aa r X i v : . [ m a t h . C V ] A p r Strongly quasi-proper maps and the f-flattningtheorem.
Daniel Barlet ∗ .07/04/15 Abstract.
We complete and precise the results of [B.13] and we prove a strongversion of the semi-proper direct image theorem with values in the space C fn ( M ) offinite type closed n − cycles in a complex space M . We describe the strongly quasi-proper maps as the class of holomorphic surjective maps which admit a meromorphicfamily of fibers and we prove stability properties of this class. In the Appendix wegive a direct and short proof of D. Mathieu’s flattning theorem (see [M.00]) for astrongly quasi-proper map which is easier and more accessible. AMS Classification 2010.
Key Words.
Finite Type Cycles, Geometrically f-Flat map, Strongly Quasi-Proper map, Stein Factorization, Semi-Proper Direct Image. ∗ Institut Elie Cartan : Alg`ebre et G´eom`etrie,Universit´e de Lorraine, CNRS UMR 7502 and Institut Universitaire de France. Contents C fn ( M ) . 3 C fn ( M ). . . . . . . . . . . . . . . . . . . . . . . 18 It is now a classical result that there exists a geometric flattening theorem for aproper surjective holomorphic map between irreducible complex spaces. This is arather simple consequence of the existence of the cycle’s space of a complex space(see [B.75] and [B-M] ch.IV section 9). This fact can be seen as the meromorphiealong the subset of “big fibers” of the “fiber map” for such a morphism. Of coursethis is a geometric version of the deep flattening theorem of H. Hironaka (see [H.75]).We consider an analogous statement for a large class of surjective morphisms be-tween irreducible complex spaces with non compact fibers: the strongly quasi-propermaps. The main problem here is the fact that the standard notion of a quasi-propermap is not strong enough for our purpose : the strict transform of a quasi-propermap by a (proper) modification of the target space is no longer quasi-proper ingeneral (see the example following the proposition 3.2.7). This comes from the factthat the quasi-properness is not sufficient to control the limits of generic fibers neara“ big fiber”. This motivate to work with the notion of strongly proper maps intro-duced in [M.00], [B.08] in an implicit way and in [B.13] explicitly.We complete and precise the study of the class “strongly quasi-proper maps” intro-duced in loc. cit. which enjoys several interesting stability properties and which isan useful tool to prove the existence of meromorphic quotients in the category ofreduced complex spaces (see [B.13]).The main results obtained here are :1. A stronger criterium for proving the SQP property and the fact that thisproperty is equivalent to the existence of a meromorphic fiber map, see theproposition 3.2.2 and the theorem 4.1.1.2. A more precise version of a the semi-proper direct image with value in thespace C fn ( M ), see the theorem 2.3.2 which uses a new analytic continuationresult given in the theorem 2.2.1.3. The strong stability theorem for SQP maps, see the theorem 3.2.12. C fn ( M ) . We suppose that the notion of analytic family of n − cycles is known and also thedefinition of the topology of the space C locn ( M ) for a complex space M (see [B.M]ch. IV section 2).Recall that a n − cycle has finite type if it has only finitely many irreducible compo-nents. The subset of finite type n − cycles in M is denoted C fn ( M ). We define thetopology on it as follows :For W := ( W , . . . , W m ) a finite set of relatively compact open sets in M let Ω( W )the subset of n − cycles C in M such that each irreducible component of | C | meetseach W i for i = 1 , . . . , m . Remark that finite intersection of some Ω( W ) is againof the form Ω( W ). Now define open sets in C fn ( M ) as union of subsets of the form U ∩ Ω( W ) where U is an open set in C locn ( M ).So the inclusion C fn ( M ) → C locn ( M ) is continuous, but it is not a homeomorphismon its image (with the induced topology). For instance, it is easy to see that acontinuous family of n − cycles ( X s ) s ∈ S in M parametrized by a Hausdorff topologi-cal space S , such that each cycle is of finite type is f-continuous (so corresponds toa continuous map S → C fn ( M ) with the topology defined above) if and only if itsset-theoretic graph | G | ⊂ S × M is quasi-proper over S ; here we use the followingdefinition of quasi-proper, valid as long as we know that the fibers of the continuousmap π : G → S are complex analytic subsets (here G ⊂ S × M is closed and thefibers are cycles in the complex space M ) :For any point s ∈ S there exists a neighborhood S of s in S and a closed S − proper subset K in π − ( S ) such that for any s ∈ S any irreducible componentof the fiber π − ( s ) meets K . Lemma 2.1.1
For any complex space M and any integer n the topology of the space C fn ( M ) has a countable basis . So this space is metrizable and then any point has a countable basis of closedneighborhoods.
Proof.
This an easy consequence of the analogous result for the topology of C locn ( M ) which is proved in [B-M] IV section 2.4:Let D be a countable dense subset in M and consider also a countable covering of M By definition a complex space is countable at infinity. by domains of charts ( U p , j p ) p ∈ N . Now for each point d ∈ D let p ( d ) be the smallestinteger p such that d ∈ U p . Define for r ∈ Q + ∗ small enough the pull-back P d,r in U p ( d ) of the polydisc with center j p ( d ) ( d ) and radius r . Then take a countable basis U ν , ν ∈ N , for the topology of C locn ( M ) and consider the family (Ω( W j ) ∩ U ν ) j,ν wherethe countable family of relatively compact open sets W j =: ( W j . . . W j µ ) runs overthe finite subsets of the countable family of the P d,r ⊂⊂ M , gives a countable basisfor the topology of C fn ( M ). (cid:4) Proposition 2.1.2
Let M be a reduced complex space and consider a sequence offinite type n − cycles ( C m ) m ≥ in M with the following properties :i) there exists a compact set K in M such that each irreducible component ofeach C m meets K .ii) The sequence ( C m ) m ≥ converges for the topology of C locn ( M ) to a n − cycle C .iii) The cycle C is in C fn ( M ) .Then the sequence ( C m ) m ≥ converges to C in the topology of C fn ( M ) . Proof.
We begin by proving the case where each C m is irreducible. Note firstthat C is not the empty n − cycle because, up to pass to a subsequence, we mayassume that for each m ≥ x m ∈ C m ∩ K in order thatthe sequence ( x m ) m ≥ converges to a point x ∈ K . Then we have x ∈ | C | .Let W := ( W , . . . , W µ ) be a finite collection of relatively compact open sets open M such that each irreducible component of | C | meets each W i for i = 1 , . . . , µ .Choose also F j , j ∈ J , a finite number of n − scales on M adapted to C and put l j := deg F j ( C ). Define the open set of C fn ( M ) containing C V := Ω( W , . . . , W µ ) ∩ j ∈ J Ω l j ( F j ) . Choose also for each i ∈ [1 , µ ] a n − scales E i := ( U i , B i , j i ) on W i adapted to C and such that deg E i ( C ) = k i ≥
1. This is possible because C has at least oneirreducible component and such an irreducible component meets each W i . So for m ≥ m the scales F j and E i will be adapted to C m and we shall have for all j ∈ J and all i ∈ [1 , µ ] l j = deg F j ( C ) = deg F j ( C m ) and deg E i ( C m ) = deg E i ( C ) = k i ≥ . This shows that for m ≥ m the unique irreducible component of C m meets each W i . So, for m ≥ m each C m for m ≥ m is in the given open set V of C fn ( M ). Sothe sequence ( C m ) m ≥ converges to C in the sense of the topology of C fn ( M ).This is enough to conclude the proof in this case.Consider now the general case. We shall prove first that it is enough to prove thefollowing claim : Claim.
For any relatively compact open set W meeting each irreducible compo-nent of C , there exists m such for m ≥ m any irreducible component of any C m meets W . The claim is enough.
Consider as before an open set V := Ω( W , . . . , W µ ) ∩ j ∈ J Ω l j ( F j )in C fn ( M ) containing C . If the claim is true we find an integer m such that for each m ≥ m we have C m ∈ Ω( W , . . . , W µ ). Now the convergence of the sequence forthe topology of C locn ( M ) gives an integer m such that for each m ≥ m we have C m ∈ ∩ j ∈ J Ω l j ( F j ). So the convergence for the topology of C fn ( M ) is proved. proof of the claim. Take any open set W ⊂ M such that any irreduciblecomponent of C meets W . If there is infinitely many m ≥ m of C m does not meet W then extract a sub-sequence (Γ p ) ofirreducible components in this sequence which converges in the topology of C locn ( M )to a non empty cycle Γ ≤ C . But any irreducible component of Γ has to meet W giving a contradiction as Γ p ∩ W = ∅ by construction. So the claim is provedand the proof is complete. (cid:4) Corollaire 2.1.3
Let M be a reduced complexe space and let A ⊂ C fn ( M ) \ {∅} be a compact subset in C locn ( M ) . Assume that the following condition is fullfilled : • There exists a compact set K ⊂ M such for any irreducible component Γ of each C ∈ A the intersection Γ ∩ K is not empty. (@) Then A is compact in C fn ( M ) . Note that conversely any compact subset in C fn ( M ) \ {∅} is compact in C locn ( M )and satisfies (@) : choose for each C ∈ A an open set U C ∩ Ω( W C ) containig C .As A is compact we can extract a finite sub-covering given by C , ..., C N and take K := ∪ Nj =1 W C j . Proof.
Note that any limit of a sequence in A for the topology of C locn ( M ) isnot the empty cycle, as it contains a point in K . Now, by assumption, for anysequence in A we may find a sub-sequence converging for the topology of C locn ( M )to a cycle in C fn ( M ). So, applying the proposition 2.1.2, we conclude that such asub-sequence converges for the topology of C fn ( M ). (cid:4) Definition 2.1.4
Let S be a reduced complex space. A map ϕ : S → C fn ( M ) will becalled holomorphic when it classifies a f-analytic family of n − cycles in M . remember that each Γ m meets the compact set K . Recall that an analytic family of n − cycles in M is f-analytic if and only if its settheoretic graph | G | ⊂ S × M is quasi-proper on S . Of course this condition impliesthat each cycle in the family is a finite type cycle. But the quasi-properness of thegraph asks more : on any compact set L in S we can find a compact set K in M such any irreducible component of any cycle parametrized by a point s ∈ L has tomeet K . Definition 2.1.5
Let U be an open set in C fn ( M ) . A continuous map f : U → T to a Banach analytic set T is called holomorphic if and only if for any reducedcomplex space S and any holomorphic map ϕ : S → U the composed map f ◦ ϕ : S → T is holomorphic. We shall say that a closed subset X in an open set U of C fn ( M ) is analytic when there exists locally on U an holomorphic map f with values in aBanach space E such that X = f − (0) .A continuous map of an analytic subset X of U to a Banach space E will be called holomorphic if it is locally induced on X by a holomorphic map of C fn ( M ) withvalues in E . The next lemma gives a first example of a closed analytic subset in C fn ( M ). Lemma 2.1.6
Let
N R := { C ∈ C fn ( M ) / C = | C |} be the subset of non reducedcycles. Then N R is a closed analytic subset in C fn ( M ) . Proof.
As the empty n − cycle is open and closed in C fn ( M ) it is enough to considernon empty cycles. Let C be any non empty cycle in C fn ( M ). Choose for eachirreducible component Γ i of C an n − scale E i := ( U i , B i , j i ) on M adapted to C such that the degree of | C | and Γ i in E i are equal to 1, and note k i := deg E i ( C ).Remark that C is reduced if and only if we have k i = 1 for each i ∈ I . Let W := ∪ I ∈ I j − i ( U i × B i ) and V := Ω( W ) ∩ (cid:0) ∩ i ∈ I Ω k i ( E i ) (cid:1) . Then a cycle C ∈ V isnot reduced if and only if there exists at least one i ∈ I such that C ∩ j − i ( U i × B i )is not reduced. As each map V → H ( ¯ U i , Sym k i ( B i )) is holomorphic, the proof isconsequence of the following claim : Claim. • The subset of H ( ¯ U ,
Sym k ( B )) corresponding to non reduced cycles in U × B is a closed analytic subset.Consider the discriminant map ∆ : Sym k ( C p ) → S k. ( k − ( C p ) defined by( x , . . . , x k ) Y ≤ i Sym k ( B )) → H ( ¯ U , S k. ( k − ( C p )) given by f ∆ ◦ f . Of course, if f ∈ H ( ¯ U , Sym k ( B )) defines a non reduced cycle in U × B we have ∆ ◦ f = 0 in H ( ¯ U , S k. ( k − ( C p )). Conversely, if ∆ ◦ f = 0 in H ( ¯ U , S k. ( k − ( C p )), then choose a point t which is not a ramification point for thereduced cycle which is the support of the cycle X associated to f . Then locally | X | is the disjoint union of h ≤ k graphs of holomorphic functions f , . . . , f h : V ( t ) → B which are two by two disjoint. If X is reduced, we have h = k and on the openneighborhood V ( t ) we have∆ ◦ f = Y ≤ i Let π : M → S be a holomorphic map between two reducedcomplex spaces. Define C fn ( π ) as the subset of C fn ( M ) of the n − cycles contained in afiber of π . Then C fn ( π ) is an analytic (closed) subset in C fn ( M ) and the obvious map p : C fn ( π ) → S is holomorphic. Proof. First we shall show that the complement of C fn ( π ) is open:take C 6∈ C fn ( π ). Then it contains two points x and y such that π ( x ) = π ( y ). Taketwo scales E := ( U, B, j ) and E ′ := ( U ′ , B ′ , j ′ ) adapted to C such that x is in thecenter j − ( U × B ) of E and y in the center ( j ′ ) − ( U ′ × B ′ ) of E ′ , small enough suchthat π ( j − ( ¯ U × ¯ B )) and π (( j ′ ) − ( ¯ U ′ × ¯ B ′ ) are disjoints. Note that the degrees of C in E and in E ′ are positive. Now a cycle C such that E and E ′ are adapted to C with the same degrees as C in these scales cannot be in C fn ( π ) because it has twopoints with different images by π . This defines an open set in the complement of C fn ( π ) containing C .To obtain a local holomorphic equation for C fn ( π ), recall the following facts :i) For any n − scale E := ( U, B, j ) on M let Ω k ( E ) be the open set in C locn ( M ) ofcycles for which E is adapted and the degree in E is k . Then the map r E,k : Ω k ( E ) → H ( ¯ U , Sym k ( B ))is holomorphic. This is a consequence of the definition of an analytic family ofcycles !ii) If for a given cycle C we have adapted scales E , . . . , E m such that any irre-ducible component of C meets the union W of the centers of the E i , i ∈ [1 , m ];then the subset of (cid:2) ∩ i ∈ [1 ,m ] Ω k i ( E i ) (cid:3) ∩ Ω( W ), where k i := deg E i ( C ), is an openset in C fn ( M ) and the holomorphic map Q i ∈ [1 ,m ] r E i ,k i is injective on this openset.Then the following lemma allows to conclude. (cid:4) Lemma 2.1.8 Let U and B be relatively compact polydiscs respectively in C n and C p , and let π : W → F be a holomorphic map of an open neighborhood W of ¯ U × ¯ B to a Banach space F . Then the subset X of H ( ¯ U , Sym k ( B )) of multiform graphscontained in a fiber of π is a closed Banach analytic subset of H ( ¯ U , Sym k ( B )) .Moreover, the map ϕ : X → F defined by sending X ∈ X to the point π ( X ) ∈ F such that | X | ⊂ π − ( π ( X )) is holomorphic. proof. Consider for each h ∈ [1 , k ] the holomorphic map N h ( π ) : Sym k ( W ) → S h ( F )given by the h − th Newton symmetric function ( z , . . . , z k ) P kj =1 π ( z j ) h , where S h ( F ) is the h − th symmetric power of F . These maps are holomorphic and for f ∈ H ( ¯ U , Sym k ( B )) we can compose the associated map ˜ f ∈ H ( ¯ U , Sym k ( W )),sending t to (( t, x ) , . . . , ( t, x k )) if f ( t ) := ( x , . . . , x k ), with ⊕ kh =1 N h ( π ) to obtain aholomorphic map Φ : H ( ¯ U , Sym k ( B )) −→ H ( ¯ U , ⊕ kh =1 S h ( F )) . Now fix a point t ∈ U and a non empty open polydisc U ′ ⊂⊂ U and consider theholomorphic mapsΨ : H ( ¯ U , ⊕ kh =1 S h ( F )) → H ( ¯ U ′ , L ( C n , ⊕ kh =1 S h ( F )))and χ : H ( ¯ U , ⊕ kh =1 S h ( F )) → ⊕ kh =2 S h ( F )defined as follows : Ψ( f ) is given by the derivative of f on ¯ U ′ and χ ( f ) is given bythe collection of the k h − .N h ( f ( t )) − (cid:0) N ( f ( t )) (cid:1) h ∈ S h ( F ) for h ∈ [2 , k ]. Thesemaps are holomorphic and the Banach analytic subset Z := Ψ − (0) ∩ χ − (0) is thesubset in H ( ¯ U , ⊕ kh =1 S h ( F )) corresponding to constant maps ¯ U → ⊕ kj =1 S h ( F )) suchthat the value is of the form k.a ⊕ k.a ⊕ · · · ⊕ k.a k for some a ∈ F . So Φ − ( Z ) isexactly the subset X of H ( ¯ U , Sym k ( B )).To conclude the proof, it is enough to remark that the holomorphic map k . (cid:0) ev ◦ Φ (cid:1) induces on X the desired map, where ev : H ( ¯ U , ⊕ kh =1 S h ( F )) → F is given by eval-uation at t of the component on S ( F ) = F . (cid:4) Note that an obvious consequence of the previous proposition is the fact that if N is a closed analytic subset in M , then C fn ( N ) is a closed analytic subset of C fn ( M ).An useful variant of this result is given by the following lemma. That is to say the Banach space generated by x h , x ∈ F , in the Banach space of continuoushomogeneous polynomials of degree h on the dual F ′ of F . Lemma 2.1.9 Let M and S be reduced complex spaces. Let ( X s ) s ∈ S be an f-analyticfamily of n − cycles of M parametrized by S . Let T be an analytic set in M . Thenthe subset Z := { s ∈ S / | X s | has an irreducible component contained in T } is a closed analytic subset in S . proof. It is enough to consider the case of an analytic family of multiform graphsin a product U × B of polydiscs, classified by an holomorphic map f : S × U → Sym k ( B ) , where T ⊂ U × B is the subset given by T := g − (0) for a holomorphic function g : U × B → C . If ˜ f : S × U → Sym k ( U × B ) is the holomorphic map associated to f keeping the U − component, then consider, for U ′ ⊂⊂ U , the holomorphic mapΦ : S → H ( ¯ U ′ , C ) given by s (cid:0) t N r ( g )[ ˜ f ( s, t )] (cid:1) where N r ( g ) : Sym k ( U × B ) → C is the norm of g .Then for U ′ non empty, the set Φ − (0) in S is the corresponding Z . As Φ is holo-morphic, this conclude the proof. (cid:4) This lemma allows to give the following corollary to the proposition 2.1.7 Corollaire 2.1.10 Let T be a closed analytic subset of the complex space M . Thenthe subset T ⊂ C fn ( M ) of cycles having an irreducible component contained in T isa closed analytic subset in C fn ( M ) . Proof. First we want to prove that the complement of T is open. Let C 6∈ T and choose a point x j , j ∈ [1 , m ], which is not in T ,, and choose in each irreduciblecomponent of the cycle C a n − scales E j , j ∈ [1 , m ], on M \ T , such that x j is inthe center of E j for each j . Then k j := deg E j ( C ) is positive for each j . Let W bethe union of the centers of the scales E j , j ∈ [1 , m ] and defined the open set U in C fn ( M ) which contains C by U := Ω( W ) ∩ h ∩ j ∈ [1 ,m ] Ω k j ( E j ) i where Ω k ( E ) denotes the open set of cycles C such E is adapted to C withdeg E ( C ) = k and Ω( W ) is the set of cycles such that any irreducible componentmeets W . Then U does not meet T and so T is closed.To prove the analyticity of T consider a cycle C in T and again choose a point x j in each irreducible component of C and for each j ∈ [1 , m ] a n − scale E j adapted to C with center containing x j . Construct the open set U as above and write in this So N r ( g )[ z , . . . , z k ] = Q kj =1 g ( z j ) for [ z , . . . , z k ] ∈ Sym k ( U × B ). E j . The (finite) union of the analytic sets so defined for each j ∈ [1 , m ] is equal to T ∩ U , concluding the proof. (cid:4) Note that this result may not be true in the case of an analytic family of cycleswhich is not f-analytic as it is shown by the following example. Example. Let M := D = { z ∈ C / | z | < } , T := { − n , n ∈ N , n ≥ } andconsider the family of 0 − cycles in D parametrized by D : X s := { − s + 1 s + m , m ∈ N , m ≥ } ∩ D for s ∈ D. We have X = T and a necessary and sufficient condition on s ∈ D in order that X s meets T is that there exists m, n ∈ N \ { , , } with n = s +1 s + m . This gives that X s meets T if and only iff s = pq with p ∈ Z , q ∈ N \ { , } and | pq | < 1. This is adense set in ] − , +1[ ! (cid:3) We shall conclude this paragraph by an elementary but useful result on the numberof irreducible components of cycles in a f-continuous family of cycles. Definition 2.1.11 Let X := P i ∈ I n i .X i be a finite type n − cycle in a complex space M , where X i are the irreducible components of | X | , n i are positive integers and I isa finite set. We define the weight of X as the integer w ( X ) := P i ∈ I n i . Of course when X is a reduced cycle, the weight of X is simply the number ofirreducible components of X . Lemma 2.1.12 Let M be a complex space and n an integer. The weight function w : C fn ( M ) → N is lower semi-continuous on C fn ( M ) . Proof. Let C be a non empty cycle in C fn ( M ) and choose on M , for each irre-ducible component Γ i of | C | , an adapted scale E i := ( U i , B i , j i ) such that we havedeg E i (Γ i ) = deg E i ( | C | ) = 1 and deg E I ( C ) = k i . Then we have w ( C ) = P i ∈ I k i .Let W := ∪ i ∈ I j − i ( U i × B i ) and define the open neighbourhood V of C in C fn ( M )as : V := Ω( W ) ∩ ( ∩ i ∈ I Ω k i ( E i )) . Then for any C ∈ V we have the inequality 1 ≤ w ( C ) ≤ P i ∈ I k i = w ( C ). This isconsequence of the fact that each irreducible component of C has to meet W , andthat the degree of C in the scale E i is equal to k i . (cid:4) Then, using the existence of a meromorphic geometrically flat Stein reduction for ameromorphic geometrically f-flat map proved in [B.13] theorem 4.2.4, we obtain thefollowing result.1 Proposition 2.1.13 Let M be a complex space, n an integer and ϕ : N → C fn ( M ) be a holomorphic map where N is an irreducible complex space such that for y ina dense Zariski open set the cycle ϕ ( y ) is reduced. Then the number of irreduciblecomponents of ϕ ( y ) is constant on a dense open set in N . (cid:4) Note that, thanks to the lemma 2.1.6, if there exists one y ∈ N such that ϕ ( y ) is areduced cycle, then there exists a dense Zariski open set N ′ in N such for all y ∈ N ′ the cycle ϕ ( y ) is reduced. Proof. Let G ⊂ N × M be the graph of the f-analytic family of n − cycles of M classified by ϕ . As the projection π : G → N is geometrically f-flat, we can applythe theorem 4.2.4 of [B.13], see also the theorem 4.1.1 in section 4. Let ν : ˜ G → G the normalization of G , q : ˜ G → Q and p : Q → N the maps given by the theorem.Then q is geometrically f-flat with irreducible general fibers and p is proper finiteand surjective. If k is the generic degree of p , then it is easy to see, using the lemma2.1.9 with the subset ν − ( T ) ⊂ ˜ G , where T ⊂ G is the subset of non normal pointsin G , that there is a dense open subset in N on which each ϕ ( y ) has exactly k irreducible components. (cid:4) .The next lemma shows that the graph of a f-analytic family of cycles in M parametrizedby an irreducible complex space is again a finite type cycle, assuming that the genericcycle is reduced. It is an easy exercice left to the reader to show that the existenceof a reduced cycle in such a family is not necessary for this result. Lemma 2.1.14 Let M be a complex space and N an irreducible complex space. Let ϕ : N → C fn ( M ) be a holomorphic map such that for some y in N the cycle ϕ ( y ) is reduced. Let G ⊂ N × M be the graph of the f-analytic family of n − cycles in M classified by the map ϕ . Then G has finitely many irreducible components. Proof. Let N ′ be the set of points in N such that ϕ ( y ) is reduced. Then theirreducibility of N and the lemma 2.1.6 imply that N ′ is an dense Zariski open setin N . Denote p : G → N the projection. Then G ′ := p − ( N ′ ) is a dense Zariskiopen set in G . Then the proposition 2.1.13 gives that the number of irreduciblecomponents of ϕ ( p ( z )) for any z in a dense open set G ′′ := p − ( N ′′ ) is constantequal to a positive integer k , where N ′′ ⊂ N ′ is a dense open set.Note I the set of irreducible components of G . For i ∈ I we shall denote G i thecorresponding irreducible component. Let Z be the analytic subset in G corre-sponding to points which are in two distinct irreducible components of G . It is aclosed analytic subset with no interior point in G and also in any irreducible com-ponent G i of G . Now for each i the image of the restriction p i of the projection p to G i contains an open dense subset N i , as this restriction is equidimensionalbetween irreducible complex spaces. Now the set I is countable, so the intersection N := N ′′ ∩ ( ∩ i ∈ I N i ) is dense by Baire’s theorem. Let N be the dense Zariski open By the remark 2 following the definition 3.1.4 each p i is in fact surjective. N such that ϕ ( y ) has no irreducible component contained in Z .Consider nowa point y in N ∩ N . Then ϕ ( p ( y )) is reduced, have k irreducible components, andeach irreducible component G i contains at least one of these components; but anyirreducible component of ϕ ( p ( y )) cannot be in two different G i because y is in N .So the set I has at most k elements. (cid:4) The main goal of this paragraph is to prove the following analytic continuationresult. Theorem 2.2.1 Let M be a complex space and n an integer. Consider a f-continuousfamily ( X s ) s ∈ S of finite type n − cycles in M parametrized by a reduced complex space S . Fix a point s in S and assume that there exists an open set M ′ in M meetingany irreducible component of | X s | and such that the family ( X s ∩ M ′ ) s ∈ S is analyticat s . Then there exists an open neighborhood S of s in S such that the family ( X s ) s ∈ S is f-analytic. Let us make explicit the situation of the previous theorem in term of classifyingmaps : we have a continuous map ϕ : S → C fn ( M ) such that the composed map r ◦ ϕ is holomorphic at s , where r : C fn ( M ) → C locn ( M ′ ) is the restriction map. Then thestatement is that ϕ is holomorphic on an open neighborhood S of s in S , assumingthat M ′ meets each irreducible component of | X s | .Remark that the map r is holomorphic so that the holomorphy at s of r ◦ ϕ is anecessary condition for the holomorphy of ϕ on an open neighborhood of s . Thetheorem says that this condition is sufficient.One key point in the proof of the previous theorem is given by the following analyticcontinuation result. Proposition 2.2.2 Let S be a reduced complex space and let U ⊂ U be two openpolydiscs in C n . Let f : S × U → C be a continuous function, holomorphic on { s } × U for each fixed s ∈ S and assume also that the restriction of f to S × U isholomorphic. Then f is holomorphic on S × U . Proof of the proposition. Consider first the case where S is smooth. As theproblem is local on S it is enough to treat the case where S is an open set in some C m .Fix then a relatively compact open polydisc P in S . The function f defines a map F : U → C ( ¯ P , C ) where C ( ¯ P , C ) is the Banach space of continuous functions on¯ P , via the formula F ( t )[ s ] = f ( s, t ) for t ∈ U et s ∈ ¯ P . The map F is holomorphic: meaning that for any holomorphic map ψ : T → C fn ( M ) of a reduced complex space T thecomposed map r ◦ ψ is holomorphic. U ⊂⊂ U with fixed s ∈ S which computes the partial derivatives in t := ( t , . . . , t n ): ∂f∂t i ( s, t ) = 1(2 iπ ) n Z ∂∂U f ( s, τ ) . dτ ∧ · · · ∧ dτ n ( τ − t ) . . . ( τ i − t i ) . . . ( τ n − t n ) ∀ t ∈ U ∀ i ∈ [1 , n ] . This shows that F is C − differentiable and its differential in t ∈ U is given by h P ni =1 F i ( t ) .h i , h ∈ C n , where F i is the map associated to the function( s, t ) ∂f∂t i ( s, t ) i ∈ [1 , n ]which is holomorphic for any fixed s ∈ S thanks to the Cauchy formula above.Let H ( ¯ P , C ) be the (closed) subspace of C ( ¯ P , C ) of continuous functions which areholomorphic on P . Our assumption implies that the restriction of F to U takesits values in this subspace. Let us show that for each point t ∈ U , F ( t ) is still in H ( ¯ P , C ): assume this is not true. Then there exists t ∈ U with F ( t ) H ( ¯ P , C ),and so, by the Hahn-Banach theorem, there exists a continuous linear form λ on C ( ¯ P , C ), vanishing on H ( ¯ P , C ) and such that λ ( F ( t )) = 0. But the function t λ ( F ( t )) is holomorphic on U and vanishes on U . So it vanishes identicallycontradicting the fact that λ ( F ( t )) = 0. So F is an holomorphic map with valuesin H ( ¯ P , C ) and f is holomorphic on S × U when S is smooth.The case where S is a weakly normal complex space is then an immediate conse-quence, as the continuity of f on S × U and the holomorphie of f on S reg × U ,obtained above, are enough to conclude.When S is a general reduced complex space the function f is then a continuousmeromorphic function on S × U which is holomorphic on S × U . So the closedanalytic subset Y ⊂ S × U of points at which f is not holomorphic has emptyinterior in each { s } × U . So the criterium 3.1.7 of analytic continuation of chapterIV in [B-M] allows to conclude. (cid:4) Remark. It is an easy exercise to weaken the hypothesis of the previous propo-sition replacing the continuity of f by the hypothesis “ f is measurable and locallybounded on S × U ”. Then replace the Banach space C ( ¯ P , C ) by the Banach spaceof bounded measurable functions on ¯ P and in the second step consider the casewhere S is normal; conclude following the same lines. (cid:3) Proof of the theorem 2.2.1. Consider the graph | G | ⊂ S × M of the f-continuous family ( X s ) s ∈ S and let A be the open set of points in | G | such that thefamily is analytic in a neighborhood. Precisely, the point ( σ, ζ ) ∈ | G | is in A if thereexist open neighborhoods S σ and U ζ respectively of σ in S and of ζ in M such thatthe family ( X s ∩ M ζ ) s ∈ S σ is analytic. Remark that, because of our assumption, A meets each irreducible component of { s } × | X s | .First, assume that there exists a smooth point of | X s | in the boundary of the set4 A ∩ ( { s } × | X s | ). Consider now such a point ( s , z ) and choose also a n − scale E := ( U, B, j ) adapted to X s satisfying the following conditions:deg E ( | X s | ) = 1 j ∗ ( X s ) = k. ( U × { } ) z ∈ j − ( U × B ) j ( z ) := ( t , . Then we have a continuous classifying map f : S × U → Sym k ( B ) where S is anopen neighborhood of s in S . The map f is holomorphic for each fixed s ∈ S . Asthe point ( s , z ) is in the boundary of the open set A ∩ ( { s } × | X s | ) of { s } × | X s | ,there exists a (non empty) polydisc U ⊂ U such that the restriction of f to S × U is holomorphic near s . So, up to shrink S , we can assume that f is holomorphic on S × U . Applying the proposition 2.2.2 to each scalar component of f , we concludethat f is holomorphic on S × U . As we can apply the previous argument to anylinear projection of U × B on U near the vertical one, we obtain also the fact that f is an isotropic map. This contradicts the fact that ( s , z ) is in the boundary of A ∩ ( { s } × | X s | ).If the boundary of A ∩ ( { s } × | X s | ) is contained in the singular set of | X s | , thenwe can apply the criterium ([B-M] ch.IV crit`ere 3.9.1) to obtain directly that A contains | X s | .So in all cases, the family ( X s ) s ∈ S is analytic at s . As the graph | G | is quasi-properon S by assumption, it is enough to apply the proposition 2.2.3 to conclude theproof. (cid:4) Proposition 2.2.3 Let M and S be a reduced complex spaces and let ( X s ) s ∈ S bea f-continuous family of n − cycles in M . Assume that this family is analytic at s .Then there exists an open neighbourhood S ′ of s in S such the family ( X s ) s ∈ S ′ is af-analytic family of n − cycles in M . Proof. The quasi-properness hypothesis on the graph of the family gives an openneighbourhood S of s in S and a relatively compact open set W in M such thatany irreducible component of any X s , s ∈ S , meets W . Now for each irreduciblecomponent Γ of | X s | choose a point z Γ ∈ Γ ∩ W , an open neighbourhood S Γ of s in S and an open neighbourhood W Γ of z Γ in W such that the family ( X s ∩ W Γ ) s ∈ S Γ is analytic. Then, as there is a finite number of Γ, the subset S ′ := ∩ Γ S Γ is an openneighbourhood of s in S , and the family ( X s ∩ V ) s ∈ S ′ is analytic where V := ∪ Γ W Γ .Then, for each point s ∈ S ′ , we may apply the first part of the proof of the theorem2.2.1 which is given above (without the present proposition) and conclude that ourfamily is holomorphic at any point s ∈ S ′ . (cid:4) Remark that the previous result is not true in general for a family of cycles whichis analytic at s and is not f − continuous in a neighbourhood of s .5 The aim of this paragraph is to give an improvement to the following importantresult given by the theorem 5.0.5 of [B.13]. First recall this result from loc. cit. Theorem 2.3.1 Let M and S be a reduced complex spaces and fix an integer n .Assume that we have a holomorphic map ϕ : S → C fn ( M ) which is semi-proper .Then the closed subset ϕ ( S ) has a natural structure of weakly normal complex spacesuch that the tautological family of n − cycles parametrized by ϕ ( S ) is an f-analyticfamily of cycles in M . Here is our improvement. Theorem 2.3.2 Let M and S be a reduced complex spaces and fix an integer n .Assume that we have a holomorphic map ϕ : S → C fn ( M ) which is semi-proper.Then the closed subset ϕ ( S ) in C fn ( M ) is analytic and the sheaf induced by the sheafof holomorphic functions on C fn ( M ) defines a structure of reduced (locally finitedimensionnal) complex space on it. This improvement of the theorem 5.0.5 of [B.13] will allow us to avoid to restrictseveral statements to weakly normal complex spaces. It is also an opportunity toprecise the proof of this delicate result.The main tool for the proof of the theorem 2.3.2 is the semi-proper direct imagetheorem of [M.00] and the theorem 2.2.1.Before going to the proof of the theorem 2.3.2 we have to give some details here ona crucial point for the use of the semi-proper direct image theorem with value in aBanach space in order to have a semi-proper direct image theorem with values in C fn ( M ) for any complex space M . This is to be compared with condition ( H ) in[M.00]. Proposition 2.3.3 Let M and S be reduced complex spaces and ϕ : S → C fn ( M ) be a holomorphic semi-proper map. Then for each C ∈ ϕ ( S ) there exists an openneighborhood V of C in C fn ( M ) and a holomorphic map f : V → U in an open set U of a Banach space, such that the map f ◦ ϕ : ϕ − ( V ) → U is semi-proper. Proof. First recall that a basis of the topology of C fn ( M ) is given by finite inter-sections of open sets of the type Ω k ( E ) and Ω( W ); here Ω k ( E ), associated to annatural integer k ≥ n − scale E on M , is defined as the subset of C ∈ C fn ( M )such that E is adapted to C and deg E ( C ) = k and the open set Ω( W ), associatedto a relatively compact open set W in M , is defined as the subset of C ∈ C fn ( M )such that each irreducible component of C meets W . This means that for any cycle C ∈ C fn ( M ) we can find an open neighborhood V of C and acompact set K in S such that ϕ ( S ) ∩ V = ϕ ( K ) ∩ V . C ∈ C fn ( M ) and, using the semi-properness of ϕ , let V := (cid:0) ∩ Jj =1 Ω( W j ) (cid:1) ∩ (cid:0) ∩ Ni =1 Ω k i ( E i ) (cid:1) an open neighborhood of C in C fn ( M ) and K ⊂ S a compact set in S such that ϕ ( S ) ∩ V = ϕ ( K ) ∩ V . (*)As ϕ ( K ) is a compact subset in C fn ( M ), there exists a relatively compact open set W in M such that for any s ∈ K any irreducible component of ϕ ( s ) meets W . Aswe can make V smaller around C keeping the condition ( ∗ ) above, we shall assumethat W is one of the W i , i ∈ [1 , M ] and that the union of the centers of the scales E i , i ∈ [1 , N ] contains the compact set ¯ W .We consider now the holomorphic map f : V ∩ ϕ ( S ) −→ N Y i =1 H ( ¯ U i , Sym k i ( B i )) ⊂ U given by f ( C ) = ( f i ) i ∈ [1 ,N ] where f i ∈ H ( ¯ U i , Sym k i ( B i )) is the classifying map of C in the scale E i := ( U i , B i , j i ) and where U is an open set in the Banach space Q Ni =1 H ( ¯ U i , E i ( k i )) such that Q Ni =1 H ( ¯ U i , Sym k i ( B i )) is a closed analytic subset of U . Then f : V → U is holomorphic and injective. We shall show that f induces anhomeomorphism of V ∩ ϕ ( S ) onto f ( V ∩ ϕ ( S )).As this map is bijective and continuous, it is enough to prove that this map is closed;if the sequence ( f ( C ν )) converges to f ( C ) in U where C ν and C are in V ∩ ϕ ( S ), weshall prove that the sequence ( C ν ) converges to C in V (that is to say in C fn ( M )).The convergence of ( f ( C ν )) to f ( C ) implies the convergence of ( C ν ∩ W ) to C ∩ W in C locn ( W ), as we assume that the union of the centers of the scales E i covers ¯ W (see [B-M] ch. IV). Write C ν = ϕ ( s ν ) where s ν is in K . Up to pass to a subsequence,we can assume that the sequence ( s ν ) converges to t ∈ K . Then ( C ν ) converges to D := ϕ ( t ) in C fn ( M ) as ϕ is continuous. Then D is in ϕ ( K ) but it is not, a priori,clear that D is in V and then equal to C . Nevertheless we have C ∩ W = D ∩ W by the uniqueness of a limit in C locn ( W ). As C and D are in ϕ ( K ), each irreduciblecomponents of C and D meets W , and this allows to conclude that D = C . Aseach converging subsequence of the sequence ( C ν ) converges to C and as we are inthe compact set ϕ ( K ) of C fn ( M ), we obtain that the sequence ( C ν ) converges to C in C fn ( M ).Now the composed map f ◦ ϕ : ϕ − ( V ) → f ( V ∩ ϕ ( S )) is semi-proper. Then, upto shrink the open set U around f ( V ∩ ϕ ( S )), the map f ◦ ϕ : ϕ − ( V ) → U is alsosemi-proper . We denote by E i ( k i ) the Zariski tangent space at k. { } of Sym k i ( C p i ). So Sym k i ( C p i ) is analgebraic subset of E i ( k i ) ; see [B-M] ch.I. The image of a semi-proper map is locally compact; so it is locally closed in U . Proof of the theorem 2.3.2. Assume now that in the previous proof we choosefor each scale E i , i ∈ [1 , N ] an open polydisc U ′ i ⊂⊂ U i such that the union of the j − i ( U ′ i × B i ) covers the compact set ¯ W and that we replace the Banach analyticset H ( ¯ U i , Sym k ( B i )) by the Banach analytic set Σ U i ,U ′ i ( k i ) which classifies the f ∈ H ( ¯ U i , Sym k ( B i )) which are isotropic on ¯ U ′ i . As the natural projection Σ U i ,U ′ i ( k i ) → H ( ¯ U i , Sym k ( B i )) is an holomorphic homeomorphism (see [B.75] chapter III or [B-M2] chapter V), and as the map ˜ f : V → N Y i =1 Σ U i ,U ′ i ( k i )is holomorphic, we obtain that the map ˜ f ◦ ϕ : ϕ − ( V ) → Q Ni =1 Σ U i ,U ′ i ( k i ) is semi-proper on its image and so there exists an open set ˜ U in the ambient Banach spaceof Q Ni =1 Σ U i ,U ′ i ( k i ) such the map ˜ f : V → ˜ U is semi-proper. So the semi-proper directimage theorem with values in a Banach open set gives that ˜ f ( V ∩ ϕ ( S )) is a locallyfinite dimensional analytic subset in ˜ U .Consider now the f-continuous family of n − cycles in M parametrized by ˜ f ( V ∩ ϕ ( S ))induced via the homeomorphism ˜ f by the tautological family on V ∩ ϕ ( S ). Theuniversal properties of the Banach analytic spaces Σ U i ,U ′ i ( k i ) imply that this familyof n − cycles in M is analytic on the open set ∪ Ni =1 j − i ( U ′ i × B i ) which contains theopen set W . As W meets each irreducible component of each cycle in V ∩ ϕ ( S ),we can apply the theorem 2.2.1 and conclude that this family is f-analytic. By theuniversal property of C fn ( M ), the corresponding classifying map˜ f − : ˜ f ( V ∩ ϕ ( S )) −→ V ∩ ϕ ( S ) ⊂ C fn ( M )is holomorphic. To conclude, it is enough to use the fact that ˜ f and ˜ f − are holo-morphic between V ∩ ϕ ( S ) and ˜ f ( V ∩ ϕ ( S )). (cid:4) Remark. We can avoid to use the full strength of the theorem 2.2.1 in the proofabove, because the semi-properness of the map ˜ f ◦ ϕ : ϕ − ( V ) → ˜ f ( V ∩ ϕ ( S )) impliesthe semi-properness of the holomorphic map( ˜ f ◦ ϕ ) × id M : | G | | ϕ − ( V ) → ˜ f ( V ∩ ϕ ( S )) × M where | G | ⊂ S × M is the set-theoretic graph of the f-analytic family classified by ϕ .Then the (finite dimensional) semi-proper direct image theorem of Kuhlmann givesthe analyticity of the set-theoretic graph of the f-continuous family parametrized by˜ f ( V ∩ ϕ ( S )), so the fact that this family is weakly holomorphic. (cid:3) Recall that we have already proved that ˜ f ( V ∩ ϕ ( S )) is a reduced (locally finite dimensional)complex space. C fn ( M ) . Terminology. • A modification between two reduced complex spaces will be always a properholomorphic map which induces an isomorphism between two dense Zariskiopen sets. • We shall say that a holomorphic map p : P → N between two irreduciblecomplex spaces is dominant if there exists an open dense subset Λ ⊂ P suchthat the restriction of p to Λ is an open map with dense image in N . When P is not irreducible, we shall say that p is dominant when its restriction to eachirreducible component of P is dominant.Remark that this implies that the restriction of p to the smooth parts of P hasgenerically maximal rank equal to dim N . Definition 2.4.1 Let π : M → N and p : P → N be two holomorphic maps where M, N, P are complex spaces. Assume that N is irreducible and that π and p aredominant. Then we shall call the strict fiber product of π and p , the map ˜ π : ˜ M := M × N,str P → P which is the restriction of the projection of the usual fiber product to the union ofthe irreducible components of M × N P which are dominant over P . Of course, the strict fiber product has two holomorphic projections on M and P which factorize its natural projection on N . They are both dominant.The reader will find easily examples of fiber products which are not equal to the cor-responding strict fiber product. For instance, if τ : M → N is a modification whichis not injective, the fiber product M × N M has at least one irreducible componentwhich is not contained in the corresponding strict fiber product which coincides withthe map τ itself.It is an easy exercise left to the reader to prove that if we assume that p : P → N is a modification and that π : M → N is dominant the projection M × N,str P → M is a modification.The definition of a meromorphic map from a reduced complex space N to C fn ( M ) isthe usual one. Definition 2.4.2 Fix a complex space M and an integer n . Let N be a reducedcomplex space and Σ ⊂ N a nowhere dense closed analytic subset in N . We shallsay that a holomorphic map ϕ : N \ Σ → C fn ( M ) is meromorphic along Σ (or moresimply that ϕ : N − −− > C fn ( M ) is meromorphic) when there exists a modification σ : N → N with center in Σ and a holomorphic map ϕ : N → C fn ( M ) extending ϕ ◦ σ . Corollaire 2.4.3 Fix a complex space M and an integer n . Let N and P be reducedcomplex spaces and let ϕ : N → P ×C fn ( M ) be a semi-proper holomorphic map. Then ϕ ( N ) is a closed analytic subset in P × C fn ( M ) which is locally of finite dimension. The proof will be an easy consequence of the theorem 2.3.2 using the followinglemma. Lemma 2.4.4 Fix a complex space M and an integer n . Let P be a reduced complexspace. Denote p : P × M → P and q : P × M → M the projections. Then the closedanalytic subset C fn ( p ) ⊂ C fn ( P × M ) is bi-holomorphic to the product P × C fn ( M ) . Proof. Denote α : C fn ( p ) → P the natural holomorphic projection (see the propo-sition 2.1.7) and β : C fn ( p ) → C fn ( M ) be the holomorphic map induced by the directimage by q . Remark that for any p − relative n − cycle the restriction of q is a holo-morphic homeomorphism on its image which is closed (so the restriction of q to sucha cycle is a homeomorphism), then the direct image theorem with parameter applies(see [B.75] or [B-M] ch.IV). So the map ( α, β ) : C fn ( p ) → P × C fn ( M ) is holomorphicand bijective. The inverse map γ , given by γ ( p, C ) := { p }× C ∈ C fn ( p ) ⊂ C fn ( P × M ),is also holomorphic thanks to the product theorem for analytic families of cycles(see also [B.75] or [B-M] ch.IV). (cid:4) Proof of the corollary. The holomorphic map ϕ defines a holomorphic map ψ : N → C fn ( P × M ) with values in C fn ( p ). The map ψ is holomorphic and semi-proper and we conclude by applying the theorem 2.3.2. (cid:4) Lemma 2.4.5 Let ϕ : N − −− > C fn ( M ) be a meromorphic map and consider amodification σ : N → N such that there exists a holomorphic map ϕ : N → C fn ( M ) extending ϕ ◦ σ . Then the holomorphic map ( σ, ϕ ) : N → N × C fn ( M ) is properand its image is the closure in N × C fn ( M ) of the graph of ϕ | N \ Σ . Proof. To prove the properness of the map ( σ, ϕ ) we have to prove that it isa closed map and that all fibers are compact. The second point is obvious as σ is proper. So consider a closed set F in N and a sequence ( y ν ) in F such that( σ ( y ν ) , ϕ ( y ν )) converges to ( x, C ) ∈ N × C fn ( M ). Take a compact neighbourhood V of x in N . As σ is proper, σ − ( V ) is compact and contains all y ν for ν largeenough. So, up to pass to a subsequence, we can assume that ( y ν ) converges to y in N . Then the sequence ( ϕ ( y ν )) converges to ϕ ( y ) in C fn ( M ). So we have y ∈ F and ( x, C ) = ( σ, ϕ )( y ). The last assertion is obvious. (cid:4) Note that even in this case the product theorem is not obvious because a n − scale on P × M adapted to a cycle like { p } × C does not necessary comes from a n − scale on M . τ of the reduced complex space ˜ N := ( σ, ϕ )( N ) on N is a proper modification and that the map ˜ ϕ : ˜ N → C fn ( M ) is holomorphic andextends ϕ ◦ τ . Moreover, for any modification σ : N → N such that there exists aholomorphic map ϕ : N → C fn ( M ) extending ϕ ◦ σ the holomorphic map ( σ , ϕ )factorizes through ˜ N , meaning that there exists a holomorphic map h : N → ˜ N such that σ = τ ◦ h and ϕ = ˜ ϕ ◦ h . Definition 2.4.6 In the situation of the lemma above the reduced complex space ˜ N ⊂ N × C fn ( M ) will be called the graph of the meromorphic map ϕ . We first explain that geometrically f-flat maps (resp. strongly quasi-proper maps)is the class of holomorphic maps admitting a holomorphic fiber map (resp. a mero-morphic fiber map) defined on the target space with values in the space of finitetype cycles of the source space (of the appropriate dimension). Then we give somebasic results for these notions including stability results in order to dispose of easycriteria ensuring that a holomorphic map is strongly quasi-proper, condition whichis not so simple to verify directly on the definition.Of course these tools are essential for the applications given in [B.13] mainly becausethey allow to use the holomorphic semi-proper direct image theorem 2.3.2 with val-ues in the space C fn ( M ) where M is any complex space and n any integer. for anexample of how these tools allow to give some results on meromorphic quotients forsome holomorphic action of a complex Lie group on a reduced complex space see[B.15]. We recall first the notion of geometrically f-flat holomorphic map. Definition 3.1.1 Let M be a pure dimensional complex space and let N be an irre-ducible complex space. Put n := dim M − dim N . Let π : M → N be a holomorphicmap. We shall say that π is a geometrically f-flat map ( GF map for short) ifthere exists a holomorphic map ϕ : N → C fn ( M ) such that for y generic in N the cycle ϕ ( y ) is reduced and such that the projection on M of the graph G of thef-analytic family of n − cycles in M classified by ϕ , is an isomorphism. M π ❆❆❆❆❆❆❆❆ ≃ / / G ⊂ N × M pr x x rrrrrrrrrrr N One y such ϕ ( y ) is reduced is enough as N is irreducible ; see lemma 2.1.6. M has finitely manyirreducible components. The restriction of π to each of these irreducible componentsis quasi-proper and equidimensional (so surjective), but it is not true that theserestrictions are GF maps ; see the example below. But it is not far to be true as weshall see thanks to the lemma 3.1.3 and the remark 2 following the definition 3.1.4 Example. Let X := { ( x, y ) ∈ C / x = y } and let f : X → C the continuousmeromorphic function defined by f ( x, y ) = x/y . Let G ± ⊂ X × C the graph of ± f and define Y := G + ∪ G − ⊂ X × C . Then the projection π : Y → X is a GFmap, as the map F : X → Sym ( C ) ≃ C given by F ( x, y ) = (0 , y ) is holomorphicand classifies the fibers of π (note that f ( x, y ) = y ). But the restriction of theprojection to the irreducible component G + of Y is not a GF map as f is notholomorphic on X . Lemma 3.1.2 Let π : M → N be a holomorphic n − equidimensional map betweena pure dimensional complex space M to an irreducible complex space N , where theinteger n is equal to dim M − dim N . Assume that there exists a holomorphic map ϕ : N → C fn ( M ) such that for any y in a dense set in N the cycle ϕ ( y ) is reduced and equal to thefiber of π . Then π is a GF map. Proof. As for y in a dense set we have ϕ ( y ) ∈ C fn ( π ) which is a closed (analytic)subset in C fn ( M ) (see the proposition 2.1.7), we have | ϕ ( y ) | ⊂ π − ( y ) for any y ∈ N .Consider the second projection pr : G → M of the graph G ⊂ N × M of thef-analytic family of n − cycles classified by ϕ . As π is an open map, there existsa dense subset of x in M such that ( π ( x ) , x ) is in G . So the holomorphic map( π, id M ) : M → N × M takes its values in G and it gives a holomorphic inverse to pr . So π is a GF map. (cid:4) Remark. A GF-map is quasi-proper, equidimensional and surjective. But a holo-morphic quasi-proper equidimensionnal (and then surjective) map between irre-ducible complex spaces is not always a GF map : for instance the (weak) nor-malization of a reduced complex space N is not a GF-map if N is not (weakly)normal. Nevertheless the next lemma explains that such a map it is not so far tobe a GF map. Lemma 3.1.3 Let π : M → N be a holomorphic map between a pure dimensionalcomplex space M and an irreducible complex space N . Let n := dim M − dim N and let ν : ˜ N → N be the normalization of N . Then the following properties areequivalent :i) π is quasi-proper and n − equidimensional. ii) The projection ˜ π : ˜ M := M × N,str ˜ N → ˜ N is a GF-map. Proof. Assume i) ; first let us prove that ˜ π is quasi-proper. Let ˜ K be a compactset in ˜ N ; as K := ν ( ˜ K ) is a compact set in N there exists a compact set L in M such that each irreducible component of a π − ( y ) for y ∈ K meets L . Then thecompact set ( L × ν − ( K )) ∩ ( M × N ˜ N ) meets any irreducible component of a ˜ π − (˜ y )for each ˜ y ∈ ν − ( K ). So ˜ π is quasi-proper as any irreducible component of a fiberof ˜ π is an irreducible component of a fiber of π . The n − equidimensionality of ˜ π isobvious. Now ˜ π : M × N,str ˜ N → ˜ N is quasi-proper and equidimensional on a normalcomplex space, so there exists a holomorphic map (see [B.M] ch.IV) ϕ : ˜ N → C fn ( ˜ M )such that | ϕ (˜ y ) | = ˜ π − ( s ) for ˜ y general in ˜ N , where we put ˜ M := M × N,str ˜ N . Usingthe analyticity of the direct image of cycles by ν the lemma 3.1.2 gives that ˜ π is aGF map.Conversely, assume ii). First we prove the n − equidimensionality of π . Assume thereexists a non normal point y ∈ N such that the fiber π − ( y ) has an irreducible com-ponent Γ of dimension strictly bigger than n . This irreducible component does notappears in a fiber of ˜ π . We can find a sequence of points in π − ( N norm ) convergingto the generic point of Γ, where N norm is the open dense set of normal points in N .As ˜ M is a finite modification of M we can assume, up to pass to a sub-sequence,that this sequence converges in ˜ M . We then find a point y in N such that the fiberof ˜ π contains the generic point of Γ. This contradicts our assumption.Let us prove the quasi-properness of π . Let K be a compact in N ; then ν − ( K )is a compact in ˜ N and so there exists a compact L in M such that any irreduciblecomponent of ˜ π − (˜ y ) for ˜ y ∈ ν − ( ˜ K ) meets ( L × ν − ( K )) ∩ ˜ M . Then any irreduciblecomponent of π − ( y ) for y ∈ K meets L . (cid:4) Definition 3.1.4 A holomorphic map π : M → N between a pure dimensionalcomplex space M to an irreducible complex space N which satisfies the conditionsof the lemma above will be called a weakly geometrically f-flat map (in short a wGF map). Remarks. 1. As in the situation of the previous lemma the projection M × N,str ˜ N → M isa modification, this implies that M has finitely many irreducible componentsbecause the lemma 2.1.14 implies that ˜ M := M × N,str ˜ N has only finitely manyirreducible components.2. If the holomorphic map π : M → N is wGF map its restriction to any irre-ducible component of M is again wGF because this restriction is equidimen-sional and quasi-proper.3. Conversely, if π : M → N is holomorphic, if M is pure dimensional with finitelymany irreducible components and if the restriction of π on each irreducible See the definition 2.4.1. M is a wGF map, then π is a wGF map, again thanks to thelemma 3.1.3.4. If π : M → N is a wGF map then for any closed irreducible analytic subset P ⊂ N the restriction π π − ( P ) : π − ( P ) → P is a wGF map.The first stability results for this notion of wGF map are given by the followinglemma. Lemma 3.1.5 Let α : M → N be a wGF map and σ : P → N be a dominantholomorphic map of an irreducible complex space P to N . The the natural projectionof the fiber product ˜ α : M × N,str P → P is a wGF map. If α is a GF map, so is ˜ α . proof. Assume first that α is geometrically f-flat; its fibers are classified by aholomorphic map ϕ : N → C fn ( M )where n := dim M − dim N . Compose this map with σ ; this gives a holomorphicmap ϕ ◦ σ which classifies a f-analytic family of n − cycles in M parametrized by P .Let G ⊂ P × M the graph of this family. As for y generic in N the cycle ϕ ( y ) isreduced and the map σ is dominant, the graph G of the family parametrized by P is reduced and given by G := { ( x, z ) ∈ M × P / x ∈ | ϕ ( σ ( z )) |} . But, as | ϕ ( y ) | = α − ( y ) for each y ∈ N , we see that G = M × N,str P and that for z in a dense set in P the fiber of the projection of G on P is the fiber of the map ˜ α .So this map is geometrically f-flat, thanks to the lemma 3.1.2.Consider now the case where α is only a wGF map. So, thanks to the lemma 3.1.3,if ν : ˜ N → N is the normalization of N , the fiber product ˜ M := M × N,str ˜ N has aprojection ˜ α on ˜ N which is a GF map. Consider now the normalization µ : ˜ P → P .We have the following commutative diagram, where ˜ σ is a lifting of σ ◦ µ to ˜ N which exists by normality of ˜ P as σ is dominant.˜ M ˜ α / / ˜ ν (cid:15) (cid:15) ˜ N ν (cid:15) (cid:15) ˜ P ˜ σ o o µ (cid:15) (cid:15) M α / / N P σ o o Let ϕ : ˜ N → C fn ( ˜ M ) the holomorphic map classifying the fibers of the GF map ˜ α .Composed with ˜ σ and with the direct image map ˜ ν ∗ for n − cycles (see [B.M] ch. IVcase of a proper map) it gives a holomorphic map ψ : ˜ P ˜ σ −→ C fn ( ˜ M ) ˜ ν ∗ −→ C fn ( M )4and it is easy to see that for z in a general subset in ˜ P its value is the reduced cycle C in M where the cycle C × { z } in M × N,str ˜ P which is the fiber of the naturalprojection p : M × N,str ˜ P → ˜ P , as we have ( M × N,str P ) × P,str ˜ P ≃ M × N,str ˜ P . So p is a GF map from the first case proved above, and we conclude the proof thanksto the lemma 3.1.3. (cid:4) Remark. Let α : M → N be a wGF map and σ : P → N any holomorphic map,it is easy to see that the projection M × N,str P → P is again a wGF map using thelemma 3.1.3 ; this generalizes the previous lemma and the remark 4 following thedefinition 3.1.4. Note that the case of a GF map is not clear because the image of σ may be inside the locus of non reduced fibers of α . Lemma 3.1.6 Let α : M → N and β : N → P two wGF maps between a puredimensional complex space M and irreducible complex spaces N and P . Then β ◦ α is a wGF map. Proof. Put m := dim M − dim N and also n := dim N − dim P . Fix an irreduciblecomponent ∆ of ( β ◦ α ) − ( z ) for some z in P . Then α induces a map α ′ : ∆ → β − ( z )where β − ( z ) has pure dimension n and α ′ has pure dimension m fibers. So ∆ hasdimension at most m + n , and, as the map β ◦ α is surjective between each irreduciblecomponent of M and the irreducible complex space P with dim M − dim P = m + n ,we obtain that ∆ has dimension m + n and also that α (∆) is dense in an irreduciblecomponent Γ of β − ( z ). So β ◦ α is equidimensional (and surjective).We shall show that it is also quasi-proper. Let z be a point in P . The quasi-properness of β implies that there exists an open neighborhood V of z in P and arelatively compact open set W in N such that any irreducible component Γ of anyfiber β − ( z ) for z ∈ V meets W . As W is relatively compact and α quasi-proper,there exists a relatively compact open set U in M such any irreducible componentΓ of a fiber α − ( y ) with y ∈ W meets U . Take now z ∈ V and ∆ an irreduciblecomponent of ( β ◦ α ) − ( z ). Let Γ be the irreducible component of β − ( z ) in which α (∆) is dense (see above). As Γ meets W the dense set α (∆) in Γ meets the nonempty open set Γ ∩ W of Γ. Let y be a point in α (∆) ∩ W such that there exists apoint x in ∆ with α ( x ) = y and such that x is a smooth point in ( β ◦ α ) − ( z ). Sucha point x exists because we may choose y to be a smooth point in β − ( z ) and alsoin the image of the open dense set in ∆ of points which are smooth in ( β ◦ α ) − ( z ).Now an irreducible component Γ of α − ( y ) containing x has to be contained in ∆as it is contained in ( β ◦ α ) − ( z ) and contains x . But such a Γ meets U as y is in W . So ∆ meets U . This gives the quasi-properness of β ◦ α . (cid:4) The next lemma will be used later on. Lemma 3.1.7 Let τ : M → M be a modification and α : M → N a holomorphicmap between irreducible complex spaces. Assume that α ◦ τ is a GF map. Then α is a GF map. Proof. Our assumption gives a holomorphic map ϕ : N → C n ( M ) which clas-sifies the fibers of τ ◦ α . Denote by Σ the center of τ . Then the corollary 2.1.10implies that for generic y ∈ N no irreducible component of | ϕ ( y ) | is contained in τ − (Σ) (see the corollary 2.1.10); then the direct image τ ∗ ( ϕ ( y )) has its supportequal to α − ( y ). This means that the composition of ϕ with the direct image of n − cycle by τ gives a holomorphic map ψ : N → C n ( M ) such that for generic y ∈ N we have ψ ( y ) = | ψ ( y ) | = α − ( y ). As α is n − equidimensional becausedim α − ( z ) ≤ dim( α ◦ τ ) − ( z ) for each z ∈ N , the lemma 3.1.2 allows to concludethat α is a GF map. (cid:4) Lemma 3.1.8 Let α : M → T and p : T → N holomorphic maps between normalcomplex spaces such that p is proper finite and surjective and p ◦ α is GF. Then α is a GF map. Proof. As N is normal, it is enough to show that α is quasi-proper and equidi-mensional. Let δ be an irreducible component of α − ( t ) for some t ∈ T . As p has finite fibers, δ is an irreducible component of ( p ◦ α ) − ( p ( t )) and so has puredimension n := dim M − dim N . So α is n − equidimensional and any irreduciblecomponent of the fiber of α at a point t ∈ T is an irreducible component of the fiberof p ◦ α at the point p ( t ).Let K be a compact subset in T ; then p ( K ) is compact and there exists a compactset L in M such that for any y ∈ p ( K ) any irreducible component γ of ( p ◦ α ) − ( y )meets L . Now an irreducible component of a fiber at a point t ∈ K is an irreduciblecomponent of a fiber of p ◦ α at the point p ( t ) ∈ p ( K ). So it meets the compact L . (cid:4) It is an simple exercice to show an analoguous result for wGF maps instead of GFmaps (and the normality assumptions can be omitted).We conclude this paragraph on GF maps by the following “embedding theorem”. Theorem 3.1.9 Let π : M → N a holomorphic GF map between irreducible com-plex spaces. Then the fiber map ϕ : N → C fn ( M ) is a proper holomorphic embedding. Proof. The first step of the proof is to show that the map ϕ is proper, that is tosay that ϕ is closed with compact fibers. But as this map is clearly injective we onlyhave to prove that for any closed subset F ⊂ N its image ϕ ( F ) is closed in C fn ( M ).So consider a sequence ( y ν ) of points in F such that the sequence ( ϕ ( y ν )) convergesto an element C ∈ C fn ( M ). Let W ⊂⊂ M a relatively compact open set in M suchthat each irreducible component of | C | meets W . As Ω( W ) is an open set in C fn ( M )containing C the cycle ϕ ( y ν ) is in Ω( W ) for ν large enough. So choose for eachsuch ν a point x ν in W ∩ π − ( y ν ). Then each x ν is in the compact set ¯ W ∩ π − ( F ).Up to pass to a sub-sequence we may assume that the sequence ( x ν ) converges to apoint x ∈ ¯ W ∩ π − ( F ). Then y := π ( x ) is in F and is the limit of the sequence ( y ν ),6so we have C = ϕ ( y ) with y ∈ F . So the map ϕ is closed and then proper.Now the theorem 2.3.2 implies that N π := ϕ ( N ) is a closed analytic subset whichis an irreducible complex space and ϕ : N → N π is a holomorphic homeomorphism.To complete the proof of the theorem we have to show that any germ of holomorphicfunction at a point y ∈ N is the pull-back of a germ of holomorphic function of N π at the point ϕ ( y ).So fix a point y ∈ N and a holomorphic germ f ∈ O N,y . Choose a smooth point x of | ϕ ( y ) | . Let E := ( U, B, j ) be a n − scale on M adapted to ϕ ( y ) such that x isin j − ( U × B ) and such that ϕ ( y ) ∩ j − ( U × B ) = k.j − ( U × { } ). Then we canassume that the holomorphic germ f ∈ O N,y is define on the open set π ( j − ( U × B ))(remember that π is open as it is a GF map). Now choose a function ρ ∈ C ∞ c ( U ) suchthat R U ρ ( t ) .dt ∧ d ¯ t = 1 /k . Define the ( n, n ) − form ω := f ( π ( j − ( t, x ))) .ρ ( t ) .dt ∧ d ¯ t on U × B ; it has a B − proper support and it is ¯ ∂ − closed. Then it induces byintegration on cycles a holomorphic function on the open set Ω k ( E ) of C fn ( M ) viathe integration map Z (cid:3) ω : H ( ¯ U , Sym k ( B )) → C given by X Z X ω. On any ϕ ( z ) = C ∈ ϕ ( N ) ∩ Ω k ( E ) this holomorphic function takes the value f ( z ),and this proves that ϕ : N → N π is an isomorphism of complex spaces. (cid:4) Note that, conversely, if ϕ : N → C fn ( M ) is a proper holomorphic embedding, anecessary and sufficient condition for ϕ to be the fiber map of a GF homorphic map π : M → N is the fact that for y generic in N the cycle ϕ ( y ) is reduced and that theprojection p : G → M of the graph G ⊂ N × M of the f-analytic family parametrizedby N is an isomorphism onto M . Of course this implies that for y = y ′ in N thecycles ϕ ( y ) and ϕ ( y ′ ) are disjoint. Recall now, for the convenience of the reader, the definition of a strongly quasi-proper map given in [B.13], using the terminology introduced above. Consider thefollowing situation : The standard situation. Let π : M → N be a holomorphic map between apure dimensional complex space M and an irreducible complex space N . Define theinteger n as n := dim M − dim N . Assume that there is a closed analytic subsetΣ ∈ N with no interior point in N such that the restriction π ′ : M \ π − (Σ) → N \ Σis a GF map. Let ϕ ′ : N \ Σ → C fn ( M ) the holomorphic map classifying the fibersof π ′ (as n − cycles in M ) and Γ ⊂ ( N \ Σ) × M the graph of this family. In the proper case ; for the generalization of Remmert’s direct image theorem with value in aopen set in a Banach space see [B-M] ch.III th. 7.3.1. see [B.75] ch. IV or [B-M] ch.IV Remark. For a quasi-proper surjective map π : M → N between irreduciblecomplex spaces, with n := dim M − dim N , there always exists a closed analyticsubset Σ ⊂ N with no interior point in N such that the following restriction of ππ ′ : M \ π − (Σ) → N \ Σ is a GF map (see the end of the proof of the proposition3.2.2 for details). Definition 3.2.1 In the standard situation described above we shall say that π is strongly quasi-proper (a SQP map for short) if and only if the closure ¯Γ of Γ in N × C fn ( M ) is proper over N . Of course we shall show that a SQP map is quasi-proper in the usual sense (see theproof of the next proposition).We begin by an improvement of the criterium given in [B.13] in order that a holo-morphic map π : M → N between irreducible complex spaces will be SQP. Proposition 3.2.2 Let h : M → N a holomorphic map between a pure dimensionalcomplex space M and an irreducible complex space N . Put n := dim M − dim N .Assume that there exists a dense subset Λ in N such that h − (Λ) is dense in M and such that for each y ∈ Λ the fiber h − ( y ) is non empty, reduced and of (pure)dimension n with finitely many irreducible components. Note γ : Λ → C fn ( M ) themap defined by γ ( y ) := h − ( y ) where the n − cycle γ ( y ) is reduced. Let Γ the graphof the map γ and ¯Γ the closure of Γ in N × C fn ( M ) . Our main assumption is nowthe following: • The natural projection τ : ¯Γ → N is proper.Then the map h is strongly quasi-proper. Proof. The empty n − cycle is open (and closed) in C fn ( M ) so any sequence con-verging to it is stationary. As Λ is dense in N and γ ( y ) = ∅ n for y ∈ Λ, any fiberof τ cannot be equal to {∅ n } . But τ (¯Γ) = N as it is closed and contains Λ. Thisimplies that h ( M ) = τ (¯Γ) = N and h is surjective.Our second step (and the main step in fact) will be the proof that the map h isquasi-proper. So fix a point y ∈ N and let V be an open relatively compact neigh-bourhood of y in N . Fix y ′ ∈ V and choose an irreducible component C of h − ( y ′ ) .Let x ′ be a (generic) point in C such that x ′ does not belong to any other irreduciblecomponent of h − ( y ′ ). Then we can choose a sequence ( x ν ) ν ≥ in h − (Λ) convergingto x ′ . For ν ≫ h ( x ν ) ∈ V so the cycles γ ( h ( x ν )) are in the compactset p ( τ − ( ¯ V )) of C fn ( M ). Up to pass to a sub-sequence, we can assume that thesequence ( γ ( h ( x ν ))) converges to a cycle δ ∈ C fn ( M ). As we have x ν ∈ γ ( h ( x ν ))for each ν we have x ′ ∈ | δ | . By compactness of the subset p ( τ − ( ¯ V )) of C fn ( M ), We mean here reduced as a cycle ; this is equivalent to the fact that the natural structure ofcomplex space on this fiber is generically reduced “in the algebraic sense”. From the surjectivity of h proved above, h − ( y ′ ) is not empty. K in M such that any irreducible component of anycycle in p ( τ − ( ¯ V )) meets K . So this is the case for each irreducible component ofany cycle γ ( h ( x ν )) and of δ . Let δ be an irreducible component of δ containing x ′ .So δ meets K . But | δ | is contained in h − ( y ′ ) because the condition for a n − cyclein C fn ( M ) to be contained in a fiber of h is a closed condition (see the proposition2.1.7). As δ is irreducible, contained in h − ( y ′ ) and contains x ′ , this implies δ ⊂ C and so C meets K . So we have proved that for any y ∈ N there exists an openneighbourhood V of y in N and a compact set K in M such that for any y ′ ∈ V and any irreducible component C of h − ( y ′ ) the intersection C ∩ K is not empty.This is the definition of the quasi-properness of the map h .Now we conclude that the subset Σ of N of y such that dim h − ( y ) > n is a closedanalytic set with no interior point because Σ is the image by h of the closed analyticsubset Z := { x ∈ M / dim x ( h − ( h ( x ))) > n } in M which is an union of irreducible components of fibers of h ; so the restrictionof h to this analytic subset is still quasi-proper. So Kuhlmann’s theorem gives theconclusion. Now let Σ be the closed analytic subset of non normal points in N and let N ′ := N \ (Σ ∪ Σ ). Then the map h ′ : h − ( N ′ ) → N ′ is quasi-properequidimensional on a normal complex space. Then we have a holomorphic map ϕ : N ′ → C fn ( M )classifying the generic fibers of h ′ . Of course, for y ∈ Λ ∩ N ′ we have | ϕ ( y ) | = γ ( y ).But the subset T of points y ∈ N ′ such that ϕ ( y ) is not a reduced cycle is a closedanalytic subset of N ′ (see the lemma 2.1.6) which has no interior point. So the densesubset Λ meets the dense open set N ′ \ T in a dense subset in N . This implies thatthe closure of the graph of ϕ in N × C fn ( M ) is equal to ¯Γ. Then, by definition, themap h is strongly quasi-proper. (cid:4) An immediate corollary of this result is a first stability result of SQP maps. Corollaire 3.2.3 Let π : M → N be a strongly quasi-proper holomorphic mapbetween irreducible complex spaces and τ : T → N a holomorphic map from an irre-ducible complex space T such that τ ( T ) is not contained in A the set of points in N having ”big” fibers for π . Then for any irreducible component τ ∗ ( M ) i of ( T × N M ) which surjects on T the pull-back τ ∗ ( π ) i : τ ∗ ( M ) i → T is strongly quasi-proper. (cid:4) We give now a characterization of SQP maps in term of GF maps, which is variantof the important theorem 2.4.4 of [B.13]; see [M.00] or the Appendix (section 4) fora complete proof. Theorem 3.2.4 Let π : M → N be a surjective holomorphic map between irre-ducible complex spaces. The map π is strongly quasi-proper if and only if thereexists a modification σ : ˜ N → N such that the strict transform ˜ π : ˜ M → ˜ N by σ ofthe map π is a GF map. Proof. Assume first that π is a SQP map. Then there exists a modification σ : ˜ N → N and a holomorphic map ˜ ϕ : ˜ N → C fn ( M ) extending holomorphically themap ϕ : N ′ → C fn ( M ) classifying the generic fibers of π (see section 4). Then definethe holomorphic map ψ : ˜ N → C fn ( ˜ M ) by ψ (˜ y ) := { ˜ y } × ˜ ϕ ( y ) as a cycle in ˜ N × M .The holomorphy is consequence of the product theorem for analytic families of cy-cles (see [B.75] or [B.M] ch.IV) and these cycles are in ˜ M := M × N,str ˜ N becausethis is true for generic ˜ y ∈ ˜ N and so for all ˜ y ∈ ˜ N by continuity of ψ . Then we canconclude that ˜ π is a GF map applying the lemma 3.1.2.Assume now that we have a modification σ : ˜ N → N such that the strict transform˜ π : ˜ M → ˜ N of π by σ is a GF map. Then the composition of the holomorphicmap ψ : ˜ N → C fn ( ˜ M ) classifying the fibers of ˜ π with the direct image of cycles bythe modification ˜ σ : ˜ M → M induced by the projection on M of the fiber product M × N,str ˜ N gives a holomorphic map ˜ ϕ : ˜ N → C fn ( M ). It is clear that, if Σ is thecenter of σ , the restriction of ˜ ϕ is the map ϕ on N \ Σ. So, applying the theorem3.1.9 we see that π is a SQP map because the closure of the graph of its fiber mapis the image of ˜ N which is proper on N . (cid:4) Remark that in the previous theorem we can replace GF by wGF thanks to lemma3.1.3. Proposition 3.2.5 A quasi-proper surjective holomorphic map between irreduciblecomplex spaces is SQP if and only if its fiber map, defined and holomorphic on adense Zariski open set in M , is meromorphic. Moreover, if π : M → N is aquasi-proper surjective map such that its fiber map is meromorphic, then the stricttransform ˜ π : ˜ M → N π of π by the modification τ : N π → N given by the graph ofthe meromorphic fiber map (see the paragraph 1.4) is a GF map. Proof. Consider a SQP holomorphic map π : M → N . Then, thanks to thetheorem 3.2.4, there exists a modification σ : ˜ N → N such that the strict trans-form ˜ π : ˜ M → ˜ N of π by σ is a GF map. So there exists a holomorphic map˜ ϕ : ˜ N → C fn ( ˜ M ) classifying the fibers of ˜ π . By composition with the direct im-age of cycles by the modification ˜ σ : ˜ M → M we obtain, thanks to the directimage theorem for cycles by a proper map (see [B.M] ch.IV), a holomorphic map ϕ := ˜ σ ∗ ◦ ˜ ϕ : ˜ N → C fn ( M ).Conversely assume that π : M → N is a quasi-proper surjective holomorphic mapbetween irreducible complex spaces such that the fiber map (defined on a denseZariski open set) is meromorphic; let τ : N π → N be the modification given by thegraph N π of the meromorphic fiber map of π (see the paragraph 1.4). Then we havea holomorphic map ˜ ϕ : N π → C fn ( M ) which extends the fiber map. Then to provethat the strict transform ˜ π : ˜ M → N π of π by τ is a GF map, consider the holomor-phic map ψ : N π → C fn ( M × N N π ) given by ˜ y ˜ ϕ (˜ y ) × { ˜ y } . It is holomorphic by It is enough to take the set of normal points in N intersected with the complement of the “bigfibers” subset as in the remark following the definition of the standard situation (see the beginingof the paragraph 3.2). y in N π the cycle ψ (˜ y ) is in ˜ M := M × N,str N π this map is holomorphic withvalues in C fn ( ˜ M ) and it is then easy to see that it is the (holomorphic) fiber map for˜ π . So ˜ π is a GF map. (cid:4) Definition 3.2.6 Let π : M → N be a SQP map. With the notations introducedabove, the map ˜ ϕ := τ ∗ ◦ ψ : N π → C fn ( M ) will be called a classifying map for thefamily of fibers of π . The following proposition summarizes some basic properties of a SQP map. Proposition 3.2.7 Let π : M → N be a strongly quasi-proper holomorphic mapbetween irreducible complex spaces. Then we havei) the map π is quasi-proper.ii) the locus Σ of “big fibers” of π , Σ := { y ∈ N / dim π − ( y ) > n } , where n := dim M − dim N , is a closed analytic subset in N .iii) Let Σ ⊂ N be the closed analytic nowhere dense subset which is the union ofthe “big fibers” subset and the subset of non normal points in N . There existsa holomorphic map ϕ : N \ Σ → C fn ( M ) classifying the generic fibers of π andthe map ϕ is meromorphic along Σ .iv) Let τ : N π → N the projection on N of the graph of the meromorphic map ϕ and ˜ ϕ : N π → C fn ( M ) its second projection. For each y ∈ N we have π − ( y ) = ∪ z ∈ τ − ( y ) | ˜ ϕ ( z ) | , that is to say that any “big fiber” of π is filled up by limits of generic fibers. Proof. The properties i) and ii) has been proved in the proof of the proposition3.2.2. The point iii) is proved in the proposition 3.2.5. To prove iv) consider a point x ∈ M and consider a sequence ( x ν ) in π − ( N \ Σ) converging to x , where N \ Σ isa dense open set where ϕ is holomorphic. This exists by irreducibility of M . Then y ν := π ( x ν ) converges to y := π ( x ) and ϕ ( y ν ), up to pass to a sub-sequence ( τ isproper), converges to a point ˜ y ∈ τ − ( x ). We have x ν ∈ | ϕ ( y ν ) | for each ν so that x is in | ˜ ϕ (˜ y ) | . (cid:4) Remark. For an equidimensional surjective holomorphic map between irreduciblecomplex spaces to be strongly quasi-proper is equivalent to be a quasi-proper map.But if a surjective holomorphic map is not equidimensional, the condition to bequasi-proper is not strong enough; in particular it is not stable by taking the stricttransform of π : M → N by a modification on N . Of course, our definition of a SQPmap will be stable by such an operation. But we shall see that this notion enjoys1many other stability properties.The reader can also see interesting applications of this notion in [B.13] in the con-struction of meromorphic quotients for a large class of non proper analytic equiva-lence relations. Example. Let Y := { (( a, b ) , ( x, y )) ∈ C × C / a.x + b.x − a .y = 0 } and let π : Y → C the map induced by the first projection. Then we have the followingpropertiesI) The (algebraic) hypersurface Y of C is irreducible (in fact normal and con-nected).II) The map π : Y → C is quasi-proper.III) The map π is not strongly quasi-proper.IV) More precisely, after blowing-up the origin in C the strict transform of π isno longer quasi-proper. Proof of i). The critical set of the polynomial P ( a, b, x, y ) if given by the fol-lowing equations2 a.x + b = 0 , a .y = 0 , x − a.y = 0 , x = 0 (1)So the subset S := { a = b = x = 0 } ∪ { x = y = b = 0 } which is one dimensionaland is contained in Y is the singular subset of Y . So the singular set of Y is exactly S . As it has codimension 2 in Y , the hypersurface Y is normal. We shall see thateach fiber of π is connected and then the fact that C × { } is a section of π impliesthat Y is connected. So Y is irreducible. Proof of ii). First we shall describe the fibers of π . For a.b = 0 the fiber π − ( a, b ) is a smooth conic containing the origin in C . For a = 0 and b = 0 thefiber π − ( a, b ) is the union of two distinct lines through the origin. For a = 0 and b = 0 the fiber π − (0 , b ) is the line x = 0 which also contains the origin. Finally thefiber π − (0 , 0) is C . So each fiber is connected and contains the origin. Then the π − proper set C × { } meets every irreducible component of any fiber of π , so thismap is quasi-proper. Proof of iii). Consider now the map f s : C → C given by f s ( b ) := ( b.s, b ) where s is a non zero complex number. Then for b = 0 the fiber of π at the point f s ( b )is given by the equation b.s.x + b.x − b .s .y = 0 and assuming that b = 0 this isequivalent to x. ( s.x + 1) − b.s .y = 0 . s = 0 fixed, the limit of this non degenerate conic when b goes to 0 is theunion of the lines { x = 0 } and { x = − /s } . And it is clear that when | s | goes to 0,the component { x = − /s } gets out of any compact set in C . This phenomenonof escape at infinity near the point (0 , 0) in the target of π implies that the map π cannot be strongly quasi-proper. Proof of iv). Consider now the blow-up τ : X → C of the (reduced) origin in C . The complex manifold X is the sub-manifold X := { (( a, b ) , ( α, β )) ∈ C × P / a.β = b.α } . It will be enough to show that the strict transform of π over the chart { β = 0 } of X is not quasi-proper to achieve our goal. So let s := α/β . Then we have coordinates( s, b ) ∈ C for this chart on X . The total transform of Y is given by the equation s.b.x + b.x − s .b .y = 0and, as the function b is not generically zero on the strict transform ˜ Y of Y by τ ,we have ˜ Y β =0 = { (cid:0) ( s, b ) , ( x, y ) (cid:1) ∈ C × C / x. ( s.x + 1) − b.s .y = 0 } . So the fiber of the strict transform ˜ π at the point ( s, 0) is the union of the two lines { x = 0 } and { x = − /s } for s = 0. Then it is clear that this map is not quasi-properas an irreducible component of the fiber at (0 , s ) , s = 0 gets out of any compact setin C when s = 0 goes to 0. (cid:3) Important remark. The previous example is algebraic . And in opposition ofthe context where the notion of a strongly quasi-proper map has been introduced(in complex analytic geometry), this notion makes sense in algebraic geometry andis, of course, related to the behaviour of the limits of generic fibers of a map in somecompactification of the map. But it is independent of the chosen compactificationand so it has to be considered also in algebraic geometry.Note that the phenomenon of “escape at infinity” happens in the algebraic setting,but not the phenomenon of “infinite breaking” , which is purely transcendental. (cid:3) In order to prove some more stability results for SQP maps, we shall use the follow-ing two lemma. Lemma 3.2.8 Let α : M → N and β : N → P be holomorphic surjective mapsbetween irreducible complex spaces and let σ : ˜ P → P be a holomorphic dominant We say that we have an infinite breaking when the limit in the sense of C locn ( M ) of a sequence( C ν ) of irreducible cycles has infinitely many irreducible components. map. Let ˜ β : ˜ N → N the strict transform of β by σ and let ˜ α : ˜ M → ˜ N the stricttransform of α by ˜ σ the projection of ˜ N to N . Then the strict transform of β ◦ α by σ is the composition of the strict transform of β by σ and the strict transform of α by ˜ σ . ˜ M τ (cid:15) (cid:15) ˜ α / / ˜ N ˜ β / / ˜ σ (cid:15) (cid:15) ˜ P σ (cid:15) (cid:15) M α / / N β / / P Proof. For the “usual” fiber product we have a canonical isomorphism M × N ( N × P ˜ P ) ≃ M × P ˜ P given by the obvious projection : the inverse is given by ( x, ˜ z ) ( x, ( α ( x ) , σ (˜ z )).Then our hypothesis allows to see that the strict transforms correspond via thisisomorphism. (cid:4) Lemma 3.2.9 Let M, N, ˜ N , P be irreducible complex spaces, let α : M → N and τ : ˜ N → N be holomorphic surjective maps and σ : P → N a dominant holomorphicmap. Note α and α the strict transforms of α by τ and σ respectively.Let P := ˜ N × N,str P , and note τ ′ : P → ˜ N and σ ′ : P → P the projections. Thennote α ′ and α ′ the strict transforms of α and α by τ ′ and σ ′ respectively. M α (cid:15) (cid:15) ˜ τ / / M α (cid:15) (cid:15) M α (cid:15) (cid:15) ˜ σ o o M × ˜ N P : : tttttttttt α ′ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ˜ N τ / / N P σ o o M × P P d d ❏❏❏❏❏❏❏❏❏❏ α ′ u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ P τ ′ ` ` ❆❆❆❆❆❆❆❆❆ σ ′ > > ⑥⑥⑥⑥⑥⑥⑥⑥⑥ Then M × ˜ N,str P is canonically isomorphic to M × P,str P . Proof. Again we have a canonical isomorphism( M × N ˜ N ) × ˜ N ( ˜ N × N P ) ≃ ( M × N P ) × P ( P × N ˜ N )given by the map (( x, ˜ y ) , (˜ y, (˜ y, z ))) (( x, z ) , ( z, ˜ y )); the inverse map is given by(( x, z ) , ( z, ˜ y )) (( x, ˜ y ) , (˜ y, (˜ y, z ))) . Again our hypothesis allows to see that the strict transforms correspond via thisisomorphism. (cid:4) Proposition 3.2.10 Let M, N, P be irreducible complex spaces, α : M → N be aholomorphic SQP map and σ : P → N a holomorphic dominant map. Then thestrict transform α : M → P of α by σ is SQP map. Proof. By assumption, there exists a modification τ : ˜ N → N such that thestrict transform α : M → ˜ N is a GF map. If τ ′ : P → ˜ N is the strict transform of σ , it is again dominant and we can apply the lemma 3.2.9 to obtain that the stricttransform of α by τ ′ is a GF map. Now, as the lemma 3.2.9 gives that α ′ ≃ α ′ theconclusion follows thanks to the lemma 3.1.7. Proposition 3.2.11 Let M, N, P be irreducible complex spaces, α : M → N be aholomorphic wGF map and β : N → P a holomorphic SQP map. Then the composedmap β ◦ α is a SQP map. Proof. Let σ : ˜ P → P a modification such that the strict transform ˜ β : ˜ N → ˜ P is a GF map. Note ˜ σ : ˜ N → N the modification induced by the projection. Thenthe strict transform ˜ α of α by σ is a wGF map, thanks to lemma 3.1.5, and thecomposed map ˜ β ◦ ˜ α is then a wGF map using the lemma 3.1.6 ; then β ◦ α si a SQPmap by the remark following the proof of the proposition 3.2.4 and we concludeusing the lemma 3.2.8. (cid:4) Remarks. 1. Note that we dont need that the map β is holomorphic in the proof above : ifwe compose a wGF map with a meromorphic strongly quasi-proper map weobtain a meromorphic strongly quasi-proper map.2. It is not true that the composition of a modification with a SQP map is againa SQP map. The example below shows that the composition of a modificationwith a GF map is not quasi-proper in general. Example. Let α : C → C the projection on the first coordinate. Then α isclearly a GF map. Consider now τ : X → C the blow-up in C of the reducedideal defining the closed analytic subset Z × { } . Then the map α ◦ τ : X → C hasa fiber at 0 which is not a finite type cycle. So this map is not quasi-proper butnevertheless equidimensional. (cid:3) We conclude this section by another important stability result for strongly quasi-proper holomorphic maps. Theorem 3.2.12 Let α : M → N be a SQP holomorphic map. Let Z ⊂ N be aclosed analytic irreducible subset in N and X an irreducible component of α − ( Z ) which is dominant on Z for the map α | X induced by α . Then the map α | X : X → Z is SQP. The proof of this result will use the following proposition. We say that a meromorphic map h : N − − → P is strongly quasi-proper if the holomorphicmap ˜ h : ˜ N → P which is the projection on P of the graph ˜ N ⊂ N × P of h is strongly quasi-proper. Proposition 3.2.13 Let α : M → N be a holomorphic GF map where M is apure dimensional complex space and N a smooth connected manifold and define n := dim M − dim N . Then for each integer q ≥ there is a holomorphic map Φ q : C q ( N ) → C fn + q ( M ) which is given by Φ q ( C ) := ( p ) ∗ ( p ∗ ( C )) , where p : G → N and p : G → M are the projections of the graph G ⊂ N × M of the f-analytic familyof fibers of α . Remarks. 1. As we know that p is an isomorphism, the map ( p ) ∗ : C fn + q ( G ) → C fn + q ( M )is also bi-holomorphic in a obvious sense (see definition 2.1.4). So we mayidentify M and G in the proof of the proposition.2. As we assume that N is smooth, the cycle p ∗ ( C ) is well defined as a cycle in G as soon as the codimension of G ∩ ( | C | × M ) in N × M is the sum ofcodimensions of C in N and of G in N × M , so is equal to dim N + dim N − q .This means that its dimension is equal to dim M − dim N + q = n + q . But asthe fibers of α have pure dimension n this is true for any q − cycle in N . Proof of the proposition 3.2.13. Thanks to the remark 2. above the holo-morphy of the map j ◦ Φ q : C q ( N ) → C locn + q ( M ), where j : C fn + q ( M ) → C locn + q ( M ) is theobvious map, is an immediate consequence of the variant for the pull-back of theintersection theorem for analytic families of cycles (see [B.75] chapter VI or [B-M 2]chapter VII). The only points to show are :1. for each compact q − cycle C ∈ C q ( N ) the ( n + q ) − cycle Φ q ( C ) is a finite typecycle;2. the map Φ q is continuous for the topology of C fn + q ( M ).Fix C ∈ C q ( N ) and choose a relatively compact open set V ⊂ N such that | C | ⊂ V .As ¯ V is compact in N there exists a relatively compact open set W in M such thatany irreducible component of the fiber α − ( y ) for any y ∈ V meets W . Take nowan irreducible component D of Φ q ( C ′ ) where C ′ ∈ C q ( N ) such that | C ′ | ⊂ V . Wehave | Φ q ( C ′ ) | = | p ∗ ( C ′ ) | = G ∩ ( | C ′ | × M ) ≃ α − ( | C ′ | ). As the restriction of α to α − ( | C ′ | ) has pure n − dimensional fibers over | C ′ | its restriction to D is a wGFmap, thanks to the remark 2 following the definition 3.1.4, and D is an union ofirreducible components of fibers of α over | C ′ | . But each such irreducible componentmeets W because | C ′ | ⊂ V . So D meets W , and then Φ q ( C ′ ) is a finite type cyclefor any such C ′ .Now take a sequence ( C ν ) ν ≥ in C q ( N ) converging to C . Then it is clear fromthe discussion above that the sequence (Φ q ( C ν )) converges to Φ q ( C ) in C locn + q ( M )and that we have a relatively compact open set W in M such that any irreduciblecomponent of any Φ q ( C ν ) meets W . So the proposition 2.1.2 implies the convergenceof (Φ q ( C ν )) to Φ q ( C ) in C fn + q ( M ). This is the continuity of Φ q at C . (cid:4) see [B.75] ch.VI. Proof of the theorem 3.2.12. We begin by the description of such an X :Let τ : ˜ N → N a modification such that there exists ϕ : ˜ N → C fn ( M ) a holomorphicmap classifying the generic fibers of α . Let ˜ Z an irreducible component of τ − ( Z )which is surjective onto Z and let Y := α − ( Z ) × Z,str ˜ Z . The projection p : Y → ˜ Z is a GF map and the projection p : Y → α − ( Z ) is proper. Let ˜ X be an irreduciblecomponent of Y and define X := p ( ˜ X ). As p is proper X is a closed analyticirreducible subset in α − ( Z ). As ˜ X is surjective on ˜ Z and contains a non emptyopen set in Y , X is an irreducible component of α − ( Z ) and is dominant on Z .Conversely, if X is an irreducible component of α − ( Z ) and is dominant on Z then X × Z,str τ − ( Z ) has an irreducible component ˜ X surjective on X ( p and τ areproper) and ˜ Z := p ( ˜ X ) is an irreducible component of τ − ( Z ) which is surjectiveon Z .Now fix X, ˜ X and ˜ Z as above. Using the geometric flattning theorem (for a propermap see [B-M] chapter IV corollary 9.3.1) and Hironaka desingularization theorem,we can find a modification σ : ˆ Z → ˜ Z such that we have the following situation :i) The proper map ˜ τ := τ ◦ σ : ˆ Z → Z is equidimensionnal with ˆ Z smooth anconnected.ii) The strict transform ˆ α : ˆ X → ˆ Z of p by σ is a GF map.iii) The projection p : ˆ X → X is proper and surjective.ˆ X (cid:15) (cid:15) ˆ α / / ˆ Z σ (cid:15) (cid:15) Y p (cid:15) (cid:15) ˜ X o o (cid:15) (cid:15) p / / ˜ Z (cid:15) (cid:15) / / ˜ N τ (cid:15) (cid:15) M α − ( Z ) o o X o o α | X / / Z / / N Then we have a holomorphic map Z → C q ( ˆ Z ), where Z is the normalization of Z ,classifying the generic fibers of ˜ τ . Composing this map with the holomorphic mapΨ q : C q ( ˆ Z ) → C fn + q ( M ) associated via the proposition 3.2.13 to the holomorphicmap ψ : ˆ Z → C fn ( M ) defined as ψ := ϕ | ˜ Z ◦ ˜ τ , gives a holomorphic mapˆ Z → C fn + q ( X ) ⊂ C fn + q ( M )which classifies the generic fibers of the restriction α | X : X → Z . This complete theproof. (cid:4) Our goal is to give a proof of the following result of D. Mathieu (see [M. 00]) usingonly the proposition 3.2.2, its corollary 3.2.3 and the generalization of Remmert’sproper direct image theorem with values in a Banach open set (see [B-M] ch.III).We shall consider a surjective holomorphic map π : M → N between irreduciblecomplex spaces which is quasi-proper (in the usual sense). So we are in the stan-dard situation described at the begining of the paragraph 3.2. With the notationsintroduced there we shall assume the condition (which is our definition of a SQPmap) : • The projection p : ¯Γ → N is proper, (@)where we recall that Γ is the graph of the holomorphic map ϕ : N \ Σ → C fn ( M )classifying a f-analytic family of cycles in M , for y generic in N \ Σ, and ¯Γ its closurein N × C fn ( M ). Theorem 4.1.1 Let π : M → N a quasi-proper surjective holomorphic map betweenirreducible complex spaces and we put n := dim M − dim N . Assume that (@) issatisfied. Then there exists a modification τ : ˜ N → N such that the strict transform ˜ π : ˜ M → ˜ N is a geometrically f-flat map. The proof has two steps.The first step is to prove that it is enough to prove the result locally on Σ. Thiswill use the proper direct image theorem with values in C fn ( M ) which is a simpleconsequence (see the theorem 2.3.2) of the generalization of Remmert’s theorem withvalues in a Banach space (see [B-M] ch.III). This step is given by the proposition4.1.2.The second step is devoted to the proof that the result is locally true on Σ. Thiswill be done by induction on the dimension of the fibers of π . The induction step isgiven by the proposition 4.1.4. Proposition 4.1.2 Assume that we have a modification τ : ˜ N → N such that theconclusion of the theorem is valid. Then ¯Γ ⊂ N × C fn ( M ) is a closed analytic subsetand the sheaf of holomorphic functions on N × C fn ( M ) induces on ¯Γ a structure ofreduced complex space. Proof. Let ˜ ϕ : ˜ N → C fn ( ˜ M ) be the holomorphic map classifying the fibers of thestrict transform ˜ π : ˜ M → ˜ N of π by the modification τ : ˜ N → N . The compositionwith the direct image τ ∗ : C fn ( ˜ M → C fn ( M ) of cycles by the proper map τ gives aholomorphic map ψ := τ ∗ ◦ ˜ ϕ . Then define χ := ( τ, ψ ) : ˜ N → N × C fn ( M ). On adense open set in ˜ N the holomorphic map χ takes its values in Γ. So, if we provethat χ is a proper map we can conclude, first that its image is ¯Γ and then using the8proper direct image in this context we shall conclude that ¯Γ is a (locally finite di-mensional) reduced complex space with the sheaf of holomorphic functions inducedfrom N × C fn ( M ).As each fiber of χ is a closed subset in a fiber of τ , the fibers are compact. Wehave now to prove that the map χ is closed. Let F be a closed set in ˜ N and choosea sequence (˜ y ν ) be a sequence in F such that the sequence ( χ (˜ y ν )) converges to( y, C ) ∈ N × C fn ( M ). Then, up to pass to a sub-sequence, we can assume that thesequence (˜ y ν ) converges to some ˜ y ∈ τ − ( y ) ∩ F by the properness of τ . Then bycontinuity of χ we have χ (˜ y ) = ( y, C ) and so χ (˜ y ) is in χ ( F ). (cid:4) Corollaire 4.1.3 Let π : M → N be a holomorphic map between irreducible com-plex spaces, and assume that we are in the standard situation. Assume also that foreach point σ in Σ there exists an open neighbourhood V σ of σ in N and a modification N σ → V σ and a holomorphic map ϕ σ : N σ → C fn ( M ) extending ϕ | V σ \ Σ . Then thereexists a unique modification τ : ˜ N → N with a holomorphic map ˜ ϕ : ˜ N → C fn ( M ) extending ϕ and with the following universal property : • For any modification τ U : N U → U of an open set U in N on which thereexists a holomorphic map ϕ U : N U → C fn ( M ) extending ϕ on U \ Σ , thereexists an unique holomorphic map θ U : N U → τ − ( U ) ⊂ ˜ N compatible withthe projections on U , such that ϕ U = ˜ ϕ ◦ θ U . Proof. This corollary is an obvious consequence of the definition of a meromor-phic map of N with values in C fn ( M ), the definition of its graph, and the universalproperty of the projection of this graph on N because the content of the previousproposition is precisely the meromorphy of ϕ along Σ. Then the universal propertygives the patching for free. (cid:4) Note that we only use the direct image theorem (with values in C fn ( M )) only in theproper case for the results above (and also for defining the graph of a meromorphicmap with values in C fn ( M )). Proposition 4.1.4 Assume that in the situation of the theorem 4.1.1 the map π satisfies (@) . Let y ∈ N and assume that dim π − ( y ) = n + k with k ≥ , thenthere exists an open neighbourhood V of y in N and a modification τ : ˜ V → V suchthat the strict transform ˜ π V : ˜ M V → ˜ V has fibers of dimension at most n + k − ,where we denote M V := π − ( V ) and π V : M V → V the restriction of π . First note that the following lemma has already been proved in the proof of theproposition 3.2.2. Lemma 4.1.5 Let π : M → N be a holomorphic map between irreducible complexspaces. Assume that there exists a closed analytic subset Σ with no interior point in N such that the restriction of π to M \ π − ( M ) → N \ Σ is quasi proper and n − equidimensional, and a holomorphic map ϕ : N \ Σ → C fn ( ˜ M ) such that for y generic in N \ Σ we have ϕ ( y ) = | ϕ ( y ) | = π − ( y ) . Assume that the condition (@) issatisfied. Then π is quasi-proper. (cid:4) Remark that, thanks to the corollary 3.2.3 we keep the hypothesis (@) for ˜ π V , andthen, thanks to the lemma 4.1.5, the map ˜ π is again quasi-proper. Remark that itis not true in general that the strict transform by a modification of a quasi-propermap is again quasi-proper (see the example following the proposition 3.2.7). Proof of the proposition 4.1.4. As π is quasi-proper, there exists only finitelymany irreducible components of π − ( y ). Let Γ , . . . , Γ N be the irreducible compo-nents of π − ( y ) which have dimension n + k . Choose on each of them a genericpoint x i , i ∈ [1 , N ], and adapted ( n + k ) − scales E , . . . , E N , E i = ( U i , B i , j i ) withthe following properties:i) The polydisc U i and B i contain the origin ; let pr i : U i × B i → U i be theprojection.ii) The point x i is in the center of E I and pr i ( j i ( x i ) = (0 , j i (Γ i ∩ j − i ( U i × B i )) = U i × { } and deg E i (Γ i ) = 1.Let V be an open neighbourhood of y in N such that π − ( V ) ∩ j − i ( ¯ U i × ∂B i ) = ∅ for each i ∈ [1 , N ]. This exists because the compact set ∪ Ni =1 j − i ( ¯ U i × ∂B i ) does notmeet π − ( y ). Let W i := π − ( V ) ∩ j − i ( U i × B i ) and consider the holomorphic map θ i : W i → V × U i given by ( π, pr i ◦ j i ). As θ − i ( y , 0) = { x i } , up to shrink V , we canassume that each map θ i is proper and finite. We know that for generic y in V theanalytic set θ i ( π − ( y ) ∩ W i ) has dimension n so cannot contains { y } × U i ; then theclosed analytic set Z i := { y ∈ V / { y } × U i ⊂ θ i ( π − ( y ) ∩ W i ) } has empty interior in V . Let I i be a coherent ideal in O V such the strict transformof π : W i → V by the blowing-up of I i has no longer a fiber of dimension ≥ n + k .See [B-M] ch.III proposition 6.1.5 for the existence of such a coherent ideal with Supp ( O V (cid:14) I i ) ⊂ Z i . So consider the blow-up τ : ˜ V → V in V of the ideal I whichis the product of the I i for i ∈ [1 , N ]. It will give a strict transform ˜ π V : ˜ M V → ˜ V of π | V by τ whose fibers has dimension at most n + k − 1, concluding the proof. (cid:4) Proof of the theorem 4.1.1. First we shall consider the case where all fibersof π have dimension at most n . The lemma 4.1.5 gives that π is quasi-properand so its image is closed. So the map π is surjective and n − equidimensional. If ν : ˜ N → N is the normalization map, then the strict transform ˜ π : ˜ M → ˜ N of π is again n − equidimensional and surjective and so we have a holomorphic map0˜ ϕ : ˜ N → C fn ( ˜ M ) classifying the fibers of ˜ π , meaning that ˜ π is a GF map.Assume now that the local version of the theorem is proved when the maximaldimension of a fiber is n + k − 1, with k ≥ 1. Then the proposition 4.1.4 shows thatthis local version is also true when the maximal dimension of a fiber is n + k . Thisproves the local version of the theorem. But then the corollary 4.1.3 shows that thelocal version of the theorem implies the theorem itself, concluding the proof. (cid:4) Remark. The corollary gives a more precise result than what we state in thetheorem 4.1.1 : there exists a natural modification of N which factorizes any mod-ification of an open set U in N such that the map ϕ U \ Σ extends holomorphicallyon it. So the map ϕ is meromorphic along Σ and the natural modification of N issimply the graph of this meromorphic map (see 2.4.6).1 • [B.75] Barlet, D. Espace analytique r´eduit des cycles analytiques complexescompacts d’un espace analytique complexe de dimension finie , Fonctions deplusieurs variables complexes, II (S´em. Franois Norguet, 1974-1975), pp.1-158. Lecture Notes in Math., Vol. 482, Springer, Berlin, 1975 • [B.78] Barlet, D. Majoration du volume des fibres g´en´eriques et forme g´eom´triquedu th´eor`eme d’aplatissement , Sem. Lelong-Sloda 78-79, Lect. Notes in Math.822, p.1-17, Springer-Verlag, Berlin. • [B.08] Barlet, D. Reparam´etrisation universelle de familles f-analytiques decycles et f-aplatissement g´eom´etrique Comment. Math. Helv. 83 (2008),p. 869-888. • [B.13] Barlet, D. Quasi-proper meromorphic equivalence relations , Math. Z.(2013), vol. 273, p. 461-484. • [B.15] Barlet, D. Meromorphic quotients for some holomorphic G − actions .Preprint in preparation. • [B-M] Barlet, D. and Magn´usson, J. Cycles analytiques en g´eom´etrie com-plexe I: Th´eor`eme de pr´eparation des cycles , Cours Sp´ecialis´es n 22, Soc. Math.France (2014). • [B-M 2] Barlet, D. and Magn´usson, J. Cycles analytiques en g´eom´etrie com-plexe II in preparation. • [H.75] Hironaka, H. Flattening theorem in complex-analytic geometry , Am. J.Math. 97 (1975), p. 503-547. • [K.64] Kuhlmann, N. ¨Uber holomorphe Abbildungen komplexer R¨aume Archivder Math. 15 (1964), p.81-90. • [K.66] Kuhlmann, N. Bemerkungen ¨uber holomorphe Abbildungen komplexerR¨aume