Let M be a connected complex manifold, O

Integrable Deformations and Frobenius Manifolds

Tang-Kai Lee 2018.6

0 Basic Notations and Definitions

Let M be a connected complex manifold, O

be the sheaf of holomorphic functions on M, and Z be a smooth hypersurface in M.

• For a holomorphic vector bundle E of rank d over M, it corresponds to a locally free sheaf of O

-modules of rank d, which will be denoted by E , and is also called a “bundle.” We define Θ

to be the one correspond- ing to the tangent bundle TM.

• O

(∗ Z ) is the smallest sheaf containing all O

( kZ )( k ∈ _Z ) as subsheaves, Ω

is the sheaf of holomor- phic k-forms on M, and Ω

(∗ Z ) : = _Ω

⊗

O

(∗ Z )

• A meromorphic bundle M on M with poles along Z is a locally free sheaf of O

(∗ Z ) -modules of finite rank.

• For a holomorphic vector bundle E on X, it corresponds to a meromorphic bundle E (∗ Z ) : = E ⊗

O

(∗ Z ) .

• A meromorphic connection ∇ on a meromorphic bundle M with poles along Z is a C-linear morphism M → _Ω

⊗

M with its entries in Ω

(∗ Z ) with respect to a local frame.

1 Frobenius Structures Induces by Infinitesimal Period Mappings

Infinitesimal period mappings provide a way to construct Frobenius structures from a family of bundles on P

with flat meromorphic connections. In this section, M is always a connected complex manifold.

1.1 Higgs Fields and the Induced Product Structures

Definition 1 (Higgs fields). Let E be a holomorphic vector bundle on M. A Higgs field on E is an O

-linear morphism

Φ : E → _Ω

⊗

E with the integrability condition Φ ∧ _Φ = 0.

For a holomorphic vector field ξ on an open subset U of M, we will write Φ

: E |

→ E |

to denote the restricted morphism contracted with ξ.

Now assume Φ : Θ

→ _Ω

⊗ _Θ

is a Higgs field on the tangent bundle TM. We can view it as a morphism Θ

⊗ _Θ

→ _Θ

and see if it is symmetric.

The Higgs field can define a product structure on TM by ( ξ, η ) 7→ ξ · η : = − _Φ

( η ) . Proposition 1. Φ is symmetric ⇔ the product · is associative and commutative.

Proof: On a local chart we can write Φ = _{∑ Φ}

⊗ dz

where Φ

: = _Φ

. Then the integrability condition is equivalent to that for all i and j, Φ

◦ _Φ

= _Φ

◦ _Φ

. Then if Φ is symmetric,

∂

· ( ∂

· ∂

) = ∂

· ( ∂

· ∂

) = _Φ

( _Φ

( ∂

) = _Φ

( _Φ

( ∂

) = ∂

· ( ∂

· ∂

) = ( ∂

· ∂

) · ∂

.

The commutativity is clear.

1.2 Residue Endomorphisms

Let F be a bundle on M of rank d : = dim M. Then it induces a bundle E : = π

F on P

× M by the canonical projection π : P

× M → M. Assume there is a flat meromorphic connection e ∇ on E with a pole of order 1 along { 0 } × M and a logarithmic pole along { _∞ } × M. We will write E

: = i

E where i

: M ' { 0 } × M ,→

P

× M and likewise for E

. Of course E

' E

' F.

• The restricted connection ∇ and the induced residue endomorphism R

on E

:

In a local chart U × M with U a neighborhood of ∞ ∈ _P

, the connection matrix of e ∇ with respect to a local frame has the form (in the chart ∞ is at z

= 0)

Ω = _Ω

^dz

z

+ ∑

i≥2

Ω

dz

,

where each Ω

has holomorphic entries. Then we define a holomorphic connection ∇ on { 0 } × M, whose local matrix representation is

i

∑

Ω

( 0, z

, · · · , z

) dz

.

Let R

: i

E → i

E be the endomorphism on E

whose local matrix representation is Ω

( 0, z

, · · · , z

) . Fact 1. Regarded as a section of the bundle Hom ( E

, E

) equipped with the natural flat connection induced by ∇ _{, R}

is a horizontal section.

• The induced Higgs field Φ and the residue endomorphism R

on E

:

In a local chart U

× M with U

a neighborhood of 0 ∈ _P

, the connection matrix of e ∇ with respect to a local frame has the form

Ω

= ¹

z

( _Ω

^dz

z

+ ∑

i≥2

Ω

dz

) ,

where each Ω

has holomorphic entries. Then we define an endomorphism-valued 1-form Φ : E

→ _Ω

⊗ E

whose local matrix representation is

i

∑

Ω

( 0, z

, · · · , z

) dz

.

Let R

: i

E → i

E be the endomorphism on E

whose local matrix representation is Ω

( 0, z

, · · · , z

) .

1.3 Infinitesimal Period Mappings and the Induced Product Structure

Following the setting in the section 1.2, we regard all the objects ∇ , Φ, R

and R

on E

. Besides, we further assume E

has a metric g and they satisfy

∇ g = _{0, Φ}

= _{Φ, R}

= R

and R

+ R

= − w · _id

_(*) for some w where (·)

is the adjoint with respect to the metric, and

∇

= 0, ∇ R

= 0, Φ ∧ _Φ = 0, [ R

, Φ ] = 0, ∇ _Φ = 0 and ∇ R

+ _Φ = [ _{Φ, R}

] . (**) For a ∇ -horizontal section ω of E

, we define the associated infinitesimal period mapping

φ

: TM → E

ξ 7→ − _Φ

( ω ) .

Such an ω is called primitive if φ

is an isomorphism, and homogeneous if ω is an eigenvector of R

. Theorem 1. If E

admits a primitive and homogeneous section ω, then φ

equips M a Frobenius structure.

Proof: Since φ

is an isomorphism, we can carry on TM the structures on E

through it.

• The torsion-free flat connection

∇ on TM: For ξ ∈ TM, we define

ω

∇ ξ : = φ

∇( φ

( ξ )) .

Indeed, fix a local ∇ -horizontal frame of E

with a local coordinate z

, · · · , z

, we can write Φ = _{∑ Φ}

⊗ dz

. Then

ω

∇

∂

= φ

(∇

( φ

( ∂

))) = φ

(∇

(− _Φ

( ω ))) , which is symmetric in i and j since ∇ _Φ = 0 and ∇ ω = 0. Thus

∇ is torsion-free.

Remark 1. Note that the torsion-freeness of

∇ is equivalent to the ∇ -horizontality of φ

since

∇ φ

( ξ, η ) = ∇

φ

( η ) + ∇

φ

( ξ ) − φ

([ ξ, η ]) .

• The commutative associative product structure with the

∇ -flat unit: For ξ and η ∈ TM, we define ξ · η : = φ

(− _Φ

( φ

( η )) .

In a local coordinate,

∂

· ∂

= φ

(− _Φ

( φ

( ∂

)) = φ

( _Φ

( _Φ

( ω ))) and the conclusion follows with Φ ∧ _Φ = 0.

Let e : = φ

( ω ) . Then

ω

∇ e = φ

(∇ ω ) = ₀ and for any ξ,

ξ · e = φ

(− _Φ

( φ

( e ))) = φ

(− _Φ

( ω )) = ξ .

• The flat metric

g:

For ξ and η on TM, we define

ω

g ( ξ , η ) : = g ( φ

( ξ ) , φ

( η )) .

Then

∇

g = _{0 since} ∇ g = _{0 and} ∇ φ

= 0. Moreover by the torsion-freeness,

∇ is the Levi-Civita connection of

g.

• The euler vector field E: Let E : = φ

( R

( ω )) and say R

ω = − qω by the homogeneity.

(1)

∇(

∇ E ) = 0:

Locally

ω

∇

( E ) = φ

(∇

( R

( ω )))

= φ

( ∂

( R

)( ω ))

= φ

(([ _Φ

, R

] − _Φ

)( ω ))

= φ

((− ₁ − q ) _Φ

( ω ) − R

( _Φ

( ω )))

= ( 1 + q ) ∂

+

R

( ∂

)

by ∇ R

+ _Φ = [ _{Φ, R}

] , where

R

: = φ

◦ R

◦ φ

. Hence

∇( E ) = ( 1 + q ) id

+

R

. By ∇ R

= _0, ∇ φ

= 0 and the torsion-freeness of

∇ , we have

ω

∇

R

( ξ, η ) = φ

(∇( R

◦ φ

)( ξ, η ))

= φ

(∇

( R

( φ

( η ))) − ∇

( R

( φ

( ξ ))) − R

( φ

([ ξ , η ])))

= φ

( R

(∇

( φ

( η )) − ∇

( φ

( ξ ))) − R

( φ

([ ξ, η ])))

= φ

( R

( φ

([ _{ξ, η} ])) − R

( φ

([ _{ξ, η} ]))) = _0.

Thus

∇

R

= 0, so

∇(

∇ E ) = 0.

We have the new relations from the old ones (*):

ω

∇

g = 0, Φ

= _Φ, (

R

)

=

R

and

R

+ (

R

)

= − w · id

where the adjoint is respect to the metric

g. Note these imply the symmetry of

∇ c where c ( ξ

, ξ

, ξ

) : =

ω

g ( ξ

· ξ

, ξ

) _.

(2) L

(

g ) = D ·

g for some D, where L means the Lie derivative:

From (1), we have

ω

∇ E = ( 1 + q ) id

+

R

. Then taking the adjoint gives

(

∇ E )

= ( 1 + q ) id

+ (

R

)

= ( 1 + q − w ) id

−

R

. Then for ξ and η in TM,

ω

g (

∇

E, η ) +

g ( ξ,

∇

E ) =

g ((

∇ E )( ξ ) , η ) +

g ((

∇ E )

( ξ ) , η ) = ( 2 + 2q − w )

g ( ξ, η ) . Let D : = ₂ + _2q − w and we have

L

(

g )( ξ , η ) = E (

g ( ξ, η )) −

g ( L

ξ , η ) −

g ( ξ , L

η )

= E (

g ( ξ, η )) −

g (

∇

ξ −

∇

E, η ) −

g ( ξ ,

∇

η −

∇

E )

= E (

g ( ξ, η )) −

g (

∇

ξ, η ) −

g ( ξ,

∇

η ) +

g (

∇

E, η ) +

g ( ξ,

∇

E )

=

∇

(

g )( ξ, η ) + ( 2 + 2q − w )

g ( ξ, η )

= D ·

g ( ξ , η )

(3) L

(·) = · where · means the product structure:

First we claim that

R

: = φ

◦ R

◦ φ

is exactly the endomorphism ξ 7→ ξ · E. Indeed, by [ R

, Φ ] = 0, ξ · E = φ

(− _Φ

( φ

( φ

( R

( ω )))) = φ

(− R

( _Φ

( ω ))) = φ

( R

( φ

( ξ ))) =

R

( ξ ) . Since ∇ φ

= 0, the old relation ∇ R

+ _Φ = [ _{Φ, R}

] gives, after composing with φ

and φ

,

ω

∇

( η · E ) − (

∇

) · E − ξ · η = ξ · (

∇

E − ( 1 + q ) η ) − (

∇

E − ( 1 + q ) ξ · η ) = ξ ·

∇

E −

∇

E.

By ∇ _Φ = 0, the above result simplifies to

L

( ξ · η ) − ( L

ξ ) · η − ξ · ( L

η ) = ξ · η.

Thus the theorem follows.

2 Universal Semisimple Frobenius Structures

We aim at establishing the following theorem.

Theorem 2 ([Dub96]). There is a one-to-one correspondence

{ semisimple simply connected Frobenius manifolds } ↔ {( B

, B

, ω

, U ) satisfying the (?) conditions } with the (?) conditions that B

is regularly semisimple, that B

+ B

= wI

for some w ∈ _{Z, that w}

is an eigenvector of B

, whose components don’t vanish on the eigenbases of B

, and that U is a simply connected open set of e X

\ _Θ

.

In the theorem, X

: = {( x

, · · · , x

) ∈ _C

| x

6= x

for all i < j } and e X

is its universal cover. Fix x

= ( x

, · · · , x

) ∈ X

and a lifted point e x

∈ X e

, i.e., π ( x _e

) = x

where π : e X

→ X

is the covering map.

Proof: Suppose we are given B

+ B

= wI

for some w ∈ _{Z, B}

= diag ( x

, · · · , x

) , thus regularly semisimple, and an eigenvector ω

of B

, all components of which are non-zero.

Theorem 3 ([Mal83]). Given such B

and B

, there exist a unique holomorphic bundle E on P

× X e

and a flat meromorphic connection ∇ with a pole of order 1 along { 0 } × X e

and a logarithmic pole along { _∞ } × X e

, such that

(1) the restriction ( E

, ∇

) of ( E, ∇) at x e

has a global frame with respect to which the matrix representation of ∇

is

( ^B

0o

z + B

) ^dz z ;

(2) for any x e ∈ X e

, the eigenvalues of the residue endomorphism R

at e x are the components of π ( x _e ) .

Theorem 4 ([Sab08]). Let X be a connected complex analytic manifold and F a holomorphic vector bundle on P

× X such that for any x ∈ X, the restriction F |

has degree 0.

(1)(The nontriviality divisor) The set

Θ : = { x ∈ M : F |

is non-trivial } is ∅, X or a hypersurface of X.

(2)(The canonical identification between the restriction to 0 and ∞) We have i

F |

' i

F |

.

Theorem 5. ([Sab17]) Let X be a simply connected complex manifold and ( F, ∇) a bundle on D × X with a pole of order 1 along { 0 } × X. Suppose R

is the residue endomorphism and ( F, ˆ ^ˆ ∇) is its associated formal bundle.

(1)(the unique decomposition) If R

is regularly semisimple, ( F, ˆ ^ˆ ∇) has a unique decomposition to line bundles

( F, ˆ ^ˆ ∇) '

j

( F ^ˆ

, ˆ ∇) .

(2)(equivalence) For line bundles, the formalism ( F, ∇) 7→ ( F, ˆ ^ˆ ∇) is an equivalence of categories.

By the theorem 3, we can, following the section 1.3, obtain ∇ and R

on E

, Φ and R

on E

. Via the theorem 4, we get a bundle E on e X

\ Θ with objects ∇ , Φ, R

and R

.

Since e X

is simply connected and ∇ is a flat connection on E

, it’s trivial and thus we can find a ∇ -horizontal ω on e X

such that ω ( x _e

) = ω

. Later we will let ω be its restriction to e X

\ _Θ.

By the theorem 3 again, the residue R

is regular semisimple everywhere, so E

, on { 0 } × X e

, can be decom- posed to a direct sum of eigenbundles of rank one, each of which can be equipped with a flat connection by the theorem 5, and hence admits a global frame. We collect these d section, forming a global frame e = { e

, · · · , e

} of E

.

Restrict the frame on e X

\ Θ, also denoted by e, and let ω

be the components of ω with respect to e. We set Θ

: = _Θ ∪

d

[

i=1

{ the zero locus of ω

}
_.

By our definition of Θ

, the sections

u

: = ω

e

form a basis of E |

Xed\_Θωo

. Then the infinitesimal period mapping associated to ω gives φ

: T ( X e

\ _Θ

) → E |

Xed\_Θωo

∂

7→ − _Φ

( ω ) = u

where the fact that − _Φ

( ω ) = u

comes from the matrix representation of Φ with respect to e, which will be explained in the proceeding sections.

Therefore, φ

is an isomorphism, and by the construction in the section 1.3, e X

\ _Θ

admits a Frobenius structure. Note that we have the unit

e = φ

( ω ) = φ

( ∑ ^u

^{) =} ∑ ^∂

and the Euler vector field

E = φ

( R

( ω )) = φ

( ∑ ^x

^u

^{) =} ∑ ^x

^∂

for the matrix representation of R

with respect to e is diag ( x

, · · · , x

) , which will also be explained.

Besides, we have

∂

· ∂

= φ

(− _Φ

( φ

( ∂

))) = φ

(− _Φ

( u

)) = φ

( δ

u

) = δ

∂

. This proves one direction of the theorem.

Remark 2. We didn’t check that the objects satisfy the condition (*) and (**), which would be clear after we

show the solvability of the Birkhoff’s problem in a family.

For the other way around, let M be a semisimple simply connected Frobenius manifold. Then we can define Φ by Φ

( η ) _: = − ξ · η, R

: = − _Φ ( E ) _{, and R}

_: = ∇ E. The semisimplicity means that at each point R

is regularly semisimple, so its eigenvalues define d functions ( x

, · · · , x

) : M → X

.

Theorem 6 ([Mal83]). Let X be a simply connected complex manifold with a fixed base point x

∈ X, λ

, · · · _{, λ}

d holomorphic functions X → C such that λ

( x ) 6= λ

( x ) _{for all i} 6= j and x ∈ X, ( E

, ∇

) _a bundle on D with a connection having a pole of order 1 at the origin, and the residue R

whose eigenvalues are λ

( x

) , · · · , λ

( x

) . Then there exists a unique bundle ( E, ∇) on D × X with a connection having a pole of order 1 along { ₀ } × X such that

(1) for any x ∈ X, R

( x ) has eigenvalues λ

( x ) , · · · , λ

( x ) , and (2) ( E, ∇)|

' ( E

, ∇

) .

By the theorem 6, we can as above construct a basis e = { e

· · · , e

} , with respect to which the matrix of Φ is exactly − dR

(also will be clarified later), i.e.,

Φ ( e

) = − dx

⊗ e

, therefore, for all i and j,

e

· e

= − _Φ

e

= L

( x

) · e

.

By the commutativity of the product, L

( x

) = 0 for i 6= j. Besides, λ

: = L

( x

) is non-vanishing:

Write the unit vector field e as e = _{∑ a}

e

. Then

e

= e

· e = e

· ∑ ^a

^e

^{= a}

^λ

^.

Thus a

λ

≡ 1 so λ

is non-vanishing.

Now we get a holomorphic map

( x

, · · · , x

) : M → X

. Since M is simply connected, it can be lifted to

f : = ( x

, · · · , x

) : M → X e

,

which is a submersion, thus an open map, so it is proper. Since it’s a proper local homeomorphism between locally compact Hausdorff spaces, it’s a covering map, with the number of sheets

[ π

( f ( M )) _{: π}

( M )] = ₁

because M is simply connected and the image of a simply connected domain under a biholomorphic map is simply connected. Thus, f is isomorphic to an open subset of e X

, and the conclusion of the theorem 2 follows.

Remark 3. Note that by the canonicality of the product structure defined above, this (· , e, E ) is independent of the choice of ω

with the non-zero condition.

3 Birkhoff’s Problem on P

In this section, X is a simply-connected complex analytic manifold of dimension n.

We write P

= U

∪ U

with U

: = _P

\{ _∞ } and U

: = _P

\{ 0 } and let τ and τ

be the coordinate on them respectively.

Let D = B

( 0 ) be an open disc in U

' C for some r > 0, and ( E, e e ∇) be a holomorphic bundle of rank d on U : = D × X, with a flat meromorphic connection e ∇ having a pole of order 1 along { 0 } × X, in the sense that its restriction to ( U

∩ D ) × X is a flat connection on the holomorphic bundle e E |

_.

Fix a point x

∈ X and let (( E e

, e ∇

) be the restriction of ( E, e e ∇) to U

: = _D × { x

} .

Definition 2 (Birkhoff’s problem). Let A ( τ ) be a d × d matrix function on D with τ

A ( τ ) having holomor- phic entries, one of which doesn’t vanish at τ = 0, for some r ≥ 0. We say that Birkhoff’s problem can be solved for A ( τ ) if there exists some P ( τ ) ∈ GL

(O

) such that the matrix B ( τ ) : = P

AP + P

P

can be written as

B ( τ ) = ^B

τ

+ · · · + ^B

τ

for some B

, · · · , B

∈ M

( _C ) _.

The following theorem says that there’s a canonical solution to Birkhoff’s problem for almost every values in P

if we are given a solution for a particular value.

Theorem 7. Assume Birkhoff’s problem can be solved for ( E e

, e ∇

) at x

, i.e., we can take a global frame e

of e E

with respect to which the connection matrix of e ∇

can be written as

Ω

= ( ^B

0o

τ + B

) ^dτ τ

for some B

and B

∈ M

( _C ) . Then there exist a hypersurface Θ of X not containing x

and a unique basis e of e E (∗( D × _Θ )) which coincides with e

at x

and in which the connection matrix of e ∇ takes the form

Ω = ( ^B

( x )

τ + B

) ^dτ

τ + ^C ( x ) τ

where B

( x ) and C ( x ) are a meromorphic matrix function and 1-form on X and holomorphic on X \ _{Θ with} B

( x

) = B

.

Proof:

First we prove the existence.

• Constructions of a bundle on P

× X by sheaves gluing:

Let D

⊆ U

be an open disc centered at ∞ such that A : = D ∩ D

6= ∅. Recall τ

is the coordinate on D

, i.e., τ

= 1/τ on A. On A × X, since ( E, e e ∇)|

is a flat bundle, it is determined by its monodromy. On the other hand, for X is simply-connected, π

( A × X ) = π

( A ) _{. Thus} ( E, e e ∇)|

is isomorphic to p

( E e

, e ∇

)|

where p is the projection A × X → A.

By the change of variable, the restriction of the trivial bundle O

with the connection matrix

−( τ

B

+ B

) ^dτ

0

τ

to A × X is isomorphic to ( E, e e ∇)|

since they are both isomorphic to p

( E e

, e ∇

)|

. Thus we can glue them up to get a bundle ( E, ∇) on P

× X with a flat meromorphic connection having poles on { 0, ∞ } × X.

• The nontriviality divisor Θ of X :

Since E |

= E

is trivial, its degree is 0 so we can apply the theorem 4 of the nontriviality divisor to X and get Θ ( X because x

6∈ _Θ.

• Extensions of a basis:

Since ∇ has a logarithmic pole along { _∞ } × X, there’s an induced holomorphic flat connection ∇

on i

E . Fact 2 (the canonical identification between the restrictions to 0 and ∞). Let π be the projection P

× X → X.

Then for E above, there exist canonical isomorphisms

E (∗ π

Θ ) ' π

i

E (∗ π

Θ ) ' π

i

E (∗ π

Θ ) where i

: { 0 } × X ,→ _P

× X and i

: { _∞ } × X ,→ _P

× X.

By the triviality of i

E and the above isomorphisms, we can extend e

to a basis e of the bundle E (∗ π

Θ ) .

• On U the connection matrix of ∇ takes the desired form.

Since the connection matrix Ω has order 1 at { 0 } × X in the basis e, it can be written as B

( x )

τ + B

( τ, x )
^dτ

τ + C

( τ, x ) + ^C ( x ) τ where B

and C

are holomorphic function and 1-form.

Note after a change of variables, at infinity the connection matrix is of the form (− B

( x ) τ

− B

( ¹

τ

, x )) ^dτ

0

τ

+ C

( ¹

τ

, x ) + C ( x ) τ

.

Since the connection has a logarithmic pole at { _∞ } × X, we have B

and C

are independent of τ

, i.e., independent of τ.

Restricted to infinity, the basis e is ∇

-horizontal, where the connection matrix is by definition C

( x ) , so C

( x ) = 0.

By the fact 1, because the endomorphism R

is horizontal, we have

∇( R

e

) = R

(∇ e

)

for i = 1, · · · , d if the basis e consists of { e

, · · · , e

} . Note the matrix representation of R

with respect to the basis is (− B

( x ) τ

− B

( x ))|

= − B

( x ) . Thus if we write B

: = the i-th column of − B

( x ) ,

∇( R

e

) = ∇( e · B

) = (∇ e ) · B

+ e · dB

and

R

(∇ e

) = (∇ e ) · B

.

where e : = ( e

· · · e

) and ∇ e : = (∇ e

· · · ∇ e

) . Thus dB

( x ) = 0 for all i, so B

is constant in x. Hence we get the desired form

( ^B

( x ) τ

+ B

) ^dτ τ

+ ^C ( x ) τ .

For the uniqueness, suppose ( e

, Θ

) and ( e

, Θ

) satisfy the theorem. Let Θ : = _Θ

∪ _Θ

and X

: = X \ _Θ.

Then we get an isomorphism

O

(∗( D × _Θ )) → O

(∗( D × _Θ ))

via the base change P. Since the induced homomorphism π

( D × X

) → π

( _P

× X

) is an isomorphism, by the proof of existence, P can be extended holomorphicly to the isomorphism between O

(∗( _P

× _Θ )) . Note if the matrix of the two connections are Ω and Ω

, we have Ω

= P

ΩP + P

dP, i.e.,

dP = _PΩ

− _ΩP.

Thus if we write P as column vectors, we can write the system as dP = (( A

+ τ

A

( x )) ^dτ

0

τ

+ D ( x ) τ

) · P near the infinity. If write P ( x ) = _∑

l≥l0

τ

P

( x ) , then we have dP

= DP

.

Suppose l

< 0, then dP

= DP

= 0, so P

≡ P

( x

) = 0 since at x

P is identity. Hence the entries of P are holomorphic. Hence P ( τ

, x ) = P

( x ) . Besides, dP

= DP

= 0, so P ( τ

, x ) = P

( x ) ≡ P

( x

) = id.

Remark 4. With respect to the basis e, the matrix representation connection takes the form above, that of R

is B

( x ) , that of R

is − B

, and that of Φ is C ( x ) . Besides, the integrability condition exactly tells

dC = 0, C ∧ C = 0, [ B

, C ] = 0, and dB

+ C = [ B

, C ] , which gives the condition (**) in the section 1.3.

We use the idea of this proof to prove the theorem 3 that was applied.

Theorem 3. Given x

= ( x

, · · · , x

) = π ( x _e

) ∈ X

and B

: = diag ( x

, · · · , x

) , B

∈ M

( _C ) , there exist a unique holomorphic bundle E on P

× X e

and a flat meromorphic connection ∇ with a pole of order 1 along { 0 } × X e

and a logarithmic pole along { _∞ } × X e

, such that

(1) the restriction ( E

, ∇

) of ( E, ∇) at x e

has a global frame with respect to which the matrix representation of ∇

_is

( ^B

o0

τ

+ B

) ^dτ τ ;

(2) for any x e ∈ X e

, the eigenvalues of the residue endomorphism R

at e x are the components of π ( _{x e} ) _. Proof:

We will again apply the theorem 6:

Theorem 6. Let X be a simply connected complex manifold with a fixed base point x

∈ X, λ

, · · · , λ

d holomorphic functions X → C such that λ

( x ) 6= λ

( x ) _{for all i} 6= j and x ∈ X, ( E

, ∇

) _{a bundle} on D with a connection having a pole of order 1 at the origin, and the residue R

whose eigenvalues are λ

( x

) , · · · , λ

( x

) . Then there exists a unique bundle ( E, ∇) on D × X with a connection having a pole of order 1 along { ₀ } × X such that

(1) for any x ∈ X, R

( x ) has eigenvalues λ

( x ) , · · · , λ

( x ) , and (2) ( E, ∇)|

' ( E

, ∇

) .

To begin with, let D be an open disk centered at the origin in P

, and let ( E

, ∇

) be the trivial bundle O

with the connection matrix of ∇

being

( ^B

o0

τ + B

) ^dτ τ .

Define λ

: = p

◦ π where p

is the projection ( x

, · · · , x

) ∈ X

7→ x

∈ C. Then by the theorem 6 we get a bundle ( E, e e ∇) on D × X e

. Then the argument of the previous proof works.

4 Universal Integrable Deformations for Birkhoff’s Problems

Let ( E

, ∇

) be a bundle on P

with a pole of order one along 0 and a logarithmic pole along ∞.

• Let X be a complex manifold. An integrable deformation of ( E

, ∇

) parametrized by ( X, x

) is a bun- dle ( E, ∇) with a flat meromorphic connection on P

× X with a pole of order one along { 0 } × X and a logarithmic pole along { _∞ } × X such that ( E, ∇)|

= ( E

, ∇

) .

• An integrable deformation ( _E, ∇) of ( _E

_, ∇

) is called complete at x

if for any other integrable deforma- tion ( E

, ∇

, x

) of ( E

, ∇

) parametrized by ( X

, x

) , there exist neighborhoods V and V

of x

and x

and an analytic map f : ( V

, x

) → ( V, x

) such that ( E

, ∇

)|

= ( id

× f )

( E, ∇)|

. Moreover, such a deformation is called universal at x

if such an f is unique.

4.1 Local Universal Deformations

Theorem 8. Let B

, B

∈ M

( _C ) and ( E

, ∇

) be the trivial bundle of rank d on P

with the connection matrix

Ω

= ( ^B

0o

τ + B

) ^dτ τ

in the canonical basis. If the matrix B

is regular, i.e., its all eigenvalues have one Jordan block, then there exists a germ of universal deformation of ( E

, ∇

) .

Proof: Inspired by the ideas of the theorem 7, we consider the system, near x

in a manifold X,

dC = 0, [ B

, C ] = 0, and dB

+ C = [ B

, C ] (***) with B

( x

) = B

. Locally dC ( x ) = 0 can be solved by C ( x ) = _dΓ ( x ) with Γ ( x

) = 0. Hence the system is equivalent to

[ B

, dΓ ] = _{0 and d} ( B

+ _Γ ) = [ B

, dΓ ] _.

Note the second condition is exactly B

= B

− _Γ + [ B

, Γ ] _{since Γ} ( x

) = 0, the system reduces to [ B

− _Γ + [ B

, Γ ] _{, dΓ} ] = 0.

• The system above is integrable on M

( _C ) : Consider vectors

ξ

= ∑

i,j

ξ

∂

∂γ

and ξ

= ∑

i,j

ξ

∂

∂γ

.

Then to show the system [ B

, dΓ ] = 0 is integrable, it suffices to verify that for any ξ

and ξ

annihilated by [ B

, dΓ ]

for all m and n, ( d [ B

, dΓ ])

( ξ

, ξ

) = 0 for all m and n also. Thus first we calculate (remember B

= B

− _Γ + [ B

, Γ ] )

d [ B

, dΓ ] = dB

∧ d Γ + d Γ ∧ dB

= − _2dΓ ∧ d Γ + ₂ [ B

, dΓ ∧ d Γ ] _.