Rank 2 Bundles with Meromorphic Connections with Poles of Poincar\'e Rank 1

Holomorphic vector bundles on $\mathbb C\times M$, $M$ a complex manifold, with meromorphic connections with poles of Poincar\'e rank 1 along $\{0\}\times M$ arise naturally in algebraic geometry. They are called $(TE)$-structures here. This paper takes an abstract point of view. It gives a complete classification of all $(TE)$-structures of rank 2 over germs $\big(M,t^0\big)$ of manifolds. In the case of $M$ a point, they separate into four types. Those of three types have universal unfoldings, those of the fourth type (the logarithmic type) not. The classification of unfoldings of $(TE)$-structures of the fourth type is rich and interesting. The paper finds and lists also all $(TE)$-structures which are basic in the following sense: Together they induce all rank $2$ $(TE)$-structures, and each of them is not induced by any other $(TE)$-structure in the list. Their base spaces $M$ turn out to be 2-dimensional $F$-manifolds with Euler fields. The paper gives also for each such $F$-manifold a classification of all rank 2 $(TE)$-structures over it. Also this classification is surprisingly rich. The backbone of the paper are normal forms. Though also the monodromy and the geometry of the induced Higgs fields and of the bases spaces are important and are considered.


Introduction
A holomorphic vector bundle H on C × M, M a complex manifold, with a meromorphic connection ∇ with a pole of Poincaré rank 1 along {0} × M and no pole elsewhere, is called a (T E)-structure. The aim of this paper is the local classification of all rank 2 (T E)-structures, over arbitrary germs (M, t 0 ) of manifolds.
Before we talk about the results, we will put these structures into a context, motivate their definition, mention their occurence in algebraic geometry, and formulate interesting problems. The rank 2 case is the first interesting case and already very rich. In many aspects it is probably typical for arbitrary rank, in some not. And it is certainly the only case where such a thorough classification is feasible.
The pole of Poincaré rank 1 along {0} × M of the pair (H, ∇) means the following. Let t = (t 1 , ..., t n ) be holomorphic coordinates on M with coordinate vector fields ∂ 1 , ..., ∂ n , and let z be the standard coordinate on C. Then ∇ ∂z σ for a holomorphic section σ ∈ O(H) of H is in z −2 O(H), and ∇ ∂ j σ is in z −1 O(H). The pole of order two along ∂ z is the first case beyond the easy and tame case of a pole of order 1, i.e. a logarithmic pole. The pole of order 1 along ∂ i gives a good variation property, a generalization of Griffiths transversality for variations of Hodge structures. It is the most natural constraint for an isomonodromic family of bundles on C with poles of order 2 at 0. So, a pole of Poincaré rank 1 is in some sense the first case beyond the case of connections with logarithmic poles.
In algebraic geometry, such connections arise naturally. A distinguished case is the Fourier-Laplace transformation (with respect to the coordinate z) of the Gauss-Manin connection of a family of holomorphic functions with isolated singularities [He03,ch. 8] [Sa02,VII]. The paper [He03] defines (T ERP )-structures, which are (T E)-structures with additional real structure and pairing and which generalize variations of Hodge structures. Also the notion (T EZP )-structure makes sense, 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 isolated singularities (and some topological well-behavedness) gives rise to 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 considered. We formulate it here as the following question: Does the (T EZP )-structure of a holomorphic function germ with an isolated singularity determine the (T EZP )-structure of the universal unfolding of the function germ? The first one is a (T E)-structure over a point t 0 . The second one is a (T E)-structure over a germ (M, t 0 ) of a manifold M. It it an unfolding of the first (T E)-structure with a primitive Higgs field. The base space M is an F-manifold with Euler field.
We explain these notions. A second (T E)-structure over a manifold 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 submanifold is isomorphic to the first (T E)-structure. 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 together with a holomorphic commutative and associative multiplication • on T M which comes equipped with the integrability condition (2.1), with a unit field e ∈ T M (with •e = id) and an Euler field E ∈ T M with Lie E (•) = • (see [HM99] or Definition 2.1). A (T E)-structure over M with primitive Higgs field induces on the base manifold M the structure of an F-manifold with Euler field (see Theorem 3.14 for details).
A result of Malgrange [Ma86] (cited in Theorem 3.16 (c)) says that a (T E)-structure over a point t 0 has a universal unfolding if the endomorphism U : K → K (here K is a vector space) is regular, i.e. it has only one Jordan block for each eigenvalue. Theorem 3.16 (b) gives a generalization from [HM04]. A special case of this generalization says that a (T E)-structure with primitive Higgs field over a germ (M, t 0 ) is its own universal unfolding (see Theorem 3.16 (a)). A supplement from [DH17] says that then the base space is a regular F-manifold (see Definition 2.4 and Theorem 2.5).
Malgrange's result makes life easy if one starts with a (T E)-structure over a point whose endomorphism U is regular. But if one starts with a (T E)-structure over a point such that U is not regular, then in general it has no universal unfolding, and the study of all its unfoldings becomes very interesting. The second half of this paper (the sections 6 -8) studies this situation in rank 2. The Torelli problem for a holomorphic function germ with an isolated singularity is similar: The endomorphism U of its (T EZP )-structure is never regular (except if the function has an A 1 -singularity), but I hope that the (T EZP )-structure determines nevertheless somehow the specific unfolding with primitive Higgs field, which comes from the universal unfolding of the original function germ. Now sufficient background is given. We describe the contents of this paper.
The short section 2 recalls the classification of the 2-dimensional germs of F-manifolds with Euler fields (Theorem 2.2 from [He02] and Theorem 2.3 from [DH20-3]). It treats also regular F-manifolds (Definition 2.4 and Theorem 2.5 from [DH17]). Section 3 recalls many general facts on (T E)-structures: their definition, their presentation by matrices, formal (T E)-structures, unfoldings and universal unfoldings of (T E)-structures, Malgrange's result and the generalization in [HM04], (T E)-structures over F-manifolds, (T E)structures with primitive Higgs fields, regular singular (T E)-structures and elementary sections, Birkhoff normal form for (T E)-structures (not all have one, Theorem 3.20 cites existence results of Plemely and of Bolibroukh and Kostov). Not written before, but elementary is a correspondence 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 the construction of moduli spaces. Theorem 3.28 (which builds on results in [HS10]) constructs such moduli spaces, but only in the case of regular singular (T E)-structures. And it starts with a good family of regular singular (T E)-structures. There are two open problems. It is not clear how to generalize this notion of a good family beyond the case of regular singular (T E)-structures. And we hope, but did not prove for rank ≥ 3, that any regular singular (T E)-structure (over M with dim M ≥ 1) is a good family of regular singular (T E)-structures. For rank 2 this is true, it follows from Theorem 8.5.
Section 4 gives the classification of rank 2 (T E)-structures over a point t 0 . There are 4 types, which we call (Sem), (Bra), (Reg) and (Log) (for semisimple, branched, regular singular and logarithmic). In the type (Sem) U has two different eigenvalues, in the type (Log) U ∈ C·id, in the types (Bra) and (Reg) U has a 2 × 2 Jordan block. In the cases when U is trace free, a (T E)-structure of type (Log) has a logarithmic pole, a (T E)-structure of type (Reg) has a regular singular, but not logarithmic 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 connection with semisimple pole of order 3 (see Lemma 4.8). The semisimple case (Sem) is not central in this paper. Therefore we do not discuss it in detail and do not introduce Stokes structures. For the other types (Bra), (Reg) and (Log), section 4 discusses normal forms and their parameters. All (T E)-structures of type (Bra) have nice Birkhoff normal forms (Theorem 4.9), but not all of type (Reg) (Theorem 4.15 and the Remarks 4.17) and type (Log) (Theorem 4.18 and the Remarks 4.20). The types (Reg) and (Log) become transparent by the use of elementary sections.
A (T E)-structure of type (Sem) or (Bra) or (Reg) over a point t 0 satisfies the hypothesis of Malgrange's result, namely, the endomorphism U : K → K is regular. Therefore it has a universal unfolding, and any unfolding of it is induced by this universal unfolding. Here life is easy.
Section 5 discusses this. Also because of this fact, the semisimple case is not central in this paper.
The sections 6 -8 are devoted to the study of (T E)-structures over a germ (M, t 0 ) such that the restriction to t 0 is a (T E)-structure of type (Log). Then the set of points over which the (T E)-structure restricts to one of type (Log) is either a hypersurface or the whole of M. In the first case, it restricts to a fixed generic type (Sem) or (Bra) or (Reg) over points not in the hypersurface. In the second case, the generic type 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 such 1-parameter families are studied. In a surprisingly direct way, constraints on the difference of the leading exponents (defined in Theorem 4.18) of the logarithmic (T E)-structure over t 0 are found, and the monodromy in the generic cases (Sem) and (Reg) turns out to be semisimple (Theorem 6.2). By Plemely's result (and our direct calculations), these cases come equipped with Birkhoff normal forms. Theorem 6.3 in the second part classifies all (T E)-structures over (M, t 0 ) with trace free U, dim M = 1, logarithmic restriction to t 0 and Birkhoff normal form. Theorem 6.7 in the third part classifies all generically regular singular (T E)-structures over (M, t 0 ) with dim M = 1, logarithmic restriction to t 0 , and whose monodromy has a 2 × 2 Jordan block. The majority of these cases has no Birkhoff normal form. The Theorems 6.3 and 6.7 overlap in the cases which have Birkhoff normal forms.
Section 7 makes the moduli spaces of marked regular singular (T E)structures from Theorem 3.28 explicit in the rank 2 cases. It builds on the classification results for the types (Reg) and (Log) in section 4. The long Theorem 7.4 describes the moduli spaces and offers 5 figures in order to make this more transparent. The moduli spaces have countably many topological components, and each component consists of an infinite chain of projective spaces which are either the projective line P 1 or the Hirzebruch surface F 2 or F 2 (which is obtained by blowing down in F 2 the unique (−2)-curve). These moduli spaces simplify in the generic case (Reg) the main proof in section 8, the proof of Theorem 8.5. Section 8 gives complete classification results, from different points of view. It has three parts. Theorem 8.1 lists all rank 2 (T E)-structures over a 2-dimensional germ (M, t 0 ) such that the restriction to t 0 has a logarithmic pole, such that the Higgs field is generically primitive, and such that the induced structure of an F-manifold with Euler field extends to all of M. Theorem 8.1 (d) offers explicit normal forms. Corollary 8.3 starts with any logarithmic rank 2 (T E)-structure over a point t 0 and lists the (T E)-structures in Theorem 8.1 (d) which unfold it.
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). Theorem 8.5 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 the generic cases (Reg) and (Bra) these are precisely those in Theorem 8.1 (d) with primitive Higgs field, but in the generic cases (Sem) and (Bra) table (8.12) contains many (T E)-structures with only generically primitive Higgs field. All the (T E)-structures in table (8.12) are universal unfoldings of themselves, also those with only generically primitive Higgs field. Almost all logarithmic (T E)-structures over a point have several unfoldings which do not induce one another. Only the logarithmic (T E)-structures over a point whose monodromy has a 2 × 2 Jordan block and whose two leading exponents coincide have a universal unfolding. This follows from Theorem 8.5 and Corollary 8.3.
The second part of section 8 starts from the 2-dimensional Fmanifolds with Euler fields and discusses how many and which (T E)structures exist over each of them. It turns out that the nilpotent F-manifold N 2 with the Euler field E = t 1 ∂ 1 + t r 2 (1 + c 3 t r−1 2 )∂ 2 for r ≥ 2 (case (2.12) in Theorem 2.3) does not have any (T E)-structure over it if c 3 = 0, and it has no (T E)-structure with primitive Higgs field over it if c 3 = 0 or r ≥ 3. But most 2-dimensional F-manifolds with Euler fields have one or countably many families of (T E)-structures with 1 or 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 typical also for higher rank. But section 9 makes one phenomenon explicit which arises only in rank ≥ 3. Section 9 presents a family of rank 3 (T E)-structures with primitive Higgs fields over a fixed 3-dimensional globally irreducible F-manifold with nowhere regular Euler field, such that the family has a functional parameter. The example is essentially due to M. Saito, it is a Fourier-Laplace transformation of the main example in a preliminary version of [SaM17] (though he considers only the bundle and connection over a 2-dimensional submanifold of the F-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 2 (m) were studied. They are of generic type (Sem). In [DH20-3] (T E)-structures over the 2-dimensional F-manifold N 2 (with all possible Euler fields) were studied. They are of generic types (Bra), (Reg) or (Log). But in [DH20-2] and [DH20-3] the focus was on (T E)-structures with primitive Higgs fields. Those with generically primitive, but not primitive Higgs fields were not considered. And the approach to the classification was very different. It relied on the formal classification of rank 2 (T )-structures in [DH20-1]. The approach here is independent of these three papers.
I would like to thank Liana David for a lot of joint work on (T E)structures.

The two-dimensional F-manifolds and their Euler fields
F -manifolds were first defined in [HM99]. Their basic properties were developed in [He02]. An overview on them and on more recent results is given in [DH20-2]. In this paper we are mainly interested in the 2-dimensional Fmanifolds and their Euler fields. They were classified in [He02].
Theorem 2.2. [He02,Theorem 4.7] In dimension 2, (up to isomorphism) the germs of F-manifolds fall into three types: (a) The semisimple germ. It is called A 2 1 , and it can be given as follows.
(2.3) (b) Irreducible germs, which (i.e. some holomorphic representatives of them) are at generic points semisimple. They form a series I 2 (m), m ∈ Z ≥3 . The germ of type I 2 (m) can be given as follows.
(M, 0) = (C 2 , 0) with coordinates t = (t 1 , t 2 ) and ∂ k : Any Euler field takes the shape (2.5) (c) An irreducible germ, such that the multiplication is everywhere irreducible. It is called N 2 , and it can be given as follows.
The family of Euler fields in (2.7) on N 2 can be reduced by coordinate changes which respect the multiplication of N 2 to a family with two continuous parameters and one discrete parameter. This classification is proved in [DH20-3]. It is recalled in Theorem 2.3. The group Aut(N 2 ) of automorphisms of the germ N 2 of an F-manifold is the group of coordinate changes of (C 2 , 0) which respect the multiplication of N 2 . It is (2.8) with λ ′ (0) = 0 and λ(0) = 0.} Theorem 2.3. Any Euler field on the germ N 2 of an F-manifold can be brought by a coordinate change in Aut(N 2 ) to a unique one in the following family of Euler fields.

(T E)-structures in general
3. 1. Definitions. A (T E)-structure is a holomorphic vector bundle on C × M, M a complex manifold, with a meromorphic connection ∇ with a pole of Poincaré rank 1 along {0} × M and no pole elsewhere.
Here we consider them together with the weaker notion of (T )-structure and the more rigid notions of a (T L)-structure and a (T LE)-structure. The structures had been considered before in [HM04], and they are related to structures in [Sa02,VII] and in [Sa05].
which satisfies the Leibniz rule, and which is flat (with respect to X ∈ T M , not with respect to ∂ z ), Equivalent: For any z ∈ C * , the restriction of ∇ to H| {z}×M is a flat holomorphic connection.
∇ is a flat connection on H| C * ×M with a pole of Poincaré rank 1 along {0} × M, so it is a map which satisfies the Leibniz rule and is flat.
such that for any z ∈ P 1 − {0}, the restriction of ∇ to H| {z}×M is a flat connection. It is called pure if for any t ∈ M the restriction H| P 1 ×{t} is a trivial holomorphic bundle on P 1 .
(d) Definition of a (TLE)-structure (H → P 1 × M, ∇): It is simultaneously a (T E)-structure and a (T L)-structure, where the connection ∇ has a logarithmic pole along {∞} × M. The (T LE)-structure is called pure if the (T L)-structure is pure.
Remarks 3.2. Here we write the data in Definition 3.1 (a)-(b) and the compatibility conditions between them in terms of matrices. Consider a (T E)-structure (H → C × M, ∇) of rank rk H = r ∈ N. We will fix the notations for a trivialization of the bundle H| U ×M for some small neighborhood U ⊂ C of 0. Trivialization means the choice of a basis v = (v 1 , ..., v r ) of the bundle H| U ×M . Also, we choose local coordinates t = (t 1 , ..., t n ) with coordinate vector fields ∂ i = ∂/∂t i on M. We write , but this dependence on t ∈ M is usually not written explicity. The flatness 0 = dΩ + Ω ∧ Ω of the connection ∇ says for i, j ∈ {1, ..., n} with i = j (3.7) (3.8) These equations split into the parts for the different powers z k for k ≥ 0 as follows (with A (3.10) In the case of a (T )-structure, B and all equations except (3.4) which contain B are dropped.
Consider a second (T E)-structure ( H → C × M, ∇) of rank r over M, where all data except M are written with a tilde. Let v and v be trivializations. A holomorphic isomorphism from the first to the second (3.11) (3.11) says more explicitly (3.13) These equations split into the parts for the different powers z k for k ≥ 0 as follows (with T (−1) := 0): The isomorphism here fixes the base manifold M. Such isomorphisms are called gauge isomorphisms. A general isomorphism is a composition of a gauge isomorphism and a coordinate change on M (a coordinate change induces an isomorphism of (T E)-structures, see Lemma 3.6).
Remark 3.3. In this paper we care mainly about (T E)-structures over the 2-dimensional germs of F-manifolds with Euler fields. For each of them except (N 2 , E = (t 1 + c 1 )∂ 1 ), the group of coordinate changes of (M, 0) = (C 2 , 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 Euler fields. ]) t 0 for t 0 ∈ M consists of formal power series k≥0 f k z k whose coefficients f k ∈ O M,t 0 have a common convergence domain. In the case of (M, The following lemma is obvious. Lemma 3.6. Let (H → C × M, ∇) be a (T E)-structure over M, and let ϕ : M ′ → M be a holomorphic map between manifolds. One can pull back H and ∇ with id ×ϕ : C × M ′ → C × M. We call the pull back ϕ * (H, ∇). It is a (T E)-structure over M ′ . We say that the pull back ϕ * (H, ∇) is induced by the (T E)-structure (H, ∇) via the map ϕ.
(ii) Here the behaviour of the (T E)-structure (H, ∇) over (M, t 0 ) = (C 2 , 0) with coordinates t = (t 1 , t 2 ) along t 1 is quite trivial. It is convenient to split it off. The next subsection does this in greater generality. (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, we mean an invariant of its formal isomorphism class.
(a) Its pole part U, that means the pair (K, U) up to isomorphism, is a formal invariant of the (T E)-structure. Especially, the holomorphic functions δ (0) := det U ∈ O M and ρ (0) := 1 r tr U ∈ O M are formal invariants. (3.17) Then the functions δ (1) and ρ (1) are independent of the choice of the basis v. The locally for any t 0 defined functions δ (1) and ρ (1) glue to global holomorphic functions δ (1) ∈ O M and ρ (1) ∈ O M . They are formal invariants. Furthermore, the function ρ (1) is constant on any component of M.
The following lemma is obvious.
(a) Consider a holomorphic function g : M → C. The trivial line bundle And, of course, with (local) coordinates t 1 on C and t ′ on M [3] , and the projection Part (c) allows to go from an arbitrary (T E)-structure to one with trace 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 an associated one with trace free pole part. If the associated one is induced by a third (T E)-structure, then the original one is induced by a closely related (T E)-structure with one parameter more. Lemma 3.11 continues Lemma 3.10.
Consider the (T E)-structure (H [2] , ∇ [2] ) from Lemma 3.10 with trace free pole part which is defined by (3.18) The basis v can be chosen such that the matrices satisfy and (3.10) for k = 1, tr(B (1) − ρ (1) 1 r ) = 0 by Lemma 3.9 and especially Start with an arbitrary basis v, consider the function consider T := e g · 1 r , and v := v · T . (3.13) gives . Therefore now suppose tr(B − zρ (1) 1 r ) = 0. (3.10) for k ≥ 3 gives tr 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 · 1 r for a suitable function h ∈ C{t}. Then B = B, A i . Such a function exists because (3.9) for k = 2 implies ∂ i tr A The Theorems 3.14 and 3.16 show in two ways that primitivity of a 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 Higgs field induces a multiplication • on T M which makes M an F -manifold. A (T E)-structure (H → C × M, ∇) with primitive Higgs field induces in addition a vector field E on M, which, together with •, makes M an F -manifold with Euler field. The multiplication •, unit field e and Euler field E (the latter in the case of a (T E)-structure), are defined by 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 universal unfolding 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 of a 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 universal unfoldings of themselves, but where the Higgs bundle is only generically primitive. Still in the examples which we consider, the base manifold is an F-manifold with Euler field globally.
Malgrange [Ma86] proved that a (T E)-structure over a point t 0 has a universal unfolding with primitive Higgs field if the endomorphism U : K t 0 → K t 0 is regular, i.e. it has for each eigenvalue only one Jordan block.  (a) An unfolding of it is a (T E)-structure (H [1] → C × (M × C l 1 , (t 0 , 0)), ∇ [1] ) over a germ (M × C l 1 , (t 0 , 0)) (for some l 1 ∈ Z ≥0 ) together with a fixed isomorphism which is the identity on M ×{0}, and an isomorphism j from the second unfolding to the pullback of the first unfolding by ϕ such that By definition of a universal unfolding in part (c), a (T E)-structure has (up to canonical isomorphism) at most one universal unfolding, because any two universal unfoldings induce one another by unique maps. . Suppose that a vector ζ t 0 ∈ K t 0 with the following properties exists.
(IC) (Injectivity condition) The map C • ζ t 0 : (GC) (Generation condition) ζ t 0 and its images under iteration of the maps U| t 0 : K t 0 → K t 0 and C X : Then a universal unfolding of the (T E)-structure over a germ (M × C l , (t 0 , 0)) (l ∈ N 0 suitable) exists. It is unique up to isomorphism. Its Higgs field is primitive.
(c) [Ma86] A (T E)-structure over a point t 0 has a universal unfolding with primitive Higgs field if the endomorphism [z 2 ∇ ∂z ] = U : K t 0 → K t 0 is regular, i.e. it has for each eigenvalue only one Jordan block. In that case, the germ of the F-manifold with Euler field which underlies the universal unfolding, is by definition (Definition 2.4) regular.
(ii) Consider the germ (M, 0) = (C 2 , 0) of a 2-dimensional F-manifold with 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 if and only if c 1 = c 2 . In the cases I 2 (m) (m ≥ 3) it does not hold. In the case of N 2 with E = t 1 ∂ 1 + g(t 2 )∂ 2 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 primitive Higgs field over a germ (M, t 0 ) of a regular F-manifold with Euler field is determined up to gauge isomorphism by the restriction of the (T E)structure to t 0 .
(iv) Lemma 3.6, Definition 3.8, Lemma 3.9, Lemma 3.10, Lemma 3.11, Lemma 3.12, Definition 3.13, Theorem 3.14 and Definition 3.15 hold 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 for formal (T E)-structures.
(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 whose restriction to {∞} × M is flat with respect to the residual connection. (iv) The problem whether a (T E)-structure over a point has an extension to a pure (T LE)-structure is a special case of the Birkhoff problem, which itself is a special case of the Riemann-Hilbert-Birkhoff problem. The book [AB94] and chapter IV in [Sa02] are devoted to these 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)-structure over a point t 0 with a logarithmic pole at z = 0, in which case it is trivial.  (b) For α ∈ C, define the finite dimensional C-vector space C α of the following global sections of H ′ , (where t = (t 1 , ..., t n ) are local coordinate and ∂ i are the coordinate vector fields). Observe z k · C α = C α+k for k ∈ Z. For each α the map (3.31) In order to see this, choose numbers α j ∈ C and elementary sections s j ∈ C α j for j ∈ {1, ..., r} such that s 1 , ..., s r form a global basis of H ′ . Then (3.33) Here (3.33) is the expansion of a j as a Laurent series in z with holomorphic coefficients a kj ∈ O U 2 in t. Then (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 (3.35) The Kashiwara-Malgrange V -filtration is given by the locally free subsheaves for r ∈ R,  (c) The endomorphism e −2πiRes 0 : K → K has the same eigenvalues as the monodromy M mon , but it might have a simpler Jordan block structure. If no eigenvalues of Res 0 differ by a nonzero integer (nonresonance condition) then e −2πiRes 0 has the same Jordan block structure as 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 0 } ) of its restriction to t 0 for any t 0 ∈ M.
(ii) In the case of a regular singular (T E)-structure over a simply connected manifold M, one can choose elementary sections s j ∈ C α j , j ∈ {1, ..., rk H}, for some α j ∈ C, such that they form a basis of H * and such the extension to {0} × M which they define, is a logarithmic (T E)-structure. Then the base change from any local basis of H to the basis (s 1 , ..., s rk H ) of this new (T E)-structure is meromorphic, so the two (T E)-structures give the same meromorphic bundle. This observation fits to the usual definition of meromorphic bundle with regular singular pole.
(iii) The property of a section to have moderate growth, is invariant under pull back. Therefore also the property of a (T E)-structure to be regular singular is invariant under pull back.
3. 6. Marked (T E)-structures and moduli spaces for them. It is easy to give a (T E)-structure (H → C × M, ∇) with nontrivial Higgs field and which is thus not the pull back of the (T E)-structure over a point, such that nevertheless the (T E)-structures over all points t 0 ∈ M are isomorphic as abstract (T E)-structures. Examples are given in Remark 7.1 (ii). The existence of such (T E)-structures obstructs the construction of nice Hausdorff moduli spaces for (T E)-structures up to isomorphism. The notion of a marked (T E)-structure hopefully remedies this. But in the moment, we have only results in the regular singular 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. Theorem 3.29 states that this functor is represented by a complex space. It builds on results in [HS10, ch. 7] Several remarks discuss what is missing in the other cases and what more we have in the regular singular rank 2 case, thanks to the Theorems 6.3, 6.7 and 8.5.
(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,  The family (H, ψ) is called good if some r ∈ R and some N ∈ N exist which satisfy (3.40) Remarks 3.27. (i) The notion of a family of marked (T E)-structures is too weak. For example, it contains the following pathological family of logarithmic (T E)-structures of rank 1 over X := C (with coordinate t) and with trivial monodromy. Write s 0 ∈ C 0 for a generating flat section. Define H by The marked (T E)-structures over all points t ∈ C * ⊂ X = C are isomorphic 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).
(ii) Theorem 3.29 gives evidence that the notion of a good family of marked regular singular (T E)-structures is useful. But it is not clear a priori whether any regular singular (T E)-structure (H → C × M, ∇) over a simply connected manifold M is a good family of marked regular singular (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 regular singular rank 2 (T E)-structures over M = C.  [HS10]. Here it is relevant that r and N with (3.39) and (3.40) imply the existence of an r 2 ∈ R with r 2 < r and (3.44) In [HS10], (T ERP )-structures are considered. (3.39) and (3.44) are demanded there. (3.40) is not demanded there explicitly, but it follows from the properties of the pairing there, and this is used in Lemma 7.2 in [HS10]. The additional conditions of (T ERP )-structures are not essential for the arguments in the proof of Lemma 7.2 and Theorem 7.3 in [HS10]. Therefore these proofs apply also here and give the statements for

Rank 2 (T E)-structures over a point
Here we will classify the rank 2 (T E)-structures over a point.
First we will treat the semisimple case (Sem). Then the cases (Bra), (Reg) and (Log) will be considered together. Lemma 4.8 will justify the names (Bra) and (Reg). Finally, the three cases (Bra), (Reg) and (Log) will be treated one after the other. The following lemma gives some first information. Its proof is straightforward.
) there), and denote its invariants from Lemma 3.9 by U, δ (0) , ρ (0) , δ (1) , ρ (1) . Then ( H → C, ∇) is of the same type (Sem) or (Bra) or (Reg) or (Nil) as (H → C, ∇). The following table characterizes of which type the The case (Sem). A (T E)-structure over a point with a semisimple endomorphism U with pairwise different eigenvalues is formally isomorphic to a socalled elementary model, and its holomorphic isomorphism class is determined by its Stokes structure. These two facts are well known. A good reference is [Sa02, II 5. and 6.]. The older reference [Ma83a] considers only the underlying meromorphic bundle, In order to formulate the result for rank 2 (T E)-structures more precisely, we need some notation.
Part (a) follows for example from [Sa02, II Theorem 5.7] together with [Sa02, II Remark 5.8] (Theorem 5.7 considers only the underlying meromorphic bundle; Remark 5.8 takes care of the holomorphic bundle). For the parts (b) and (c), one needs to deal in detail with the Stokes structure. We will not do it here, as the semisimple case is not central 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), applies to these (T E)-structures. Such a (T E)-structure has a unique universal unfolding. The parameters (α 1 , α 2 , s 1 , s 2 ) are constant, the parameters (u 1 , u 2 ) are local coordinates on the base space. The base space is an F-manifold of type A 2 1 with Euler field E = u 1 e 1 + u 2 e 2 . See Remark 5.3 (iii).
(ii) We do not offer normal forms for the (T E)-structures in Theorem 4.4 for three reasons: (1) As said in (i), the (T E)-structures above unfold uniquely to (T E)-structures over germs of F-manifolds. In that sense they are easy to deal with. (2) It looks difficult to write down normal forms.
(3) Normal forms should be considered together with the Stokes parameters, and the corresponding Riemann-Hilbert map from the space of monodromy data (α 1 , α 2 , s 1 , s 2 ) to a space of parameters for normal forms should be studied. This is a nontrivial project, which does 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, and the relations between them, (4.10) 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 (4.14) Then the formal invariants δ (0) , ρ (0) , δ (1) and ρ (1) of Lemma 3.9 are given by We are in the case (Bra) if b Consider T ∈ GL 2 (C{z}) and the new basis v = v · T and its matrix Then B is determined by (3.13), which is We will use this quite often in order to construct or compare normal forms. The following immediate corollary of the proof of Lemma 3.11 provides a reduction of b 1 .
Corollary 4.7. The base change matrix T = e g · C 1 with g as in (3.20) , (4.20) From now on we will work in this section only with bases v with b 1 = 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) = 1 2 tr U = 0). See the Lemmata 3.10 and 3.11 for the relation to the general 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 but not 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) 0 ⊗ C{z} C{z}[z −1 ] by the map C → C, x → x 4 = z, is the space of germs at 0 of sections of a meromorphic bundle on C with a meromorphic connection with an order 3 pole at 0 with semisimple pole part with eigenvalues κ 1 and κ 2 = −κ 1 with − 1 4 κ 2 1 = δ (1) ∈ C * . Thus κ 2 1 is a formal invariant of the (T E)-structure of type (Bra).
Proof: Consider a C{z}-basis v of O(H) 0 such that its matrix B is as in (4.13) and such that b 1 = zρ (1) . This is possible by Corollary 4.7 and the assumption ρ (0) = 0. As U is nilpotent, but not 0, b (1) 4 = 0, and consider the pullback of the (T E)structure by the map C → C, x → x 4 = z. Then dz z = 4 dx x and z∂ z = 1 4 x∂ x and One sees a pole of order 3 with matrix 4(b (1) 4 E) of the pole part. It is tracefree and has the eigenvalues κ 1 and κ 2 = −κ 1 with κ 2 1 = 4b (1) 4 ∈ C * . This shows the claims in the case b (1) (1) 4 = 0, and consider the pullback of the (T E)structure by the map C → C, x → x 2 = z. Then dz z = 2 dx x and z∂ z = 1 2 x∂ x and One sees a logarithmic pole. Therefore also the sections v 1 and v 2 have moderate growth, and the (T E)-structure is regular singular.
4. 4. The case (Bra). The following theorem gives complete control on the (T E)-structures over a point of the type (Bra). Here Eig(M mon ) ⊂ C is the set of eigenvalues of the monodromy of such a (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 3.9 and the set Eig(M mon ) are invariants of its isomorphism class. Together they form a complete set of invariants. That means, the isomorphism class of the (T E)-structure is determined by these invariants.
(b) Any such (T E)-structure has a C{z}-basis v of O(H) 0 such that its matrix is in Birkhoff normal form, and more precisely, the matrix B has the shape Remarks 4.10. (i) Because of part (a), two Birkhoff normal forms as in (4.25) with data ( (1) 4 = 1. But in view of the (T E)-structures in the 4th case in Theorem 6.3 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) on the existence of a Birkhoff normal form is satisfied.
Suppose on the contrary that the Then that subspace is generated by a section σ = 0 with z∇ ∂z σ = h(z)σ for some function h ∈ C{z}[z −1 ].
Let v be a C{z}-basis of O(H) 0 with matrix B as in (4.13) with (1) 4 = 0 holds because of (4.17), and because of it σ = v 1 or σ = v 2 is impossible. Therefore we can choose σ = The first line minus g times the second line gives the equation 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) is satisfied. 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 a Birkhoff normal form can be chosen with b 1 = ρ (0) + zρ (1) because of Corollary 4.7. Then it has the shape (4.29) (4.19) gives (4.30) can be chosen such that b (1) 2 = 0. Then the Birkhoff normal form B has the shape in (4.25).
Suppose now that B has this shape, (4.31) Consider the new basis v = v · T and its matrix B where We are searching for coefficients τ Under these constraints, (4.19) gives The determinant of the 3 × 3 matrix is −2b (1) 4 = 0. Therefore the system (4.36) has a unique solution (τ Iterating this construction, one finds that one can change the matrix B in (4.25) by a holomorphic base change to a matrix B with for any k ∈ Z.
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 set Eig(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)-structure with a logarithmic pole at ∞. From its pole part at ∞ and Theorem 3.23 (c) one reads off Together with Step 2, this shows only if in Remark 4.10 (i) and all statements in Theorem 4.9.
Corollary 4.11. The monodromy of a (T E)-structure over a point of the type (Bra) has a 2 × 2 Jordan block if its eigenvalues coincide (equivalently, if b (1) 3 ∈ 1 2 Z for some (or any) Birkhoff normal form in Theorem 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 any Birkhoff normal form in Theorem 4.9 (b) b (1) 3 ∈ 1 2 Z, and one can choose a Birkhoff normal form with b The induced pure (T LE)structure has at ∞ a logarithmic pole, and its residue endomorphism 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 × 2 Jordan block.
In the case b (1) Again, the pole at ∞ is logarithmic. Now the nonresonance condition in Theorem 3.23 (c) is satisfied. Because of b (0) 2 = 0, the monodromy has a 2 × 2 Jordan block.
For (T E)-structures of the type (Bra), formal isomorphism is coarser than holomorphic isomorphism.
(a) The set Eig(M mon ) and the equivalent set {±b Proof: Part (a) follows from part (b). For the proof of part (b), we have to find T ∈ GL 2 (C[[z]]) such that T , B in (4.25) and for k ≥ 1, This is equivalent to One can choose τ 3 ∈ (± 1 4 + 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 1 , f 2 ) with monodromy given by (4.42) The eigenvalues are ±ie −2πiα (1) . Choose numbers t 1 ∈ C and t 2 ∈ C * . The following basis of H ′ is univalued.
(4.43) t 2 ). Consider the rank 2 bundle H ′ → C * × M with flat connection and flat multivalued basis f = (f 1 , f 2 ) with monodromy given by (4.45) The basis v in (4.43) is univalued. The matrices A 1 , A 2 and B in its connection 1-form Ω as in (3.4)-(3.6) are given by (4.44) and 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 × {0} is a (T E)-structure of type (Log).
Theorem 4.15. Consider a regular singular, but not logarithmic, rank 2 (T E)-structure (H → C, ∇) over a point. Associate to it the data in the notations 4.14.
In the case N mon = 0 and λ 1 = λ 2 (then β 1 = β 2 ), we suppose min(deg z b 11 , deg z b 12 ) ≤ min(deg z b 21 , deg z b 22 ). If it does not hold a priori, we can exchange s 1 and s 2 .
In the case α 1 − α 2 ∈ N, t 2 is obviously independent of the choice of s 1 .
Corollary 4.16 is an immediate consequence of Theorem 4.15.
Corollary 4.16. The set of regular singular, but not logarithmic, rank 2 (T E)-structures over a point is in bijection with the set The first set parametrizes the cases with N mon = 0, the second and third set parametrize the cases with N mon = 0. Theorem 4.15 describes the corresponding (T E)-structures.
Remarks 4.17. The connection matrices for the special bases in Theorem 4.15 can be written down easily.
(a) The case N mon = 0: There exist unique numbers α 1 , α 2 with e −2πiα j = λ j and the following property: There exist elementary sec- (4.61) The isomorphism class of the (T E)-structure is uniquely determined by the information N mon = 0 and the set {α 1 , α 2 }. The numbers α 1 and α 2 are called leading exponents.
Proof: First, (a) and (b) are considered together. By Theorem 3.23 (a), O(H) 0 is generated by two elementary sections s 1 ∈ C α 1 and s 2 ∈ C α 2 for some numbers α 1 and α 2 . The numbers α 1 and α 2 are the eigenvalues of the residue endomorphism. So, they are unique. This finishes already the proof of part (a).
The first set parametrizes the cases with N mon = 0, the second set parametrizes the cases with N mon = 0.  The basis in (4.61): (4.64) The basis in (4.63): (4.65) The basis (s 1 , s 2 ) gives a Birkhoff normal form in the cases N mon = 0 and in the cases (N mon = 0 & α 1 = α 2 ). In the cases (N mon = 0 & α 1 − α 2 ∈ N), a Birkhoff normal form does not exist.

Rank 2 (T E)-structures over germs of regular F-manifolds
This section discusses unfoldings of (T E)-structures over a point t 0 of type (Sem) or (Bra) or (Reg). Here Malgrange's unfolding result Theorem 3.16 (c) applies. It provides a universal unfolding for the (T E)-structure over t 0 . Any unfolding is induced by the universal unfolding. 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 a point of type (Log). Section 8 treats arbitrary such unfoldings. Section 6 prepares this. It treats 1-parameter unfoldings with trace free pole parts of logarithmic (T E)-structures over a point.
If one starts with a (T E)-structure with primitive Higgs field over a germ (M, t 0 ) of a regular F-manifold, then the endomorphism U| t 0 : Vice versa, if one starts with a (T E)-structure over a point t 0 with a regular endomorphism U : K t 0 → K t 0 , then it unfolds uniquely to a (T E)-structure with primitive Higgs field over a germ of a regular F-manifold by Malgrange's result Theorem 3.16 (c). The germ of the regular F-manifold is uniquely determined by the isomorphism class of U : K t 0 → K t 0 (i.e. its Jordan block structure). And the (T E)structure is uniquely determined by its restriction to t 0 .
The following statement on the rank 2 cases is an immediate consequence of Malgrange's unfolding result Theorem 3.16 (c), the classification 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)) and the 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 0 except those of type (Log), the endomorphism U : K t 0 → K t 0 is regular. The (T E)-structure has a unique universal unfolding. This unfolding has a primitive Higgs field. Its base space is a germ (M, t 0 ) = (C 2 , 0) of an F-manifold with Euler field and is as follows: In the case of (Bra) or (Reg), a coordinate change brings E to the form t 1 ∂ 1 + ∂ 2 .
(b) Any unfolding of a rank 2 (T E)-structure over t 0 with regular endomorphism U : K t 0 → K t 0 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, in rank 2, it is easy to give it explicitly. The following lemma offers one way.
If U| t 0 is regular and rank H = 2, then the Higgs field of the (T E)-structure H (unf ) is everywhere primitive. Therefore then M is an F-manifold with Euler field. The Euler field is E = t 1 ∂ 1 + t 2 ∂ 2 .
(e) If U| t 0 is regular and rank H = 2, the (T E)-structure over the germ (M, t 0 ) is the universal unfolding of the one over t 0 .

(e) follows from (d) and Malgrange's result Theorem 3.16 (c).
Remarks 5.3. (i) In the cases (Reg) we will see the universal unfoldings again in section 7, in the Remarks 7.2. In a first step in the Remarks 7.1, the value t 2 in the normal form in the Remarks 4.17 is turned into a parameter in P 1 . The Remarks 7.2 add another parameter t 1 in C. Then the Higgs field becomes primitive and the base space C × P 1 becomes a 2-dimensional F-manifold with Euler field. For each t 0 ∈ C × C * , the (T E)-structure over t 0 is of type (Reg), and the (T E)structure over the germ (M, t 0 ) is a universal unfolding of the one over t 0 .
(ii) In the cases (Bra), the following formulas give a universal unfolding over (C 2 , 0) of any (T E)-structure of type (Bra) over the point 0 (see Theorem 4.9 for their classification), such that the Euler field is 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 0 ) = (C, 0) of a logarithmic rank 2 (T E)-structure over t 0 will be considered. Invariants of it will be defined. Theorem 6.2 gives constraints on these invariants and shows that the monodromy is semisimple if the generic type is (Sem) or (Bra).
By Theorem 3.20 (a) (which is trivial in our case because of the logarithmic pole at z = 0 of the (T E)-structure over t 0 ) and Remark 3.19 (iii), the (T E)-structure has a Birkhoff normal form, i.e. an extension to a pure (T LE)-structure, if its monodromy is semisimple. Subsection 6. 2: All pure (T LE)-structures over (M, t 0 ) = (C, 0) with trace free pole part and with logarithmic restriction to t 0 are classified in Theorem 6.3. These comprise all with semisimple monodromy and thus all with generic types (Sem) or (Bra). Subsection 6. 3: All (T E)-structures over (M, t 0 ) = (C, 0) with trace free pole part and with logarithmic restriction over t 0 whose monodromies have a 2×2 Jordan block are classified in Theorem 6.7. Their generic types are (Reg) or (Log) because of Theorem 6.2. Most of them have no Birkhoff normal forms. The intersection with Theorem 6.3 is small and consists of those which have Birkhoff normal forms.
The Theorems 6.3 and 6.7 together give all unfoldings with trace free pole parts over (M, t 0 ) = (C, 0) of logarithmic rank 2 (T E)-structures over t 0 .

1. Numerical invariants for such (T E)-structures. The next definition 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 0 ), ∇) be a (T E)-structure with trace free pole part over (M, t 0 ) = (C, 0) (with coordinate t) whose restriction over t 0 = 0 is logarithmic. Let M ⊂ C be a neighborhood of 0 on which the (T E)-structure is defined. On M − {0} it has a fixed type, (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 in terms of (non)vanishing of δ (0) , δ (1) ∈ tC{t} and U: For the generic types (Sem), (Bra) and (Reg), define k 1 ∈ N by For the generic types (Sem) and (Bra) define k 2 ∈ Z by k 2 := deg t δ (0) − k 1 for the generic type (Sem), deg t δ (1) − k 1 for the generic type (Bra). (6. 2) The following theorem gives for the generic type (Bra) and part of the generic type (Sem) restrictions on the eigenvalues of the residue endomorphism of the logarithmic pole at z = 0 of the (T E)-structure over t 0 = 0. And it shows that the monodromy is semisimple if the generic type is (Sem) or (Bra).
(a) Suppose that the generic type is (Sem). (i) Then k 2 ≥ k 1 . (ii) If k 2 > k 1 then the eigenvalues of the residue endomorphism of the logarithmic pole at z = 0 of the (T E)-structure over t 0 are ρ (1) ± k 1 −k 2 2(k 1 +k 2 ) . Their difference is smaller than 1. Especially, the eigenvalues of the monodromy are different, and the monodromy is semisimple.
(b) Suppose that the generic type is (Bra). (i) Then k 2 ∈ N.
(ii) The eigenvalues of the residue endomorphism of the logarithmic pole at z = 0 of the (T E)-structure over t 0 are ρ (1) ± −k 2 2(k 1 +k 2 ) . Their difference is smaller than 1. Especially, the eigenvalues of the monodromy are different, and the monodromy is semisimple.
(ii) Suppose k 2 > k 1 . By a linear change of the basis v, we can arrange that k We can make a coordinate change in t such that afterwards for an arbitrarily chosen γ ∈ C * . Then a diagonal base change leads to a basis which is again called v with matrices which are again called A and B with b (0) (6.10) Now the vanishing of the coefficients in C{t} of C 2 · z 0 , C 2 · z 1 , D · z 0 , D · z 1 and E · z 1 in (6.5)-(6.7) tells the following.
With respect to the basis v| (0,0) of K (0,0) , the matrix of the residue endomorphism of the logarithmic pole at z = 0 of the (T E)-structure over t 0 = 0 is It is semisimple with the eigenvalues ρ (1) ± b (1) 3,0 , whose difference is smaller than 1. The monodromy is semisimple with the two different eigenvalues exp(−2πi(ρ (1) ± b (1) 3,0 )). (iii) Suppose k 2 = k 1 . As in the proof of (ii), we can make a coordinate change in t and then obtain a C{t, z}-basis for an arbitrarily chosen γ ∈ C * . Now the constant base change matrix The vanishing of the coefficients in C{t} of C 2 · z 0 , E · z 0 , D · z 1 , C 2 · z 1 and E · z 1 in (6.5)-(6.7) tells the following.
With respect to the basis v| (0,0) of K (0,0) , the matrix of the residue endomorphism of the logarithmic pole at z = 0 of the (T E)-structure over t 0 = 0 is It is diagonal with the eigenvalues ρ (1) ±b 3,0 . Therefore the monodromy has the eigenvalues exp(−2πi(ρ (1) ± b (1) , the eigenvalues of the residue endomorphism do not differ by a nonzero integer. Because of Theorem 3.23 (c), then the monodromy is semisimple.
We will show that the monodromy is also in the cases b (1) 3,0 ∈ 1 2 Z−{0} semisimple, by reducing these cases to the case b (1) 3,0 ∈ 1 2 Z <0 can be reduced to this case by exchanging v 1 and v 2 . We will construct a new (T E)-structure over (M, t 0 ) = (C, 0) with the same monodromy and again with trace free pole part and of generic type (Sem) with logarithmic restriction over t 0 , but where B (1) (0) is replaced by Applying this sufficiently often, we arrive at the case b 3,0 = 0, which has semisimple monodromy.
The basis v := v · 1 0 0 z of H ′ := H| C * ×(M,t 0 ) in a neighborhood of (0, 0) defines a new (T E)-structure over (M, 0) because of z∇ ∂t v = v z −1 a 2 C 2 + a 3 D + za 4 E and a (0) 2 = 0, (6.14) Of course, it has the same monodromy. The restriction over t 0 = 0 has a logarithmic pole at z = 0 because b (1) Its generic type is still (Sem). Its numbers k 1 and k 2 satisfy The assumption k 1 < k 2 would lead together with part (ii) to two different eigenvalues of the monodromy, a contradiction. Therefore k 1 = k 2 = k 1 . Thus we are in the same situation as before, with b (1) 4 = γ 2 t k 1 +k 2 for an arbitrarily chosen γ ∈ C * . Then a diagonal base change leads to a basis which is again called v with matrices which are again called A and B with b (0) (6.17) The vanishing of the coefficients in C{t} of C 2 · z 0 , D · z 0 , C 2 · z 1 , D · z 1 and E · z 2 in (6.5)-(6.7) tells the following.
3,0 . This shows With respect to the basis v| (0,0) of K (0,0) , the matrix of the residue endomorphism of the logarithmic pole at z = 0 of the (T E)-structure over t 0 = 0 is It is semisimple with the eigenvalues ρ (1) ± b (1) 3,0 , whose difference is smaller than 1. The monodromy is semisimple with the two different eigenvalues exp(−2πi(ρ (1) ± b were subject of Definition 6.1 and Theorem 6.2. They gave their generic type and invariants (k 1 , k 2 ) ∈ N 2 (for the generic types (Sem) and (Bra)) and k 1 ∈ N (for the generic type (Reg)). Theorem 6.3 will give an invariant k 1 ∈ 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. Theorem 6.3. Any pure rank 2 (T LE)-structure over (M, t 0 ) = (C, 0) with trace free pole part and with logarithmic restriction over t 0 has after a suitable coordinate change in t a unique Birkhoff normal form in the following list. Here the Birkhoff normal form consists of two matrices A and B which are associated to a global basis v of H whose restriction to {∞} × (M, t 0 ) is flat with respect to the residual connection along {∞} × (M, t 0 ), via z∇ ∂t v = vA and z 2 ∇ ∂z v = vB. The matrices have the shape (1) 4 E does not turn up, resp. b (1) 4 = 0). The left column of the following list gives the generic type of the underlying (T E)-structure and, depending on the type, the invariant k 1 ∈ N or the invariants k 1 , k 2 ∈ N from Definition 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 3 ∈ R ≥0 ∪ H, α 4 ∈ C − {−1}, k 1 ∈ N and k 2 ∈ N are invariants in some cases. In the first 6 cases, a Before the proof, several remarks on these Birkhoff normal forms are made. The proof is given after the Remarks 6.6.
Remarks 6.4. (i) The matrix B(0) = zB (1) (0) is the matrix of the logarithmic pole at z = 0 of the restriction over t 0 = 0 of the (T E)structure. In all cases except the 6th case and the 9th case, it is , so it is diagonal. In these cases the monodromy is semisimple with eigenvalues exp(−2πi(ρ (1) ± b (1) 3 )). In the 6th case and the 9th case, this matrix is z(ρ (1) C 1 + C 2 ). Then the matrix and the monodromy have a 2×2 Jordan block, and the monodromy has the eigenvalue exp (−2πiρ (1) ). In all cases, the leading exponents (defined in Theorem 4.18) of the logarithmic (T E)-structure over t 0 are called α 0 1 & α 0 2 , and they are 3 . (6.21) The 6th and 9th cases turn up again in Theorem 6.7. See the Remarks 6.8 (iv)-(vi).
(6.31) (ii) These formulas (6.31) show for the 1st to 7th cases in the list in Theorem 6.3 the following: The pull back via ϕ : (C, 0) → (C, 0) with ϕ(s) = s n for some n ∈ N of such a (T E)-structure with invariants (k 1 , k 2 ) or k 1 is a (T E)-structure in the same case where the invariants (k 1 , k 2 ) or k 1 are replaced by ( k 1 , k 2 ) = (nk 1 , nk 2 ) or k 1 = nk 1 , and where all other invariants coincide with the old invariants.
(iii) The following table says which of the (T E)-structures in the 1st to 7th cases in the list in Theorem 6.3 are not induced by other such (T E)-structures.
(iv) In the 8th and 9th cases, the (T E)-structure is induced by its restriction over t 0 via the map ϕ : (M, t 0 ) → {t 0 }, so it is constant along M.
Remarks 6.6. In the 2nd and 4th cases in the list in Theorem 6.3, another C{t, z}-basis v of O(H) 0 with nice matrices A and B is v = v · T with (6.33) In the 2nd case Proof of Theorem 6.3: Consider any pure (T LE)-structure over (M, t 0 ) = (C, 0) with trace free pole part and with logarithmic restriction to t 0 . Choose a global basis v of H whose restriction to {∞} × (M, t 0 ) is flat with respect to the residual connection along {∞} × (M, t 0 ). Its matrices A and B with z∇ ∂t v = vA and z 2 ∇ ∂z v = vB have because of (3.18) (in Lemma 3.11) the shape (6.19) and (1) j ∈ C. They satisfy the relations (3.28) (and, equivalently, (6.5)-(6.7)), so, more explicitly, First we consider the cases when all a j ∈ tC{t} and because of the differential equations (6.38). Then B = zB (1) , and it is clear that this matrix can be brought to the form B = zρ (1) C 1 + zα 3 D or B = zρ (1) C 1 + zC 2 by a constant base change. α 3 ∈ C can be replaced by −α 3 , so α 3 ∈ R ≥0 ∪ H is unique. This gives the last two cases in the list. There the generic type is (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 4 = 0, and the generic type is (Log). If b (0) j = 0, then the generic type is (Sem) or (Bra) or (Reg). Consider for a moment the cases when the residue endomorphism of the logarithmic pole at z = 0 of the (T E)-structure over t 0 is semisimple. By Theorem 6.1, these cases include the generic types (Sem) and (Bra). Then a linear base change gives b (1) 4 = 0, so that the 3 × 3-matrix in (6.38) is diagonal. Then denote β j : 4,β 4 · t β 4 ) = 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) (1) 4 = 0. In the cases k 2 > k 1 , by Theorem 6.2, b (1) 3 is up to the sign unique, and we can choose it to be b (1) (possibly by exchanging v 1 and v 2 ). In the cases k 2 = k 1 we write We can change its sign and get a unique α 3 ∈ R ≥0 ∪ H. We make a suitable coordinate change in t and obtain b (0) 4 = 0, β 2 + β 4 = k 1 + k 2 ) or (b (0) 3 = 0, 2β 3 = k 1 + k 2 ) (or both). In both cases (6.40) gives (6.43) In the cases k 2 > k 1 , we have β 2 < β 3 < β 4 . Then (6.41) and the relation (6.8) imply b In the case k 2 − k 1 > 0 even, a linear coordinate change in t and a diagonal base change allow to reduce the triple (b In the case k 2 − k 1 > 0 odd, we have β 3 / ∈ N, so b In the cases k 2 = k 1 and α 3 = 0, as in the proof of Theorem 6.2 (a)(iii), a base change with constant coefficients leads to b (possibly by exchanging v 1 and v 2 ). We make a suitable coordinate change in t and obtain b as in (6.41). The nonvanishing δ (1) = 0 and deg t δ (1) = k 1 + k 2 say 4,β 4 ) ∈ (C * ) 3 to the triple (1, 1, −1). We obtain the normal form in the list in Theorem 6.3.
Generic type (Reg): First we consider the case when the residue endomorphism of the logarithmic pole at z = 0 of the (T E)-structure over t 0 is semisimple. Then a linear base change gives b (1) And a suitable coordinate change in t gives b  (1) 3 = 0. Then β 2 = β 3 = β 4 = 1/γ, and this is equal to k 1 , as β j ∈ N for at least one j. Write α 4 := b (1) 3 = 0. Then (6.46) still holds. By a base change with constant coefficients, we can obtain b 2 ∈ N. A coordinate change in t leads to a (0) 2 = k 1 t k 1 −1 . We obtain the normal form in the seventh case in the list in Theorem 6.3. 6. 3. Generically regular singular (T E)-structures over (C, 0) with logarithmic restriction over t 0 = 0 and not semisimple monodromy. The only 1-parameter unfoldings with trace free pole part of logarithmic (T E)-structures over a point, which are not covered by Theorem 6.3, have generic type (Reg) or (Log) and not semisimple monodromy. This follows from Theorem 6.2 and Theorem 3.20 (a). These (T E)-structures are classified in Theorem 6.7. Some of them are in the 6th or 9th case in Theorem 6.3, but most are not.
The leading exponents of the (T E)-structures over t = 0 come from Theorem 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 are independent of t and are still called α 1 & α 2 . Recall α 1 − α 2 ∈ Z ≥0 . The leading exponents of the logarithmic (T E)-structure over t 0 = 0 from Theorem 4.18 (b) are now called α 0 1 & α 0 2 . Recall α 0 1 − α 0 2 ∈ Z ≥0 . Precisely one of the three cases (I), (II) and (III) in the following table holds.
Before the proof, some remarks are made.
(iii) The generic type is (Reg) in the cases (I) and (II). In these cases the (T E)-structure is induced by the special cases of (6.49) respectively (6.50) with f = t via the map ϕ = f : (C, 0) → (C, 0).
(vi) The overlap of the (T E)-structures in Theorem 6.3 and in Theorem 6.7 is as follows.
We can write with some suitable g 1 , g 2 , g 3 , g 4 ∈ C{t, z}. This shows 1 + zC{t, z}v . It is contained in the first line of (6.66), and therefore also in the third line of (6.66). But this leads to a contradiction, when we compare the coefficient of s 0 2 . Here observe 3 + 1.
This contradiction shows that case (IV ) is impossible.

Marked regular singular rank 2 (T E)-structures
The regular singular rank 2 (T E)-structures over points were subject of the subsections 4. 5 and 4. 6, those over (C, 0) were subject of Theorem 6.3 and Remark 6.4 (iv) and of Theorem 6.7 and the Remarks 6.8.
First we will consider in the Remarks 7.1 (i)+(ii) regular singular rank 2 (T E)-structures over P 1 , which arise naturally from the Theorems 4.15 and 4.18. The (T E)-structures over the germs (P 1 , 0) and (P 1 , ∞) 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 1 extends to a rank 2 (T E)-structure of generic type (Reg) or (Log) over C × P 1 with primitive Higgs field. With Theorem 3.14, the base manifold C × P 1 obtains a canonical structure as F-manifold with Euler field. For each t 0 ∈ C × C * , the (T E)-structure over the germ (C × P 1 , t 0 ) is a universal unfolding of its restriction over t 0 . For each t 0 ∈ C × {0, ∞}, the (T E)-structure over the germ (C ×P 1 , t 0 ) will reappear in the Theorems 8.1, 8.5 and 8.6. See Remark 7.2 (i)+(ii).
Then we will observe in Corollary 7.3 that any marked regular singular (T E)-structure is a good family of marked regular singular (T E)structures (over points) in the sense of Definition 3.26 (b). 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 subject of Theorem 3.29. The parameter space P 1 of each (T E)-structure over P 1 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 described in Theorem 7.4.
(iii) Let (H → C × (M, t 0 ), ∇) be a regular singular unfolding of a regular singular, but not logarithmic rank 2 (T E)-structure over t 0 . Because of part (ii), it is induced by the (T E)-structure (H (4) , ∇ (4) ) via a map (M, t 0 ) → (M (4) , t (4) ) for some t (4) ∈ {0} × C * . Because it is regular singular, the image of the map is in {0} × C * ⊂ {0} × M (3) . As there the leading exponents are constant, they are also constant on the unfolding (H, ∇). . The restriction of the (T E)-structure over the germ (∆, t 0 ) is in the case N mon = 0 isomorphic to one in the 5th case in 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 1 (t 0 ), α 2 (t 0 )) of the (T E)-structure over t 0 are either (α gen 1 + 1, α gen 2 ) or (α gen 1 , α gen 2 + 1), because of table (6.27) in Remark 6.4 (iv) and because of the definition of the cases (I) and (II) in Theorem 6.7.
(iii) We have here a notion of horizontal directions which is similar to that for classifying spaces of Hodge structures. There it comes from Griffiths transversality. Here it comes from the part of the pole of Poincaré rank 1, which says that the covariant derivatives ∇ ∂ j along vector fields on the base space see only a pole of order 1.
In the cases of the F α 1 ,α 2 2 with α 1 − α 2 ∈ N − {1, 2}, the horizontal directions are the tangent spaces to the fibers of the P 1 fibration. In the cases of F α 1 ,α 1 −2 2 and F α 1 ,α 1 −1 2 , the horizontal directions contain these tangent spaces. But on points in the (−2)-curve in F α 1 ,α 1 −2 2 and on the singular point in F α 1 ,α 1 −1 2 , any direction is horizontal. The sections 5 and 8 together treat all rank 2 (T E)-structures over germs (M, t 0 ) of manifolds. Section 5 treated the unfoldings of (T E)structures of types (Sem) or (Bra) or (Reg) over t 0 . Section 8 will treat the unfoldings of (T E)-structures of type (Log) over t 0 .
It builds on section 6, which classified the unfoldings with trace free pole parts over (M, t 0 ) = (C, 0) of a logarithmic rank 2 (T E)-structure over t 0 and on section 7, which treated arbitrary regular singular rank 2 (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 with trace free pole parts and vice versa. Subsection 8. 1 gives the classification results. Subsection 8. 2 extracts from them a characterization of the space of all (T E)-structures with generically primitive Higgs fields over a given germ of a 2dimensional F-manifold with Euler field. Subsection 8. 3 gives the proof of Theorem 8.5.
First we characterize in Theorem 8.1 the 2-parameter unfoldings of rank 2 (T E)-structures of type (Log) over a point such that the Higgs field is generically primitive and induces an F-manifold structure on the underlying germ (M, t 0 ) of a manifold. Theorem 8.1 is a rather immediate implication of Theorem 6.3 and Theorem 6.7 together with the 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 a point t 0 all unfoldings within the set of (T E)-structures in Theorem 8.1 (a). The proof consists of inspection of the explicit classification in Theorem 8.1 (d).
Theorem 8.5 is the main result of this section. It lists a finite subset of the unfoldings in Theorem 8.1 (d) with the following property: Any unfolding of a rank 2 (T E)-structure of type (Log) over a point is induced by a (T E)-structure in this list. The (T E)-structures in the list 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 Theorem 6.7.
Finally, Theorem 8.6 lists the rank 2 (T E)-structures over a germ (M, t 0 ) of a manifold such that the Higgs field is primitive (so that (M, t 0 ) becomes a germ of an F-manifold with Euler field) and the restriction over t 0 is of type (Log). This list turns out to be a sublist of the one in Theorem 8.5. Theorem 8.6 follows easily from Theorem 8. There is a unique rank 2 (T E)-structure (H [3] → C × (C, 0), ∇ [3] ) over (C, 0) (with coordinate t 2 ) with trace free pole part, with nonvanishing Higgs field and with logarithmic restriction over t 2 = 0 such that (O(H), ∇) arises from (O (H [3] ), ∇ [3] ) as follows. There are coordinates t = (t 1 , t 2 ) on (M, t 0 ) such that (M, t 0 ) = (C 2 , 0) and a constant c 1 ∈ C such that where pr 2 : (M, t 0 ) → (C, 0), (t 1 , t 2 ) → t 2 (see Lemma 3.10 (a) for E (t 1 +c 1 )/z ). The (T E)-structure (H, ∇) is of type (Log) over (C × {0}, 0) and of one generic type (Sem) or (Bra) or (Reg) or (Log) over (C × C * , 0). ) over (C, 0) with trace free pole part, nonvanishing Higgs field and logarithmic restriction over 0 are classified in Theorem 6.3 and Theorem 6.7. They are in suitable coordinates the first 7 of the 9 cases in the list in Theorem 6.3 and the cases (6.49) and (6.50) with f = 1 k 1 t k 1 for some k 1 ∈ N in Theorem 6.7. (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 1 , t 2 ) such that (M, t 0 ) = (C 2 , 0), and there is a C{t, z}-basis v of O(H) 0 whose matrices A 1 , A 2 , B ∈ M 2×2 (C{t, z}) with z∇ ∂ i v = vA i , z 2 ∇ ∂z v = vB are in the following list of normal forms. The normal form is unique. Always Always M is an F-manifold with Euler field in one of the normal forms in Theorem 2.2 and 2.3 (in the case (i) the product ∂ 2 •∂ 2 is only almost in the normal form in Theorem 2.2; in the case (iii) with α 4 = −1 the Euler field is only almost in the normal form in Theorem 2.3).
Remark 8.2. The other normal forms in Remark 6.6 for the generic type (Sem) with k 2 − k 1 ∈ 2N and for the generic type (Bra) give the following other normal forms. In both cases, the formulas for A 1 = C 1 , γ, B, the F-manifold and E are unchanged, only the matrix A 2 changes. For the generic type (Sem) with k 2 − k 1 ∈ 2N, A 2 becomes For the generic type (Bra), A 2 becomes (c) Consider a rank 2 (T E)-structure (H [3] → C × (C, 0), ∇ [3] ) (with coordinate t 2 on (C, 0)) with trace free pole part and with logarithmic restriction over t 2 = 0. If it admits an extension to a pure (T LE)structure, it is contained in Theorem 6.3. If not, then it is contained in 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 ( ) ⊗ E (t 1 +c 1 )/z explicit. The cooordinate t and the matrix A in Theorem 6.3 and in Remark 6.8 (iv) become now t 2 and A 2 . Here the matrices in the 6th case in Theorem 6.3 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 1 /2 2 if α 1 = α 2 (⇒ case (II) and (6.54)) and to f = t k 1 2 if α 1 − α 2 ∈ N (case (I) and (6.53) or case (II) and (6.54)). The new matrix B is (−t 1 − c 1 )C 1 plus the matrix B in Theorem 6.3 and in Remark 6.8 (iv).
In the normal forms in Remark 6.8 (iv) we replaced α 1 and α 2 by ρ (1) and α 4 as follows, follows from inspection of the normal forms in part (d).
(a) Consider a (T E)-structure as in (a). Choose coordinates t = (t 1 , t 2 ) on (M, t 0 ) such that (M, t 0 ) = (C 2 , 0) and the germ of the Fmanifold is in a normal form in Theorem 2.2 (especially e = ∂ 1 ) and the Euler field has the form E = (t 1 + c 1 )∂ 1 + g(t 2 )∂ 2 for some c 1 ∈ C and some g(t 2 ) ∈ C{t 2 }.
Choose any C{t, z}-basis v of O(H) 0 and consider its matrices We make a base change with the matrix T ∈ GL 2 (C{t, z}) which is the unique solution of the differential equation Then the matrices A 1 , A 2 , B of the new basis v = vT satisfy 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).
Theorem 8.6. The set of rank 2 (T E)-structures with primitive (not just generically primitive) Higgs field over a germ (M, t 0 ) of an Fmanifold and with restriction of type (Log) over t 0 is (after the choice of suitable coordinates) the proper subset of those in the list (8.12) in Theorem 8.5 which satisfy k 1 = 1 respectively k 1 = 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 field over a germ (M, t 0 ) of an F-manifold and with restriction of type (Log) over t 0 consists by Theorem 8.1 (a)+(d) of those (T E)-structures in Theorem 8.1 (d) which satisfy A 2 (t 2 = 0) / ∈ C · C 1 . This holds if and only if k 1 = 1 respectively k 1 = 1 ( k 1 = 1 if the generic type is (Reg) and N mon = 0), and then A 2 (t 2 = 0) ∈ {−γC 2 , −γD, C 2 }. 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 the generic cases (Reg) and (Log). is the universal unfolding of its restriction over t 0 , and it is its own universal unfolding. So then B 1 = B 2 = B 3 , and the classification of the (T E)-structures over points in section 4 determines this space B 1 .
In the case of A 2 1 with E = (u 1 + c 1 )e 1 + (u 2 + c 2 )e 2 with c 1 = c 2 , any (T E)-structure over t 0 is of type (Sem). Theorem 4.4 tells that then B 1 is connected and 4-dimensional. The parameters are the two regular singular exponents and two Stokes parameters.
In the case of N 2 with E = (t 1 + c 1 )∂ 1 + ∂ 2 , any (T E)-structure over t 0 is either of type (Bra) or of type (Reg). Then B 1 has one component for type (Bra) and countably many components for type (Reg).
Corollary 4.16 gives the countably many components for type (Reg). One is 1-dimensional, the others are 2-dimensional.  , t 0 ), •, e, E − c 1 ∂ 1 )). Therefore we can and will restrict to the cases with E| t 0 = 0.
In the cases A 2 1 and I 2 (m), the Euler field with E| t 0 = 0 is unique on (M, t 0 ), therefore we do not write it down.
The 2 continuous parameters are the regular singular exponents of the (T E)-structures at generic points in M.
So, (N 2 , E) with c 3 ∈ C * does not allow (T E)-structures over it, and (N 2 , E) with c 3 = 0 and r ≥ 3 does not allow (T E)-structures over it with primitive Higgs field. If B j (N 2 , E) = ∅ then B j (N 2 , E) ∼ = C and the continuous parameter is ρ (1) in Theorem 8.1 (d) (iii).
Remarks 8.8. (i) Theorem 8.1 (d) (i) tells how many (T E)-structures exist over the F-manifold with Euler field I 2 (m), such that the Higgs bundle is generically primitive and induces this F-manifold structure. There are [ m 2 ] many holomorphic families from the different choices of (k 1 , k 2 ) ∈ N 2 with k 2 ≥ k 1 and k 1 + k 2 = m. They have 2 parameters if m is even and 1 parameter if m is odd, compare (8.17) and (8.18). For each I 2 (m), only one of these families consists of (T E)-structures with primitive Higgs fields.
(ii) Consider m ≥ 3. Write M = C 2 for the F-manifold I 2 (m) in Theorem 2.2, and M [log] = C × {0} 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 2 ] many Stokes structures which give (T E)-structures on M −M [log] . Because of (i), all these (T E)-structures extend holomorphically over M [log] , and they give the [ m 2 ] holomorphic families of (T E)-structures on I 2 (m) in (i).
(iii) Especially remarkable is the case A 2 1 = I 2 (2). There Theorem 8.1 (a)+(d)(i) implies directly that each holomorphic (T E)-structure over A 2 1 with generically primitive Higgs field has primitive Higgs field and is an elementary model (Definition 4.3), so it has trivial Stokes structure.
(iv) This result is related to much more general work in [CDG17] and [Sa19] on meromorphic connections over the F-manifold A n 1 near points where some of the canonical coordinates coincide. Let us restrict to the special case of a neighborhood of a point where all canonical coordinates coincide. This generalizes the germ at 0 of A 2 1 to the germ at 0 of A n 1 . [CDG17, Theorem 1.1] and [Sa19, Theorem 3] both give the triviality of the Stokes structure. Though their starting points are slightly restrictive. [CDG17] starts in our notation from pure (T LE)-structures with primitive Higgs fields. The step before in the case of A 2 1 , passing from a (T E)-structure over A 2 1 to a pure (T LE)-structure, is done essentially in our Theorem 6.2 (a)(iii). Our argument for the triviality of the Stokes structure is then contained in the proof of Theorem 6.3.
[Sa19] starts in our notation from (T E)-structures which are already formally isomorphic to sums n i=1 E u i /z z α i . Then it is shown that they are also holomorphically isomorphic to such sums. In this special case, Corollary 5.7 in [DH20-2] give this implication, too.
So, we obtain a regular singular (T E)-structure on M ′ with primitive Higgs field. The F-manifold structure on M ′ is given by e = ∂ 1 and ∂ 2 • ∂ 2 = 0, so it is N 2 , and the Euler field is E = t 1 ∂ 1 + (α 1 − α 2 )t 2 2 ∂ 2 . F-manifold and Euler field extend from M ′ to M, but not the (T E)structure.

=⇒ and
(3) ⇐= are obvious from the normal form in the 3rd case in Theorem 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 isomorphic to an elementary model in Definition 4.3 (see also Remark 8.8 (ii)). This shows ⇐⇒ is a consequence of the invariance of the Stokes structure within isomonodromic deformations.
If gcd(k 0 1 , k 0 2 ) = 1, then the generic type is (Sem), k 2 − k 1 ∈ 2N, k 0 2 − k 0 1 ∈ 2N, and the invariant ζ of (H, ∇)| C×(∆,t 0 ) is ζ = 0. But then the regular singular exponents α 1 and α 2 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 of the (T E)-structure (H, ∇). Now ζ = 0 implies k In the cases with N mon = 0, table (8.12) offers one of generic type (Sem) with k 1 = k 2 = 1 (and some with k 2 > k 1 if α 1 −α 2 ∈ Q∩(−1, 1)) and one or two of generic type (Reg), see table (8.11). In the cases with N mon = 0, table (8.12) offers two of generic type (Reg) if the leading exponents α 1 and α 2 satisfy α 1 −α 2 ∈ N, and one if they satisfy α 1 = α 2 , compare also the figures 4 and 5 in Theorem 7.4 (b). Therefore the inducing (T E)-structure in table (8.12) is not unique except for the case N mon = 0 and α 1 = α 2 , if the original (T E)-structure has the form ϕ * (O(H M. Saito presents in [SaM17] a family of Gauss-Manin connections with a functional parameter. In the arXiv paper [SaM17], the bundle has rank n, but in a preliminary version it has rank 3 and is more transparent. Here we translate the rank 3 example by a Fourier-Laplace transformation into a family of (TE)-structures with primitive Higgs fields over a fixed 3-dimensional globally irreducible F-manifold with an Euler field, such that the F-manifold with Euler field is nowhere regular. The family of (TE)-structures has a functional parameter h(t 2 ) ∈ C{t 2 }.
In the following, we write down a (TE)-structure of rank 3 on a manifold M = C 3 with coordinates t 1 , t 2 , t 3 . The restriction to {t ∈ C 3 | t 1 = 0} = {0} × C 2 is a FL-transformation of Saito's example. The parameter t 1 and this F-manifold are not considered in [SaM17].
The sections v 1 , v 2 , v 3 define also an extension H → P 1 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 a flat 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 2 ) = const. But probably similar higher dimensional examples allow also an extension to pure (TLEP)-structures while keeping the functional freedom. This would give families of Frobenius manifolds with Euler fields with functional freedom on a fixed F-manifold with Euler field.
In the example above, t 1 , t 2 , t 3 are flat coordinates and t 1 = t 1 , t 2 , t 3 = t 3 are generalized canonical coordinates (in which the multiplication has simple formulas).