Foliated Structure of The Kuranishi Space and Isomorphisms of Deformation Families of Compact Complex Manifolds
aa r X i v : . [ m a t h . C V ] D ec FOLIATED STRUCTURE OF THE KURANISHISPACE AND ISOMORPHISMS OF DEFORMATIONFAMILIES OF COMPACT COMPLEX MANIFOLDS(STRUCTURE FEUILLET´EE DE L’ESPACE DEKURANISHI ET ISOMORPHISMES DE FAMILLES DED ´EFORMATIONS DE VARI´ET´ES COMPACTES COMPLEXES)
Laurent Meersseman
October 8th, 2010
Abstract.
Consider the following uniformization problem. Take two holomorphic(parametrized by some analytic set defined on a neighborhood of 0 in C p , for some p >
0) or differentiable (parametrized by an open neighborhood of 0 in R p , forsome p >
0) deformation families of compact complex manifolds. Assume theyare pointwise isomorphic, that is for each point t of the parameter space, the fiberover t of the first family is biholomorphic to the fiber over t of the second family.Then, under which conditions are the two families locally isomorphic at 0? In thisarticle, we give a sufficient condition in the case of holomorphic families. We showthen that, surprisingly, this condition is not sufficient in the case of differentiablefamilies. We also describe different types of counterexamples and give some elementsof classification of the counterexamples. These results rely on a geometric study ofthe Kuranishi space of a compact complex manifold.Consid´erons le probl`eme d’uniformisation suivant. Prenons deux familles ded´eformation holomorphes (param´etr´ees par un ensemble analytique d´efini dans unvoisinage de 0 dans C p pour p >
0) ou diff´erentiables (param´etr´ees par un voisinagede 0 dans R p pour p >
0) de vari´et´es compactes complexes. Supposons les ponctuelle-ment isomorphes, c’est-`a-dire, que pour tout point t de l’espace des param`etres, lafibre en t de la premi`ere famille est biholomorphe `a la fibre en t de la deuxi`emefamille. Sous quelle(s) condition(s) les deux familles sont-elles localement isomor-phes en 0? Dans cet article, nous donnons une condition suffisante dans le cas defamilles holomorphes. Nous montrons ensuite que, de fa¸con surprenante, la conditionn’est pas suffisante dans le cas des familles diff´erentiables. Nous d´ecrivons ´egalementplusieurs types de contre-exemples et donnons quelques ´el´ements de classificationsde ces contre-exemples. Ces r´esultats reposent sur une ´etude g´eom´etrique de l’espacede Kuranishi d’une vari´et´e compacte complexe.1991 Mathematics Subject Classification . 32G07, 57R30.
Key words and phrases.
Deformations of Complex Manifolds, Foliations, Uniformization (d´e-formations de vari´et´es complexes, feuilletages, uniformisation).This work was elaborated and written during a one-year visit as a CNRS member at thePacific Institute for Mathematical Sciences and the University of British Columbia, Vancouver,BC. I would like to thank the PIMS and the UBC Math Department for their hospitality, as wellas the CNRS for giving me the opportunity of this visit.This work was partially supported by grant FABER from the Conseil R´egional de Bourgogneand by project COMPLEXE (ANR-08-JCJC-0130-01) from the Agence Nationale de la Recherche.Typeset by
AMS -TEX LAURENT MEERSSEMAN
Introduction
This article deals with the problem of giving a useful criterion to ensure thattwo holomorphic (respectively differentiable) deformation families are isomorphicas families. This takes the form of the following uniformization problem. Let i = 1 , π i : X i → U respectively π i : X i → V be two holomorphic (respectively differentiable) families of compact complex man-ifolds parametrized by some analytic set U defined on a neighborhood of 0 in C p ,for some p > V of 0 in R p , for some p > pointwise isomorphic , that is, for all t ∈ U (respectively t ∈ V ), the fiber X ( t ) = π − ( { t } ) is biholomorphic to the fiber X ( t ) = π − ( { t } ).Then the question is Question 1.
Under which hypotheses are the families X and X locally isomorphicat ? By locally isomorphic , we mean that there exists an open neighborhood W of 0 ∈ U (respectively in V ), and a biholomorphism Φ (respectively a CR-isomorphism)between X ( W ) = π − ( W ) and X ( W ) = π − ( W ) such that the following diagramis commutative. X ( W ) Φ −−−−→ X ( W ) π y y π W −−−−−−→ Identity W We are also interested in the following broader problem.
Question 2.
Under which hypotheses are the families X and X locally equivalentat ? By locally equivalent , we mean that there exist open neighborhoods W and W of0 ∈ U (respectively in V ), a biholomorphism φ between W and W (respectively adiffeomorphism) and a biholomorphism Φ (respectively a CR-isomorphism) between X ( W ) = π − ( W ) and X ( W ) = π − ( W ) such that the following diagram iscommutative. X ( W ) Φ −−−−→ X ( W ) π y y π W −−−−→ φ W In other words, X and X are locally equivalent at 0 if φ ∗ X and X are locallyisomorphic for some local biholomorphism φ of U (respectively V ) fixing 0.Fix a family X . In this paper, we shall say that this family has the localisomorphism property (at 0), respectively has the local equivalence property (at 0)if every other family X which is pointwise isomorphic to it is locally isomorphic toit (at 0), respectively locally equivalent (at 0).It is known since Kodaira-Spencer (see [K-S2], [We] and Section V.1 of thisarticle) that there exist pointwise isomorphic families of primary Hopf surfaces SOMORPHISMS OF DEFORMATION FAMILIES 3 which are not locally isomorphic, both in the differentiable and the holomorphiccases.On the other hand, the classical Fischer-Grauert Theorem [F-G], can be restatedas follows. Let X be a compact complex manifold and U be an open neighborhoodof 0 in some C p . Then every trivial family X × U has the local isomorphismproperty. This works also for differentiable families. Indeed the proof given in[F-G] for holomorphic families is easily adapted to the differentiable case, the coreof the proof being Theorem 6.2 of [K-S1] which is valid both for differentiable andholomorphic families.Moreover, J. Wehler proved in [We] that, over a smooth base, holomorphic fam-ilies of compact complex tori (in any dimension) as well as holomorphic families ofcompact manifolds with negatively curved holomorphic curvature (this implies thatthey are Kobayashi hyperbolic) have the local isomorphism property. This time,the proofs do not adapt to the differentiable case.Observe that in the two previous cases, the function h ( t ), that is the dimensionof the cohomology group H ( X t , Θ t ) (where Θ t is the sheaf of holomorphic vectorfields along X t ) is constant for all t ∈ U . It is equal to n in the case of n -dimensionaltori, and to 0 in the case of negatively curved manifolds. Wehler asks in theintroduction of [We] if this condition is sufficient to have the local isomorphismproperty.In this paper, we prove that this is the case, even over a singular base. Namely, Theorem 3. If U is reduced and if the function h is constant for all the fibers ofa holomorphic deformation family π : X → U , then X has the local isomorphismproperty. We then give examples (both in the differentiable and holomorphic setting) offamilies not having the local equivalence property, as well as of locally equivalentbut not locally isomorphic families. We classify these counterexamples into twotypes, and we give in Theorem 4 a complete classification of 1-dimensional familiesof type II not having the local equivalence property.Coming back to the search for a criterion, we prove that, surprisingly, things arecompletely different in the differentiable case.
Theorem 5.
There exist differentiable families of 2-dimensional compact complextori parametrized by an interval that are pointwise isomorphic but not locally iso-morphic at a given point.
To solve the uniformization problems stated above, we first study the geometryof the Kuranishi space K of a compact complex manifold X . We show in Theorem1 that it has a natural holomorphic foliated structure: two points belonging tothe same leaf correspond to biholomorphic complex structures. More precisely, K admits an analytic stratification such that each piece of the induced decomposition(see Section III for more details) is foliated. The leaves are complex manifolds, butthe transverse structure of the foliation may be singular (this happens when theKuranishi space is singular).The foliation may be of dimension or of codimension zero. In Theorem 2, weprove that there exists leaves of positive dimension (that is the foliation has positive LAURENT MEERSSEMAN dimension on some piece of the decomposition) if and only if the function h is notconstant in the neighbourhood of 0 in K (0 representing the central point X ). Inparticular, in many examples, the foliation is a foliation by points.Although Theorems 3, 4 and 5 on the uniformization problems are not strictlyspeaking a consequence of Theorems 1 and 2 on the foliated structure of K , thegeometric picture of K they bring played an essential role in the understandingand resolution of the problem. The key ingredients to prove the Theorems aresome trivial remarks on diffeomorphisms of the Kuranishi space (see Section II),the Fischer-Grauert Theorem [F-G], a result of Namba [Na] and a fundamentalproposition proved by Kuranishi in [Ku2].We end the article with a discussion of the relationship between the uniformiza-tion problem and the universality of the Kuranishi space. I. Notations and background
Let X be a compact complex manifold. We denote by X diff the underlyingsmooth manifold and by J the corresponding complex operator.A (holomorphic) deformation family of X is a proper and flat projection π froma C -analytic space X (possibly non-reduced) over an analytic set U defined on anopen neighborhood of 0 in some C p . A differentiable deformation family (see [K-S1]) is a smooth submersion π from a smooth manifold X endowed with a Levi-flatintegrable almost CR-structure over an open neighborhood V of 0 in some R p ,whose level sets are tangent to the CR-structure. If the almost CR-structure on X is not supposed to be integrable, one has a differentiable deformation family ofalmost-complex structures of X .In the three cases, the central fiber X = π − ( { } ) is assumed to be biholomor-phic to X . Sometimes, we consider marked deformation families of X , that is wefix a precise holomorphic identification i : X → X .Let us recall some features of the construction of the Kuranishi space follow-ing [Ku1]. The set of almost complex structures close to J is identified with aneighbourhood A of 0 in the space A of (0 , X with values in T , . Inparticular, 0 represents the complex structure J we started with (here and in thesequel, the topology used on spaces of sections of a vector bundle over X is inducedby some Sobolev norms, see [Ku1] for more details).Put a hermitian metric h on X . Then we have a ¯ ∂ -operator on A p , the spaceof (0 , p )-forms with values in T , , a formal adjoint operator δ with respect to theinduced hermitian product on A p and a Laplace-like operator (cid:3) . Let SH denotethe set of δ -closed forms in A . Kuranishi proves in [Ku1] Proposition K1.
For A small enough, there exists a neighborhood B of in SH and an application Ξ from A to B mapping an almost complex structure α onto a δ -closed representant Ξ( α ) . Moreover, if α ( t ) is a smooth family of almost complexstructures, then so is Ξ( α ( t )) . By representant, we mean that Ξ( α ) and α induce isomorphic almost complexstructures on X diff . SOMORPHISMS OF DEFORMATION FAMILIES 5
Then Kuranishi defines a holomorphic map G from A to A and proves thatit is a biholomorphism between a special subset of A (containing in particular allintegrable almost complex structures close enough to 0) and a neighborhood W of 0 in H , the (finite-dimensional) space of harmonic forms in A ; or, using theDolbeault isomorphisms, in H ( X, Θ).The
Kuranishi space K is the set of integrable structures in W . It is an analyticset. The Kuranishi family K is the family obtained by endowing each fiber X diff ×{ α } of X diff × K → K with the corresponding complex structure encoded by α .Moreover, it is a marked family.The Kuranishi family is complete for smooth deformation families as well as forholomorphic deformation families (unmarked and marked), not only at 0 but alsoat all points of K (shrinking K if necessary). Hence every deformation family X of X is locally isomorphic to the pull-back of K by some map f from its base spaceto the Kuranishi space. By abuse of notation, we write X = f ∗ W . If X has amarking, then we ask the pull-back to respect the markings.The Kuranishi family is versal at 0, i.e. complete in the previous sense withthe additional property that its Zariski tangent space has dimension equal to thedimension of H ( X, Θ). This last property may not be true at points differentfrom 0. The versality property is equivalent to the following. Given a holomorphic marked deformation family π : X → U , in the writing X = f ∗ K , then f may not beunique, but its differential at 0 is. The same property holds for smooth deformationfamilies. It must be stressed that this property is related to the markings. It isusually lost when dealing with unmarked families. The marking is necessary inorder to prevent from reparametrizing the family by an automorphism which actsnon-trivially in the central fiber. Remark.
In the previous setting, if f is unique, then K is called universal . Theuniversality property does not hold for any Kuranishi space, see [Wa] and SectionV.5. Remark.
The versality property of K at 0 implies its unicity (as a germ), see [Ca,Proposition 5.3]. Remark.
The Kuranishi space may be not reduced at every point [Mu]. This ex-plains why one also considers holomorphic families over a non-reduced base. In thiscontext, we would like to point out the following subtle point. If X is a holomor-phic family over a reduced base, or is a differentiable family, then, in the writing X = f ∗ K , the morphism f is in fact a morphism from the base U of X to K red , thereduction of K . Hence the fact that K is reduced or not is not relevant for thesefamilies. But, if U is non-reduced, then such a f is not completely determined byits image in K red ; one has also to specify the value of its differential at each point.Anyway, keeping in mind this difference, it is still true that such a family is obtainedby pull-back from the Kuranishi family and that the differential of the pull-backmap at 0 is unique; that is, the notions of completeness and versality remain thesame. In particular, from the previous remark, one deduces that if a deformationfamily of X over some reduced analytic space A is versal for differentiable families,then it is diffeomorphic to the reduction of the Kuranishi space of X .Finally, we will make an intensive use of the following Proposition LAURENT MEERSSEMAN
Proposition K2. If h is constant on K , there exists a neighborhood U of in K and a neighborhood U of the identity in the group of diffeomorphisms of X diff suchthat, for all couples ( t, t ′ ) of distinct points of U , we have α t ′ ≡ f ∗ α t for some diffeomorphism f = ⇒ f
6∈ U
Here we have α t = G − ( t ) (respectively α t ′ = G − ( t ′ )). To fully understand thisstatement, recall that the Kuranishi family is constructed as a families of complexoperators. Hence every fiber K t is naturally defined as ( X diff , α t ). This crucialProposition is proved by Kuranishi in [Ku2] and used to show that h constantimplies the universality of K . We will discuss this at the end of the article. II. Preliminary remarks on diffeomorphisms of the Kuranishi family
Let us begin with some definitions.
Definition. A diffeomorphism of the Kuranishi space K is a bijective map φ fromsome open neighborhood of 0 in its reduction K red onto some open neighborhoodof 0 in K red such that(i) It sends a complex structure onto an isomorphic complex structure.(ii) Both φ and φ − are restrictions to K red ∩ W ′ ⊂ W ′ ⊂ W ⊂ H ( X, Θ) of asmooth map of W ′ ⊂ W for some W ′ .Such a diffeomorphism is generally, but not always, assumed to fix 0. Noticethat such a map is smooth in the sense of [Ku1]. Definition. A diffeomorphism of the Kuranishi family K is a continuous map F from some open neighborhood of K in K to K such that(i) F descends as a diffeomorphism f of K .(ii) the restriction of F to any fiber of K → K is a biholomorphism.(iii) F is CR in the following sense. Since K → K is a flat morphism, it is locallyisomorphic at each point to an open set of D dim K × K . Representing F locally as amap between two such charts, we ask it to be holomorphic in the D dim K -variables,and smooth in the other variables (in the sense of the previous definition).Notice that, even when F fixes the central fiber, we do not ask it to respect themarkings .In the same way, we define automorphisms of K as isomorphisms of some openneighborhood of 0 in the analytic space K (and not K red this time) generally fixing 0(and thus as restrictions to K of local isomorphisms of W at 0); and automorphisms of K as local isomorphisms of K descending as automorphisms of K . Finally, allthese definitions apply with trivial changes to other deformation families of X thanthe Kuranishi family. Remark.
In the definition of a diffeomorphism (and an automorphism) of K , weconsider K as an analytic space of a C -vector space, and not as a set of complexoperators. This explains why a diffeomorphism of K may not lift to a diffeomor-phism of K . This lifting problem is very close to the local isomorphism problem.Indeed, we will give a criterion for lifting an automorphism of K in Corollary 4, as SOMORPHISMS OF DEFORMATION FAMILIES 7 a consequence of the criterion to have the local isomorphism property. And we alsogive in Lemma 6 an example of an automorphism of K which does not lift.In the second part of this Section, we deal with the problem of extending adiffeomorphism (respectively an automorphism) of the central fiber K to a diffeo-morphism (respectively an automorphism) of K . Let us make first the followingtrivial remark. Let φ be an automorphism of X . Via the marking of K , we considerit as an automorphism of K . The family ( K , K ) with the new identification φ ◦ i is a new versal family for X , hence, by unicity, there exists an automorphism Φ of K fixing 0 and extending φ .The two following Lemmas are trivial but of fundamental importance for thesequel. Part (i) of the first one is even weaker than the previous statement but it hasthe advantage to admit slight generalizations stated in Lemma 2 and Proposition1. Lemma 1. (i) Let φ be an automorphism of X . Then there exists a diffeomorphism Φ of K extending φ .(ii) Let φ be a diffeomorphism of X diff such that φ ∗ belongs to the set A ofProposition K1. Then there exists a diffeomorphism Φ of K extending φ .Proof. In both cases, see φ as a diffeomorphism of X diff . Then it satisfies φ ∗ φ ∗ φ ∗ : α ∈ A φ ∗ α ∈ A with 0 as fixed point or with 0 sent close to 0. Consider the following composition W ′ ⊂ W ⊂ H G − −−−→ A φ ∗ −→ A −→ SH G −→ H taking W ′ small enough to have φ ∗ ( W ′ ) ⊂ A .This gives ˜ φ , a map from W ′ ⊂ W to H . This ˜ φ respects the almost complexstructures, that is sends an almost complex structure onto one which is isomorphic.Hence, it induces naturally a smooth map from K to K that we denote by Φ.Consider the image K ′ ⊂ K red of Φ. Since Φ respects the complex structures,we have that K red is diffeomorphic to Φ ∗ K ′ = Φ ∗ K red . Hence, the analytic set K ′ is versal for differentiable families, so, as remarked at the end of Section I, K ′ and K red must be equal (as germs of 0) and Φ must be a diffeomorphism. (cid:3) Lemma 2.
Let φ be an automorphism of X isotopic to the identity. Then thereexists a diffeomorphism Φ of K extending φ isotopic to the identity. Moreover, ifwe fix an isotopy φ t between φ and the identity on X , then the isotopy between Φ and the identity of K can be assumed to be equal to φ t when restricted to X .Proof. Apply the proof of Lemma 1 to each member of the isotopy φ t . We thusobtain a family Φ t of diffeomorphisms of K extending φ t for all t . By compacityof [0 , t can be defined on a same open neighborhood of 0. Finallysmoothness in t comes from Proposition K1. (cid:3) Of course, the important point in Lemma 2 is that Φ is isotopic to the identity.We draw from these lemmas an important consequence.
LAURENT MEERSSEMAN
Proposition 1.
Assume that the function h is not constant at ∈ K . Then, wecan find φ , an automorphism of X isotopic to the identity, such that(i) it extends as a diffeomorphism Φ of K isotopic to the identity.(ii) the projection of every such extension on K gives a diffeomorphism of K whosegerm at is not the germ of the identity.Proof. The first step of the proof is a very classical argument. Assume withoutloss of generality that K is reduced. The function h is known to be upper semi-continuous. The assumption that it is not constant in a neighborhood of 0 impliesthen that it has a strict maximum at 0. Take a basis of H ( K , Θ ). If every vectorfield of this basis could be extended to a vector field of the family K (tangent to thefibers, globally smooth and holomorphic along the fibers), at a point t ∈ K closeenough to 0, they all would form a free family of dimension h (0) of H ( K t , Θ t ),contradicting the inequality h ( t ) < h (0).Hence, there exists a global vector field ξ on K which cannot be extended as avector field of K tangent to the fibers. Let φ be the corresponding automorphism of K isotopic to the identity obtained by exponentiation for small time. By Lemma2, there exists a diffeomorphism Φ of K extending φ isotopic to the identity, proving(i).Now, for every such choice of Φ, the induced diffeomorphism of K cannot be theidentity, even in germ, otherwise the global vector field of K obtained by differen-tiating Φ would be tangent to the fibers and would extend ξ . Contradiction whichproves the Proposition. (cid:3) A classical result of [K-S1] states that, if the function h is constant in a smoothdeformation family π : X → V , then every automorphism of X isotopic to theidentity can be extended as a diffeomorphism of the family X which is the identityon V . Automorphisms of X which does not extend as automorphisms of K that arethe identity on K are usually called obstructed automorphisms . Proposition 1 tellsus that obstructed automorphisms extend as diffeomorphisms of K with non-trivialprojection on K , but isotopic to the identity. III. Foliated structure of the Kuranishi space1. Local submersions.
Let t be a point of K corresponding to the complex manifold X t = ( X diff , J t ).Denote by ( K ) t the space K but with base point t and not 0 and by ( K ) t thecorresponding deformation family of X t (choosing some identification maps). Thefamily ( K ) t → ( K ) t is complete at t , but not always versal. On the other hand, let K ( t ) be the Kuranishi space of X t , and K ( t ) the corresponding versal family. Wethus have a sequence of pointed analytic spaces( S ) ( K ( t ) , i t −→ (( K ) t , t ) s t −→ ( K ( t ) , K ( t ) I t −→ ( K ) t S t −→ K ( t ) SOMORPHISMS OF DEFORMATION FAMILIES 9
And ( S ) is the restriction of the sequence (defined on neighborhoods of the basepoints) ( H ( X t , Θ t ) , ˜ ı t −→ ( H ( X , Θ ) , α t ) ˜ s t −→ ( H ( X t , Θ t ) , Lemma 3.
The map ˜ s t is a submersion at α t .Proof. Since K ( t ) is versal at 0, the composition ı t ◦ s t is a local isomorphism at 0.So is ˜ ı t ◦ ˜ s t . Hence ˜ s t is a submersion at J t . (cid:3) Apply the submersion Theorem to ˜ s t . This gives a diagram V ⊂ ( H ( X , Θ ) , J t ) ˜ s t −−−−→ W ⊂ ( H ( X t , Θ t ) , local biholomorphism y x natural projection W × B −−−−−→ identity W × B where B is the unit euclidean ball of C p for p = h (0) − h ( t ) and h ( t ) denotes thedimension of H ( X t , Θ t ).This submersion allows to locally foliate H ( X , Θ ) in a neighborhood of α t .The leaves correspond to deformation families of almost complex structures whichare pull-back by ˜ s t of a constant family. In other words, the points of a same leafall define the same almost complex structure up to isomorphism.When we restrict to K , we obtain the diagram of analytic spaces( K ) t s t −−−−→ K ( t ) local biholomorphism y x natural projection K ( t ) × B −−−−−→ identity K ( t ) × B that defines a local foliated structure of K .We aim at gluing those local foliations together into a global foliation. Thisbrings some problems since the induced foliations in two arbitrarily close points maybe of different dimensions. To overcome this problem, it is necessary to decomposethe space K .
2. Decomposition of K . Decompose K as a disjoint union( D ) K = K min ⊔ . . . ⊔ K max where K i denotes the set of points t of K such that the dimension of K ( t ) is i .Observe that the completeness of K at each point implies that 0 belongs to K max .Set Z j = ⊔ i ≤ j K i . The sequence Z min ⊂ . . . ⊂ Z max = K is a stratification of K .We want to show that this decomposition is analytic in the sense that, for allmin ≤ i < max, the set Z i is an analytic subset of Z i +1 and thus of K . Lemma 4.
The decomposition ( D ) of K is analytic.Proof. Assume first that K is reduced. Definemin ≤ c ≤ max E c = { t ∈ K | h ( t ) ≥ c } These sets forms an analytic stratification of K , see [Gr]. Call G = ( G c ) theassociated decomposition.On the other hand, denoting by G c the family of complex structures with base G c , the results of [Gr] show that the group H ( G c , Θ) is isomorphic to a locallyfree sheaf over G c whose stalk at t is H ( X t , Θ t ). The set of integrable structuresin this sheaf is given as the zero set Z t of a field of quadratic forms in the fiberswhich is analytic in t .This allows us to analytically stratify each piece G c by d ≤ c G c,d = { t ∈ G c | codim Z t ≤ d } Then the E c respectively the G c,d are analytic sets of K respectively of G c ,whereas G c ⊂ E c is a quasi-analytic open set.Observe now that the completeness property of a Kuranishi space at each point close enough from the base point implies that the function t ∈ K dim K ( t ) isupper semi-continuous for the standard topology. Since G c,d is an analytic set of G c and since the Zariski closure of G c , that is E c , is equal to its closure for thestandard topology, this proves that the Zariski closure G c,d of each G c,d in G c = E c is just G c,d = { t ∈ E c | codim Z t ≤ d } We now just have to setmin ≤ i ≤ max F i = ∪ c − d ≥ i G c,d to obtain an analytic stratification of K whose associated decomposition F i \ F i +1 ( i > min) and F min is the decomposition D .If K is not reduced, we just perform the previous stratification on its reduction,then we obtain the decomposition ( D ) by putting on each F i the multiplicity in-duced from K . Of course, we may be forced to add more pieces if some componentcontains an analytic subspace of higher multiplicity (think of a double point insidea line; then K has two components, whereas K red has just one). (cid:3) Let t ∈ K i for some i . By definition, i t and s t respect the decompositions of K and K ( t ), that is i t ( K ( t ) max ) = K i et s t ( K i ) = K ( t ) max It is thus possible to restrict the submersions s t to a piece of the decompositionto obtain the following diagram( K i ) t s t −−−−→ K ( t ) maxlocal biholomorphism y x natural projection K ( t ) max × B −−−−−→ identity K max t × B In the sequel, s t will always denote the restricted submersion from K i to thepiece K ( t ) max , for t ∈ K i . SOMORPHISMS OF DEFORMATION FAMILIES 11
3. Foliated structure of K . Using the submersion s t , we will define a foliated structure on each piece K i . Definition.
Let X be an analytic space. A transversally singular foliation ofdimension p on X is given by an open covering ( U α ) of X red and local isomorphisms φ α : U α → B × Z α (for B the unit ball of C p and Z α a reduced analytic space) suchthat the changes of charts φ αβ ≡ φ β ◦ ( φ α ) − preserve the plaques B × { pt } .Thus a transversally singular foliation is a lamination with a transverse structureof an analytic space. The choice for this somewhat unusual name (rather thananalytic lamination or something analogous) comes from the fact that we find morejudicious to reserve the word lamination to a situation where the total space hasno analytic structure.Now, given such a foliation, one may define the leaves as in the classical case bygluing the plaques. Hence the leaves are holomorphic manifolds. Remark also thatthe germ of the analytic space Z α is the same along a fixed leaf.Starting from a non-reduced space X , one may consider such a foliation on itas a foliation of its reduction with holomorphic submanifolds as leaves; or onemay endow X itself with a ”non-reduced” foliation whose leaves are non-reducedholomorphic submanifolds. Both points of view are equivalent.We may state Theorem 1.
Let K be the Kuranishi space of a manifold X . Consider the decom-position D of K .Then each piece K i admits a transversally singular foliation F i locally definedby the submersions s t of Section 2. Notice that two points belonging to the same leaf correspond to the same complexmanifold (up to biholomorphism). Notice also that the Kuranishi family K admitsan induced decomposition in pieces K i , each of these pieces being foliated. By [F-G], the leaf of K i corresponding to t ∈ K i is a locally trivial fibre bundle with fibre X t over the leaf of K i through t . And the foliation of K (respectively K ) extendsto a holomorphic foliation, respectively an almost-complex foliation (this time inthe classical sense, that is with smooth transverse structure) of W and respectively W using the submersions S t . To simplify the exposition, we will always considerthe foliation of K . Proof.
Take a covering of a piece K i by open sets where a submersion s t is welldefined. Using the submersion Theorem, we obtain foliated charts modelled on aproduct of a ball of dimension (dim K − i ) with an analytic space of dimension i .Now, the changes of charts respect the leaves, since they have an intrinsic geometricdefinition: the leaf through t ∈ K i is the maximal connected subset of K i of pointscorresponding to the complex manifold X t up to biholomorphism. (cid:3) Remark that this proof adapts immediately to the case of W , with the onlydifference that X t may just be an almost complex manifold. Notation and Definition.
We denote by F the global foliation of Theorem 1,that is the union of the foliations F i .We say that the foliation F is trivial if each foliation F i is a foliation by points. The foliation F is trivial if and only if the decomposition ( D ) has a single piece,that is if and only if the dimension of K ( t ) is constant near 0. On the other hand,observe that F may be of codimension 0 (that is each F i has codimension 0 in K i )with a non-trivial decomposition (see the examples below).
4. Examples. (i) Let X be a complex torus of dimension n . Following [K-S2], the Kuranishi spaceof X may be represented by a neighborhood of 0 in C n and it is versal at eachpoint. Therefore, the decomposition ( D ) has a unique piece and the foliation F istrivial.However, observe that given a well-chosen compact complex torus, there existsan infinite sequence of points of K corresponding to this torus [K-S2, p. 413]. Thisshows that two points belonging to different leaves of F may nevertheless definethe same complex structure.(ii) Let X be the Hirzebruch surface F . Its Kuranishi space may be representedby a unit 1-dimensional disk whose non-zero points all correspond to P × P , see[Ca]. The decomposition is K = K ⊔ K = { } ⊔ D ∗ and both foliations have codimension zero.(iii) Let X be the Hopf surface obtained from C \ { (0 , } by taking the quotientby the group h i generated by the homothety ( z, w ) · ( z, w ). Its Kuranishispace is described in [K-S2]. It may be represented by a neighborhood of the matrix2Id in K = { A ∈ GL ( C ) | | Tr A | > , | ∆( A ) | = | (Tr A ) − A | < } A point A of K corresponds to the Hopf surface C \ { (0 , } / h A i . If A is a mul-tiple of the identity, then the corresponding Kuranishi space K ( A ) has dimensionfour; in other words K is versal along the set∆ = { λ Id | | λ | > } However, if A is not a multiple of the identity, the dimension of K ( A ) drops to2. Thus we decompose K into two pieces K = ∆ and K = K \ ∆On the other hand, consider the map φ : A ∈ K (Tr A, ∆( A )) ∈ C Let ( σ, δ ) be a point of C with | σ | > | δ | <
1. If δ is different from zero, allpoints of φ − ( σ, δ ) correspond to the same Hopf surface. If δ is zero, the same istrue for all points of φ − ( σ, δ ) except for σ/ · Id, which corresponds to a differentHopf surface. Notice that in this case, the level set φ − ( { ( σ, δ ) } ) is singular at σ/ · Id.As a consequence of all that, the foliation F is a foliation by points, whereas thefoliation F is a non-trivial one, which is given by the level sets of the submersion φ restricted to K . It has dimension and codimension two. SOMORPHISMS OF DEFORMATION FAMILIES 13
IV. Non-triviality criterion for F The aim of this section is to prove the following result.
Theorem 2.
Let K be the Kuranishi space of X and let F be the foliation of K constructed in section III.Then F is trivial if and only if h is a constant function on K . This leads to the following corollary.
Corollary 1.
The Kuranishi space K is versal at all points if and only h is aconstant function on K .Proof of Corollary 1. Combine Theorem 2 and the remark after the definition oftriviality for F . (cid:3) Let us proceed to the proof of Theorem 2.
Proof of Theorem 2.
Assume h constant on K and assume at the same time that F is non-trivial. Thus there exists a piece K i ⊂ K whose foliation F i has positive-dimensional leaves. So there exist non-constant smooth paths c : [0 , → K suchthat the induced family C = c ∗ K has all fibers biholomorphic. Choose such anon-constant path staying inside the neighborhood U appearing in Proposition K2.Now Fischer-Grauert Theorem [F-G] implies that C is the trivial family; in otherwords there exists ( φ t ) t ∈ [0 , an isotopy such that(i) φ ≡ Id.(ii) For all t ∈ [0 , φ t ) ∗ α c (0) = α c ( t ) .For t small enough, we have φ t in U , violating Proposition K2. Contradiction.The foliation is trivial.Reciprocally, assume h non-constant. Then by Proposition 1, there exists anautomorphism φ of X isotopic to the identity all of whose extensions as a diffeomor-phism of K does not project onto the identity of K on any neighborhood of 0. LetΦ be one of these extensions; still by Proposition 1, recall that Φ may be chosen iso-topic to the identity. Let Φ t be the isotopy. All that means that there exist points x ∈ K arbitrary close to 0 such that the path t ∈ [0 , Φ t ( x ) ∈ K is a non-constant path. But Fischer-Grauert Theorem may be geometrically reformulatedas follows. Lemma 5.
The Kuranishi space of a compact complex manifold does not containa non-constant path passing through all of whose points correspond to X .Proof. Assume the contrary and consider the family associated to such a non-constant path. By [F-G], its Kodaira-Spencer map is zero at every point. On theother hand, K is versal at each point of the path. This is due to the fact that atany such point t , the Zariski dimension of K is greater than h ( t ) by completeness.Since h ( t ) is equal to h (0), they must be equal yielding the versality of K at t .Now all that means that this non-constant path should be parametrized by a mapwhose derivative at each point is zero. Contradiction. (cid:3) As a consequence, for such a point x , the space ( K ) x cannot be the Kuranishispace K ( x ). That is it is not versal at x . But we already noticed in III.3 that thisis enough to prove that the foliation is non-trivial. (cid:3) Finally, note that:
Corollary 2.
The stratum K max (or, in the non-reduced case, the union of thestrata corresponding to the stratum K max of K red ) is the set of points t ∈ K suchthat h ( t ) = h (0) .Proof. Assume that K is reduced. Let H = { t ∈ K | h ( t ) = h (0) } This is an analytic space by [Gr] (recall that h ( t ) ≥ h (0) implies equality).Arguing exactly as in the first part of the proof of Theorem 2, we show that F istrivial on H . So H is included in K max .Conversely, assume that h is not constant on K max . Then arguing as in thesecond part of the proof of Theorem 2, we show that F max is non trivial. Contra-diction.The non-reduced case can be treated in a similar way. (cid:3) Remark.
Indeed, although Proposition K2 is stated for the complete Kuranishispace K , a quick look at the proof shows that it is valid in restriction to any subset V ⊂ K where h is constant equal to h (0). V. The isomorphism and equivalence problems
We refer to the introduction for the definition of these two problems. Let us givetwo more definitions.
Definitions.
Let X and X be two families which are pointwise isomorphic butnot locally isomorphic at 0. Then we say that they form a type (II) counterexample(to the isomorphism property) if there exist f, g : ( U, −→ ( K, X = f ∗ K and X = g ∗ K .(ii) There exist U ⊂ U and U ⊂ U such that f ( U ) and g ( U ) are equal.And we say that they form a type (I) counterexample if we cannot find f and g as above.Same definitions are valid for the equivalence problem. Roughly speaking, a type(II) counterexample is a counterexample obtained by reparametrization, whereas atype (I) counterexample relies on the particular geometric structure of the Kuranishispace.
1. Counterexamples.
A counterexample of type (II) for the two problems (in both differentiable andholomorphic cases) can be found in [K-S2] (and explained in [We]). Start with theHopf surface C \ { (0 , } / h i . We use the notations of Section III.4.(iii). Define X as the family corresponding to an embedding disk (respectively an interval) inthe closure of the two-dimensional leaf φ − (4 , SOMORPHISMS OF DEFORMATION FAMILIES 15
Definition.
A holomorphic (respectively differentiable) jumping family is a family π : X → U (respectively π : X → V ) such that(i) it is trivial outside 0, but the central fiber X is not biholomorphic to the genericfiber.(ii) The Kodaira-Spencer map at 0 is not zero.Notice that the Kodaira-Spencer map ρ of our family X is not zero at 0, since itis an embedding at 0. This follows from the versality property of K at 0. Considernow the ramified covering z ∈ D z ∈ D or more generally z n ∈ D or respectively t ∈ I t ∈ I . Then define X as the pull-back of X by thisapplication. By the “chain-rule for the Kodaira-Spencer map”, we have at 0 ρ (cid:18) ∂∂z (cid:19) = ρ (cid:18) Jac ( z z n ) · ∂∂z (cid:19) = ρ (0) = 0respectively ρ (cid:18) ∂∂z (cid:19) = ρ (cid:18) Jac ( t t ) · ∂∂z (cid:19) = ρ (0) = 0so X is not isomorphic, nor equivalent, to X .Of course, this construction can be generalized starting from any jumping family(for example, one can take the jumping family with the Hirzebruch surface F ascentral fiber and P × P as generic fiber; this shows that such counterexamplesexist even for projective manifolds). So we state: Proposition 2.
A (holomorphic or differentiable) jumping family has neither thelocal isomorphism, nor the local equivalence property.
It is important for the sequel to observe that the function h is not constant ina jumping family (cf [Gri]).We give now a type (I) counterexample for the two problems, in both differen-tiable and holomorphic cases. Although it can be obtained easily from the treatmentof Hopf surfaces in [K-S2], we do not know of any reference where it is described.Once again, we use the results and the notations of Section III.4.(iii). Consider X = C \ { (0 , } × D / (cid:28)(cid:18) t t t (cid:19) , t (cid:29) X = C \ { (0 , } × D / (cid:28)(cid:18) t t t (cid:19) , t (cid:29) Replacing D by I in the definition of X and X , one obtains a differentiablecounterexample.We claim that X and X are pointwise isomorphic but not locally equivalentat 2Id, and finally that they have distinct image in K . The last point is a direct consequence of the fact that, since the families are embedded, same image wouldimply locally isomorphic.Now, an elementary computation shows that for t = 0, the fibers ( X ) t and( X ) t are biholomorphic and conjugated by P ( t ) = (cid:18) ± t q ± t − (cid:19) where q is any complex number (we assume without loss of generality that P has de-terminant one). Since ( X ) = ( X ) and since none of these conjugating matricesextend at 0, we are done for the isomorphism problem. Finally, for the equiva-lence problem, just observe that if t and t ′ are distinct and both different from 0,then ( X ) t and ( X ) t are not biholomorphic (look at the traces). Hence, in thiscase, there is no difference between the isomorphism problem and the equivalenceproblem.Notice that h is not constant along these families, dropping from 4 (at 0) to 2.Let us give now examples of locally equivalent but not locally isomorphic families.We still use the Kuranishi space of the Hopf surface described in Section III, 4, (iii).The key point is given by the following Lemma. Lemma 6.
The map A ∈ K → t A ∈ K is an automorphism of K fixing Id whichdoes not lift to an automorphism of K .Proof. This is clear for K using the fact that A and t A are conjugated. On theother hand, assume that this automorphism lifts to an automorphism of K . Then,in a neighborhood of 2Id, it would be possible to find a family of invertibles matrices P ( A ) depending holomorphically on A such that t A = P − ( A ) · A · P ( A )where we assume without loss of generality that P ( A ) has determinant equal toone. Straightforward computations show that we must have P ( A ) = (cid:18) α ± i ± i (cid:19) for A = (cid:18) t (cid:19) t ∈ C where α is any complex number and can be chosen independently of t . And wemust also have P ( A ) = (cid:18) α
00 1 /α (cid:19) for A = (cid:18) t (cid:19) t ∈ C where α is any non-zero complex number and can be chosen independently of t .Since these two families do not have any common limit where t goes to zero, weare done. (cid:3) Now let X be the Kuranishi family and let X be obtained by pull-back by thetransposition map. So the two families are locally equivalent by definition. Now,by Lemma 6, they are not locally isomorphic.Observe that this trick gives only type (II) counterexamples. SOMORPHISMS OF DEFORMATION FAMILIES 17
2. Holomorphic families.
In this section, we prove
Theorem 3.
Let π : X → U be a holomorphic family of deformations. If U is re-duced and if h is constant in a neighborhood of , then it has the local isomorphismproperty. Notice the immediate Corollaries.
Corollary 3.
Let X be a compact complex manifold such that h is constant on itsKuranishi space X . Then any holomorphic deformation family of X with reducedbase has the local isomorphism property at . Corollary 4.
Let π : X → U be a holomorphic family. Assume that h is constantand that U is reduced. Then every automorphism of U lifts to an automorphism of X .Proof of Corollary 4. Let f be an automorphism of U , then X and f ∗ X are point-wise isomorphic. So are locally isomorphic by Theorem 3. And this means that f lifts. (cid:3) On the other hand, recall that we gave in Lemma 6 an example of an automor-phism of a reduced Kuranishi space which does not lift.
Proof of Theorem 3.
Assume h constant in a neighborhood of 0. Assume first that π is a 1-dimensional family parametrized by the unit disk. Let π ′ : X ′ → D be apointwise isomorphic family. Let f, g : D −→ K such that X = f ∗ K and X ′ = g ∗ K . We may assume without loss of generality that(i) The maps f and g are defined on the whole disk (otherwise shrink and uni-formize).(ii) The families X and X ′ are equal to f ∗ K and g ∗ K , not only isomorphic (otherwisereplace).We assume also without loss of generality that K is reduced, since f and g mapin fact onto K red .Call D the image of f , and D ′ that of g . If D or D ′ is reduced to a point, then allthe fibers of X and of X ′ are biholomorphic and Fischer-Grauert Theorem impliesthat both D and D ′ are reduced to a point. Both families are locally trivial, hencelocally isomorphic. So we may assume that D and D ′ are disks.Choose ( φ t ) t ∈ D a family of pointwise biholomorphisms φ t : K f ( t ) −→ K g ( t ) Lemma 7.
There exists a dense subset of D such that, for each t in this subset,there exists a sequence ( t n ) n ∈ N with all t n different from t and with ( φ t n ) convergingto φ t in Sobolev norms as n goes to infinity.Proof. It is inspired from [F-G]. Since the set D is uncountable, the sequence ( φ t ) t ∈ D contains an accumulation point for the Sobolev topology. This comes from the fact that Dif f ( X diff ) endowed with the Sobolev topology contains a countable densesequence (compare with [Bourbaki, Topologie G´en´erale, Chapitre 10, Th´eor`eme 3.1]which gives a proof for the topology of uniform convergence). Moreover, given anyopen set U ′ of D the same is true for the subset ( φ t ) t ∈ U ′ . Hence the claim. (cid:3) As a consequence, fix a neighborhood V of 0 in K . Then Lemma 7 implies thatthere exists a sequence ( t n ) n ∈ N ∈ V converging to some point t ∞ of V with φ t n converging to φ t ∞ in Sobolev norms.Assume that f ( t ∞ ) is equal to g ( t ∞ ) and that φ t ∞ is the identity. By PropositionK2, assuming V small enough, this would mean that we must have n ≥ n f ( t n ) = g ( t n )for n big enough. Now, this implies that f − g is a holomorphic function on thedisk with a non-discrete set of zeros, hence f ≡ g and we are done. Observe that K is naturally embedded in the vector space H ( X, Θ) as an analytic set, hence thedifference f − g is meaningful as holomorphic map from D to H ( X, Θ).In the general case, things become more complicated, but the previous patterncan be used as a guideline to proceed. Consider the embedded family K | D ′ → D ′ and write K | D ′ = ( D ′ × X diff , α )where the complex operator α t ∈ A turns { t } × X diff into the complex manifold K ( t ).Remark that we have a diffeomorphism φ − t ∞ : K g ( t ∞ ) −→ K f ( t ∞ ) Remark also that, by Corollary 2, the set K is versal at both f ( t ∞ ) and g ( t ∞ ). Soby Lemma 1, there exists a diffeomorphism (Ψ , ψ ) of K defined on a neighborhoodof f ( t ∞ ) which extends φ − t ∞ . To simplify the notations, we identify in this proof apoint t of K and the integrable almost-complex operator α t = G − ( t ) defining K t .With this convention, ψ is constructed as a composition of( φ − t ∞ ) ∗ : A −→ A with the map Ξ of Proposition K1. This gives us a new realization h ≡ ψ ◦ g : U ′ ⊂ D −→ K defined on a neighborhood U ′ of t ∞ such that X ′ is locally isomorphic to h ∗ K at t ∞ .But now we can make use of Proposition K2. The sequenceΨ ◦ φ t n : K f ( t n ) −→ K h ( t n ) converges in Sobolev norms toΨ ◦ φ t ∞ : K f ( t ∞ ) Identity −−−−−→ K h ( t ∞ ) = K f ( t ∞ ) SOMORPHISMS OF DEFORMATION FAMILIES 19 hence f and h take the same values not only at t ∞ but also at every t n for n big enough. Moreover, still by Proposition K2, and since we assumed that K isreduced, the map h is the unique map such that X ′ is locally isomorphic to h ∗ K at t ∞ (provided K is based and marked at f ( t ∞ ) and provided a marking of X ′ is fixed at t ∞ and asked to be preserved). Since the family X ′ is a holomorphicfamily, h must be holomorphic. So as before we have f ≡ h . Remark.
This is just another way of saying that K is universal with respect tofamilies with h constant equal to h (0). Proposition K2 was proved by Kuranishito have this type of result.We claim that X ′ is isomorphic to h ∗ K over the whole disk D , and not only overa neighborhood of t ∞ in D .This can be proven as follows. Let U ′ ⊂ D be the maximal subset of D such that X ′| U ′ is isomorphic to h ∗| U ′ K . Let t ∈ D and let c be a path in D joining c (0) = t ∞ to c (1) = t . We will prove that t is in U ′ .The problem that could appear is that ( φ − t ∞ ) ∗ f ( c ), which is a path in A , is notfully included in the domain of definition A of Ξ. Let K ′ ⊂ A be the Kuranishispace of X ′ f ( c (1)) based at ( φ − t ∞ ) ∗ f ( c ) (which is reduced since K is reduced). LetΞ ′ be the map of Proposition K1 defined in a neighborhood A ′ of ( φ − t ∞ ) ∗ f ( c (1))in A . For simplicity, assume that the whole path ( φ − t ∞ ) ∗ f ( c ) is included in A ∪ A ′ . Take a point s ∈ [0 ,
1] such that ( φ − t ∞ ) ∗ f ( c ( s )) lies in the intersection of A and A ′ . Then there exists a local isomorphism between the pointed analytic sets( K, Ξ(( φ − t ∞ ) ∗ f ( c ( s ))) and ( K ′ , Ξ ′ (( φ − t ∞ ) ∗ f ( c ( s ))) since these two spaces are versalfor X ′ f ( c ( s )) . And this isomorphism can be chosen in such a way that the imageof Ξ(( φ − t ∞ ) ∗ f ( c )) is sent to Ξ ′ (( φ − t ∞ ) ∗ f ( c )) in a neighborhood of s , still by theuniversality property.Let us sum up. We can glue K and K ′ to obtain an analytic space ˜ K such that h extends as ˜ h along c in such a way that X ′ is isomorphic to ˜ h ∗ ˜ K along the fullpath c . Still by universality, in our case, ˜ h must be equal to h , so that t is in U . Inparticular, observe that the image of ˜ h stays in K ⊂ ˜ K . So the claim is proved.Now, we have X ′ ≃ h ∗ K = f ∗ K ≃ X on the whole disk (the symbol ≃ meaning isomorphic). In other words, X and X ′ are locally isomorphic in a neighborhood of 0. This proves the Theorem for1-dimensional families.Let us now assume that the families X and X ′ are p -dimensional. We will usegeneral arguments (already used in [We], though not exactly in the same way) topass from the one-dimensional to the general case.By a Theorem of Namba [Na, Theorem 2], the union H of pointwise holomor-phic maps from X t to X ′ t for all t can be endowed with a structure of a reducedanalytic space such that the natural projection map p : H → U is holomorphic andsurjective. Moreover, the topology of H is that of uniform convergence.Let S ⊂ H be the subset of pointwise isomorphisms. It is an open set of H so areduced analytic space with a holomorphic (still surjective in this particular case) projection map p . This openness property can be shown as follows. Given φ , anisomorphism between X t and X ′ t for a fixed t , every ψ close enough from φ in thetopology of uniform convergence is a local isomorphism at each point. We just haveto prove now that ψ must be bijective. Forgetting the complex structures we can see φ and ψ as maps of X diff , using differentiable trivializations. Since X diff is compactand ψ locally bijective, ψ is surjective. Besides, still by compacity, there exists afinite open covering of X diff such that any map close enough from φ is injectivewhen restricted to any member of this covering. Assume ψ is not globally injective.Then, we could find a sequence of non-injective maps ψ n converging uniformly onto φ . So there would be two sequences of points ( x n ) and ( y n ) such that n ∈ N x n = y n ψ n ( x n ) = ψ n ( y n )By compacity of X diff , they will converge to some points x and y such that φ ( x ) = φ ( y ), hence x = y . This clearly contradicts the previous property of localinjectivity of all φ n on a fixed covering.To finish the proof of Theorem 3, it is enough to show that p : S → U has a localholomorphic section at 0. But now, we conclude from what we did for 1-dimensionalfamilies that p has local holomorphic sections at 0 along every embedded disk D in U . Fix one of these local sections, say σ . Take another such section σ ′ . Then σ and σ ′ differ by composition (at the source) by an automorphism of X and bycomposition (at the target) by an automorphism of X ′ . If these automorphismsbelong to the connected component of the identity, since U is reduced and h isconstant, both extend locally as automorphisms of the nearby fibers [Gr]. But thismeans exactly that, composing σ ′ with these extensions, one may assume withoutloss of generality that σ ′ takes the same value at 0 as σ . Using this trick and takingaccount that the number of connected components of the automorphism group of X is countable, we see that there exist local sections with the same value at 0 foralmost every disk embedded in U passing through 0.Now, by a Proposition of Grauert and Kerner [G-K], there exists an analyticembedding of a neighborhood S of σ (0) in S i : S −→ D dim p − ( { } ) × U such that the following diagram commutes S i −−−−→ D dim p − ( { } ) × U p y y U −−−−−→ Identity U Observe that the dimension of p − ( { } ) is h (0) and that, since h is constant, i ( p − ( s )) is an open set of D h (0) × { s } . On the other hand, by what preceeds, p ( S ) must be equal to an open neighborhood of 0 in U (because U is reduced). Asa consequence, i is a local isomorphism, which exactly means that X and X ′ arelocally isomorphic at 0. (cid:3) SOMORPHISMS OF DEFORMATION FAMILIES 21
Remark.
The last strategy (using Namba’s Theorem and so on) cannot be useddirectly to obtain the result for 1-dimensional families. Indeed, it is not possibleto exclude the case of p − ( { } ) being isolated from the other fibers, so that in thediagram above, the image p ( S ) reduces to 0. The only fact that can be provendirectly is that, if we know that there exists a sequence φ t n : X t n −→ X ′ t n converging uniformly to some φ : X → X ′ , then the two families are locallyisomorphic at 0. This is just because, in this case, p ( S ) is an analytic set of D (weare in the 1-dimensional case) containing an infinite sequence ( t n ) accumulating on0. So p ( S ) must contain an open neighborhood of 0. Now, we obtained the sameconclusion using Proposition K2. Remark.
In the non-reduced case, Theorem 3 is false, as shown by the followingeasy example. Consider the upper half-plane H of C as the parameter space ofelliptic curves. Let H → H the versal (at each point) associated family. Choose apoint τ ∈ H . Take U to be the double point U = { t = 0 | t ∈ C } Let π : X → U be the constant family obtained by pull-back by a constant mapfrom U to H (with value τ ). Now, since U is not reduced, there exists also non-constant morphisms from U to H . Let f be the unique such morphism sendingthe single point of U to τ and the vector ∂/∂t of the Zariski tangent space of U to the horizontal unit vector of H based at τ . Define X as f ∗ H . Then X and X are obviously pointwise isomorphic, but they are not locally isomorphic, bycomputation of their Kodaira-Spencer map. It is 0 for X , and not zero for X .
3. Type (II)-counterexamples to the equivalence problem.
We derive now a characterization of type (II) counterexamples to the equivalenceproblem in the one-dimensional case.
Theorem 4.
The following statements are equivalent.(i) The one-dimensional holomorphic families π : X → D and π ′ : X ′ → D form atype (II) counterexample to the equivalence problem.(ii) Both are obtained from the same jumping family π ′′ : X ′′ → D by pull-backs bysome maps. Moreover, the degrees of these maps (as ramified coverings of D ) aredifferent.Proof. Assume that X (respectively X ′ ) are obtained from the Kuranishi space of X by pull-back by some map f (respectively h ). Call D the image of f and D ′ that of h . Shrinking the domains of definition if necessary to have the same image D ′′ ⊂ D ∩ D ′ and uniformizing at the source and at the target by unit disks, weobtain the following diagram D uniform. −−−−−→ f − ( D ′′ ) ⊂ D f −−−−→ D ′′ uniform. −−−−−→ D Id y y Id D uniform. −−−−−→ h − ( D ′′ ) ⊂ D h −−−−→ D ′′ uniform. −−−−−→ D To simplify, we still denote by f (respectively by h ) the composition of the toparrows (respectively of the bottom arrows). Moreover, we denote by π ′′ : X ′′ → D the target family and replace X (respectively X ′ ) by f ∗ X ′′ (respectively h ∗ X ′′ ).We may assume without loss of generality that f and g are unramified coveringsover D ∗ of respective degrees n and m . So we have [Fo, Theorem 5.11] z ∈ D f ( z ) = z n g ( z ) = z m changing the uniformizing maps at the source by a rotation if necessary.If m and n are equal, then X and X ′ are locally equivalent at 0. So assume n > m .Now, from the one hand, by definition of the pull-back, for all t ∈ D , the fibers X t and X ′′ t n are biholomorphic, as well as X ′ t and X ′′ t m . And from the other hand,the assumption for the families of being pointwise isomorphic means in this newsetting that there exists Φ : ( U ⊂ D , −→ ( U ⊂ D , X t and X ′ φ ( t ) are biholomorphic. Hence by transitivity, X t and X φ − ( t n/m ) are biholomorphic for every choice of a determination of t n/m .Observe that this is valid for t belonging to a sufficiently small neighborhood U ′ of0 in D . Set C = { t ∈ D | | t | = λ } for λ a fixed real number, which is supposed small enough to have C ⊂ U ′ . Lemma 8.
Let t ∈ C . Then the closure of the set E t = { t ∈ D | X t is biholomorphic to X t } contains C .Proof of Lemma 8. Assume first that φ is equal to a · Id for a non-zero. Defining α k for k ∈ N by induction ( α = a − α k +1 = α − · α n/mk (we choose a determination of α k α n/mk for each k ), we have that X t and X α k · ( t n/m ) k are biholomorphic. In particular, all the points of { t exp(2 iπl ( m/n ) k ) | k > , l ∈ Z } correspond to X t proving the density of E t in C .Now, if φ is not a homothety, it admits a Taylor expansion φ ( t ) = at + higher order terms SOMORPHISMS OF DEFORMATION FAMILIES 23
Besides, one has that t ∈ E t as soon as t n/m = t n/m , or( φ − ( t n/m )) n/m = ( φ − ( t n/m )) n/m or more generally (cid:16) φ − (cid:0) . . . ( φ − ( t n/m )) n/m . . . (cid:1) n/m (cid:17) n/m = (cid:16) φ − (cid:0) . . . ( φ − ( t n/m )) n/m . . . (cid:1) n/m (cid:17) n/m Using the Taylor expansion of φ together with the fact that n/m >
1, we obtainthat the sequence (cid:16) φ − (cid:0) . . . ( φ − ( t n/m )) n/m . . . (cid:1) n/m (cid:17) n/m α k · ( t n/m ) k tends to 1 as k goes to infinity. In this expression, the determinations of the n/m -thpower are chosen at each step according to the choices made for α k .This means that, given t = t exp(2 iπl ( m/n ) k )for some fixed k > l ∈ Z , and given any ǫ >
0, there exists t ′ ∈ D which is ǫ -close to t such that t ′ belongs to E t . This is enough to conclude that the closureof E t contains C . (cid:3) Hence, there exists a dense subset of points corresponding to X t in any annulusaround the circle | z | = | t | . Following [Gr], the function h is constant on a Zariskiopen subset of D . So we may assume that it is constant on D ∗ . That means thatthe differentiable family of deformations parametrized by | z | = | t | is a regular one. Proposition 3.
All points of the circle C = {| z | = | t |} in D correspond to thesame complex manifold X t .Proof. This is a step by step adaptation of the proof of [F-G]. We will prove thatthe Kodaira-Spencer map is zero for a dense subset of points, hence by regularityfor all points, so that all points correspond to the same manifold X t . We willexplain in details how to modify the proof of [F-G] so that it generalizes to thiscase, but will refer freely to [F-G] for the common parts.Choose ǫ >
0. Choose also an annulus A around C . Choose finally a differen-tiable trivialization T : π − ( { s ∈ A | | s − t | < ǫ } ) −→ X t with T t ≡ Id.For every t in C , define a diffeomorphism ˜ α t from X t to X t as follows. First,choose some t ′ ∈ A such that(i) We have | t ′ − t | < min( | t − t | , ǫ ). (ii) The parameter t ′ belongs to the set E t ∩ π − ( { s ∈ A | | s − t | < ǫ } )This is possible by Lemma 8. By what preceeds, there exists a biholomorphism β t between X t ′ and X t . Define ˜ α t ≡ β t ◦ T − t ′ First notice that the set E = { t ∈ C | ∃ ( t n ) n ∈ N ∈ C such that ( ˜ α t n ) uniformly converges to ˜ α t } is dense in C . This comes from the fact that the set of continuous maps from X to π − ( C ) is of countable type, see [F-G] and the appendix.Let t ∈ E . Without loss of generality, we may assume that there exists a finiteset of submersion charts U i ⊂ π − ( { s ∈ A | | s − t | < ǫ } ) ψ i −−−−→ C dim X × { s ∈ A | | s − t | < ǫ } π y y { s ∈ A | | s − t | < ǫ } −−−−−→ Identity { s ∈ A | | s − t | < ǫ } covering π − ( { s ∈ A | | s − t | < ǫ } ).Set α n ≡ ˜ α t n ◦ α − t . Then the sequence ( α n ) converges uniformly to the identityof X t . Let ( V j ) be a covering of π − ( { s ∈ A | | s − t | < ǫ } ) by relatively compactopen sets with smooth boundaries such that there exists a refining map r and aninteger n satisfying ∀ n ≥ n , α n ( V j ) ⊂ U r ( j ) First, assume for simplicity that h ( t ) = 0. Let x ∈ V i ∩ X t and let ( z, t ) be thecoordinates in the chart ψ r ( i ) . For n ≥ n , define ξ ni ( x ) = ( ψ − r ( i ) ) ∗ ( ψ r ( i ) ◦ α n ( x ) − ψ r ( i ) ( x ))= ( ψ − r ( i ) ) ∗ ( z ( α n ( x )) − z ( x ) , t ( α n ( x )) − t ( x ))= ( ψ − r ( i ) ) ∗ ( z ( α n ( x )) − z ( x ) , t n − t )where ( ψ − r ( i ) ) ∗ denotes the pushforward of a vector field by the differential of ψ − r ( i ) .This gives a smooth vector field on V i ∩ X t which is transverse to X t (since the t -coordinate is non-zero). Now, let M n = max i sup x ∈ V i ∩ X t k ( ψ r ( i ) ) ∗ ξ ni ( x ) k for some choice of a norm on C dim X t × R . Notice that M n is positive since it isbigger than | t n − t | ; and that it is finite because of the finiteness of the number ofcharts and because of the relative compactness of the V i . SOMORPHISMS OF DEFORMATION FAMILIES 25
Lemma 9.
The sequence /M n ( ξ ni ) converges uniformly to a holomorphic vectorfield ξ i on V i ∩ X t .Proof of Lemma 9. Let η ni = 1 /M n ( ψ r ( i ) ) ∗ ( ξ ni ). It is a uniformly bounded sequenceof functions on D i = ψ r ( i ) ( V i ∩ X t ). If we prove that is an equicontinuous sequence,then Ascoli’s Theorem will ensure the uniform convergence.Now, for all j between 1 and dim X , the sequence¯ ∂ j η ni ≡ ∂∂ ¯ z j η ni is uniformly convergent to zero since we have¯ ∂ j η ni = 1 M n ( ¯ ∂ j ( z ( α n ◦ ψ − r ( i ) )) − ¯ ∂ j z, ¯ ∂ j ( t ( α n ◦ ψ − r ( i ) )))and since α n tends uniformly to the identity, hence z ( α n ◦ ψ − r ( i ) ) tends uniformlyto z and t ( α n ◦ ψ − r ( i ) )) to t .On the other hand, deriving with respect to z j the Bochner-Martinelli formulafor η ni , one obtains, for k = dim X t , ∂ j η ni ( z ) = k !(2 iπ ) k (cid:16)Z ∂D i k X ν =1 (( − ν η ni )( ζ ) ( ¯ ζ ν − ¯ z ν )( ¯ ζ j − ¯ z j ) | ζ − z | k +1 d ¯ ζ [ ν ] ∧ dζ + Z D i k X ν =1 (( − ν ¯ ∂ ν η ni )( ζ ) ( ¯ ζ ν − ¯ z ν )( ¯ ζ j − ¯ z j ) | ζ − z | k +1 d ¯ ζ [ ν ] ∧ dζ (cid:17) where d ¯ ζ [ ν ] = d ¯ ζ ∧ . . . ∧ d ¯ ζ ν − ∧ d ¯ ζ ν +1 ∧ . . . ∧ d ¯ ζ n .Since ( η ni ) is a uniformly bounded sequence and since ( ¯ ∂ j η ni ) is uniformly conver-gent to zero, we obtain that ( ∂ j η ni ) is also a uniformly bounded sequence. So ( η ni )is Lipschitz with a Lipschitz constant independant of n . Therefore it is equicontin-uous.Finally, since ( ¯ ∂ j η ni ) is uniformly convergent to zero, the limit is automaticallyholomorphic. (cid:3) Following [F-G], it is easy to prove that these ξ i glue together to define a globalnon-zero holomorphic vector field ξ on X t . This vector field must be transverse to X t for we assumed h ( t ) = 0. Hence, the Kodaira-Spencer map at t is zero.If h ( t ) is not zero, one has first to modify each α n by composition with a finitenumber of well-chosen automorphisms of X t . The construction of the holomorphicvector field ξ is then exactly the same. And finally one uses the special propertiesof this new sequence of ( α n ) to prove that ξ cannot be tangent to X t . Details areexactly the same as in [F-G].As a consequence, one obtains that the family over C has zero Kodaira-Spencermap on a dense subset of points, hence on C as it is a regular family. And applyingTheorem 6.2 of [K-S1], one has that this family is locally trivial at every point. In particular, all the fibers correspond to the same compact complex manifold up tobiholomorphism. (cid:3) But the existence of such a circle of biholomorphic fibers forces the foliation ofSection III to be non-trivial. From the previous proof, we deduce that all the circles z = | t | of X correspond to a unique complex structure, say X t . Fix such a t differentfrom 0. This implies that the intersection of D with the leaf of the foliation passingthrough t contains a circle of points. Since the foliation is holomorphic, this meansthat a neighborhood of this circle corresponds to X t . Let s be in the boundary ofthis neighborhood. Then the same argument shows that a neighborhood of the circle | z | = | s | lies in the leaf passing through s . Now, the two previous neighborhoodsmust have non-empty intersection which implies that X s and X t are biholomorphic.We conclude from that that all the points of D ∗ correspond to X t . Hence, byFischer-Grauert Theorem, X ′′ must be a jumping family.To prove the converse, we need to refine the argument given in the proof ofProposition 2. Consider the local Kodaira-Spencer map of X at 00 ∈ U ⊂ D H ( U, Θ) ρ X −−→ H ( X | U , Θ)which represents the obstruction to lifting a holomorphic vector field in the base U ⊂ D to the family X | U = π − ( U ). The direct limit of ρ X for U smaller andsmaller gives the pointwise Kodaira-Spencer map used in the proof of Proposition2 and which represents the pointwise first-order obstruction to this lifting problem.But we can also define a pointwise ( p + 1)-th order obstruction for any p ∈ N andany ξ ∈ H ( U, Θ) by taking the p -jet of ρ X ( ξ ) at 0 (jet as local sections of Θ) andpassing to the direct limit. This defines a ( p + 1)-th order Kodaira-Spencer map J p ( T D ) ρ ( p ) X −−→ H ( X , Θ ( p ) )where J p ( T D ) is the vector space of p -jets at 0 of holomorphic vector fields of D defined in a neighborhood of 0 and Θ ( p ) is the bundle of p -jets of holomorphicsections of Θ (cf [Wa]).Since the local Kodaira-Spencer map satisfies a chain-rule property, so does ρ ( p ) X ,Hence, starting from X , X ′ pull-backs of X ′′ by maps f and g , we obtain thefollowing equality ρ ( p ) X (cid:18) ∂∂t (cid:19) = ρ ( p ) X ′′ (cid:18) f ∗ ( j po ( ∂∂t )) (cid:19) and ρ ( p ) X ′ (cid:18) ∂∂t (cid:19) = ρ ( p ) X ′′ (cid:18) g ∗ ( j p (( ∂∂t )) (cid:19) with f ∗ (respectively g ∗ ) denoting the action of f (respectively g ) on p -jets of vectorfields. Now if f has degree n and g degree m , the above ( p + 1)-th obstruction of X vanishes for p < n and does not vanish for p = n , whereas that of X ′ vanishes for p < m and does not vanish for p = m . Hence, if m and n are different, the families X and X ′ are not locally isomorphic at 0. (cid:3) Of course, this is no more true for higher-dimensional families. Starting from twopointwise isomorphic but not locally isomorphic one-dimensional jumping families,one can take their products with a fixed family and obtain type (II) counterexampleswhich are not coming from jumping families.
SOMORPHISMS OF DEFORMATION FAMILIES 27
4. Differentiable families.
Things are completely different for differentiable families. In fact, we have
Theorem 5. (i) Let π : X → V be a real analytic family. If h is constant along the family, thenit has the local isomorphism property.(ii) Some differentiable families π : X → I of 2-dimensional compact complex torido not have the local isomorphism property.Moreover, there exist counterexamples of type (I) among families of 2-dimensio-nal compact complex tori.Proof. (i) This is exactly the same proof as that of Theorem 3. For the 1-dimensionalpart, we observe that the only properties of holomorphic maps used are propertiesof analytic functions. For the passage to higher dimension, it is enough to embedpointwise isomorphic families X and X ′ in holomorphic families X C → U and X ′ C → U with constant h . For example, one may take for U the reduction ofthe stratum K max . Then the only difference is that the map p : S → U given byNamba’s Theorem may not be surjective. But the same argument shows that ithas a holomorphic section at 0 defined on an analytic subspace of U containing V . Remark.
The same proof shows that if two differentiable families over V are point-wise isomorphic and locally isomorphic along each path of V containing 0, thenthey are locally isomorphic.(ii) Because of (i), a smooth family of tori not having the local isomorphism propertyat 0 must be flat at 0.Recall [K-S2] that the open set M = { A ∈ M ( C ) | det( ℑ A ) > } is a versal (and even universal) deformation space for every 2-dimensional compactcomplex torus. A point A = ( A , A ) of M corresponds to the quotient of C bythe lattice generated by (1 ,
0) (0 , A A Notice that every torus can be obtained as such a quotient. Two different points A and B of M define the same torus up to biholomorphism if and only if there exists γ = (cid:18) γ γ γ γ (cid:19) ∈ SL ( Z ) such that B = A · γ = ( γ + Aγ ) − ( γ + Aγ )Finally, h is constant equal to 4 (given by the translations), so the condition ofTheorem 3 is satisfied.Let Ω = (cid:18) i i (cid:19) and Ω( t ) = (cid:18) i + t b ( t ) c ( t ) i + t (cid:19) t ∈ R and let X t be the corresponding tori. The smooth functions b and c satisfy1. They are smoothly flat at zero, i.e. all their derivatives at zero are zero.2. We have b (0) = c (0) = 0 and b ( t ) > c ( t ) > t different from zero. The path Ω in M defines a differentiable family of 2-dimensional compact com-plex tori centered at X . Define Ω ≡ Ω andΩ ( t ) = ( Ω ( t ) if t ≤ t Ω ( t ) if t ≥ is also a smooth path.We claim that the corresponding families X → Ω and X → Ω are pointwiseisomorphic but not locally isomorphic at 0.First note that, for all t , t Ω ( t ) = (cid:18) (cid:19) · Ω ( t ) · (cid:18) (cid:19) = Ω ( t ) · γ for γ = ∈ SL ( Z )That implies that, for t >
0, the map( z, w ) ∈ C ( w, z ) ∈ C descends as a biholomorphism between X ( t ) and X ( t ). So the families are point-wise isomorphic.On the other hand, for a generic lattice, it is well-known that the automorphismgroup of a torus is generated by translations and by − Id. Indeed, for this particularchoice of matrices Ω( t ), it is straightforward that this is the case if the numbers i + t , b ( t ), c ( t ), their squares and all the products of two of them are linearly independentover Q . Hence, for generic t , the tori X ( t ) and X ( t ) have no other automorphismsthan these ones. This allows to find sequences ( t ′ n ) n ∈ N of negative numbers and( t ′′ n ) n ∈ N of positive numbers converging to 0 such that(i) For each n , up to translations, the only biholomorphisms between X ( t ′ n ) and X ( t ′ n ) are the projection of ± Id on C .(ii) For each n , up to translations, the only biholomorphisms between X ( t ′′ n ) and X ( t ′′ n ) are the projection of ± (cid:18) (cid:19) on C .Suppose now that X and X are locally isomorphic at 0. Then there would exista family (Φ t ) of biholomorphisms of C (for t in a neighborhood of 0) such that(i) It is smooth in t .(ii) Every Φ t descends as a biholomorphism between X ( t ) and X ( t ).But, by what preceeds, at t ′ n the map Φ t must be ± Id up to a translation factor,whereas at t ′′ n , it must be ± (cid:18) (cid:19) up to a translation factor. Since these two SOMORPHISMS OF DEFORMATION FAMILIES 29 sequences do not converge to the same type of limit when n goes to infinity, wearrive to a contradiction. The families Ω and Ω are not locally isomorphic at 0.On the other hand, the previous family still has the local isomorphism propertywhen restricted to ( −∞ ,
0] and [0 , ∞ ). Nevertheless, it is easy to modify it in orderto have a counterexample even when restricted to ( −∞ ,
0] and [0 , ∞ ).Start with the same path Ω as before, but this time assume that the functions b and c satisfy1. There exists a sequence ( t n ) n ∈ N of positive numbers converging to 0 such that b and c are zero and flat at all t n .2. We have b and c even.3. We have b ( t ) = c ( t ) for t positive and not belonging to the sequence ( t n ).For example, let h ( t ) = ( t ≤ − /t ) otherwiseand f : t ∈ R X p ∈ Z h ( t + p ) · h ( − t − p + 1) ∈ R and finally b ≡ αh ( | − | ) · f (log | − | ) b ≡ βh ( | − | ) · f (log | − | )for β = α . In this case, we have ( t n ) = exp( − n ).The path Ω in M defines a differentiable family of 2-dimensional compact com-plex tori centered at X . Define Ω ≡ Ω andΩ ( t ) = ( Ω ( t ) if | t | ∈ [ t n , t n +1 ] for some n t Ω ( t ) if | t | ∈ [ t n − , t n ] for some n That implies that, for t ∈ [ t n − , t n ] for some n , the map( z, w ) ∈ C ( w, z ) ∈ C descends as a biholomorphism between X ( t ) and X ( t ). In particular, it defines anautomorphism of X and of X ( t n ) = X ( t n ) for all n . This proves the pointwiseisomorphism between the fibers.On the other hand, as in the previous example, one can find sequences ( t ′ n ) n ∈ N and ( t ′′ n ) n ∈ N of positive numbers converging to 0 such that(i) For each n , we have t ′ n ∈ [ t n , t n +1 ] and, up to translations, the only biholo-morphisms between X ( t ′ n ) and X ( t ′ n ) are the projection of ± Id on C .(ii) For each n , we have t ′′ n ∈ [ t n − , t n ] and, up to translations, the only biholo-morphisms between X ( t ′′ n ) and X ( t ′′ n ) are the projection of ± (cid:18) (cid:19) on C .This is enough to prove that these two families, when seen as families over [0 , ∞ ),are not locally isomorphic at 0. Since the functions b and c are even, the same istrue over ( −∞ , (cid:3) In the differentiable case, it seems difficult to give a sufficient condition to havethe local isomorphism property, except for the following trivial one.
Proposition 4.
Let X be a compact complex manifold. Suppose that K is a localmoduli space for X (that means that two different points of X corresponds to twonon-biholomorphic manifolds). Then every holomorphic (over a reduced base) aswell as differentiable deformation family of X has the local isomorphism property.Proof. In this case, given any deformation family X of X , the map from the param-eter space of X to K is uniquely determined by the pointwise complex structure ofthe fibers. (cid:3)
5. Universality.
Let us finish this section by a comparison between our uniformization problemsand the problem of universality of the Kuranishi space.
Proposition 5.
Let K be the Kuranishi space of some compact complex manifold X . Then the following statements are equivalent.(i) The space K is universal for differentiable families.(ii) The space K is universal for holomorphic families over a reduced base.(iii) The foliation of K described in Section III is trivial.(iv) The function h is constant on K .Proof. The equivalence ( iii ) ⇐⇒ ( iv ) is given by Theorem 2. The implications( iv ) ⇒ ( i ) and ( iv ) ⇒ ( ii ) are immediate consequences of Proposition K2. Indeed,it is used in [Ku2] to prove that. The converse ( ii ) ⇒ ( iv ) is proved in [Wa] and[Wa2]. Indeed, (ii) can be replaced by: the space K red is universal for holomorphicfamilies over a reduced base. Now, from [Wa, Proposition 4.2], we have that itis the case if and only a certain extension problem (called the second extensionproblem in [Wa]) is solvable for K red . Then one uses [Wa2, p. 349] to conclude.So we just need to prove ( i ) ⇒ ( iv ). Assume h non-constant. Then, by Propo-sition 1, there exists an automorphism φ of X isotopic to the identity such that anyextension as a diffeomorphism of K does not project onto the identity of K . LetΦ be such an extension. Still by Proposition 1, we may assume that Φ is isotopicto the identity. Let (Φ t ) t ∈ [0 , be such an isotopy, Φ being the identity map. Nowset Ψ( − ) = Φ λ ( − ) ( − ), for some smooth function λ : K → [0 ,
1] satisfying(i) λ (0) = 0.(ii) det Jac λ = 0.For a good choice of λ , the map Ψ is a local diffeomorphism at 0. Indeed, adirect computation shows thatJac Ψ = Id + Jac λ · ∂ Φ t ∂t | t =0 so it is enough to take k Jac λ k very small.Recall now that Φ t may be chosen so that, for all t , its germ at 0 does not projectas the germ of the identity (see the proof of Lemma 2 and Proposition 1). Fromthis, we deduce that the germ of Ψ at X is not the identity, even if Ψ | X is theidentity of X . In other words, one can find a path c in K passing through 0 whoseimage by Ψ is different from c . But this means that the family corresponding to c is SOMORPHISMS OF DEFORMATION FAMILIES 31 locally isomorphic to the family corresponding to Ψ( c ), with the same identificationat 0. Hence K is not universal for differentiable families. (cid:3) Consider now the case where K is non-reduced. For example, assume K = { t = 0 | t ∈ C } is the double point. Consider the trivial family X = K × C −→ U = K × C Assume that X has an automorphism α = exp ξ isotopic to the identity with theadditional property that its action on H ( X , Θ ) is non trivial. Then the family(exp( t · ξ )) t ∈ C defines an automorphism of X red which is the identity on X and projects onto theidentity of U red = C .Now the crucial point is that it also defines an automorphism F of X which isthe identity on X . But F projects onto a non-trivial morphism f of U . Indeed, f is still the identity on U red . But its differential is not the identity for t = 0. It maybe identified with the action of exp( t · ξ ) on H ( X t , Θ t ) ≃ T t U . Hence X can beobtained as a pull-back of K by the map s : (0 , ∂/∂z, t ) ∈ U = K × C (0 , ∂/∂z ) ∈ K but also as a pull-back of K by s ◦ f . These two morphisms respect the marking at0 but are different, disproving the universality of K .Observe that this argument can easily be adapted to the case where K is arbitrarybut non-reduced.Although we do not know of such a precise example (Mumford’s example in [Mu]has no automorphisms isotopic to the identity), it makes very plausible that Propo-sition 5 (especially the equivalence between (ii) and (iv)) is not true for holomorphicfamilies over a non-reduced base.Observe that Theorems 3 and 5 compared to Corollary 4 show that, surprisingly,the local isomorphism problem is fundamentally different from the universalityproblem. In this last problem, there is no difference between the differentiable caseand the (reduced) holomorphic case. References [Ca] F. Catanese,
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Ein Theorem der analytischen Garbentheorie und die Modulra¨ume kom-plexer Strukturen , Publ. Math. IHES (1960).[G-K] H. Grauert, H. Kerner, Deformationen von Singularit¨aten komplexer Ra¨ume , Math.Ann. (1964), 236–260.[Gri] P. Griffiths,
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On deformations of complex analytic structures I , Ann. ofMath. (1958), 328–402.[K-S2] K. Kodaira, D.C. Spencer, On deformations of complex analytic structures II , Ann. ofMath. (1958), 403–466.[Ku1] M. Kuranishi, New Proof for the Existence of Locally Complete Families of ComplexStructures , Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, Berlin, 1965,pp. 142–154.[Ku2] M. Kuranishi,
A note on families of complex structures , Global Analysis, Papers inhonor of K. Kodaira, Princeton University Press, Princeton, 1969, pp. 309–313.[Mu] D. Mumford,
Further Pathologies in Algebraic Geometry , Amer. J. Math. (1962),642–648.[Na] M. Namba, On Deformations of Automorphism Groups of Compact Complex Mani-folds , Tˆohoku Math. J. (1974), 237–283.[Wa] J.J. Wavrik, Obstructions to the existence of a space of moduli , Global Analysis, Papersin honor of K. Kodaira, Princeton University Press, Princeton, 1969, pp. 403–414.[Wa2] J.J. Wavrik,
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Laurent MeerssemanI.M.B.Universit´e de BourgogneB.P. 4787021078 Dijon CedexFrance
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