In the cases with N mon = 0, table (8.12) offers one of generic type(Sem) with k = k = 1 (and some with k > k if α − α ∈ Q ∩ ( − , N mon = 0, table (8.12) offers two of generic type (Reg) if the leadingexponents α and α satisfy α − α ∈ N , and one if they satisfy α = α ,compare also the figures 4 and 5 in Theorem 7.4 (b). Therefore theinducing ( T E )-structure in table (8.12) is not unique except for thecase N mon = 0 and α = α , if the original ( T E )-structure has theform ϕ ∗ ( O ( H [5] ) , ∇ [5] ) ⊗ E − ρ (0) /z .In the other cases, the proof of part (a) shows the uniqueness of the( T E )-structure ( H [3] , ∇ [3] ). The uniqueness of ( H [3] , ∇ [3] ) gives alsothe uniqueness of ( H [4] , ∇ [4] ) in the first paragraph of the proof of part(a).(c) This follows from the uniqueness in part (b) (cid:3) A family of rank 3 ( T E ) -structures with a functionalparameter M. Saito presents in [SaM17] a family of Gauss-Manin connectionswith a functional parameter. In the arXiv paper [SaM17], the bundlehas rank n , but in a preliminary version it has rank 3 and is moretransparent.Here we translate the rank 3 example by a Fourier-Laplace trans-formation into a family of (TE)-structures with primitive Higgs fieldsover a fixed 3-dimensional globally irreducible F-manifold with an Eulerfield, such that the F-manifold with Euler field is nowhere regular. Thefamily of (TE)-structures has a functional parameter h ( t ) ∈ C { t } .In the following, we write down a (TE)-structure of rank 3 on amanifold M = C with coordinates t , t , t . The restriction to { t ∈ C | t = 0 } = { } × C is a FL-transformation of Saito’s example.The parameter t and this F-manifold are not considered in [SaM17]. ANK 2 (
T E )-STRUCTURES 95
There the base space has only the two parameters t and t . Choosean arbitrary function h ( t ) ∈ C { t } with h ′′ (0) = 0.Let H ′ → C ∗ × M be a holomorphic vector bundle with flat connec-tion with trivial monodromy and basis of global flat sections s , s , s .Define an extension to a vector bundle H → C × M using the followingholomorphic sections of H ′ , which also form a basis of sections of H ′ : v := e t /z · ( zs + t · zs + h ( t ) · zs + t · z s ) , (9.1) v := e t /z · ( z s + h ′ ( t ) · z s ) ,v := e t /z · z s . Denote v := ( v , v , v ). Denote ∂ t j := ∂ j . Then z ∇ ∂ v = v · , (9.2) z ∇ ∂ v = v · h ′′ ( t ) 0 ,z ∇ ∂ v = v · ,z ∂ z v = v · − t · + t + z · v · . Write ∂ := ( ∂ , ∂ , ∂ ). The pole parts give the multiplication ◦ onthe F-manifold and the Euler field E by ∂ ◦ ∂ = ∂ · , (9.3) ∂ ◦ ∂ = ∂ · h ′′ ( t ) 0 ,∂ ◦ ∂ = ∂ · ,E ◦ ∂ = − ∂ · − t · + t , so E = t · ∂ − t · ∂ . One can introduce a new coordinate system ( e t , e t , e t ) = ( t , e t , t ) onthe germ ( M,
0) with ∂ e t = 1 p h ′′ ( t ) · ∂ . (9.4)Denote e ∂ j := ∂ e t j and e ∂ := ( e ∂ , e ∂ , e ∂ ) = ( ∂ , e ∂ , ∂ ). Introduce also thenew section e v := 1 p h ′′ ( t ) · v , (9.5)and the new basis e v = ( e v , e v , e v ) = ( v , e v , v ) of the given (TE)-structure. Then z ∇ e ∂ e v = e v · , (9.6) z ∇ e ∂ e v = e v · + e v · ∂ √ h ′′ ( t )
00 0 0 ,z ∇ e ∂ e v = e v · ,z ∂ z e v = e v · − t · + t + z · e v · . In the new coordinates the multiplication becomes simpler and inde-pendent of the choice of h ( t ) (as long as h ′′ ( t ) = 0): e ∂ ◦ e ∂ = e ∂ · , (9.7) e ∂ ◦ e ∂ = e ∂ · , e ∂ ◦ e ∂ = e ∂ · ,E ◦ e ∂ = − e ∂ · − t · + t , so E = t · e ∂ − t · e ∂ . This is the nilpotent F-manifold for n = 3 in [DH17, Theorem 3]. Butthe Euler field here is different from the one in [DH17, Theorem 3]. The ANK 2 (
T E )-STRUCTURES 97 endomorphism E ◦ here is not regular, but has only the one eigenvalue t and has for t = 0 one Jordan block of size 2 × × t = 0.The sections v , v , v define also an extension b H → P such that the(TE)-structure extends to a pure (TLE)-structure.Furthermore v satisfies all properties of the section ζ in Theorem 6.6(b) in [DH20-2]. Thus the F-manifold with Euler field is enriched to aflat F-manifold with Euler field (Definition 3.1 (b) in [DH20-2]).If we try to introduce a pairing which would make it into a pure(TLEP)-structure, we get a constraint h ′′ ( t ) = const . But probablysimilar higher dimensional examples allow also an extension to pure(TLEP)-structures while keeping the functional freedom. This wouldgive families of Frobenius manifolds with Euler fields with functionalfreedom on a fixed F-manifold with Euler field.In the example above, t , t , t are flat coordinates and e t = t , e t , e t = t are generalized canonical coordinates (in which the multiplication hassimple formulas). References [AB94] D. Anosov, A.A. Bolibruch: The Riemann-Hilbert-Birkhoff problem. As-pects of Mathematics, vol. (1994), Vieweg 1994.[CDG17] G. Cotti, B. Dubrovin, D. Guzzetti: Isomonodromy deformations atan irregular singularity with coalescing eigenvalues. Duke Math. J. (2019), 967–1108.[DH17] L. David and C. Hertling: Regular F -manifolds: initial conditions andFrobenius metrics. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. XVII (2017), 1121–1152.[DH20-1] L. David and C. Hertling: (T)-structures over 2-dimensional F-manifolds: formal classification. Annali di Matematica Pura ed Applicata(1923 -) (2020), 1221–1242.[DH20-2] L. David and C. Hertling: Meromorphic connections over F-manifolds.Accepted for publication in: Integrability, Quantization, and Geometry (edsI. Krichever, S. Novikov, O. Ogievetsky, S. Shlosman), Proc. of Symp. inPure Math., AMS, to appear. Also arXiv:1911.03331v1, 58 pages;[DH20-3] L. David and C. Hertling: (
T E )-structures over the 2-dimensionalglobally nilpotent F -manifold Accepted for publication in Revue RoumaineMath. Pure Appliq.. Also arXiv:2001.01063v1, 53 pages.[He02] C. Hertling: Frobenius manifolds and moduli spaces for singularities. Cam-bridge Tracts in Mathematics, vol. , Cambridge University Press, 2002.[He03] C. Hertling: tt ∗ geometry, Frobenius manifolds, their connections, andthe construction for singularities. J. Reine Angew. Math. (2003), 77–161.[HM99] C. Hertling, Yu. Manin: Weak Frobenius manifolds. Int. Math. Res.Notices , 277–286. [HM04] C. Hertling, Yu. Manin: Meromorphic connections and Frobenius mani-folds. In: Frobenius manifolds, quantum cohomology, and singularities (eds.C. Hertling and M. Marcolli), Aspects of Mathematics E 36, Vieweg 2004,113–144.[HS11] C. Hertling and C. Sabbah: Examples of non-commutative Hodge struc-tures. Journal of the Inst. of Math. Jussieu (3) (2011), 635–674.[HHP10] C. Hertling, L. Hoevenaars, H. Posthuma: Frobenius manifolds, pro-jective special geometry and Hitchin systems. J. reine angew. Math. (2010), 117–165.[HS10] C. Hertling and Ch. Sevenheck: Limits of families of Brieskorn latticesand compactified classifying spaces. Advances in Mathematics (2010),1155–1224.[Ma83a] B. Malgrange: La classification des connexions irr´eguli`eres `a une vari-able. In: S´eminaire de’l ENS, Math´ematique et Physique, 1979–1982,Progress in Mathematics vol. 37, Birkh¨auser, Boston 1983, pp. 381–399.[Ma83b] B. Malgrange: Sur les d´eformations isomonodromiques, I, II. In:S´eminaire de’l ENS, Math´ematique et Physique, 1979–1982, Progress inMathematics vol. 37, Birkh¨auser, Boston 1983, pp. 401–438.[Ma86] B. Malgrange: Deformations of differential systems, II. J. RamanujanMath. Soc. (1986), 3–15.[Sa02] C. Sabbah: D´eformations isomonodromiques et vari´et´es de Frobenius.Savoirs Actuels, EDP Sciences, Les Ulis, 2002, Math´ematiques. Englishtranslation: Isomonodromic deformations and Frobenius manifolds. Uni-versitext, Springer and EDP Sciences, 2007.[Sa05] C. Sabbah: Polarizable twistor D -modules. Ast´erisque (2005),vi+208 pages.[Sa19] C. Sabbah: Integrable deformations and degenerations of some irregularsingularities. Accepted for publication in Publ. RIMS Kyoto Univ.. AlsoarXiv:1711.08514v3, 35 pages.[SaM17] M. Saito: Deformations of abstract Brieskorn lattices. arXiv:1707.07480v4, 15 pages. Lehrstuhl f¨ur algebraische Geometrie, Universit¨at Mannheim, B6,26, 68159 Mannheim, Germany
E-mail address ::