The case (2.5), $E=t_{1}\partial_{1}$: Here each $(TE)$-structure has generic
type (Log) and semisimple monodromy. Here

$\displaystyle
B_{1}({\mathcal{N}}_{2},E)\cong\dot{\bigcup\limits_{k_{1}\in{\mathbb{N}}}}{\mathbb{C}},$
$\displaystyle
B_{2}({\mathcal{N}}_{2},E)=B_{3}({\mathcal{N}}_{2},E)\cong{\mathbb{C}},\qquad\text{here}\quad
k_{1}=1.$

The continuous parameter is $\rho^{(1)}$ in Theorem 8.1$(d)$ (iv) or,
equivalently, one of the two residue eigenvalues (which are
$\rho^{(1)}\pm\frac{1}{2}$).

The case (2.7),
$E=t_{1}\partial_{1}+t_{2}^{r}\big{(}1+c_{3}t_{2}^{r-1}\big{)}$ for some
$r\in{\mathbb{Z}}_{\geq 2}$ and some $c_{3}\in{\mathbb{C}}$: Here each
$(TE)$-structure has generic type (Reg) and satisfies $N^{\rm mon}\neq 0$.
Here

$\displaystyle
B_{1}({\mathcal{N}}_{2},E)\begin{cases}=\varnothing,&\text{if}\quad
c_{3}\in{\mathbb{C}}^{*},\\\ \cong{\mathbb{C}}&\text{if}\quad
c_{3}=0,\end{cases}$ $\displaystyle
B_{2}({\mathcal{N}}_{2},E)=B_{3}({\mathcal{N}}_{2},E)\begin{cases}=\varnothing,&\text{if}\quad
c_{3}\in{\mathbb{C}}^{*}\quad\text{or}\quad r\geq 3,\\\
=B_{1}({\mathcal{N}}_{2},E)&\text{if}\quad c_{3}=0\quad\text{and}\quad
r=2.\end{cases}$

So, $({\mathcal{N}}_{2},E)$ with $c_{3}\in{\mathbb{C}}^{*}$ does not allow
$(TE)$-structures over it, and $({\mathcal{N}}_{2},E)$ with $c_{3}=0$ and
$r\geq 3$ does not allow $(TE)$-structures over it with primitive Higgs field.
If $B_{j}({\mathcal{N}}_{2},E)\allowbreak\neq\varnothing$ then
$B_{j}({\mathcal{N}}_{2},E)\cong{\mathbb{C}}$ and the continuous parameter is
$\rho^{(1)}$ in Theorem 8.1$(d)$ $(iii)$.

The case (2.6), $E=t_{1}\partial_{1}+c_{2}t_{2}\partial_{2}$ for some
$c_{2}\in{\mathbb{C}}^{*}$: This is a rich case. Here we decompose
$B_{j}=B_{j}({\mathcal{N}}_{2},E)$ as

$\displaystyle B_{j}=B_{j}^{({\rm Reg}),0}\ \dot{\cup}\ B_{j}^{({\rm
Reg}),\neq 0}\ \dot{\cup}\ B_{j}^{({\rm Bra})},$

where the first set contains $(TE)$-structures of generic type (Reg) with
$N^{\rm mon}=0$, the second set contains $(TE)$-structures of generic type
(Reg) with $N^{\rm mon}\neq 0$, and the third set contains $(TE)$-structures
of generic type (Bra). Then

$\displaystyle B_{1}^{({\rm
Reg}),0}\cong\dot{\bigcup\limits_{k_{1}\in{\mathbb{N}}}}{\mathbb{C}},\qquad
B_{2}^{({\rm Reg}),0}=B_{3}^{({\rm Reg}),0}\cong{\mathbb{C}},$ $\displaystyle
B_{1}^{({\rm Reg}),\neq 0}\begin{cases}=\varnothing&\text{if}\quad
c_{2}\in{\mathbb{C}}^{*}\setminus{\mathbb{Q}}^{*},\\\
\cong\dot{\bigcup}_{(k_{1},\alpha_{4})\in{\mathbb{N}}\times{\mathbb{Z}}\colon
k_{1}c_{2}=1+\alpha_{4}}{\mathbb{C}}&\text{if}\quad
c_{2}\in{\mathbb{Q}}^{*},\end{cases}$ $\displaystyle B_{2}^{({\rm Reg}),\neq
0}=B_{3}^{({\rm Reg}),\neq 0}\begin{cases}=\varnothing&\text{if}\quad
c_{2}\in{\mathbb{C}}\setminus{\mathbb{Z}},\\\ \cong{\mathbb{C}}&\text{if}\quad
c_{2}\in{\mathbb{Z}}\setminus\\{0\\},\end{cases}$ $\displaystyle B_{1}^{({\rm
Bra})}=B_{2}^{({\rm Bra})}=B_{3}^{({\rm Bra})}=\varnothing\qquad\text{if}\quad
c_{2}^{-1}\in{\mathbb{C}}^{*}\setminus{\mathbb{Z}}_{\geq 2},$
$\displaystyle\left.\\!\\!\\!\begin{array}[]{l}B_{1}^{({\rm
Bra})}\cong\dot{\bigcup}_{(k_{1},k_{2})\in{\mathbb{N}}^{2}\colon
k_{1}+k_{2}=c_{2}^{-1}}{\mathbb{C}},\\\ B_{2}^{({\rm
Bra})}\cong\dot{\bigcup}_{(k_{1},k_{2})\in{\mathbb{N}}^{2}\colon
k_{1}+k_{2}=c_{2}^{-1},\gcd(k_{1},k_{2})=1}{\mathbb{C}},\\\ B_{3}^{({\rm
Bra})}\cong{\mathbb{C}},\quad\text{here
}(k_{1},k_{2})=\big{(}1,c_{2}^{-1}-1\big{)}\end{array}\\!\\!\right\\}\qquad\text{if}\quad
c_{2}^{-1}\in{\mathbb{Z}}_{\geq 2}.$

######  Remarks 8.8.

  1. $(i)$

Theorem 8.1$(d)$ $(i)$ tells how many $(TE)$-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
$\big{[}\frac{m}{2}\big{]}$ many holomorphic families from the different
choices of $(k_{1},k_{2})\allowbreak\in{\mathbb{N}}^{2}$ with $k_{2}\geq
k_{1}$ and $k_{1}+k_{2}=m$. They have 2 parameters if $m$ is even and 1
parameter if $m$ is odd, compare (8.21) and (8.22). For each $I_{2}(m)$, only
one of these families consists of $(TE)$-structures with primitive Higgs
fields.

  2. $(ii)$

Consider $m\geq 3$. Write $M={\mathbb{C}}^{2}$ for the $F$-manifold $I_{2}(m)$
in Theorem 2.2, and $M^{\rm[log]}={\mathbb{C}}\times\\{0\\}$ for the subset of
points where the multiplication is not semisimple. Over these points the
restricted $(TE)$-structures are of type (Log). We checked that there are
$\big{[}\frac{m}{2}\big{]}$ many Stokes structures which give
$(TE)$-structures on $M\setminus M^{\rm[log]}$. Because of $(i)$, all these
$(TE)$-structures extend holomorphically over $M^{\rm[log]}$, and they give
the $\big{[}\frac{m}{2}\big{]}$ holomorphic families of $(TE)$-structures on
$I_{2}(m)$ in $(i)$.

  3. $(iii)$

Especially remarkable is the case $A_{1}^{2}=I_{2}(2)$. There Theorem
8.1$(a){+}(d)$ $(i)$ implies directly that each holomorphic $(TE)$-structure
over $A_{1}^{2}$ with generically primitive Higgs field has primitive Higgs
field and is an elementary model (Definition 4.4), so it has trivial Stokes
structure.

  4. $(iv)$

This result is related to much more general work in [3] and [22] on
meromorphic connections over the $F$-manifold $A_{1}^{n}$ 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 $A_{1}^{2}$ to the germ at 0 of $A_{1}^{n}$.

[3, Theorem 1.1] and [22, Theorem 3] both give the triviality of the Stokes
structure. Though their starting points are slightly restrictive. [3] starts
in our notation from pure $(TLE)$-structures with primitive Higgs fields. The
step before in the case of $A_{1}^{2}$, passing from a $(TE)$-structure over
$A_{1}^{2}$ to a pure $(TLE)$-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.

[22] starts in our notation from $(TE)$-structures which are already formally
isomorphic to sums $\bigoplus_{i=1}^{n}{\mathcal{E}}^{u_{i}/z}z^{\alpha_{i}}$.
Then it is shown that they are also holomorphically isomorphic to such sums.
In this special case, Corollary 5.7 in [7] give this implication, too.

  5. $(v)$

In $(ii)$ we stated that in the case of $I_{2}(m)$ with $m\geq 3$, each
$(TE)$-structure on $M\setminus M^{\rm[log]}$ with primitive Higgs field
extends holomorphically to $M$. In the case of ${\mathcal{N}}_{2}$ this does
not hold in general. For example, start with the flat rank $2$ bundle
$H^{\prime}\to{\mathbb{C}}^{*}\times M$, where $M={\mathbb{C}}^{2}$ (with
coordinates $t=(t_{1},t_{2})$) with semisimple monodromy with two different
eigenvalues $\lambda_{1}$ and $\lambda_{2}$. Choose
$\alpha_{1},\alpha_{2}\in{\mathbb{C}}$ with ${\rm e}^{-2\pi{\rm
i}\alpha_{j}}=\lambda_{j}$. Let $s_{j}\in C^{\alpha_{j}}$ be generating
elementary sections. Define the new basis

$\displaystyle\underline{v}=(v_{1},v_{2})=\big{(}{\rm
e}^{t_{1}/z}\big{(}s_{1}+{\rm e}^{-1/t_{2}}s_{2}\big{)},{\rm
e}^{t_{1}/z}(zs_{2})\big{)}$

on $H^{\prime}|_{M^{\prime}}$, where
$M^{\prime}:=M\setminus{\mathbb{C}}\times\\{0\\}$. Then

$\displaystyle z\nabla_{\partial_{1}}\underline{v}=\underline{v}\cdot C_{1},$
$\displaystyle z\nabla_{\partial_{2}}\underline{v}=\underline{v}\cdot
t_{2}^{-2}{\rm e}^{-1/t_{2}}C_{2},$ $\displaystyle
z^{2}\nabla_{\partial_{z}}\underline{v}=\underline{v}\cdot\bigg{(}{-}t_{1}C_{1}+(\alpha_{2}-\alpha_{1}){\rm
e}^{-1/t_{2}}C_{2}+z\begin{pmatrix}\alpha_{1}&0\\\
0&\alpha_{2}+1\end{pmatrix}\bigg{)}.$

So, we obtain a regular singular $(TE)$-structure on $M^{\prime}$ with
primitive Higgs field. The $F$-manifold structure on $M^{\prime}$ is given by
$e=\partial_{1}$ and $\partial_{2}\circ\partial_{2}=0$, so it is
${\mathcal{N}}_{2}$, and the Euler field is
$E=t_{1}\partial_{1}+(\alpha_{1}-\alpha_{2})t_{2}^{2}\partial_{2}$.
$F$-manifold and Euler field extend from $M^{\prime}$ to $M$, but not the
$(TE)$-structure.

###  8.3 Proof of Theorem 8.5

$(a)$ Let $\big{(}H\to{\mathbb{C}}\times\big{(}M,t^{0}\big{)},\nabla\big{)}$
be an unfolding of a $(TE)$-structure of type (Log) over $t^{0}$. The
$(TE)$-structure
$\big{(}H^{[2]}\to{\mathbb{C}}\times\big{(}M,t^{0}\big{)},\nabla^{[2]}\big{)}$
in Lemma 3.10$(c)$ with
$\big{(}{\mathcal{O}}\big{(}H^{[2]}\big{)},\nabla^{[2]}\big{)}=({\mathcal{O}}(H),\nabla)\otimes{\mathcal{E}}^{-\rho^{(0)}/z}$
has trace free pole part. Lemma 3.10$(d)$ and $(e)$ apply. Because of them, it
is sufficient to prove that the $(TE)$-structure
$\big{(}H^{[2]},\nabla^{[2]}\big{)}$ is induced by a $(TE)$-structure
$\big{(}H^{[3]}\to{\mathbb{C}}\times\big{(}M^{[3]},t^{[3]}\big{)},\nabla^{[3]}\big{)}$
over $\big{(}M^{[3]},t^{[3]}\big{)}=({\mathbb{C}},0)$ via a map
$\varphi\colon\big{(}M,t^{0}\big{)}\to\big{(}M^{[3]},t^{[3]}\big{)}$, where
the $(TE)$-structure $\big{(}H^{[3]},\nabla^{[3]}\big{)}$ is one of the
$(TE)$-structures in the 1st to 7th cases in Theorem 6.3 or one of the
$(TE)$-structures in the cases (I) or (II) in Theorem 6.7 with invariants as
in table (8.20). Then the $(TE)$-structure
$\big{(}H^{[4]},\nabla^{[4]}\big{)}$ which is constructed in Lemma 3.10$(d)$
from $\big{(}H^{[3]},\nabla^{[3]}\big{)}$ is one of the $(TE)$-structures in
Theorem 8.1$(d)$ with invariants as in table (8.20), and it induces by Lemma
3.10$(e)$ the $(TE)$-structure $(H,\nabla)$.

From now on we suppose $\rho^{(0)}=0$, so
$(H,\nabla)=\big{(}H^{[2]},\nabla^{[2]}\big{)}$. We consider the invariants
$\delta^{(0)},\delta^{(1)}\in{\mathcal{O}}_{M,t^{0}}$ and ${\mathcal{U}}$ and
the four possible generic types (Sem), (Bra), (Reg) and (Log), which are
defined by the following table, analogously to Definition 6.1,

(Sem) | (Bra) | (Reg) | (Log)
---|---|---|---
$\delta^{(0)}\neq 0$ |  $\delta^{(0)}=0$, $\delta^{(1)}\neq 0$ |  $\delta^{(0)}=\delta^{(1)}=0$, ${\mathcal{U}}\neq 0$ | ${\mathcal{U}}=0$

First we treat the generic types (Reg) and (Log), then the generic type (Sem)
and (Bra).

Generic types $($Reg$)$ and $($Log$)$. Then the $(TE)$-structure $(H,\nabla)$
is regular singular. We can use the results in Section 7 (which built on
Theorems 6.3 and 6.7). Choose a marking for the $(TE)$-structure $(H,\nabla)$.
Then by Remark 7.5$(i)$, there is a unique map
$\varphi\colon\big{(}M,t^{0}\big{)}\to M^{(H^{{\rm ref},\infty},M^{\rm
ref}),{\rm reg}}$ which maps $t\in M$ to the point in the moduli space over
which one has up to isomorphism the same marked $(TE)$-structure as over $t$.
The map $\varphi$ is holomorphic. By Remark 7.5$(i){+}(ii)$ it maps
$\big{(}M,t^{0}\big{)}$ to one projective curve which is isomorphic to
$M^{(3),0,\alpha_{1},\alpha_{2}}$ or $M^{(3),\neq 0,\alpha_{1},\alpha_{2}}$ or
$M^{(3),0,\alpha_{1},{\rm log}}$. The $(TE)$-structure $(H,\nabla)$ is induced
by the $(TE)$-structure over this curve via the map $\varphi$. The point
$t^{0}$ is mapped to $0$ or $\infty$ in the cases
$M^{(3),0,\alpha_{1},\alpha_{2}}$ or $M^{(3),\neq 0,\alpha_{1},\alpha_{2}}$
(not 0 in the case $M^{(3),\neq 0,\alpha_{1},\alpha_{1}}$) as the
$(TE)$-structure over $t^{0}$ is logarithmic. The germs at $0$ and $\infty$ in
$M^{(3),0,\alpha_{1},\alpha_{2}}$ and $M^{(3),\neq 0,\alpha_{1},\alpha_{2}}$
(not 0 in the case $M^{(3),\neq 0,\alpha_{1},\alpha_{1}}$) and the germ at any
point $t_{2}^{(3)}$ in $M^{(3),0,\alpha_{1},{\rm log}}$ are contained in table
(8.20). 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 $M$ of the germ $\big{(}M,t^{0}\big{)}$, and we choose
on it coordinates $t=(t_{1},\dots,t_{m})$ (with $m=\dim M$) with $t^{0}=0$. We
denote by $M^{\rm[log]}$ the analytic hypersurface

$\displaystyle
M^{\rm[log]}:=\begin{cases}\big{(}\delta^{(0)}\big{)}^{-1}(0),&\text{if the
generic type is (Sem)},\\\\[4.30554pt]
\big{(}\delta^{(1)}\big{)}^{-1}(0),&\text{if the generic type is
(Bra)}.\end{cases}$

It contains $t^{0}$. Choose a disk $\Delta\subset M$ through $t^{0}$ with
$\Delta\setminus\\{t^{0}\\}\subset M\setminus M^{\rm[log]}$. The restricted
$(TE)$-structure $(H,\nabla)|_{{\mathbb{C}}\times(\Delta,t^{0})}$ has the same
generic type as the $(TE)$-structure $(H,\nabla)$. The restricted
$(TE)$-structure $(H,\nabla)|_{{\mathbb{C}}\times(\Delta,t^{0})}$ is
isomorphic to a $(TE)$-structure in the cases 1, 2, 3 or 4 in Theorem 6.3.

The parameters of the restricted $(TE)$-structure
$(H,\nabla)|_{{\mathbb{C}}\times(\Delta,t^{0})}$ are given in the following
table:

Generic type | Parameters
---|---
(Sem) |  $k_{1},k_{2}\in{\mathbb{N}}$ with $k_{2}\geq k_{1}$, $\rho^{(1)}\in{\mathbb{C}}$,
| $\begin{cases}\zeta\in{\mathbb{C}}&\text{if}\ k_{2}-k_{1}\in
2{\mathbb{N}},\\\ \alpha_{3}\in{\mathbb{R}}_{\geq
0}\cup{\mathbb{H}}&\text{if}\ k_{1}=k_{2}\end{cases}$
(Bra) |  $k_{1},k_{2}\in{\mathbb{N}}$, $\rho^{(1)}\in{\mathbb{C}}$

There is a unique pair $\big{(}k_{1}^{0},k_{2}^{0}\big{)}\in{\mathbb{N}}^{2}$
with $(k_{1},k_{2})\in{\mathbb{Q}}_{>0}\cdot\big{(}k_{1}^{0},k_{2}^{0}\big{)}$
and with the conditions in table (8.29),

Generic type and invariants | Conditions
---|---
(Sem$)\colon\ k_{2}-k_{1}>0$ odd | $\gcd\big{(}k^{0}_{1},k^{0}_{2}\big{)}=1$
(Sem$)\colon\ k_{2}-k_{1}\in 2{\mathbb{N}}$, $\zeta=0$ | $\gcd\big{(}k^{0}_{1},k^{0}_{2}\big{)}=1$
(Sem$)\colon\ k_{2}-k_{1}\in 2{\mathbb{N}}$, $\zeta\neq 0$ |  $k^{0}_{2}-k^{0}_{1}\in 2{\mathbb{N}}$, $\gcd\left(k^{0}_{1},\frac{k_{1}^{0}+k^{0}_{2}}{2}\right)=1$
(Sem$)\colon k_{2}=k_{1}$ | $k^{0}_{2}=k^{0}_{1}=1$
(Bra) | $\gcd\big{(}k^{0}_{1},k^{0}_{2}\big{)}=1$
(8.29)

In fact, it is the pair $\big{(}k_{1}^{0},k_{2}^{0}\big{)}\in{\mathbb{N}}^{2}$
of minimal numbers which satisfies

$\displaystyle(k_{1},k_{2})\in{\mathbb{N}}\cdot\big{(}k_{1}^{0},k_{2}^{0}\big{)}$

and which satisfies in the case (Sem) with $k_{2}-k_{1}\in 2{\mathbb{N}}$ and
$\zeta\neq 0$ additionally $k_{2}^{0}-k_{1}^{0}\in 2{\mathbb{N}}$.

We denote by
$\big{(}H^{[3]}\to{\mathbb{C}}\times\big{(}M^{[3]},t^{[3]}\big{)},\nabla^{[3]}\big{)}$
the $(TE)$-structure over $\big{(}M^{[3]},t^{[3]}\big{)}=({\mathbb{C}},0)$
which has $\big{(}k_{1}^{0},k_{2}^{0}\big{)}$ instead of $(k_{1},k_{2})$, but
which has the same other parameters as the restricted $(TE)$-structure
$(H,\nabla)|_{{\mathbb{C}}\times(\Delta,t^{0})}$.

We have seen in Remarks 6.5$(ii)$ and $(iii)$ that the restricted
$(TE)$-structure $(H,\nabla)|_{{\mathbb{C}}\times(\Delta,t^{0})}$ is induced
by the $(TE)$-structure $\big{(}H^{[3]},\nabla^{[3]}\big{)}$ via the branched
covering
$\varphi^{\Delta}\colon(\Delta,t^{0})\to\big{(}M^{[3]},t^{[3]}\big{)}$ with
$\varphi^{\Delta}(\tau)=\tau^{k_{1}/k_{1}^{0}}$. Here $\tau$ denotes that
coordinate on $\Delta$ with which
$(H,\nabla)|_{{\mathbb{C}}\times(\Delta,t^{0})}$ can be brought to a normal
form in the cases 1, 2, 3 and 4 in Theorem 6.3.

It rests to extend $\varphi^{\Delta}$ to a map $\varphi\colon M\to M^{[3]}$
such that $(H,\nabla)$ is induced by $\big{(}H^{[3]},\nabla^{[3]}\big{)}$ via
this map $\varphi$.

######  Claim 8.9.

There exists a unique holomorphic function $\varphi\in{\mathcal{O}}_{M}$ with

$\displaystyle\varphi|_{\Delta}=\varphi^{\Delta},$ (8.30)
$\displaystyle\delta^{(0)}=-\varphi^{k_{1}^{0}+k_{2}^{0}}\qquad\text{if the
generic type is $($Sem$)$},$ (8.31)
$\displaystyle\delta^{(1)}=\frac{k_{2}^{0}}{k_{1}^{0}+k_{2}^{0}}\cdot\varphi^{k_{1}^{0}+k_{2}^{0}}\qquad\text{if
the generic type is $($Bra$)$}.$ (8.32)

###### Proof.

Choose any point $t^{[1]}\in M^{\rm[log]}$ and any disk $\Delta^{[1]}$ through
$t^{[1]}$ with $\Delta^{[1]}\setminus\big{\\{}t^{[1]}\big{\\}}\subset
M\setminus M^{\rm[log]}$. In order to show the existence of a function
$\varphi\in{\mathcal{O}}_{M}$ with (8.31) respectively (8.32), it is
sufficient to show that $\delta^{(0)}|_{\Delta^{[1]}}$ respectively
$\delta^{(1)}|_{\Delta^{[1]}}$ has at $t^{[1]}$ a zero of an order which is a
multiple of $k_{1}^{0}+k_{2}^{0}$.

The restricted $(TE)$-structure
$(H,\nabla)|_{{\mathbb{C}}\times(\Delta^{[1]},t^{[1]})}$ has the same generic
type as $(H,\nabla)$ and is isomorphic to a $(TE)$-structure in the cases 1,
2, 3 or 4 in Theorem 6.3. Its invariants $k_{1}$ and $k_{2}$ are here called
$k_{1}^{[1]}$ and $k_{2}^{[1]}$, in order to distinguish them from the
invariants of $(H,\nabla)|_{(\Delta,t^{0})}$. We want to show

$\displaystyle\big{(}k_{1}^{[1]},k_{2}^{[2]}\big{)}\in{\mathbb{N}}\cdot\big{(}k_{1}^{0},k_{2}^{0}\big{)}.$
(8.33)

We did not say much about the Stokes structure. Here we need the following
properties of it, if the generic type is (Sem):

$\displaystyle k_{2}=k_{1}$
$\displaystyle\quad\stackrel{{\scriptstyle(1)}}{{\iff}}(H,\nabla)|_{{\mathbb{C}}\times\\{t^{[2]}\\}}\text{
has trivial Stokes structure for
}t^{[2]}\in\Delta\setminus\big{\\{}t^{0}\big{\\}}$
$\displaystyle\quad\stackrel{{\scriptstyle(2)}}{{\iff}}(H,\nabla)|_{{\mathbb{C}}\times\\{t^{[2]}\\}}\text{
has trivial Stokes structure for
}t^{[2]}\in\Delta^{[1]}\setminus\big{\\{}t^{[1]}\big{\\}}$
$\displaystyle\quad\stackrel{{\scriptstyle(3)}}{{\iff}}k_{2}^{[1]}=k_{1}^{[1]}.$

$\stackrel{{\scriptstyle(1)}}{{\Longrightarrow}}$ and
$\stackrel{{\scriptstyle(3)}}{{\Longleftarrow}}$ 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 $t\in{\mathbb{C}}^{*}$ in the 1st and 2nd case in Theorem 6.3
are not holomorphically isomorphic to an elementary model in Definition 4.4
(see also Remark 8.8$(ii)$). This shows
$\stackrel{{\scriptstyle(1)}}{{\Longleftarrow}}$ and
$\stackrel{{\scriptstyle(3)}}{{\Longrightarrow}}$. The equivalence
$\stackrel{{\scriptstyle(2)}}{{\iff}}$ is a consequence of the invariance of
the Stokes structure within isomonodromic deformations.

In the generic type (Sem) with $k_{1}=k_{2}$ we have also
$k_{2}^{[1]}=k_{1}^{[1]}$ and $k_{2}^{0}=k_{1}^{0}=1$, and thus especially
(8.33).

Now consider the cases with $k_{2}>k_{1}$. This comprises the generic type
(Bra) and gives in the generic type (Sem) also $k_{2}^{0}>k_{1}^{0}$ and
$k_{2}^{[1]}>k_{1}^{[1]}$. So
$(H,\nabla)|_{{\mathbb{C}}\times(\Delta^{[1]},t^{[1]})}$ is in the 1st, 2nd or
4th case in Theorem 6.3. The number $b_{3}^{(1)}$ in Theorem 6.3 is uniquely
determined by the properties
$b_{3}^{(1)}\in{\mathbb{Q}}\,\cap\big{]}\frac{-1}{2},0\big{[}$ and
$\operatorname{Eig}(M^{\rm mon})=\big{\\{}\exp\big{(}{-}2\pi{\rm
i}(\rho^{(1)}\pm b_{3}^{(1)}\big{)}\big{)}\big{\\}}$ (see Remark 6.4$(i)$ for
the second property). Therefore

$\displaystyle\frac{k_{1}^{0}-k_{2}^{0}}{2\big{(}k_{1}^{0}+k_{2}^{0}\big{)}}=\frac{k_{1}-k_{2}}{2(k_{1}+k_{2})}=b_{3}^{(1)}=\frac{k_{1}^{[1]}-k_{2}^{[1]}}{2\big{(}k_{1}^{[1]}+k_{2}^{[1]}\big{)}}\qquad\text{in
the case (Sem)},$
$\displaystyle\frac{-k_{2}^{0}}{2\big{(}k_{1}^{0}+k_{2}^{0}\big{)}}=\frac{-k_{2}}{2(k_{1}+k_{2})}=b_{3}^{(1)}=\frac{-k_{2}^{[1]}}{2\big{(}k_{1}^{[1]}+k_{2}^{[1]}\big{)}}\qquad\text{in
the case (Bra)}.$ (8.34)

This implies
$\big{(}k_{1}^{[1]},k_{2}^{[1]}\big{)}\in{\mathbb{Q}}_{>0}\cdot\big{(}k_{1}^{0},k_{2}^{0}\big{)}$.
In the cases with $\gcd\big{(}k_{1}^{0},k_{2}^{0}\big{)}=1$ (8.33) follows.

If $\gcd\big{(}k_{1}^{0},k_{2}^{0}\big{)}\neq 1$, then the generic type is
(Sem), $k_{2}-k_{1}\in 2{\mathbb{N}}$, $k_{2}^{0}-k_{1}^{0}\in 2{\mathbb{N}}$,
and the invariant $\zeta$ of $(H,\nabla)|_{{\mathbb{C}}\times(\Delta,t^{0})}$
is $\zeta\neq 0$. However, then the regular singular exponents $\alpha_{1}$
and $\alpha_{2}$ of the restriction of the $(TE)$-structure $(H,\nabla)$ over
points in $M\setminus M^{\rm[log]}$ are invariants of the $(TE)$-structure
$(H,\nabla)$. By (6.11) and (8.34) also $\zeta$ is an invariant of the
$(TE)$-structure $(H,\nabla)$. Now $\zeta\neq 0$ implies
$k_{2}^{[1]}-k_{1}^{[1]}\in 2{\mathbb{N}}$. Again (8.33) follows.

Equations (6.8) and (6.12) imply that $\delta^{(0)}|_{\Delta^{[1]}}$
respectively $\delta^{(1)}|_{\Delta^{[1]}}$ has at $t^{[1]}$ a zero of an
order which is a multiple of $k_{1}^{0}+k_{2}^{0}$. Therefore a function
$\varphi\in{\mathcal{O}}_{M}$ with (8.31) respectively (8.32) exists.

Equations (6.8) and (6.12) for
$(H,\nabla)|_{{\mathbb{C}}\times(\Delta,t^{0})}$ tell

$\displaystyle\delta^{(0)}|_{\Delta}=-\tau^{k_{1}+k_{2}}=-\big{(}\tau^{k_{1}/k_{1}^{0}}\big{)}^{k_{1}^{0}+k_{2}^{0}}=-\big{(}\varphi^{\Delta}\big{)}^{k_{1}^{0}+k_{2}^{0}}\qquad\text{in
the case (Sem),}$
$\displaystyle\delta^{(1)}|_{\Delta}=\frac{k_{2}}{k_{1}+k_{2}}\tau^{k_{1}+k_{2}}=\frac{k_{2}^{0}}{k_{1}^{0}+k_{2}^{0}}\big{(}\tau^{k_{1}/k_{1}^{0}}\big{)}^{k_{1}^{0}+k_{2}^{0}}=\frac{k_{2}^{0}}{k_{1}^{0}+k_{2}^{0}}\big{(}\varphi^{\Delta}\big{)}^{k_{1}^{0}+k_{2}^{0}}\quad\
\text{in the case (Bra).}$

Therefore a function $\varphi$ as in the claim exists and is unique. ∎

Now compare the $(TE)$-structures $(H,\nabla)$ and
$\varphi^{*}\big{(}H^{[3]},\nabla^{[3]}\big{)}$ over $M$. Both extend to pure
$(TLE)$-structures. For $\varphi^{*}\big{(}H^{[3]},\nabla^{[3]}\big{)}$, one
uses the pull back $\varphi^{*}\big{(}\underline{v}^{[3]}\big{)}$ of the basis
$\underline{v}^{[3]}$ which gives for $\big{(}H^{[3]},\nabla^{[3]}\big{)}$ the
Birkhoff normal form in Theorem 6.3. For $(H,\nabla)$ one starts with the
analogous basis $\underline{v}^{\Delta}$ for $H|_{{\mathbb{C}}\times\Delta}$
which gives for $(H,\nabla)|_{{\mathbb{C}}\times\Delta}$ the Birkhoff normal
form in Theorem 6.3. It has a unique extension $\underline{v}$ to
${\mathbb{C}}\times M$ which still yields a Birkhoff normal form. Compare
Remark 3.19$(ii)$ for this.

Remarks 6.5$(ii)$ and $(iii)$ (or simply the Birkhoff normal forms in Theorem
6.3) tell that the map
$\big{(}\varphi^{*}\underline{v}^{[3]}\big{)}|_{{\mathbb{C}}\times\Delta}\mapsto\underline{v}^{\Delta}=\underline{v}|_{{\mathbb{C}}\times\Delta}$
is an isomorphism of pure $(TLE)$-structures.

Now consider a point $t^{[2]}\in\Delta\setminus\big{\\{}t^{0}\big{\\}}$ and
its image $t^{[4]}:=\varphi\big{(}t^{[2]}\big{)}\in
M^{[3]}\setminus\big{\\{}t^{[3]}\big{\\}}={\mathbb{C}}\setminus\\{0\\}$. Over
the germ $\big{(}M^{[3]},t^{[4]}\big{)}$, the $(TE)$-structure
$\big{(}H^{[3]},\nabla^{[3]}\big{)}$ is the part with trace free pole part of
a universal unfolding of
$\big{(}H^{[3]},\nabla^{[3]}\big{)}|_{{\mathbb{C}}\times\\{t^{[4]}\\}}$.
Therefore in a neighborhood $U\subset M$ of $t^{[2]}$, the $(TE)$-structure
$(H,\nabla)|_{{\mathbb{C}}\times U}$ is induced by
$\big{(}H^{[3]},\nabla^{[3]}\big{)}|_{{\mathbb{C}}\times(M^{[3]},t^{[4]})}$
via a map $\widetilde{\varphi}\colon U\to M^{[3]}$. We can choose it such that

$\displaystyle\widetilde{\varphi}|_{\Delta}=\varphi^{\Delta}.$ (8.35)

Equations (6.8) and (6.12) tell

$\displaystyle\delta^{(0)}|_{U}=-(\widetilde{\varphi})^{k_{1}^{0}+k_{2}^{0}}\qquad\text{in
the case (Sem)},$ (8.36)
$\displaystyle\delta^{(1)}|_{U}=\frac{k_{2}^{0}}{k_{1}^{0}+k_{2}^{0}}(\widetilde{\varphi})^{k_{1}^{0}+k_{2}^{0}}\qquad\text{in
the case (Bra)}.$ (8.37)

Equations (8.35)–(8.37) and Claim 8.9 imply
$\widetilde{\varphi}=\varphi|_{U}$. Therefore the matrices in Birkhoff normal
form for the basis $\underline{v}$ of $(H,\nabla)$ coincide on
${\mathbb{C}}\times U$ with the matrices in Birkhoff normal form for the basis
$\varphi^{*}\big{(}\underline{v}^{[3]}\big{)}$ of
$\varphi^{*}\big{(}H^{[3]},\nabla^{[3]}\big{)}$. As all matrices are
holomorphic on ${\mathbb{C}}\times M$, they coincide pairwise on
${\mathbb{C}}\times M$. Therefore the pure $(TLE)$-structure $(H,\nabla)$ with
basis $\underline{v}$ is isomorphic to the pure $(TLE)$-structure
$\varphi^{*}\big{(}H^{[3]},\nabla^{[3]}\big{)}$ with basis
$\varphi^{*}\big{(}\underline{v}^{[3]}\big{)}$. This finishes the proof of
part $(a)$ of Theorem 8.5.

$(b)$ If the original $(TE)$-structure $(H,\nabla)$ has the form
$\varphi^{*}\big{(}{\mathcal{O}}(H^{[5]}),\nabla^{[5]}\big{)}\otimes{\mathcal{E}}^{-\rho^{(0)}/z}$
then the $(TE)$-structure $\big{(}H^{[2]},\nabla^{[2]}\big{)}$ with trace free
pole part which was associated to $(H,\nabla)$ at the beginning of the proof
of part $(a)$, has the form
$\varphi^{*}\big{(}{\mathcal{O}}(H^{[5]}),\nabla^{[5]}\big{)}$.

Then any $(TE)$-structure $\big{(}H^{[3]},\nabla^{[3]}\big{)}$ over
$\big{(}M^{[3]},t^{[3]}\big{)}=({\mathbb{C}},0)$ works, whose restriction over
$t^{[3]}$ is the given logarithmic $(TE)$-structure
$\big{(}H^{[5]},\nabla^{[5]}\big{)}$.

In the cases with $N^{\rm mon}=0$, table (8.20) offers one of generic type
(Sem) with $k_{1}=k_{2}=1$ (and some with $k_{2}>k_{1}$ if
$\alpha_{1}-\alpha_{2}\in{\mathbb{Q}}\cap(-1,1)$) and one or two of generic
type (Reg), see table (8.10). In the cases with $N^{\rm mon}\neq 0$, table
(8.20) offers two of generic type (Reg) if the leading exponents $\alpha_{1}$
and $\alpha_{2}$ satisfy $\alpha_{1}-\alpha_{2}\in{\mathbb{N}}$, and one if
they satisfy $\alpha_{1}=\alpha_{2}$, compare also Figures 7.4 and 7.5 in
Theorem 7.4$(b)$. Therefore the inducing $(TE)$-structure in table (8.20) is
not unique except for the case $N^{\rm mon}\neq 0$ and
$\alpha_{1}=\alpha_{2}$, if the original $(TE)$-structure has the form
$\varphi^{*}\big{(}{\mathcal{O}}(H^{[5]}),\nabla^{[5]}\big{)}\otimes{\mathcal{E}}^{-\rho^{(0)}/z}$.

In the other cases, the proof of part $(a)$ shows the uniqueness of the
$(TE)$-structure $\big{(}H^{[3]},\nabla^{[3]}\big{)}$. The uniqueness of
$\big{(}H^{[3]},\nabla^{[3]}\big{)}$ gives also the uniqueness of
$\big{(}H^{[4]},\nabla^{[4]}\big{)}$ in the first paragraph of the proof of
part $(a)$.

$(c)$ This follows from the uniqueness in part $(b)$. ∎

##  9 A family of rank 3 $\boldsymbol{(TE)}$-structures
with a functional parameter

M. Saito presents in [23] a family of Gauss–Manin connections with a
functional parameter. In the arXiv paper [23], 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
$h(t_{2})\in{\mathbb{C}}\\{t_{2}\\}$.

In the following, we write down a $(TE)$-structure of rank 3 on a manifold
$M={\mathbb{C}}^{3}$ with coordinates $t_{1}$, $t_{2}$, $t_{3}$. The
restriction to
$\big{\\{}t\in{\mathbb{C}}^{3}\,|\,t_{1}=0\big{\\}}=\\{0\\}\times{\mathbb{C}}^{2}$
is a FL-transformation of Saito’s example. The parameter $t_{1}$ and this
$F$-manifold are not considered in [23]. There the base space has only the two
parameters $t_{2}$ and $t_{3}$. Choose an arbitrary function
$h(t_{2})\in{\mathbb{C}}\\{t_{2}\\}$ with $h^{\prime\prime}(0)\neq 0$.

Let $H^{\prime}\to{\mathbb{C}}^{*}\times M$ be a holomorphic vector bundle
with flat connection with trivial monodromy and basis of global flat sections
$s_{1}$, $s_{2}$, $s_{3}$. Define an extension to a vector bundle
$H\to{\mathbb{C}}\times M$ using the following holomorphic sections of
$H^{\prime}$, which also form a basis of sections of $H^{\prime}$:

$\displaystyle v_{1}:={\rm e}^{t_{1}/z}\cdot\big{(}zs_{1}+t_{2}\cdot
zs_{2}+h(t_{2})\cdot zs_{3}+t_{3}\cdot z^{2}s_{3}\big{)},$ $\displaystyle
v_{2}:={\rm e}^{t_{1}/z}\cdot\big{(}z^{2}s_{2}+h^{\prime}(t_{2})\cdot
z^{2}s_{3}\big{)},\qquad v_{3}:={\rm e}^{t_{1}/z}\cdot z^{3}s_{3}.$

Denote $\underline{v}:=(v_{1},v_{2},v_{3})$. Denote
$\partial_{t_{j}}:=\partial_{j}$. Then

$\displaystyle z\nabla_{\partial_{1}}\underline{v}=\underline{v}\cdot{\bf
1}_{3},$ $\displaystyle
z\nabla_{\partial_{2}}\underline{v}=\underline{v}\cdot\begin{pmatrix}0&0&0\\\
1&0&0\\\ 0&h^{\prime\prime}(t_{2})&0\end{pmatrix}\\!,$ $\displaystyle
z\nabla_{\partial_{3}}\underline{v}=\underline{v}\cdot\begin{pmatrix}0&0&0\\\
0&0&0\\\ 1&0&0\end{pmatrix}\\!,$ $\displaystyle
z^{2}\partial_{z}\underline{v}=\underline{v}\cdot\left({-}t_{1}\cdot{\bf
1}_{3}+\begin{pmatrix}0&0&0\\\ 0&0&0\\\
t_{3}&0&0\end{pmatrix}\right)+z\cdot\underline{v}\cdot\begin{pmatrix}1&0&0\\\
0&2&0\\\ 0&0&3\end{pmatrix}\\!.$

Write $\underline{\partial}:=(\partial_{1},\partial_{2},\partial_{3})$. The
pole parts give the multiplication $\circ$ on the $F$-manifold and the Euler
field $E$ by

$\displaystyle\partial_{1}\circ\underline{\partial}=\underline{\partial}\cdot{\bf
1}_{3},$
$\displaystyle\partial_{2}\circ\underline{\partial}=\underline{\partial}\cdot\begin{pmatrix}0&0&0\\\
1&0&0\\\ 0&h^{\prime\prime}(t_{2})&0\end{pmatrix}\\!,$
$\displaystyle\partial_{3}\circ\underline{\partial}=\underline{\partial}\cdot\begin{pmatrix}0&0&0\\\
0&0&0\\\ 1&0&0\end{pmatrix}\\!,$ $\displaystyle
E\circ\underline{\partial}=-\underline{\partial}\cdot\left(-t_{1}\cdot{\bf
1}_{3}+\begin{pmatrix}0&0&0\\\ 0&0&0\\\
t_{3}&0&0\end{pmatrix}\right)\\!,\qquad\text{so}\quad
E=t_{1}\cdot\partial_{1}-t_{3}\cdot\partial_{3}.$

One can introduce a new coordinate system
$\big{(}\widetilde{t}_{1},\widetilde{t}_{2},\widetilde{t}_{3}\big{)}=\big{(}t_{1},\widetilde{t}_{2},t_{3}\big{)}$
on the germ $(M,0)$ with

$\displaystyle\partial_{\widetilde{t}_{2}}=\frac{1}{\sqrt{h^{\prime\prime}(t_{2})}}\cdot\partial_{2}.$

Denote $\widetilde{\partial}_{j}:=\partial_{\widetilde{t}_{j}}$ and
$\widetilde{\underline{\partial}}:=\big{(}\widetilde{\partial}_{1},\widetilde{\partial}_{2},\widetilde{\partial}_{3}\big{)}=\big{(}\partial_{1},\widetilde{\partial}_{2},\partial_{3}\big{)}$.
Introduce also the new section

$\displaystyle\widetilde{v}_{2}:=\frac{1}{\sqrt{h^{\prime\prime}(t_{2})}}\cdot
v_{2},$

and the new basis
$\widetilde{\underline{v}}=\big{(}\widetilde{v}_{1},\widetilde{v}_{2},\widetilde{v}_{3}\big{)}=\big{(}v_{1},\widetilde{v}_{2},v_{3}\big{)}$
of the given $(TE)$-structure. Then

$\displaystyle
z\nabla_{\widetilde{\partial}_{1}}\underline{\widetilde{v}}=\underline{\widetilde{v}}\cdot{\bf
1}_{3},$ $\displaystyle
z\nabla_{\widetilde{\partial}_{2}}\underline{\widetilde{v}}=\underline{\widetilde{v}}\cdot\begin{pmatrix}0&0&0\\\
1&0&0\\\
0&1&0\end{pmatrix}+\underline{\widetilde{v}}\cdot\begin{pmatrix}0&0&0\\\
0&\partial_{2}\frac{1}{\sqrt{h^{\prime\prime}(t_{2})}}&0\\\
0&0&0&\end{pmatrix}\\!,$ $\displaystyle
z\nabla_{\widetilde{\partial}_{3}}\underline{\widetilde{v}}=\underline{\widetilde{v}}\cdot\begin{pmatrix}0&0&0\\\
0&0&0\\\ 1&0&0\end{pmatrix}\\!,$ $\displaystyle
z^{2}\partial_{z}\underline{\widetilde{v}}=\underline{\widetilde{v}}\cdot\left(-t_{1}\cdot{\bf
1}_{3}+\begin{pmatrix}0&0&0\\\ 0&0&0\\\
t_{3}&0&0\end{pmatrix}\right)+z\cdot\underline{\widetilde{v}}\cdot\begin{pmatrix}1&0&0\\\
0&2&0\\\ 0&0&3\end{pmatrix}\\!.$

In the new coordinates the multiplication becomes simpler and independent of
the choice of $h(t_{2})$ (as long as $h^{\prime\prime}(t_{2})\neq 0$):

$\displaystyle\widetilde{\partial}_{1}\circ\underline{\widetilde{\partial}}=\underline{\widetilde{\partial}}\cdot{\bf
1}_{3},$
$\displaystyle\widetilde{\partial}_{2}\circ\underline{\widetilde{\partial}}=\underline{\widetilde{\partial}}\cdot\begin{pmatrix}0&0&0\\\
1&0&0\\\ 0&1&0\end{pmatrix}\\!,$
$\displaystyle\widetilde{\partial}_{3}\circ\underline{\widetilde{\partial}}=\underline{\widetilde{\partial}}\cdot\begin{pmatrix}0&0&0\\\
0&0&0\\\ 1&0&0\end{pmatrix}\\!,$ $\displaystyle
E\circ\underline{\widetilde{\partial}}=-\underline{\widetilde{\partial}}\cdot\left({-}t_{1}\cdot{\bf
1}_{3}+\begin{pmatrix}0&0&0\\\ 0&0&0\\\ t_{3}&0&0\end{pmatrix}\right)\\!,$
$\displaystyle\text{so}\quad
E=t_{1}\cdot\widetilde{\partial}_{1}-t_{3}\cdot\widetilde{\partial}_{3}.$

This is the nilpotent $F$-manifold for $n=3$ in [4, Theorem 3]. However, the
Euler field here is different from the one in [4, Theorem 3]. The endomorphism
$E\circ$ here is not regular, but has only the one eigenvalue $t_{1}$ and has
for $t_{3}\neq 0$ one Jordan block of size $2\times 2$ and one Jordan block of
size $1\times 1$ and is semisimple for $t_{3}=0$.

The sections $v_{1}$, $v_{2}$, $v_{3}$ define also an extension
$\widehat{H}\to{\mathbb{P}}^{1}$ such that the $(TE)$-structure extends to a
pure $(TLE)$-structure.

Furthermore $v$ satisfies all properties of the section $\zeta$ in Theorem
6.6(b) in [7]. Thus the $F$-manifold with Euler field is enriched to a flat
$F$-manifold with Euler field (Definition 3.1(b) in [7]).

If we try to introduce a pairing which would make it into a pure
$(TLEP)$-structure, we get a constraint $h^{\prime\prime}(t_{2})={\rm const}$.
However, 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
$\widetilde{t}_{1}=t_{1}$, $\widetilde{t}_{2}$, $\widetilde{t}_{3}=t_{3}$ are
generalized canonical coordinates (in which the multiplication has simple
formulas).

### Acknowledgements

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – 242588615. I would like to thank Liana David for a lot
of joint work on $(TE)$-structures.

## References

  * [1] Anosov D.V., Bolibruch A.A., The Riemann–Hilbert problem, Aspects of Mathematics, Vol. 22, Friedr. Vieweg & Sohn, Braunschweig, 1994.
  * [2] Babbitt D.G., Varadarajan V.S., Formal reduction theory of meromorphic differential equations: a group theoretic view, Pacific J. Math. 109 (1983), 1–80.
  * [3] Cotti G., Dubrovin B., Guzzetti D., Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, Duke Math. J. 168 (2019), 967–1108, arXiv:1706.04808.
  * [4] David L., Hertling C., Regular $F$-manifolds: initial conditions and Frobenius metrics, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 1121–1152, arXiv:1411.4553.
  * [5] David L., Hertling C., $(T)$-structures over two-dimensional $F$-manifolds: formal classification, Ann. Mat. Pura Appl. (4) 199 (2020), 1221–1242, arXiv:1811.03406.
  * [6] David L., Hertling C., $(TE)$-structures over the irreducible 2-dimensional globally nilpotent $F$-manifold germ, Rev. Roumaine Math. Pures Appl. 65 (2020), 235–284, arXiv:2001.01063.
  * [7] David L., Hertling C., Meromorphic connections over F-manifolds, in Integrability, Quantization, and Geometry. I. Integrable systems, Proceedings of Symposia in Pure Mathematics, Vol. 103, Editors S. Novikov, I. Krichever, O. Ogievetsky, S. Shlosman, Amer. Math. Soc., Providence, RI, 2021, 171–216, arXiv:1912.03331.
  * [8] Fabry E., Sur les intégrales des équations différentielles lineaires à coefficients rationels, Ph.D. Thesis, Paris, 1885.
  * [9] Hertling C., Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, Vol. 151, Cambridge University Press, Cambridge, 2002.
  * [10] Hertling C., $tt^{*}$ geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77–161, arXiv:math.AG/0203054.
  * [11] Hertling C., Hoevenaars L., Posthuma H., Frobenius manifolds, projective special geometry and Hitchin systems, J. Reine Angew. Math. 649 (2010), 117–165, arXiv:0905.3304.
  * [12] Hertling C., Manin Yu., Weak Frobenius manifolds, Int. Math. Res. Not 1999 (1999), 277–286, arXiv:math.QA/9810132.
  * [13] Hertling C., Manin Yu., Unfoldings of meromorphic connections and a construction of Frobenius manifolds, in Frobenius Manifolds, Aspects Math., Vol. 36, Friedr. Vieweg, Wiesbaden, 2004, 113–144, arXiv:math.AG/0207089.
  * [14] Hertling C., Sabbah C., Examples of non-commutative Hodge structures, J. Inst. Math. Jussieu 10 (2011), 635–674, arXiv:0912.2754.
  * [15] Hertling C., Sevenheck C., Limits of families of Brieskorn lattices and compactified classifying spaces, Adv. Math. 223 (2010), 1155–1224, arXiv:0805.4777.
  * [16] Malgrange B., La classification des connexions irrégulières à une variable, in Mathematics and Physics (Paris, 1979/1982), Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983, 381–399.
  * [17] Malgrange B., Sur les déformations isomonodromiques. I. II. Singularités irrégulières, in Mathematics and physics (Paris, 1979/1982), Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983, 401–438.
  * [18] Malgrange B., Deformations of differential systems. II, J. Ramanujan Math. Soc. 1 (1986), 3–15.
  * [19] Sabbah C., Introduction to algebraic theory of linear systems of differential equations, in Éléments de la théorie des systèmes différentiels. $\mathcal{D}$-modules cohérents et holonomes (Nice, 1990), Travaux en Cours, Vol. 45, Hermann, Paris, 1993, 1–80.
  * [20] Sabbah C., Polarizable twistor $\mathcal{D}$-modules, Astérisque 300 (2005), vi+208 pages, arXiv:math.AG/0503038.
  * [21] Sabbah C., Isomonodromic deformations and Frobenius manifolds. An introduction, Universitext, Springer, London, 2008.
  * [22] Sabbah C., Integrable deformations and degenerations of some irregular singularities, Publ. Res. Inst. Math. Sci., to appear, arXiv:1711.08514.
  * [23] Saito M., Deformations of abstract Brieskorn lattices, arXiv:1707.07480.
  * [24] Varadarajan V.S., Linear meromorphic differential equations: a modern point of view, Bull. Amer. Math. Soc. (N.S.) 33 (1996), 1–42.