DDEFORMATIONS OF FREE AND LINEAR FREEDIVISORS
MICHELE TORIELLI
Abstract.
We investigate deformations of free and linear free di-visors. We introduce a complex similar to the de Rham complexwhose cohomology calculates deformation spaces. This cohomol-ogy turns out to be zero for all reductive linear free divisors andto be constructible for Koszul and weighted homogeneous free di-visors.
Contents
1. Introduction 12. Basic notions 33. Deformation theory for free divisors 63.1. Admissible and linearly admissible deformations 63.2. The complexes C • and C • Introduction
In this article, we develop some ideas of a deformation theory forgerms of free and linear free divisors. Free divisors were introducedby K. Saito in [14] and linear free divisors by R.-O. Buchweitz and D.Mond in [2]. Free divisors are quite fundamental in singularity the-ory, for example, the discriminants of the versal unfoldings of isolatedhypersurfaces and complete intersection singularities are always freedivisors.
Date : November 13, 2018. a r X i v : . [ m a t h . AG ] S e p MICHELE TORIELLI
A reduced divisor D = V ( f ) ⊂ C n is free if the sheaf Der( − log D ) := { δ ∈ Der C n | δ ( f ) ∈ ( f ) O C n } of logarithmic vector fields is a locallyfree O C n -module, where Der C n denote the space of vector fields on C n . It is linear if, furthermore, Der( − log D ) is globally generatedby a basis consisting of vector fields all of whose coefficients, withrespect to the standard basis ∂/∂x , . . . , ∂/∂x n of the space Der C n , arelinear functions. The simplest example is the normal crossing divisor,but the main source of examples, motivating Saito’s definition, hasbeen deformation theory, where discriminants and bifurcation sets arefrequently free divisors.These objects have been studied for the past 30 years but there isstill a lot to learn and discover about them. One interesting fact isthat there are no examples of linear free divisors in non-trivial family.One possible approach is to deform this object in such way that eachfiber of the deformation is a (linear) free divisor and that the singularlocus is deformed flatly. However, not much is known on the behaviorof (linear) free divisors under these kind of deformations.The aim of this article is to describe the spaces of infinitesimal de-formations and obstructions of a germ of a (linear) free divisor andto perform calculations for some concrete examples. It turns out thatthe property of being a free divisor for a hypersurface D has a stronginfluence on its deformations, in fact all free divisors D ⊂ C n , with n ≥
3, are non-isolated singularities and so their space of first orderinfinitesimal deformations is infinite dimensional, but in what followswe will show examples of free divisors which have a finite dimensionalversal deformation space as free divisors.We now give an overview on the paper. The first part recalls thenotions of free and linear free divisors, and describes some of theirproperties. In the second, we define the notion of (linearly) admissibledeformations for a germ of a (linear) free divisor and we introduce acomplex similar to the de Rham complex whose cohomology calculatesdeformations spaces. In this section we also prove our main result:
Theorem A.
All germs of reductive linear free divisors are formallyrigid.
This is equivalent saying that for a germ of a reductive linear freedivisor, there are no non-trivial families, at least on the level of formalpower series.Then, we analyse the weighted homogeneous case and we prove oursecond result:
Theorem B. If ( D, ⊂ ( C n , is a germ of a weighted homogeneousfree divisor, then it has a hull, i.e., it has a formally versal deformation. EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 3
In the last part, we describe some properties of this cohomology andwe prove our third result:
Theorem C. If ( D, ⊂ ( C n , is a germ of a Koszul free divisor suchthat we can put a logarithmic connection on Der C n and Der( − log D ) ,then it has a hull. This theory owes a lot to the theory of deformations of Lagrangiansingularities as developed in [17], [16] and [18].The material in this article is part of the author Ph.D. thesis [19].
Acknowledgements . The author is grateful to Michel Granger, DavidMond, Luis Narv´aez Macarro, Brian Pike, Miles Reid, Christian Seven-heck and Duco van Straten for helpful discussions on the subject of thisarticle. We thank the anonymous referee of the Annales de l’InstitutFourier for a careful reading of our draft versions and a number of veryhelpful remarks. 2.
Basic notions
Fix coordinates x , . . . , x n on C n . Definition 2.1.
A reduced divisor D = V ( f ) ⊂ C n is called free if thesheaf Der( − log D ) := { δ ∈ Der C n | δ ( f ) ∈ ( f ) O C n } of logarithmic vector fields is a locally free O C n -module. Definition 2.2.
Let D = V ( f ) ⊂ C n be a reduced divisor. Then for q = 0 , . . . , n , we define the sheaf Ω q (log D ) := { ω ∈ Ω q C n [ (cid:63)D ] | f ω ∈ Ω q C n , f dω ∈ Ω q +1 C n } of q -forms with logarithmic poles along D . Note that by definition, Ω (log D ) = Ω C n and Ω n (log D ) = 1 f Ω n C n . Lemma 2.3. ([14], Lemma 1.6)
By the natural pairing
Der p ( − log D ) × Ω p (log D ) −→ O C n ,p defined by ( δ, ω ) (cid:55)→ δ · ω, each module is the O C n ,p -dual of the other. Corollary 2.4. Ω p (log D ) and Der p ( − log D ) are reflexive O C n ,p -modules.In particular, when n = 2 , then Ω p (log D ) and Der p ( − log D ) are free O C ,p -modules. Definition 2.5.
A free divisor D is linear if there is a basis for Γ( C n , Der( − log D )) as C [ C n ] -module consisting of vector fields all of whose coefficients, withrespect to the standard basis ∂/∂x , . . . , ∂/∂x n of the space Der C n , arelinear functions, i.e. they are all homogeneous polynomials of degree . MICHELE TORIELLI
Remark 2.6.
With respect to the standard grading of
Der C n , i.e., deg x i = 1 and deg ∂/∂x i = − for every i = 1 , . . . , n , such vectorfields have weight zero. Definition 2.7.
We denote by
Der( − log D ) the finite dimensionalLie subalgebra of Der( − log D ) consisting of the weight zero logarithmicvector fields. There is a nice criterion to understand easily if a divisor is free ornot:
Proposition 2.8. (SAITO’S CRITERION) ([14], Theorem 1.8) i) Thehypersurface D ⊂ C n is a free divisor in the neighbourhood of a point p if and only if (cid:86) n Ω p (log D ) = Ω np (log D ) , i.e. if there exist n elements ω , . . . , ω n ∈ Ω p (log D ) such that ω ∧ · · · ∧ ω n = α dx ∧ · · · ∧ dx n f where α is a unit. Then the set of forms { ω , . . . , ω n } form a basis for Ω p (log D ) . Moreover, we have Ω qp (log D ) = (cid:77) i < ···
In the notation of Proposition 2.8, the matrix [ χ , . . . , χ n ] is called a Saito matrix . Lemma 2.10. ([14], Lemma 1.9)
Let δ i = (cid:80) nj =1 a ji ( x ) ∂/∂x j , i =1 , . . . , n , be a system of holomorphic vector fields at p such that(1) [ δ i , δ j ] ∈ (cid:80) nk =1 O C n ,p δ k for i, j = 1 , . . . , n ;(2) det( a ji ) = f defines a reduced hypersurface D .Then for D = { f ( x ) = 0 } , δ , . . . , δ n belong to Der p ( − log D ) , andhence { δ , . . . , δ n } is a free basis of Der p ( − log D ) . There is also an algebraic version of Saito’s criterion that does notrefer to vector fields directly but characterizes the Taylor series of thefunction f defining a free divisor: EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 5
Proposition 2.11. ([2], Proposition 1.3)
A formal power series f ∈ R = C [[ x , . . . , x n ]] defines a free divisor, if it is reduced, i.e. squarefree,and there is an n × n matrix A with entries from R such that det A = f and ( ∇ f ) A ≡ (0 , . . . ,
0) mod f, where ∇ f = ( ∂f /∂x , . . . , ∂f /∂x n ) is the gradient of f , and the lastcondition just expresses that each entry of the vector ( ∇ f ) A is divisibleby f in R. The columns of A can then be viewed as the coefficients ofa basis, with respect to the derivations ∂/∂x i , of the logarithmic vectorfields along the divisor f = 0 . Example 2.12.
The normal crossing divisor D = { x · · · x n = 0 } ⊂ C n is a linear free divisor; Der( − log D ) has basis x ∂/∂x , . . . , x n ∂/∂x n .Up to isomorphism it is the only example among hyperplane arrange-ments, see [12] , Chapter 4. Remark 2.13.
Let D ⊂ C n be a divisor defined by a homogeneous poly-nomial f ∈ C [ x , . . . x n ] of degree n. Then for each δ ∈ Der( − log D ) ,there is a n × n matrix A with entries in C , such that δ = xA∂ t , where ∂ t is the column vector ( ∂/∂x , . . . , ∂/∂x n ) t . Remark 2.14.
Let D ⊂ C n be a free divisor. D is a linear if and onlyif Der( − log D ) = O C n · Der( − log D ) . Definition 2.15.
Let D = V ( f ) ⊂ C n be a linear free divisor. Definethe subgroup G D := { A ∈ GL n ( C ) | A ( D ) = D } = { A ∈ GL n ( C ) | f ◦ A ∈ C · f } with identity component G ◦ D and Lie algebra g D . Lemma 2.16. ([8], Lemma 2.1) G ◦ D is an algebraic subgroup of GL n ( C ) and g D = { A | xA t ∂ t ∈ Der( − log D ) } . Definition 2.17.
Let D ⊂ C n be a linear free divisor. We call D reductive if g D is a reductive Lie algebra. From § Lemma 2.18.
Let D = V ( f ) ⊂ C n be a reductive linear free divisor.Then Aut( f ) ⊂ SL n ( C ) . This means that if χ ∈ Ann( D ) := { δ ∈ Der( − log D ) | δ ( f ) = 0 } then trace( δ ) = 0 . Example 2.19. i) The normal crossing divisor of Example 2.12is a reductive linear free divisor because g D = C n . MICHELE TORIELLI ii) Consider the divisor D = V (( y + xz ) z ) ⊂ C . This is a linearfree divisor because we can take the matrix A = x x − yy y zz − z as its Saito matrix. Moreover, if we consider σ the second col-umn of A , i.e. σ = 4 x∂/∂x + y∂/∂y − z∂/∂z , we have that σ ∈ Ann( D ) and trace( σ ) = 3 and hence by Lemma 2.18, D isa non-reductive linear free divisor. Lemma 2.20. ([8], Lemma 3.6, (4))
Let D ⊂ C n be a linear freedivisor. If g D is reductive then G ◦ D is reductive as algebraic group. Definition 2.21.
Let S be a complex space. Then Der C n × S/S is the setof vector fields on C n × S without components in the S direction. It isa submodule of Der C n × S . Definition 2.22.
Let S be a complex space and let D ⊂ C n × S be a di-visor. Then Der( − log D/S ) := { δ ∈ Der( − log D ) | δ ∈ Der C n × S/S } =Der( − log D ) ∩ Der C n × S/S . Remark 2.23.
Der C n × S/S and
Der( − log D/S ) are both coherent sheavesof O C n × S -modules. Deformation theory for free divisors
The aim of this section is to introduce the notion of (linearly) ad-missible deformation for germs of (linear) free divisors and then studyinfinitesimal ones in order to prove that reductive linear free divisorsare formally rigid.3.1.
Admissible and linearly admissible deformations.Definition 3.1.
Let ( D,
0) = ( V ( f ) , ⊂ ( C n , be a germ of a free di-visor and let ( S, s ) be a complex space germ. An admissible deformation of ( D, over ( S, s ) consists of a flat morphism φ : ( X, x ) −→ ( S, s ) ofcomplex space germs, where ( X, x ) ⊂ ( C n × S, (0 , s )) , together with anisomorphism from ( D, to the central fibre of φ, ( D, −→ ( X s , x ) :=( φ − ( s ) , x ) , such that (1) Der( − log X/S ) / m S,s
Der( − log X/S ) = Der( − log D ) where m S,s is the maximal ideal of O S,s .Moreover, if ( D, is linear, we define a linearly admissible defor-mation of ( D, over ( S, s ) as an admissible deformation of ( D, over EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 7 ( S, s ) such that there exists a basis of Der( − log X/S ) as O C n × S, (0 ,s ) -module consisting of vector fields all of whose coefficients are linear in x , . . . , x n . Definition 3.2.
In Definition 3.1, ( X, x ) is called the total space , ( S, s ) the base space and ( X s , x ) ∼ = ( D, the special fibre of the (lin-early) admissible deformation. We can write a (linearly) admissible deformation as a commutativediagram(2) ( D, (cid:15) (cid:15) (cid:31) (cid:127) i (cid:47) (cid:47) ( X, x ) φ (cid:15) (cid:15) {∗} (cid:31) (cid:127) (cid:47) (cid:47) ( S, s )where i is a closed embedding mapping ( D,
0) isomorphically onto( X s , x ). We will denote a (linearly) admissible deformation by( i, φ ) : ( D, (cid:31) (cid:127) i (cid:47) (cid:47) ( X, x ) φ (cid:47) (cid:47) ( S, s ) . Definition 3.3.
Given two (linearly) admissible deformations ( i, φ ) : D (cid:44) → X −→ S and ( j, ψ ) : D (cid:44) → Y −→ T , of D over S and T respectively.A morphism of (linearly) admissible deformations from ( i, φ ) to ( j, ψ ) is a morphism of the diagram (2) being the identity on D −→ {∗} .Hence, it consists of two morphisms ( τ, σ ) such that the following dia-gram commutes D (cid:110)(cid:78) i (cid:126) (cid:126) (cid:15) (cid:111) j (cid:32) (cid:32) X φ (cid:15) (cid:15) τ (cid:47) (cid:47) Y ψ (cid:15) (cid:15) S σ (cid:47) (cid:47) T Definition 3.4.
Two (linearly) admissible deformations over the samebase space S are isomorphic if there exists a morphism ( τ, σ ) with τ anisomorphism and σ the identity map. We denote by
Art the category of local Artin rings with residue field k and by Set the category of pointed sets with distinguished element ∗ . MICHELE TORIELLI
Definition 3.5.
Let ( D, ⊂ ( C n , be a germ of a free divisor. Definethe functor FD D : Art −→ Set by setting FD D ( A ) := (cid:26) Isomorphism classes of admissibledeformations of ( D, over Spec A (cid:27) . If ( D, ⊂ ( C n , is a germ of a linear free divisor, we define similarlythe functor LFD D : Art −→ Set by setting
LFD D ( A ) := (cid:26) Isomorphism classes of linearlyadmissible deformations of ( D, over Spec A (cid:27) . Theorem 3.6.
Let ( D, ⊂ ( C n , be a germ of a free divisor. Thenthe functor FD D satisfies Schlessinger’s conditions (H1) and (H2) from [15] . Moreover, if ( D, is linear, then also the functor LFD D satisfiesconditions (H1) and (H2).Proof. Let A (cid:48) −→ A and A (cid:48)(cid:48) −→ A be maps in Art such that thelatter is a small extension, see Definition 1.2 from [15]. Consider now X ∈ FD D ( A ) , X (cid:48) ∈ FD D ( A (cid:48) ) and X (cid:48)(cid:48) ∈ FD D ( A (cid:48)(cid:48) ). Define Y :=( D, O X (cid:48) × O X O X (cid:48)(cid:48) ), by [15], Lemma 3.4, it is flat over A (cid:48) × A A (cid:48)(cid:48) and itis an element of FD D ( A (cid:48) × A A (cid:48)(cid:48) ). Hence the map τ A (cid:48) ,A (cid:48)(cid:48) ,A of ( H
1) issurjective.We want to show now that τ A (cid:48) ,A (cid:48)(cid:48) ,A is a bijection in the case A (cid:48)(cid:48) = k [ (cid:15) ]and A = k . Let W ∈ FD D ( A (cid:48) × A A (cid:48)(cid:48) ) restrict to X (cid:48) and X (cid:48)(cid:48) , then we canchoose immersions q (cid:48) : X (cid:48) (cid:44) → W and q (cid:48)(cid:48) : X (cid:48)(cid:48) (cid:44) → W . Since these mapsare all compatible with the immersions from D , they agree with thechosen maps u (cid:48) : X (cid:44) → X (cid:48) and u (cid:48)(cid:48) : X (cid:44) → X (cid:48)(cid:48) , since in this case X = D .Now by the universal property of fibered product of rings, there is amap Y −→ W compatible with the above maps. Since Y and W areboth flat over A (cid:48) × A A (cid:48)(cid:48) , and the map becomes an isomorphism whenrestricted to D , we find that, by [10], Exercise 4.2, Y is isomorphic to W and hence they are equal as elements of FD D ( A (cid:48) × A A (cid:48)(cid:48) ).The previous proof works similarly also for the functor LFD D . (cid:3) Proposition 3.7.
Let ( D, ⊂ ( C n , be a germ of a free divisor.Then in any admissible deformation the singular locus of ( D, is de-formed in a flat way.Proof. Let f ∈ O C n , be a defining equation for ( D,
0) and let φ : ( X, x ) −→ ( S, s ) be a admissible deformation of ( D, − log D )can be seen as a relation among f, ∂f /∂x , . . . , ∂f /∂x n and similarly,any element of Der( − log X/S ) can be seen as a relation among
F, ∂F/∂x , . . . , ∂F/∂x n ,where F ∈ O C n × S, (0 ,s ) is a defining equation for ( X, x ). The require-ment (1) of Definition 3.1 implies then that any relation among f, ∂f /∂x , . . . , ∂f /∂x n lifts to a relation among F, ∂F/∂x , . . . , ∂F/∂x n and this is equivalent EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 9 to the deformation of the singular locus of ( D,
0) being flat. See [9],Chapter I, Proposition 1.91. (cid:3)
Proposition 3.8.
In the situation of Definition 3.1, requirement (1) implies that
Der( − log X/S ) is a locally free O C n × S, (0 ,s ) -module of rankn.Proof. By Proposition 3.7, the singular locus of ( D,
0) is deformedflatly and so O C n × S, (0 ,s ) /I is a flat O S,s -module and represents a de-formation of O C n , /I , where I = ( F, ∂F/∂x , . . . , ∂F/∂x n ) and I =( f, ∂f /∂x , . . . , ∂f /∂x n ). Hence, a free resolution of O C n , /I lifts toa free resolution of O C n × S, (0 ,s ) /I . Because ( D,
0) is free, then a freeresolution of O C n × S, (0 ,s ) /I looks like0 (cid:47) (cid:47) O n C n × S, (0 ,s ) (cid:47) (cid:47) O n +1 C n × S, (0 ,s ) ( F,∂F/∂x ,...,∂F/∂x n ) (cid:47) (cid:47) O C n × S, (0 ,s ) (cid:47) (cid:47) O C n × S, (0 ,s ) /I (cid:47) (cid:47) − log X/S )with the syzygy module of (
F, ∂F/∂x , . . . , ∂F/∂x n ), and hence, it islocally free of rank n . (cid:3) Remark 3.9.
In our theory, we require more than only that each fiberis a free divisor. In fact, let ( D, ⊂ ( C n , be a singular free divisorwith a quasi-homogeneous equation f . Then we can consider ( X,
0) =( V ( f − t ) , ⊂ ( C n × C , and φ the projection on ( C , . In this caseeach fiber is a free divisor but this is not an admissible deformation of ( D, .Proof. Because f is quasi-homogeneous, we can take χ, σ , . . . , σ n − asa basis of Der( − log D ), where χ = (cid:80) ni =1 α i x i ∂/∂x i with α , . . . , α n ∈ C is the Euler vector field and σ , . . . , σ n − annihilate f . Hence χ ( f ) = (cid:80) ni =1 α i x i ∂f /∂x i = f . Notice that because ( X,
0) is non-singular, itis a free divisor in ( C n × C ,
0) and so we can take as Saito matrix for( X, A = · · · · · · · · · · · · ∂f /∂x ∂f /∂x · · · ∂f /∂x n f − t Let λ i be the vector field represented by the i -th column of A . Considernow the vector fields σ ∗ i = σ i seen as a vector field in C n × C and τ i = tλ i + ∂f /∂x i λ n +1 − ∂f /∂x i (cid:80) nj =1 α j x j λ j . Clearly, σ ∗ i ( f − t ) = σ i ( f ) = 0and so σ ∗ i ∈ Der( − log X/ C ). Similarly, τ i ∈ Der( − log X/ C ) because τ i ∈ Der( − log X ) and its coefficient of ∂/∂t is equal to t∂f /∂x i + ∂f /∂x i ( f − t ) − ∂f /∂x i (cid:80) nj =1 α j x j ∂f /∂x j = ∂f /∂x i ( f − χ ( f )) = 0. Thisimplies that we have an inclusion (cid:104) σ ∗ , . . . , σ ∗ n − , τ , . . . , τ n (cid:105) ⊂ Der( − log X/ C ).However, because σ , . . . , σ n − are the generators of Ann( f ) := { δ ∈ Der( − log D ) | δ ( f ) = 0 } , then any element of Der( − log X/ C ) that is alinear combination of λ , . . . , λ n is a linear combinations of σ ∗ , . . . , σ ∗ n − .Consider now an element of Der( − log X/ C ) that can be written as alinear combination of the λ i involving λ n +1 . Because it is independentof ∂/∂t , then the coefficient of λ n +1 is forced to be in the Jacobian idealof f . Because t appear only in λ n +1 , this implies that, modulo the σ ∗ i ,it is a linear combination of τ , . . . , τ n . Hence σ ∗ , . . . , σ ∗ n − , τ , . . . , τ n generate Der( − log X/ C ).Because f is singular, ∂f /∂x i ∈ ( x , . . . , x n ) for all i = 1 , . . . , n andso each τ i has weight bigger than zero, i.e. deg( ∂f /∂x i α j x j ) − deg( x j ) >
0. This tells us that the Euler vector field χ / ∈ Der( − log X/ C ) / m C , Der( − log X/ C )because χ has weight zero and is not a linear combination of σ , . . . , σ n − . (cid:3) Remark 3.10. If f is non-singular, then the deformation defined inthe previous Remark is an admissible deformation.Proof. We can suppose f = x and we can take as Saito matrix x · · ·
00 1 0 · · ·
00 0 1 · · · · · · By a similar argument as the proof of the previous Remark, Der( − log X/ C )is generated by the columns of the matrix x − t · · ·
00 1 0 · · ·
00 0 1 · · · · · ·
10 0 0 · · · and hence the requirement (1) of the Definition 3.1 is fulfilled. (cid:3) Remark 3.11.
Let ( i, φ ) : ( D, (cid:44) → ( X, x ) −→ ( S, s ) be a (linearly)admissible deformation. Then it is a trivial (linearly) admissible defor-mation if and only if it is trivial as deformation of ( D, as complexspace germ. EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 11
Definition 3.12.
The complex space T (cid:15) consists of one point with localring C [ (cid:15) ] = C + (cid:15) · C , (cid:15) = 0 , that is, C [ (cid:15) ] = C [ t ] / ( t ) , where t is anindeterminate. Thus T (cid:15) = Spec( C [ t ] / ( t )) . Definition 3.13. An infinitesimal (linearly) admissible deformation of a germ of a (linear) free divisor ( D, ⊂ ( C n , is a (linearly)admissible deformation of ( D, over T (cid:15) . Definition 3.14.
Let ( D, ⊂ ( C n , be a germ of a free divisor.Then F T ( D ) := FD D ( C [ t ] / ( t )) . Similarly if ( D, is linear, then LF T ( D ) := LFD D ( C [ t ] / ( t )) . Proposition 3.15. ([9], Chapter II, Proposition 1.5)
Consider a com-mutative diagram of complex space germs ( X , x ) f (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) (cid:47) ( X, x ) f (cid:15) (cid:15) ( S , s ) (cid:31) (cid:127) (cid:47) (cid:47) ( S, s ) where the horizontal maps are closed embeddings. Assume that f fac-tors as ( X , x ) (cid:31) (cid:127) i (cid:47) (cid:47) ( C n , × ( S , s ) p (cid:47) (cid:47) ( S , s ) with i a closed embedding and p the second projection. Then thereexists a commutative diagram ( X , x ) f (cid:38) (cid:38) (cid:127) (cid:95) i (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) (cid:47) ( X, x ) f (cid:120) (cid:120) (cid:127) (cid:95) i (cid:15) (cid:15) ( C n , × ( S , s ) p (cid:15) (cid:15) (cid:31) (cid:127) (cid:47) (cid:47) ( C n , × ( S, s ) p (cid:15) (cid:15) ( S , s ) (cid:31) (cid:127) (cid:47) (cid:47) ( S, s ) with i a closed embedding and p the second projection. That is, theembedding of f over ( S , s ) extends to an embedding of f over ( S, s ) . Corollary 3.16.
Any (linearly) admissible deformation of ( D,
0) =( V ( f ) , ⊂ ( C n , over a complex space germ ( S, s ) is of the form ( X, (0 , s )) = ( V ( F ) , (0 , s )) ⊂ ( C n × S, (0 , s )) , for some unfolding F of f with φ just the projection on ( S, s ) . Remark 3.17.
Any infinitesimal (linearly) admissible deformation of ( D,
0) = ( V ( f ) , ⊂ ( C n , is of the form ( X,
0) = ( V ( f + (cid:15) · f (cid:48) ) , ⊂ ( C n × T (cid:15) , , for some f (cid:48) ∈ O C n , , where φ is just the projection on T (cid:15) . By Remark 3.11 and Chapter II, 1.4 from [9], we have the following:
Remark 3.18.
An infinitesimal (linearly) admissible deformation ( X,
0) =( V ( f + (cid:15) · f (cid:48) ) , −→ T (cid:15) is trivial if and only if there is an isomorphism O C n × T (cid:15) , / ( f ) ∼ = O C n × T (cid:15) , / ( f + (cid:15) · f (cid:48) ) which is the identity modulo (cid:15) and which is compatible with the in-clusion of O T (cid:15) in O C n × T (cid:15) , . Such an isomorphism is induced by anautomorphism ϕ of O C n × T (cid:15) , , mapping x j (cid:55)→ x j + (cid:15)σ j ( x ) and (cid:15) (cid:55)→ (cid:15) such that ( ϕ ∗ f ) = ( f ( x + (cid:15) · σ ( x ))) = ( f + (cid:15) · f (cid:48) ) , where x = ( x , . . . , x n ) and σ = (cid:80) nj =1 σ j ∂/∂x j . We now prove a relative Saito’s Lemma in order to be able to charac-terise an (linearly) admissible deformation by logarithmic vector fields.
Lemma 3.19.
Let ( S, s ) be a complex space germ with an embedding ( S, s ) ⊂ ( C r , and let t = ( t , . . . , t r ) be coordinates on the ambientspace ( C r , . Let ( X, x ) ⊂ ( C n × S, (0 , s )) be a (linearly) admissibledeformation of a germ of a (linear) free divisor ( D, ⊂ ( C n , andlet h p = 0 be a reduced equation for ( X, x ) , locally at p = ( x , t ) ∈ ( C n × S, (0 , s )) . Suppose δ (cid:48) i = (cid:80) nj =1 a ji ( x, t ) ∂/∂x j ∈ Der p ( − log X/S ) , ∀ i = 1 , . . . , n , then det( a ji ) ∈ ( h p ) O C n × S,p .Proof.
Suppose that det( a ji ) does not vanish at p , hence it does notvanish in a small neighbourhood U of p . This implies that δ (cid:48) , . . . , δ (cid:48) n are linearly independent in U . Consider now the fibre X t . We havethat (cid:101) δ (cid:48) i = (cid:80) nj =1 a ji ( x, t ) ∂/∂x j ∈ Der( − log X t ) and are linearly inde-pendent, but this implies that X t is n -dimensional, contradicting thefact that ( X, x ) is a flat (linearly) admissible deformation of ( D, n − (cid:3) Proposition 3.20.
Let ( S, s ) be a complex space germ with an embed-ding ( S, s ) ⊂ ( C r , and let t = ( t , . . . , t r ) be coordinates on the ambi-ent space ( C r , . Let ( X, x ) ⊂ ( C n × S, (0 , s )) be a (linearly) admissibledeformation of a germ of a (linear) free divisor ( D, ⊂ ( C n , andlet h p = 0 be a reduced equation for ( X, x ) , locally at p = ( x , t ) ∈ ( C n × S, (0 , s )) . Then there exist δ (cid:48) , . . . , δ (cid:48) n ∈ Der p ( − log X/S ) with δ (cid:48) i = (cid:80) nj =1 a ji ( x, t ) ∂/∂x j , such that det( a ji ) is a unit multiple of h p .Proof. By Proposition 3.8, Der p ( − log X/S ) is a free O C n × S,p -moduleof rank n . Since Der( − log X/S ) is coherent, there exists a neighbour-hood U of p such that Der( − log X/S ) | U is free. Let δ (cid:48) , . . . , δ (cid:48) n be abasis of Der( − log X/S ) | U with δ (cid:48) i = (cid:80) nj =1 a ji ( x, t ) ∂/∂x j . By Lemma EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 13 a ji ) = gh p , where g is a holomorphic function on U . Since ∂/∂x , . . . , ∂/∂x n is a basis for p ∈ U \ X , then g does not vanish on U \ X . At a smooth point p ∈ X , we can suppose X = V ( x ) andhence, we may choose as a basis of Der( − log X/S ) on X reg ∩ U thevector fields x ∂/∂x , . . . , ∂/∂x n . Thus g does not vanish anywhere on U \ ( U ∩ X sing ), but because codim U ( U ∩ X sing ) >
1, then g does notvanish anywhere on U and so it is a unit. (cid:3) Lemma 3.21.
Let R be a commutative ring, A and B be two n × n matrices and a , . . . , a n be the columns of A . Then n (cid:88) i =1 det[ a , . . . , a i − , Ba i , a i +1 , . . . , a n ] = trace( B ) det( A ) . Proof.
It is know that if we consider a n × n matrix C with columns c , . . . , c n , then d A det( C ) = n (cid:88) i =1 det[ a , . . . , a i − , c i , a i +1 , . . . , a n ] , where d is the tangent map. Then we have the following equalities n (cid:88) i =1 det[ a , . . . , a i − , Ba i , a i +1 , . . . , a n ] = d A det( BA ) = ddt (det( A + tBA )) | t =0 =det( A ) ddt (det( I + tB )) | t =0 = det( A ) d I det( B ) = det( A ) trace( B ) . (cid:3) Lemma 3.22.
Let ( S, s ) be a complex space germ with an embedding ( S, s ) ⊂ ( C r , and let t = ( t , . . . , t r ) be coordinates on the ambientspace ( C r , . Consider ( D,
0) = ( V ( f ) , ⊂ ( C n , a germ of a (lin-ear) free divisor such that Der x ( − log D ) is generated by δ , . . . , δ n .Let δ (cid:48) i = (cid:80) nj =1 a ji ( x, t ) ∂/∂x j , i = 1 , . . . , n , be a system of holomorphicvector fields at ( x , s ) ∈ ( C n × S, (0 , s )) such that(1) δ (cid:48) i | C n ,x = δ i for all i = 1 , . . . , n ;(2) [ δ (cid:48) i , δ (cid:48) j ] ∈ (cid:80) nk =1 O C n × S, ( x ,s ) δ (cid:48) k for i, j = 1 , . . . , n ;(3) det( a ji ) = h defines a reduced hypersurface X .Then for X = { h ( x, t ) = 0 } , δ (cid:48) , . . . , δ (cid:48) n belongs to Der ( x ,s ) ( − log X/S ) , { δ (cid:48) , . . . , δ (cid:48) n } is a free basis of Der ( x ,s ) ( − log X/S ) and X is a (linearly)admissible deformation of ( D, over ( S, s ) . Proof.
First of all we need to show that each δ (cid:48) k ∈ Der ( x ,s ) ( − log X/S ).We have the following equalities δ (cid:48) k ( h ) = δ (cid:48) k (det[ δ (cid:48) , . . . , δ (cid:48) n ]) = n (cid:88) j =1 det[ δ (cid:48) . . . , δ (cid:48) j − , δ (cid:48) k ( δ (cid:48) j ) , δ (cid:48) j +1 , . . . , δ (cid:48) n ] == n (cid:88) j =1 det[ δ (cid:48) . . . , δ (cid:48) j − , [ δ (cid:48) k , δ (cid:48) j ] + δ (cid:48) j ( δ (cid:48) k ) , δ (cid:48) j +1 , . . . , δ (cid:48) n ] == n (cid:88) j =1 det[ δ (cid:48) . . . , δ (cid:48) j − , [ δ (cid:48) k , δ (cid:48) j ] , δ (cid:48) j +1 , . . . , δ (cid:48) n ]+ n (cid:88) j =1 det[ δ (cid:48) . . . , δ (cid:48) j − , δ (cid:48) j ( δ (cid:48) k ) , δ (cid:48) j +1 , . . . , δ (cid:48) n ] . By 2, det[ δ (cid:48) . . . , δ (cid:48) j − , [ δ (cid:48) k , δ (cid:48) j ] , δ (cid:48) j +1 , . . . , δ (cid:48) n ] ∈ ( h ) O C n × S, ( x ,s ) for all j =1 , . . . , n , and so the first part of the last equality is in ( h ) O C n × S, ( x ,s ) .Furthermore, if we consider the matrices A = [ δ (cid:48) , . . . , δ (cid:48) n ] and B =( ∂a ik /∂x j ) i,j =1 ,...,n , we can apply Lemma 3.21 and obtain n (cid:88) j =1 det[ δ (cid:48) . . . , δ (cid:48) j − , δ (cid:48) j ( δ (cid:48) k ) , δ (cid:48) j +1 , . . . , δ (cid:48) n ] = n (cid:88) i =1 ∂a ik ∂x i h ∈ ( h ) O C n × S, ( x ,s ) . This shows that δ (cid:48) k ( h ) ∈ ( h ) O C n × S, ( x ,s ) and so δ (cid:48) k ∈ Der ( x ,s ) ( − log X/S ),for all k = 1 , . . . , n .Notice now that by 1 and 3, h | C n ,x = f . Moreover, by 1Der x ( − log D ) ⊂ Der ( x ,s ) ( − log X/S ) / m S,s
Der ( x ,s ) ( − log X/S ) . Consider σ ∈ Der ( x ,s ) ( − log X/S ) such that σ | C n ,x / ∈ Der x ( − log D ).But σ ( h ) = αh for some α ∈ O C n × S, ( x ,s ) . Hence ( σ ( h )) | C n ,x = σ | C n ,x ( f ) = α | C n ,x f and so σ | C n ,x ∈ Der x ( − log D ), but this is a con-tradiction. Hence Der x ( − log D ) = Der ( x ,s ) ( − log X/S ) / m S,s
Der ( x ,s ) ( − log X/S )and so X is a (linearly) admissible deformation of ( D,
0) over (
S, s ).Consider σ ∈ Der ( x ,s ) ( − log X/S ). Then we want to prove that σ ∈ (cid:80) ni =1 O C n × S, ( x ,s ) δ (cid:48) i . By Cramer’s rule, h∂/∂x j ∈ (cid:80) ni =1 O C n × S, ( x ,s ) δ (cid:48) i forall j = 1 , . . . , n , hence we can consider hσ = (cid:80) ni =1 f i δ (cid:48) i , for some f i ∈O C n × S, ( x ,s ) . By Lemma 3.19, we have that det[ δ (cid:48) , . . . , δ (cid:48) i − , σ, δ (cid:48) i +1 , . . . , δ (cid:48) n ] ∈ ( h ) O C n × S, ( x ,s ) . Thus h det[ δ (cid:48) , . . . , δ (cid:48) i − , σ, δ (cid:48) i +1 , . . . , δ (cid:48) n ]= det[ δ (cid:48) , . . . , δ (cid:48) i − , hσ, δ (cid:48) i +1 , . . . , δ (cid:48) n ]= det[ δ (cid:48) , . . . , δ (cid:48) i − , f i δ (cid:48) i , δ (cid:48) i +1 , . . . , δ (cid:48) n ]= f i det[ δ (cid:48) , . . . , δ (cid:48) n ] = f i h ∈ ( h ) O C n × S, ( x ,s ) . Thus f i ∈ ( h ) O C n × S, ( x ,s ) for all i . This show that σ = (cid:80) ni =1 ( f i /h ) δ (cid:48) i ∈ (cid:80) ni =1 O C n × S, ( x ,s ) δ (cid:48) i . (cid:3) EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 15
Notice that if we consider S to be a reduced point, then the previousLemma is the same statement of Lemma 2.10.We can now state and prove the main result of the section: Theorem 3.23.
Let ( D,
0) = ( V ( f ) , ⊂ ( C n , be a germ of a freedivisor and δ , . . . , δ n a set of generators for Der( − log D ) . Any elementof F T ( D ) can be represented by n classes ˜ δ , . . . , ˜ δ n ∈ Der C n / Der( − log D ) such that the O C n × T (cid:15) , -module generated by δ (cid:48) = δ + (cid:15) · ˜ δ , . . . , δ (cid:48) n = δ n + (cid:15) · ˜ δ n is closed under Lie brackets. If the deformation is linearlyadmissible, then the coefficients of all ˜ δ i , in any representation of anelement of F T ( D ) , must be linear functions too.Proof. Let (
X, x ) ⊂ ( C n × T (cid:15) ,
0) be an infinitesimal (linearly) ad-missible deformation of ( D, X,
0) = ( V ( f + (cid:15) · f (cid:48) ) , ⊂ ( C n × T (cid:15) , X,
0) is the total space of an infinitesimal (linearly) admissible de-formation of ( D,
0) implies that there exists an n × n matrix A ( (cid:15) ) withcoefficients in C [ x , . . . , x n , (cid:15) ] / ( (cid:15) ) such that det A ( (cid:15) ) = ( f + (cid:15) · f (cid:48) ).But (cid:15) = 0 implies that we can write A ( (cid:15) ) = B + (cid:15) · C , where B and C are n × n matrices with coefficients in C [ x , . . . , x n ]. Hence f = det A (0) = det B and so B is a Saito matrix for ( D, δ i as the columns of B and ˜ δ i as the columns of C and this proves that the Lie algebra Der( − log X/T (cid:15) ) is generated by δ + (cid:15) · ˜ δ , . . . , δ n + (cid:15) · ˜ δ n as required. Because Der( − log X/T (cid:15) ) is a Liealgebra, then [ δ (cid:48) i , δ (cid:48) j ] ∈ Der( − log X/T (cid:15) ) for all i, j = 1 , . . . , n , but then[ δ (cid:48) i , δ (cid:48) j ] ∈ (cid:80) nk =1 O C n × S, ( x ,s ) δ (cid:48) k for i, j = 1 , . . . , n .We now consider the classes of ˜ δ , . . . , ˜ δ n modulo Der( − log D ), be-cause if ˜ δ , . . . , ˜ δ n ∈ Der( − log D ), then f (cid:48) ∈ ( f ) O C n , and hence, by[9], Chapter II, 1.4, the deformation is trivial.On the other hand, let ˜ δ , . . . , ˜ δ n ∈ Der C n / Der( − log D ) be n classesof vector fields such that the O C n × T (cid:15) , -module generated by δ + (cid:15) · ˜ δ , . . . , δ n + (cid:15) · ˜ δ n is closed under Lie brackets. The determinant of thematrix of coefficients [ δ + (cid:15) · ˜ δ , . . . , δ n + (cid:15) · ˜ δ n ] is equal to f + (cid:15) · f (cid:48) and so by Lemma 3.22 it is enough to show that this determinant isreduced. First, noticed that for (cid:15) = 0 the determinant is equal to f and hence is reduced. Now, reducedness is an open property and sothe result holds.The last part of the statement is trivial. (cid:3) The complexes C • and C • . We recall here the notion of thecomplex of Lie algebroid cohomology in the case of Der( − log D ), see[13] for the general theory. Definition 3.24.
Let C • be the complex with modules C p := H om O C n ( p (cid:94) Der( − log D ) , Der C n / Der( − log D )) and differentials ( d p ( ψ ))( δ ∧ · · · ∧ δ p +1 ) := p +1 (cid:88) i =1 ( − i [ δ i , ψ ( δ ∧ · · · ∧ (cid:98) δ i ∧ · · · ∧ δ p +1 )]++ (cid:88) ≤ i Notice that C = Der C n / Der( − log D ) and the map d is defined by d : C −→ H om O C n (Der( − log D ) , Der C n / Der( − log D )) σ (cid:55)→ ( δ (cid:55)→ [ δ, σ ]) . We recall now the definition of the complex of Lie algebra cohomol-ogy from [11]. Definition 3.26. Let C • be the complex defined by C p := Hom C ( p (cid:94) Der( − log D ) , (Der C n / Der( − log D )) ) and the differentials ( d p ( ψ ))( δ ∧ · · · ∧ δ p +1 ) := p +1 (cid:88) i =1 ( − i [ δ i , ψ ( δ ∧ · · · ∧ ˆ δ i ∧ · · · ∧ δ p +1 )]++ (cid:88) ≤ i Infinitesimal admissible deformations.Theorem 3.27. Let ( D, ⊂ ( C n , be a germ of a free divisor. Thenthe germ at the origin of the first cohomology sheaf of the complex C • is isomorphic to F T ( D ) , i.e. H ( C • ) ∼ = F T ( D ) . Proof. To prove that we can identify H ( C • ) with F T ( D ), two thingshave to be checked: we must first identify the elements of ker( d : C −→C ) with admissible deformations of ( D, d : C −→ C is the collection of trivial admissible defor-mations of ( D, n classes of vector fields˜ δ , . . . , ˜ δ n ∈ Der C n / Der( − log D ) such that the O C n × T (cid:15) , -module gen-erated by the elements δ + (cid:15) · ˜ δ , . . . , δ n + (cid:15) · ˜ δ n is closed under Liebrackets.Take an element ψ ∈ ker( d ), which means that ψ ([ δ, ν ]) − [ δ, ψ ( ν )] + [ ν, ψ ( δ )] = 0 in Der C n / Der( − log D )for all δ, ν ∈ Der( − log D ). Then ψ corresponds to the admissibledeformation given by the O C n × T (cid:15) , -module L generated by δ + (cid:15) · ψ ( δ ) , . . . , δ n + (cid:15) · ψ ( δ n ) . By C -linearity of the Lie brackets, L is closed under Lie brackets ifand only if for any two elements δ + (cid:15) · ψ ( δ ) , ν + (cid:15) · ψ ( ν ) ∈ L we have[ δ + (cid:15) · ψ ( δ ) , ν + (cid:15) · ψ ( ν )] ∈ L , which is equivalent to F := [ δ, ν ] + (cid:15) · ([ δ, ψ ( ν )] − [ ν, ψ ( δ )]) ∈ L . Consider G := [ δ, ν ] + (cid:15) · ψ ([ δ, ν ]) which is an element of L , so thecondition F ∈ L is equivalent to G − F ∈ L , that is ψ ([ δ, ν ]) − [ δ, ψ ( ν )] + [ ν, ψ ( δ )] ∈ Der( − log D ) . This means exactly that ψ ∈ ker( d ).Let us consider now an infinitesimal admissible deformation ( X, 0) =( V ( f + (cid:15) · f (cid:48) ) , − log X/T (cid:15) ) = (cid:104) δ + (cid:15) · ψ ( δ ) , . . . , δ n + (cid:15) · ψ ( δ n ) (cid:105) for some ψ ∈ ker( d ). By Remark 3.18, f + (cid:15) · f (cid:48) is trivial if and only if ( ϕ ∗ f ) = ( f ( x + (cid:15) · σ ( x ))) = ( f + (cid:15) · f (cid:48) ),for some ϕ ∈ Aut( C n × T (cid:15) ). In this situation, the module of vectorfields generated by ϕ ∗ (Der( − log D )) is equal to Der( − log X/T (cid:15) ), i.e. (cid:104) D ϕ − ( x ) ϕ ( δ ( ϕ − ( x ))) , . . . , D ϕ − ( x ) ϕ ( δ n ( ϕ − ( x ))) (cid:105) = (cid:104) δ + (cid:15) · ψ ( δ ) , . . . , δ n + (cid:15) · ψ ( δ n ) (cid:105) , where for h : X −→ Y , then D x h : T x X −→ T h ( x ) Y is the tangent map.Because we can consider each vector field on C n also as a map from C n into itself, we have the following equalities D ϕ − ( x ) ϕ ( δ i ( ϕ − ( x ))) = D x − (cid:15) · σ ( x ) ϕ ( δ i ( x − (cid:15) · σ ( x ))) = = δ i ( x − (cid:15) · σ ( x )) + (cid:15) · D x − (cid:15) · σ ( x ) σ ( δ i ( x − (cid:15) · σ ( x ))) == δ i ( x ) − (cid:15) · ( D x δ i ( σ ( x )) − D x − (cid:15) · σ ( x ) σ ( δ i ( x ))) == δ i ( x ) + (cid:15) · ( D x σ ( δ i ( x )) − D x δ i ( σ ( x ))) = δ i ( x ) + (cid:15) · [ σ, δ i ]( x )and that tells us that ψ ( δ i ) = [ σ, δ i ], i.e. ψ ∈ image( d ). (cid:3) Lemma 3.28. Let D ⊂ C n be a free divisor. Then Der( − log D ) isa self-normalising Lie subalgebra of Der C n . That is, if we consider χ ∈ Der C n such that [ χ, δ ] ∈ Der( − log D ) for all δ ∈ Der( − log D ) ,then χ ∈ Der( − log D ) .Proof. By the definition of Der( − log D ), it is enough to show that ifwe consider p ∈ D a smooth point, then χ ( p ) ∈ T p D . Without lossof generality, we can suppose that at p the divisor D is defined by theequation x = 0, that its Saito matrix is[ δ , · · · , δ n ] = x · · · 00 1 0 · · · 00 0 1 · · · · · · and that χ ( p ) = (cid:80) ni =1 a i ∂/∂x i with a i ∈ O C n ,p . In this way, we havereduced the problem to proving that a ∈ ( x ) O C n ,p .By hypothesis, [ χ, δ ] ∈ Der p ( − log D ) for all δ ∈ Der p ( − log D ), inparticular [ χ, δ ] = a ∂/∂x − (cid:80) ni =1 x ∂a i /∂x ∂/∂x i = ( a − x ∂a /∂x ) ∂/∂x − (cid:80) ni =2 x ∂a i /∂x ∂/∂x i ∈ Der p ( − log D ). Hence, ( a − x ∂a /∂x ) ∈ ( x ) O C n ,p and so a ∈ ( x ) O C n ,p as required. (cid:3) In a similar way we can prove the following: Lemma 3.29. Let D ⊂ C n be a linear free divisor. Then Der( − log D ) is a self-normalising Lie subalgebra of (Der C n ) . Proposition 3.30. H ( C • ) = 0 .Proof. Consider σ ∈ H ( C • ) = ker( d ). Hence, [ − , σ ] is the zero map,i.e. for all δ ∈ Der( − log D ) we have that [ δ, σ ] ∈ Der( − log D ). Thenby Lemma 3.28, σ ∈ Der( − log D ). (cid:3) Proposition 3.31. Let ( D, ⊂ ( C n , be a germ of a smooth divisor.Then F T ( D ) = 0 . EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 19 Proof. We can suppose f = x and we can take as Saito matrix thematrix S = [ δ , . . . , δ n ] = x · · · 00 1 0 · · · 00 0 1 · · · · · · . Moreover, we can represent an element of C as the column of the n × n matrix S + (cid:15) · T , where T is the matrix T = [˜ δ , . . . , ˜ δ n ] = g g · · · g n · · · · · · and g i = g i ( x , . . . , x n ) ∈ O C n , .Because [ δ i , δ j ] = 0 for every i, j = 1 , . . . , n , then the element S + (cid:15) · T is in the kernel of d if and only if g i = − ∂g /∂x i for all i = 2 , . . . , n .To show that this element is zero in cohomology, it is enough to find σ ∈ C = Der C n / Der( − log D ) such that [ σ, δ i ] = ˜ δ i for all i = 1 , . . . , n ,i.e. S + (cid:15) · T is in the image of d . Consider σ = g ∂/∂x , then it isthe element we are looking for. (cid:3) Proposition 3.32. Let ( D, ⊂ ( C n , be the germ of the normalcrossing divisor. Then F T ( D ) = 0 .Proof. Let f = x · · · x n be a defining equation for D . We can take asSaito matrix S = [ δ , . . . , δ n ] = x · · · x · · · · · · x n . Moreover, we can represent an element of C as columns of the n × n matrix S + (cid:15) · T , where T is the matrix T = [˜ δ , . . . , ˜ δ n ] = g , g , · · · g ,n g , g , · · · g ,n ... ... ... g n, g n, · · · g n,n and g i,j = g i,j ( x , . . . , ˆ x i , . . . , x n ) ∈ O C n .Because [ δ i , δ j ] = 0 for every i, j = 1 , . . . , n , then the element rep-resented by S + (cid:15) · T is in the kernel of d if and only if A i,j = − [ δ i , ˜ δ j ] + [ δ j , ˜ δ i ] ∈ Der( − log D ) for all i, j = 1 , . . . , n . Let us sup-pose that i < j , then A i,j = − x i ∂g ,j /∂x i ... g i,j ... − x i ∂g j,j /∂x i ... − x i ∂g n,j /∂x i + x j ∂g ,i /∂x j ... x j ∂g i,i /∂x j ... − g j,i ... x j ∂g n,i /∂x j Now, A i,j ∈ Der( − log D ) for all i, j = 1 , . . . , n if and only if A i,j = 0 ifand only if T = g , − x ∂g , /∂x · · · − x n ∂g , /∂x n − x ∂g , /∂x g , · · · − x n ∂g , /∂x n ... ... ... − x ∂g n,n /∂x − x ∂g n,n /∂x · · · g n,n . To show that this element is zero in cohomology, it is enough to find σ ∈ C = Der C n / Der( − log D ) such that [ σ, δ i ] = ˜ δ i for all i = 1 , . . . , n ,i.e. S + (cid:15) · T is in the image of d . Consider σ = g , ... g n,n then it is the element we are looking for. (cid:3) Remark 3.33. There exist free divisors such that F T ( D ) (cid:54) = 0 .Proof. Consider f = xy ( x − y )( x + y ) ∈ C [ x, y ] and the germ of a freedivisor ( D, 0) = ( V ( f ) , ⊂ ( C , 0) with Saito matrix A = (cid:20) x y x y − y (cid:21) . To find an infinitesimal admissible deformation for ( D, 0) we have tofind a non zero element α ∈ H ( C • ) = F T ( D ) . Let α be defined bythe columns of the following matrix B = (cid:20) xy − y (cid:21) . this is an element of H ( C • ) that describes the infinitesimal admissibledeformation X = V ( xy ( x − y )( x + (1 + (cid:15) ) y )) = V ( f + (cid:15) ( x y − xy )) ⊂ C × T (cid:15) . This infinitesimal admissible deformation is non-trivial because EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 21 it is a non-trivial deformation of f as a germ of function because x y − xy is not in the Jacobian ideal of f , see [9], Chapter II, 1.4. (cid:3) Infinitesimal linearly admissible deformations.Theorem 3.34. Let ( D, ⊂ ( C n , be a germ of a linear free divisor.Then the germ at the origin of the first cohomology sheaf of the complex C • is isomorphic to LF T ( D ) , i.e. H ( C • ) ∼ = LF T ( D ) . Proof. This is a consequence of Theorem 3.27 and the second part ofTheorem 3.23. (cid:3) Corollary 3.35. Let ( D, ⊂ ( C n , be a germ of a linear free divisor.Then the functor LFD D satisfies Schlessinger condition (H3) from [15] .Proof. This is a consequence of the previous Theorem and of the factthat the cohomology of a finite dimensional Lie algebra is finite dimen-sional. (cid:3) Corollary 3.36. Let ( D, ⊂ ( C n , be a germ of a linear free divisor.Then LFD D has a hull.Proof. This is a consequence of Theorem 2.11 from [15], Theorem 3.6and the previous Corollary. (cid:3) Proposition 3.37. H ( C • ) = 0 .Proof. Like the proof of Proposition 3.30 but using Lemma 3.29. (cid:3) Definition 3.38. Let M be a vector space and let g be a Lie algebra.A representation of g in M is a homomorphism (cid:37) of g in gl ( M ) . In what follows, we will refer both to the homomorphism (cid:37) and tothe vector space M as representations of g . Remark 3.39. LF T ( D ) is the first Lie algebra cohomolgy of Der( − log D ) with coefficients in the non-trivial representation (Der C n / Der( − log D )) . We collect now some results from [7], [11] and [20] about Lie algebrasand Lie algebra cohomology, that will allow us to compute LF T ( D )more easily in the case of germs of reductive linear free divisors. Proposition 3.40. ([7], Corollary 1.6.4) Let g be a reductive Lie al-gebra and let (cid:37) be a finite dimensional representation of g . Then thefollowing condition are equivalent(1) (cid:37) is semisimple;(2) for all a in the centre of g , (cid:37) ( a ) is semisimple. We will use the following celebrated theorem of Hochschild and Serre Theorem 3.41. ([11], Theorem 10) Let g be a reductive Lie algebraof finite dimension over C . Let M be a finite dimensional semisimplerepresentation of g such that M g = (0) , where M g is the submodule of M on which g acts trivially. Then H n ( g , M ) = 0 for all n ≥ . In order to apply the previous theorem, we need the following: Lemma 3.42. Let D ⊂ C n be a reductive linear free divisor. Then allthe elements in the centre of Der( − log D ) are diagonalizable.Proof. By definition g D = { A | xA t ∂ t ∈ Der( − log D ) } is a reductiveLie algebra and hence by Lemma 2.20 and by Lemma 3.6, (2) of [8], G ◦ D is a reductive Lie group. Hence by definition, the centre Z G ◦ D of G ◦ D is composed of semisimple transformations. Moreover, the Lie algebraof the identity component of Z G ◦ D coincides with Z g D the centre of g D and hence it is composed of diagonalizable elements. (cid:3) Proposition 3.43. Let D ⊂ C n be a reductive linear free divisor.Then the representation of Der( − log D ) in (Der C n / Der( − log D )) issemisimple.Proof. This is a consequence of Proposition 3.40 and Lemma 3.42. (cid:3) Theorem 3.44. Let ( D, ⊂ ( C n , be a germ of a reductive linearfree divisor. Then LF T ( D ) = 0 .Proof. By Lemma 3.29, (Der C n / Der( − log D )) Der( − log D ) = 0 and henceby Theorem 3.41, LF T ( D ) = 0. (cid:3) Corollary 3.45. Let ( D, ⊂ ( C n , be a germ of a reductive linearfree divisor. Then it is formally rigid. The statement of Theorem 3.44 is false if we consider non-reductivegerms of linear free divisors. In fact, Brian Pike suggested us thefollowing Example 3.46. Consider f = x ( x − x x x + x x + 2 x x x − x x ) ∈ C [ x , . . . , x ] as a defining equation of the germ of a linearfree divisor ( D, ⊂ ( C , . Then we can consider the Saito matrix x x x x x x x x x − x x x − x x − x x . Consider σ = 16 x ∂/∂x +11 x ∂/∂x +6 x ∂/∂x + x ∂/∂x − x ∂/∂x ,then σ ∈ Ann( D ) and trace( σ ) = 30 , hence, by Lemma 2.18, ( D, isthe germ of a non-reductive linear free divisor. EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 23 To find an infinitesimal linearly admissible deformation for ( D, we have to find a non-zero element α ∈ H ( C • ) = LF T ( D ) . Let α bedefined by the columns of the following matrix x − x . This is an element of H ( C • ) that describes the infinitesimal linearlyadmissible deformation X = V ( x ( x (1+ (cid:15) ) − x x x + x x +2 x x x − x x )) = V ( f + (cid:15) ( x x )) ⊂ C × T (cid:15) . This infinitesimal linearly ad-missible deformation is non-trivial because it is a non-trivial deforma-tion of f as a germ of function, in fact x x / ∈ J ( D ) . Moreover, onecan check, via a long Macaulay computation, that LF T ( D ) is -dimensional and this element is one of its generators. For more detailssee [19] , Appendix C.1. The weighted homogeneous case.Proposition 3.47. Let ( D, ⊂ ( C n , be a germ of a free divisordefined by a weighted homogeneous polynomial of degree k . Then anelement of F T ( D ) can be represented by f (cid:48) ∈ C [ x , . . . , x n ] k , where C [ x , . . . , x n ] k is the space of polynomial of weighted degree k .Proof. Let f be a defining equation for ( D, f is weightedhomogeneous, then there exists χ ∈ Der( − log D ) such that χ ( f ) = f .Consider ( X, x ) an infinitesimal admissible deformation of ( D, f + (cid:15) · f (cid:48) ,where f (cid:48) ∈ O C n , . Suppose that f (cid:48) is weighted homogeneous of degree β . Because ( X, x ) is admissible, it means that χ lifts and so thereexists χ (cid:48) ∈ Der C n such that ( χ + (cid:15) · χ (cid:48) )( f + (cid:15) · f (cid:48) ) = (1 + (cid:15) · α )( f + (cid:15) · f (cid:48) )and so χ (cid:48) ( f ) + χ ( f (cid:48) ) = αf + f (cid:48) , for some α ∈ O C n , . Because f (cid:48) is weighted homogeneous of degree β , then χ ( f (cid:48) ) = βf (cid:48) . Hence, theprevious expression becomes ( χ (cid:48) − α ) f = (1 − β ) f (cid:48) . However, ( χ (cid:48) − α ) f lies in the Tyurina ideal of f which is equal to the Jacobian idealof D due to the quasi-homogeneity of f and so (1 − β ) f (cid:48) is in theJacobian ideal of D . If f (cid:48) is in the Jacobian ideal, then the admissibledeformation is trivial, by [9], Chapter II, 1.4, otherwise β = 1 and so f (cid:48) is of weighted degree k .If f (cid:48) is not weighted homogeneous, we can apply the previous argu-ment to each of its weighted homogeneous parts. (cid:3) Lemma 3.48. Let ( D, ⊂ ( C n , be a germ of a free divisor definedby a weighted homogeneous polynomial. Then a basis of F T ( D ) canbe chosen to be made of monomials.Proof. This is because we have a good C ∗ -action. (cid:3) Corollary 3.49. Let ( D, ⊂ ( C n , be a germ of a free divisor de-fined by a weighted homogeneous polynomial of degree k with non-zeroweights ( a , . . . , a n ) . Then dim C F T ( D ) ≤ dim C C [ x , . . . , x n ] k /J ( D ) ∩ C [ x , . . . , x n ] k , where J ( D ) is the Jacobian ideal of D .Proof. It is a consequence of Lemma 3.48, Proposition 3.47 and that J ( D ) defines only trivial deformations. (cid:3) Corollary 3.50. Let ( D, ⊂ ( C n , be a germ of a free divisor definedby a weighted homogeneous polynomial. Then FD D has a hull.Proof. By Corollary 3.49, condition (H3) from [15] is satisfied. Thenthe result follows from Theorem 3.6 and Theorem 2.11 from [15]. (cid:3) Because each germ of a linear free divisor ( D, ⊂ ( C n , 0) is definedby a homogeneous equation of degree n , we have the following: Corollary 3.51. Let ( D, ⊂ ( C n , be a germ of a linear free divisor.Then FD D has a hull. By Corollary 2.4, every reduced curve is a free divisor. Then: Theorem 3.52. Let ( D, ⊂ ( C , be a reduced curve germ definedby a weighted homogeneous polynomial of degree k . Then F T ( D ) ∼ = C [ x, y ] k /J ( D ) ∩ C [ x, y ] k . Proof. Let f be a defining equation for ( D, f is weightedhomogeneous, then there exists χ ∈ Der C n such that χ ( f ) = f . Let δ = ∂f /∂x∂/∂y − ∂f /∂y∂/∂x . Because D has an isolated singularity,then δ, χ form a basis of Der( − log D ).By Proposition 3.47, we know that if ( X, x ) is an infinitesimal ad-missible deformation of ( D, 0) defined by f + (cid:15) · f (cid:48) , then f (cid:48) ∈ C [ x, y ] k .On the other hand, let f (cid:48) ∈ C [ x, y ] k , then consider ( X, 0) definedby f + (cid:15) · f (cid:48) = F , then it is an infinitesimal admissible deformationbecause both δ and χ lift. In fact, we can consider δ (cid:48) = ∂F/∂x∂/∂y − ∂F/∂y∂/∂x and χ as elements of Der( − log X/T (cid:15) ).We have to go modulo J ( D ) ∩ C [ x, y ] k to avoid trivial admissibledeformations. (cid:3) Remark 3.53. The previous Theorem is false in higher dimension. EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 25 Proof. Consider f = 4 x y − x z +27 y − xy z +128 x z − z ∈ C [ x, y, z ]. It is weighted homogeneous of degree 12 with weights (2 , , D, ⊂ ( C , C C [ x, y, z ] /J ( D ) ∩ C [ x, y, z ] = 3 but F T ( D ) = 0. (cid:3) Corollary 3.54. Let ( D, ⊂ ( C , be a germ of a free divisor definedby a homogeneous polynomial of degree k . Then dim C F T ( D ) = k − if k ≥ , and is zero otherwise.Proof. If k = 1, then J ( D ) = C [ x, y ] and if k = 2, then J ( D ) = ( x, y )and so, by Theorem 3.52, in both cases F T ( D ) = 0.Let us suppose now that k ≥ 3. We have that dim C C [ x, y ] k = k + 1and that J ( D ) ∩ C [ x, y ] k gives us 4 relations: x∂f /∂x, x∂f /∂y, y∂f /∂x, y∂f /∂y .Because ( D, 0) is an isolated singularity, then ∂f /∂x, ∂f /∂y form aregular sequence and so the Koszul relation generates the relations be-tween the partial derivative of f . Because the Koszul relation is ofdegree k − > 1, then x∂f /∂x, x∂f /∂y, y∂f /∂x, y∂f /∂y are linearlyindependent. Hence, dim C C [ x, y ] k /J ( D ) ∩ C [ x, y ] k = k + 1 − k − (cid:3) Example 3.55. (1) Consider f = xy ( x − y )( x + y ) ∈ C [ x, y ] and let ( D, 0) = ( V ( f ) , ⊂ ( C , . Then F T ( D ) is -dimensionaland it is generated by x y .(2) Consider f = x + y ∈ C [ x, y ] and the germ of a free divisor ( D, 0) = ( V ( f ) , ⊂ ( C , . A direct computation shows that F T ( D ) = 0 and so it is formally rigid. Remark 3.56. Let ( D, ⊂ ( C n , be a germ of a free divisor definedby a weighted homogeneous polynomial. Then we can compute the co-homology of C • degree by degree, because each module and map involvedis degree preserving. Theorem 3.57. Let ( D, ⊂ ( C n , be a germ of a free divisor definedby a weighted homogeneous polynomial. Then F T ( D ) ∼ = ( H ( C • ) ) ,where ( H ( C • ) ) is the weight zero part of H ( C • ) .Proof. Let f be a defining equation for ( D, 0) weighted homogeneousof degree k and let ( X, x ) be an infinitesimal admissible deformationof ( D, X, x ) has definingequation f + (cid:15) · f (cid:48) , with f (cid:48) weighted homogeneous of degree k .Because Der( − log D ) is a graded module, we can consider δ , . . . , δ n ∈ Der( − log D ) a weighted homogeneous basis. By Proposition 3.20,Der(log X/T (cid:15) ) is generated by δ + (cid:15) · ˜ δ , . . . , δ n + (cid:15) · ˜ δ n such that thedeterminant of their coefficients is f + (cid:15) · f (cid:48) . Because f and f (cid:48) are both weighted homogenous of the same degree, then each ˜ δ i is weighted ho-mogeneous of the same degree as δ i , for all i = 1 , . . . , n .As seen in the proof of Theorem 3.27, there exists ψ ∈ C such that ψ ( δ i ) = ˜ δ i . So by the previous argument ψ is a weight-preserving mapand so represents an element of ( H ( C • ) ) . (cid:3) Corollary 3.58. Let ( D, ⊂ ( C n , be a germ of a linear free divisor.Then F T ( D ) ∼ = LF T ( D ) .Proof. It is clear that ( H ( C • ) ) = H ( C • ) . (cid:3) Corollary 3.59. Let ( D, ⊂ ( C n , be a germ of a reductive linearfree divisor. Then it is formally rigid also as free divisor.Proof. This is a consequence of Theorem 3.44 and Corollary 3.58. (cid:3) Properties of the cohomology Constructibility of the cohomology. As we have seen in theprevious section, the cohomology of the complex C • plays an impor-tant role in the theory of admissible deformations for a germ of a freedivisor. From Schlessinger’s Theorem 2.11 from [15], we know that themain point in proving the existence of a hull is the finiteness of thiscohomology. The following subsection is devoted to study this problem. Definition 4.1. Let X be a n -dimensional complex manifold. We de-note by D X the sheaf of differential operators on X and by G r F • ( D X ) the sheaf on T ∗ X of graded rings associated with the filtration F • bythe order of σ ( P ) the principal symbol of a differential operator P. Definition 4.2. Let D ⊂ C n be a divisor defined by the ideal I . Wedefine the V -filtration relative to D on D C n by V Dk ( D C n ) := { P ∈ D C n | P ( I j ) ⊂ I j − k ∀ j ∈ Z } for all k ∈ Z , where I j = O C n when j is negative. Similarly, we define V Dk ( D C n ,x ) := { P ∈ D C n ,x | P ( f j ) ⊂ f j − k ∀ j ∈ Z } , where f is a local equation for D at x . If there is no confusion, wedenote V Dk ( D C n ) and V Dk ( D C n ,x ) simply by V k ( D C n ) and V k ( D C n ,x ) , re-spectively. Definition 4.3. A logarithmic differential operator is an element of V ( D C n ) . Remark 4.4. We have Der( − log D ) = Der C n ∩V ( D C n ) = G r F • ( V ( D C n )) , EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 27 and F ( V ( D C n )) = O C n ⊕ Der( − log D ) . Proof. The first equality comes directly from the definitions. The sec-ond one is a consequence of the fact that F ( D C n ) = O C n ⊕ Der C n . (cid:3) Definition 4.5. Let M be a O C n -module. A connection on M withlogarithmic poles along D or a logarithmic connection on M , is ahomomorphism over C ∇ : M −→ Ω (log D ) ⊗ M , that verifies Leibniz’s identity ∇ ( hm ) = dh ⊗ m + h ∇ ( m ) for any h ∈ O C n and m ∈ M , where d is the exterior derivative over O C n . For any q ∈ N we will denote Ω q (log D ) ⊗ M by Ω q (log D )( M ) . Definition 4.6. Let M be a O C n -module with ∇ a logarithmic con-nection. We can define the following left O C n -linear morphism ∇ (cid:48) : Der( − log D ) −→ E nd C ( M ) δ (cid:55)→ ∇ δ where ∇ δ ( m ) := (cid:104) δ, ∇ ( m ) (cid:105) . Remark 4.7. The morphism ∇ (cid:48) verifies Leibniz’s condition ∇ δ ( hm ) = δ ( h ) m + h ∇ δ ( m ) for any δ ∈ Der( − log D ) , h ∈ O C n and m ∈ M . Remark 4.8. Given a left O C n -linear morphism ∇ (cid:48) : Der( − log D ) −→ E nd C ( M ) verifying Leibniz’s condition, we define ∇ : M −→ Ω (log D )( M ) with ∇ ( m ) the element of Ω (log D )( M ) = H om O C n (Der( − log D ) , M ) such that ∇ ( m )( δ ) = ∇ (cid:48) ( δ )( m ) . Definition 4.9. A logarithmic connection ∇ is integrable if, for each δ, δ (cid:48) ∈ Der( − log D ) , it verifies ∇ [ δ,δ (cid:48) ] = [ ∇ δ , ∇ δ (cid:48) ] , where [ , ] represents the Lie bracket in Der( − log D ) and the commu-tator in E nd C ( M ) . Example 4.10. Consider M = Der C n / Der( − log D ) . Then we canintroduce on M the integrable logarithmic connection defined by ∇ δ :=[ δ, − ] . If we take M = Der( − log D ) or Der C n , then ∇ δ = [ δ, − ] doesnot in general define a connection on M because it is not O C n -linearin δ . Proposition 4.11. ([3], Corollary 2.2.6) Let D ⊂ C n be a free divisorand let M be a O C n -module. An integrable logarithmic connection on M gives rise to a left V ( D C n ) -module structure on M and vice versa. We now explain a condition that allows us to put a structure of V ( D C n )-module on Der( − log D ) and Der C n .Fix D ⊂ C n a free divisor and δ i = (cid:80) nj =1 a ij ∂/∂x j , i = 1 , . . . , n abasis for Der( − log D ), where a ij ∈ O C n for i, j = 1 , . . . , n . We knowthat Der( − log D ) forms a Lie subalgebra of Der C n , hence we can write[ δ i , δ j ] = n (cid:88) k =1 b ijk δ k where b ijk ∈ O C n for all i, j, k = 1 , . . . , n and similarly we can write[ δ i , ∂/∂x j ] = n (cid:88) k =1 c ijk ∂∂x k where c ijk ∈ O C n for all i, j, k = 1 , . . . , n . In this way we obtain the dataof 2 n matrices B i = ( b ijk ) and C i = ( c ijk ) of holomorphic function on C n . Let us write δ · ∂ := [ δ, ∂ ] for any derivation ∂ and any logarithmicderivation δ . Then we have δ i · δ t = B i δ t , ≤ i ≤ n and δ i · ∂ t = C i ∂ t , ≤ i ≤ n where δ = ( δ , . . . , δ n ) and ∂ = ( ∂/∂x , . . . , ∂/∂x n ). Lemma 4.12. For i, j = 1 , . . . , n we have that δ i ( C j ) − δ j ( C i ) + [ C j , C i ] = n (cid:88) k =1 b ijk C k if and only if n (cid:88) k =1 a kr ∂ ( b ijk ) ∂x l = 0 , ∀ i, l, r = 1 , . . . , n. EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 29 Proof. We first notice that by definition c ijk = − ∂ ( a ik ) /∂x j . The firstequality is an equality between matrices, hence we can check it entryby entry. Let 1 ≥ l, r ≥ 0. We now check the entry ( l, r ). In this casethe expression becomes − δ i ( ∂ ( a jr ) ∂x l ) + δ j ( ∂ ( a ir ) ∂x l ) + n (cid:88) k =1 ∂ ( a jk ) ∂x l ∂ ( a ir ) ∂x k + − n (cid:88) k =1 ∂ ( a ik ) ∂x l ∂ ( a jr ) ∂x k = − n (cid:88) k =1 b ijk ∂ ( a kr ) ∂x l . Consider now the Jacobi identity[[ δ i , δ j ] , ∂∂x l ] + [[ δ j , ∂∂x l ] , δ i ] + [[ ∂∂x l , δ i ] , δ j ] = 0 . The coefficient of ∂/∂x r of the previous expression is δ i ( ∂ ( a jr ) ∂x l ) − δ j ( ∂ ( a ir ) ∂x l ) − n (cid:88) k =1 ∂ ( a jk ) ∂x l ∂ ( a ir ) ∂x k ++ n (cid:88) k =1 ∂ ( a ik ) ∂x l ∂ ( a jr ) ∂x k − n (cid:88) k =1 b ijk ∂ ( a kr ) ∂x l − n (cid:88) k =1 a kr ∂ ( b ijk ) ∂x l = 0 . Hence, the first equality is satisfied if and only if n (cid:88) k =1 a kr ∂ ( b ijk ) ∂x l = 0 . (cid:3) Proposition 4.13. We can define a structure of left V ( D C n ) -moduleon Der C n if n (cid:88) k =1 a kr ∂ ( b ijk ) ∂x l = 0 , ∀ i, l, r = 1 , . . . , n. Proof. To define a structure of left V ( D C n )-module on Der C n , we definethe action of δ i on any derivation ∂ by δ i • ∂ := [ δ i , ∂ ] , or in other words δ i • ∂ t := C i ∂ t , ≤ i ≤ n. The structure just introduced is a V ( D C n )-module structure if andonly if ( δ i δ j − δ j δ i ) • ∂ t = ( n (cid:88) k =1 b ijk δ k ) • ∂ t . An easy computation shows us that this is true if and only if δ i ( C j ) − δ j ( C i ) + [ C j , C i ] = n (cid:88) k =1 b ijk C k hence we can conclude by Lemma 4.12. (cid:3) Remark 4.14. Notice that the action on Der C n of any logarithmicderivation δ = (cid:80) nk =1 β k δ k is given by δ • ∂ t = n (cid:88) k =1 β k C k ∂ t . Lemma 4.15. For i, j = 1 , . . . , n , n (cid:88) k =1 a lk ∂ ( b ijr ) ∂x k = 0 , ∀ i, l, r = 1 , . . . , n if and only if δ i ( B j ) − δ j ( B i ) + [ B j , B i ] = n (cid:88) k =1 b ijk B k . Proof. This is similar to the proof of Lemma 4.12. (cid:3) Proposition 4.16. We can define a structure of left V ( D C n ) -moduleon Der( − log D ) if n (cid:88) k =1 a lk ∂ ( b ijr ) ∂x k = 0 , ∀ i, l, r = 1 , . . . , n. Proof. As the proof of Proposition 4.13. (cid:3) Corollary 4.17. Let D ⊂ C n be a linear free divisor. Then Der C n and Der( − log D ) are left V ( D C n ) -modules.Proof. In this case b ijk ∈ C and so the two previous conditions aretrivially fulfilled. (cid:3) Corollary 4.18. Let D ⊂ C be a free divisor defined by a weightedhomogeneous equation. Then Der C and Der( − log D ) are left V ( D C ) -modules.Proof. Because D is defined by f a weighted homogenous equationand because Der( − log D ) is a free O C -module of rank 2, then we canchoose χ, δ as a basis of Der( − log D ), where χ is an Euler vector fieldand δ ( f ) = 0. Then [ χ, δ ] = αδ , where α ∈ C and so all the b ijk ∈ C .Hence the two previous conditions are trivially fulfilled. (cid:3) EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 31 Definition 4.19. Define the complex Ω • (log D )(Der C n / Der( − log D )) := Ω • (log D ) ⊗ O C n (Der C n / Der( − log D )) with differentials ∇ p : Ω p (log D )(Der C n / Der( − log D )) −→ Ω p +1 (log D )(Der C n / Der( − log D )) given by ∇ p ( ω ⊗ δ ) := dω ⊗ δ + ( − p ω ∧ ∇ ( δ ) , where d is the usual exterior derivative on Ω • (log D ) and ∇ ( δ ) is theelement of Ω (log D ) ⊗ O C n (Der C n / Der( − log D )) such that [ ν, δ ] = ν · ∇ ( δ ) for all ν ∈ Der( − log D ) . Theorem 4.20. There is an isomorphism of complexes of sheaves ofcomplex vector spaces between Ω • (log D )(Der C n / Der( − log D )) and C • ,defined by γ p : Ω p (log D )(Der C n / Der( − log D )) −→ C p γ p ( ω ∧ · · · ∧ ω p ⊗ δ )( δ ∧ · · · ∧ δ p ) := det( ω i · δ j ) ≤ i,j ≤ p δ. Proof. Applying Theorem 3.2.1 from [3] in our case, we deduce thatthere is an isomorphism ψ • between the complexΩ • (log D )(Der C n / Der( − log D ))and the dual of the logarithmic Spencer complex H om V ( D C n ) ( V ( D C n ) ⊗ O C n • (cid:94) Der( − log D ) , Der C n / Der( − log D )) . The isomorphism is defined locally by ψ p (( ω ∧ · · · ∧ ω p ) ⊗ δ )( P ⊗ ( δ ∧ · · · ∧ δ p )) := P · det( ω i · δ j ) ≤ i,j ≤ p · δ. On the other hand, we can write ψ p = λ p ◦ γ p , where λ p is the isomor-phism λ p : C p −→ H om V ( D C n ) ( V ( D C n ) ⊗ O C n p (cid:94) Der( − log D ) , Der C n / Der( − log D )) , defined by λ p ( α )( P ⊗ ( δ ∧ · · · ∧ δ p )) := P · α ( δ ∧ · · · ∧ δ p ) . A direct computation shows that λ p commutes with the differentialsand hence defines an isomorphism of complexes.Hence, also γ p is an isomorphism and commutes with the differentialsand therefore defines an isomorphism of complexes. (cid:3) Definition 4.21. Let D ⊂ C n be a divisor. We say that D is a Koszulfree divisor at x if it is free at x and if there exists a basis δ , . . . , δ n of Der x ( − log D ) such that the sequence of symbols σ ( δ ) , . . . , σ ( δ n ) isregular in G r F • ( D C n ) x . If D is a Koszul free divisor at every point, wesimply say that it is a Koszul free divisor. Notice that for a free divisor D , to be Koszul is equivalent to beingholonomic in the sense of Definition 3.8 from [14], i.e. the logarithmicstratification of D is locally finite. See [8], Theorem 7.4. Example 4.22. (1) ([4], Example 2.8, 3)) Each reduced divisor D ⊂ C is Koszul free.(2) The normal crossing divisor of Example 2.12 is Koszul free.(3) ([4], Example 2.8, 5)) Consider the free divisor D = V (2 z − x z + 2 x z + 2 xy z − x y − y ) ⊂ C with Saitomatrix A = [ δ , δ , δ ] = y x − z x z − x xy y − xy y − xz z . Then the sequence of symbols σ ( δ ) , σ ( δ ) , σ ( δ ) is regular in Gr F • ( D C n ) .(4) ([4], Example 4.2) Consider the free divisor D = V ( xy ( x + y )( y + xz )) ⊂ C with Saito matrix x x y − y − z ( x + y ) xz + y Then D is not Koszul free. Theorem 4.23. Let ( D, ⊂ ( C n , be a germ of a Koszul free di-visor such that (cid:80) nk =1 a kl ∂ ( b ijk ) /∂x r = 0 for i, j, l, r = 1 , . . . , n and (cid:80) nl =1 a kl ∂ ( b ijr ) /∂x l = 0 , for i, j, k, r = 1 , . . . , n . Then all H i ( C • ) areconstructible sheaves of finite dimensional complex vector spaces.Proof. Write E = Der C n / Der( − log D ), E = Der C n and E = Der( − log D ).Using the assumptions, we deduce from Proposition 4.13 and 4.16, thatwe can consider the short exact sequence0 −→ E −→ E −→ E −→ V ( D C n )-module E . By twisting with O C n [ D ],we find another V ( D C n )-resolution0 −→ E [ D ] −→ E [ D ] −→ E [ D ] −→ . EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 33 By [5], Proposition 1.2.3, the complexes D C n L ⊗ V ( D C n ) E i [ D ], for i = 1 , D C n L ⊗ V ( D C n ) E [ D ] through the above resolution as D C n ⊗ V ( D C n ) E [ D ] −→ D C n ⊗ V ( D C n ) E [ D ] . By [5], Proposition 1.2.3, the above complex is holonomic in each de-gree and we deduce that R H om D C n ( O C n , D C n L ⊗ V ( D C n ) E [ D ]) is con-structible. By Theorem 4.20 and by noticing that the isomorphismof [5], Corollary 3.1.5, is true for any V ( D C n )-module, we have thefollowing isomorphisms C • ∼ = Ω • (log D )( E ) ∼ = R H om D C n ( O C n , D C n L ⊗ V ( D C n ) E [ D ])and hence, we can conclude. (cid:3) Corollary 4.24. Let ( D, ⊂ ( C n , be a germ of a Koszul free divisorsuch that (cid:80) nk =1 a kl ∂ ( b ijk ) /∂x r = 0 and (cid:80) nk =1 a lk ∂ ( b ijr ) /∂x k = 0 , for i, j, l, r = 1 , . . . , n . Then FD D has a hull.Proof. By Theorem 4.23, condition (H3) from [15] is satisfied. Thenthe result follows from Theorem 3.6 and Theorem 2.11 from [15]. (cid:3) Corollary 4.25. Let ( D, ⊂ ( C , be a germ of a free divisor definedby a weighted homogeneous equation. Then FD D has a hull.Proof. By Example 4.22, ( D, 0) is Koszul. Because ( D, 0) is definedby f a weighted homogenous equation then we can choose χ, δ as abasis of Der( − log D ), where χ is an Euler vector field and δ ( f ) = 0.Then [ χ, δ ] = αδ , where α ∈ C and so all the b ijk ∈ C . Hence all thehypothesis of previous Corollary are fulfilled. (cid:3) Corollary 4.26. Let ( D, ⊂ ( C n , be a germ of a Koszul linear freedivisor. Then all H i ( C • ) are constructible sheaves of finite dimensionalcomplex vector spaces.Proof. This follows from Theorem 4.23 and the fact that if ( D, 0) islinear then b ijk ∈ C . (cid:3) The author is not aware if there exists a subclass of the Koszul freedivisor that fulfil the assumptions of Theorem 4.23. However, we knowthat not all Koszul free divisor satisfies them. A direct computationshows that the last Koszul free divisor described in Example 4.22 doesnot fulfil them.Moreover, the author thinks that the approach used to put a loga-rithmic connection on Der C n and Der( − log D ) is a particular case ofthe notion of integrability up to homotopy, see [1]. Propagation of Deformations. In this final subsection, we provea result which highlights the difference between the theory of admissibledeformations and the classical deformation theory of singularities.We suppose that ( D, ⊂ ( C n , 0) is a germ of a free divisor suchthat there exists a germ of a free divisor ( D (cid:48) , ⊂ ( C n − , 0) such that( D, 0) = ( D (cid:48) × C , D in C [[ x , . . . , x n − ]]. Theorem 4.27. (Corollary 4.37) There is an isomorphism of sheaves π − H i ( C • D (cid:48) ) ∼ = H i ( C • D ) where π : ( D, −→ ( D (cid:48) , is the projection on the first factor of ( D, 0) = ( D (cid:48) × C , . In particular, we have π − F T ( D (cid:48) ) ∼ = F T ( D ) . Observe that in the ordinary deformation theory of singularities,if ( D, 0) = ( D (cid:48) × C , 0) and T D (cid:48) , is non-zero then T D, is infinitedimensional. See [9], Chapter II, 1.4. Lemma 4.28. In this situation Der( − log D ) = (Der( − log D (cid:48) ) ⊗ O C n − , O C n , ) ⊕ O C n , ∂∂x n and Der C n / Der( − log D ) = Der C n − / Der( − log D (cid:48) ) ⊗ O C n − , O C n , . Hence, if δ ∈ Der( − log D ) , it can be written as δ = ( δ (cid:48) , h∂/∂x n ) , where δ (cid:48) ∈ Der( − log D (cid:48) ) ⊗ O C n − , O C n , and h ∈ O C n , . To distinguish between the complexes for ( D, 0) and for ( D (cid:48) , 0) wewill denote them respectively by ( C • D , d • D ) and ( C • D (cid:48) , d • D (cid:48) ). Proposition 4.29. There is an isomorphism (cid:37) : p (cid:94) Der( − log D ) −→ ( O C n , ⊗ O C n − , p (cid:94) Der( − log D (cid:48) )) ⊕ ( O C n , ⊗ O C n − , p − (cid:94) Der( − log D (cid:48) ))( δ ∧ · · · ∧ δ p ) = ( δ (cid:48) , h ∂∂x n ) ∧ · · · ∧ ( δ (cid:48) p , h p ∂∂x n ) (cid:55)→ ( δ (cid:48) ∧ · · · ∧ δ (cid:48) p , p (cid:88) k =1 ( − p − k h k δ (cid:48) ∧ · · · ∧ (cid:98) δ (cid:48) k ∧ · · · ∧ δ (cid:48) p ) EFORMATIONS OF FREE AND LINEAR FREE DIVISORS 35 Proof. This is because (cid:86) p O C n , = 0 for p ≥ R is a commutative ring and A and B are R -modules, then p (cid:94) ( A ⊕ B ) = (cid:77) i + j = p ( i (cid:94) A ⊗ R j (cid:94) B ) . (cid:3) As a consequence Corollary 4.30. With the hypotheses of Proposition 4.29 C pD = H om O C n, ( O C n , ⊗ O C n − , p (cid:94) Der( − log D (cid:48) ) , Der C n / Der( − log D )) ⊕H om O C n ( O C n , ⊗ O C n − , p − (cid:94) Der( − log D (cid:48) ) , Der C n / Der( − log D )) == C pD (cid:48) ⊗ O C n − , O C n , ⊕ C p − D (cid:48) ⊗ O C n − , O C n , . Remark 4.31. It is possible to write an element Γ ∈ C pD for p > as Γ = ( ψ, φ ) = ( (cid:88) i ≥ x in ψ i , (cid:88) i ≥ x in φ i ) = (cid:88) i ≥ x in ( ψ i , φ i ) with ψ i ∈ C pD (cid:48) and φ i ∈ C p − D (cid:48) . By Remark 4.31, we can describe the differentials Corollary 4.32. The differentials have the following expression ˜ d p : H om O C n, ( O C n , ⊗ O C n − , p (cid:94) Der( − log D (cid:48) ) , Der C n / Der( − log D )) −→−→ H om O C n, ( O C n , ⊗ O C n − , p +1 (cid:94) Der( − log D (cid:48) ) , Der C n / Der( − log D )) where ( ˜ d p ( ψ ))( σ ∧ · · · ∧ σ p +1 ) := (cid:88) i ≥ x in ( d pD (cid:48) ( ψ i ))( σ ∧ · · · ∧ σ p +1 ) . Proposition 4.33. The differential on C • D is given by d pD : C pD −→ C p +1 D ( ψ, φ ) (cid:55)→ ( ˜ d p ( ψ ) , ˜ d p − ( φ ) + ( − p +1 [ ∂∂x n , ψ ( − )]) . Proof. Consider Γ = ( ψ, φ ) ∈ C pD . We want now to compute d pD (Γ) ∈C p +1 D . By Remark 4.31, we need to check it only on Der( − log D (cid:48) ),hence ( d pD (Γ))( σ ∧ · · · ∧ σ p +1 , ν ∧ · · · ∧ ν p ) =(3) = ( d pD (Γ))( σ ∧ · · · ∧ σ p +1 ) + ( d pD (Γ))( ν ∧ · · · ∧ ν p ) , where σ i , ν j ∈ Der( − log D (cid:48) ).We now look at the first part of the right hand side of the previousequality ( d pD (Γ))( σ ∧ · · · ∧ σ p +1 ) == p +1 (cid:88) i =1 ( − i [ σ i , Γ( σ ∧ · · · ∧ (cid:98) σ i ∧ · · · ∧ σ p +1 )]++ (cid:88) ≤ i Moreover (cid:88) ≤ i We can rewrite the differential as d pD : C pD −→ C p +1 D ( ψ, φ ) (cid:55)→ (cid:88) i ≥ x in ( d pD (cid:48) ( ψ i ) , d p − D (cid:48) ( φ i ) + ( − p +1 ( i + 1) ψ i +1 ) . Definition 4.35. We define the morphism J to be the inclusion J : H om O C n − , ( p (cid:94) Der( − log D (cid:48) ) , Der C n − / Der( − log D (cid:48) )) = C pD (cid:48) (cid:44) →C pD = H om O C n, ( O C n , ⊗ O C n − , p (cid:94) Der( − log D (cid:48) ) , Der C n / Der( − log D )) ⊕H om O C n, ( O C n , ⊗ O C n − , p − (cid:94) Der( − log D (cid:48) ) , Der C n / Der( − log D )) == C pD (cid:48) ⊗ O C n − O C n , ⊕ C p − D (cid:48) ⊗ O C n − , O C n , ψ (cid:55)→ x n ( ψ, . All the previous work was devoted proving that in order to computethe cohomology of D it is enough to compute that of D (cid:48) : Theorem 4.36. The morphism J is a quasi-isomorphism.Proof. It is enough to show that the cokernel of J is acyclic. Considerthen Γ = (cid:88) i ≥ x in ( ψ i , φ i ) + (0 , φ ) ∈ coker( J )and suppose that Γ ∈ ker( d pD ). Then we have d pD (cid:48) ( ψ i ) = 0 and d pD (cid:48) ( φ i ) =( − p ( i + 1) ψ i +1 for all i ≥ 0. 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