RECTILINEARIZATION OF SUB ANALYTIC SETS AS A CONSEQUENCE OF LOCAL MONOMIALIZATION 6 1 STEVEN DALECUTKOSKY 0 2 Abstract. We give a new proof of the rectilinearization theorem of Hironaka. We n deduce rectilinearization as a consequence of our theorem on local monomialization for a complex and real analytic morphisms. J 1 1 In this paper we deduce Hironka’s rectilinearization theorem [20] as an application of ] our theorem on local monomialization for complex and real analytic morphisms [14]. G A Definition 0.1. Suppose that ϕ : Y → X is a morphism of complex or real analytic . manifolds, and p ∈Y. We will say that the map ϕ is monomial at p if there exist regular h at parameters x1,...,xm,xm+1,...,xt in OXan,ϕ(p) and y1,...,yn in OYan,p and cij ∈ N such m that n [ ϕ∗(xi)= Yyjcij for 1 ≤ i≤ m 1 j=1 v 2 with rank(c ) = m and ϕ∗(x ) = 0 for m < i≤ t. ij i 8 We will say that ϕ is monomial on Y if there exists an open cover of Y by open sets 4 U which are isomorphic to open subsets of Cn (or Rn) and an open cover of X by open 2 k 0 sets Vk which are isomorphic to open subsets of Ct (or Rt) such that ϕ(Uk) ⊂ Vk for all i . and there exist c (k) ∈ N such that 1 ij 0 n 6 1 ϕ∗(xi) = Yyjcij(k) for 1≤ i ≤ m : v j=1 i X with rank(c ) = m and ϕ∗(x ) = 0 for m < i ≤ t, and where x and y are the respective ij i i j r coordinates on Ct and Cn (or Rt and Rn). a A local blow up of an analytic space X is a morphism π : X′ → X determined by a triple (U,E,π) where U is an open subset of X, E is a closed analytic subspace of U and π is the composition of the inclusion of U into X with the blowup of E. Hironaka [21] and [20] introduced the notion of an´etoile on a complex analytic space Y to generalize a valuation of a function field of an algebraic variety. On any local blowup Y of Y, an ´etoile e determines a unique point on Y called the center of e. We say that a 1 1 sequence of local blowups Y of a complex analytic space Y is in an ´etoile e on Y if e has 1 a center on Y . 1 We now state the local monomialization theorem for complex analytic morphisms. Theorem 0.2. (Local Monomialization, Theorem 1.2 [14]) Suppose that ϕ : Y → X is a morphism of reduced complex analytic spaces and e is an ´etoile over Y. Then there exists partially supported by NSF. 1 a commutative diagram of complex analytic morphisms Y →ϕe X e e β ↓ ↓ α ϕ Y → X such that β ∈ e, the morphisms α and β are finite products of local blow ups of nonsingular analytic sub varieties, Y and X are nonsingular analytic spaces and ϕ is a monomial e e e analytic morphism. There exists a nowhere dense closed analytic subspace F of X such that X \F → X e e e e is an open embedding and ϕ−1(F ) is nowhere dense in Y . e e e Local monomialization theorems for real analytic morphisms are also proven in [14]. Local monomialization along an arbitrary valuation is proven for morphisms of algebraic varieties in characteristic zero in [10], [11] and [12]. Counterexamples to local monomial- ization for a morphism of characteristic p > 0 algebraic varieties is given in [13]. In Theorem 0.3 below, we show that Hironaka’s rectilinearization theorem, which was originallyprovenin[20]canbededucedfromlocalmonomializiation, Theorem0.2. Besides local monomialization, ourproofusescomplexification ofrealanalytic morphisms(Section 1 [20]), resolution of singularities of analytic spaces (Hironaka [18], [19], Aroca, Hironaka and Vicente [3] and Bierstone and Milman [6]), the Tarski Seidenberg Theorem, the fact that rectilinearization is true for semi analytic sets (this is a consequence of resolution of singularities, Hironaka [20]) and the fact that the natural map from the vouˆte ´etoil´ee of a complex analytic space X to X is proper (Hironaka, [21] and [20]). Hironaka’sproof(inTheorem7.1[20])makesessentialuseofthelocalflatteningtheorem (Hironaka, Lejeune and Teissier [22] and Section 4 of [20]) and the fiber cutting lemma (Lemma 7.3.5 [20]) to reduce to consideration of proper finite morphisms. In our proof, these arguments are replaced by the local monomialization theorem, Theorem 0.2, and the reductions of Theorems 3.1 and Theorem 3.2 of this paper. Some other notable proofs of rectilinearization are by Denef and Van Den Dries [17], Bierstone and Milman [8] and Parusinski[26]. Hironaka deduces Lojasiewicz’s inequalities for sub analytic sets from rectilinearization in [20]. The related concept (to monomialization) of toroidalization for morphisms of algebraic varieties ([1] and [2]) is used by Denef to prove p-adic quantifier elimination and related results [15], [16]. Some papers on topics related to this article are Teissier [27], Cano [9], Panazzolo [25], Lichtin, [23] and [24], Belotto [4] and Belotto, Bierstone, Grandjean and Milman [5]. Theorem 0.3. (Rectilinearization) Let X be a smooth connected real analytic space and let A be a sub analytic subset of X. Let p ∈ X and let n = dimX. Then there exist a finite number of real analytic morphisms π : V → X which are finite sequences of local α α blowups over X and induce an open embedding of an open dense subset of V into X such α that: 1) Each V is isomorphic to Rn, α 2) There exist compact neighborhoods K in V such that ∪ (K ) is a compact neigh- α α α α borhood of p in X, 3) For each α, π−1(A) is union of quadrants in Rn. α Semi analytic and sub analytic sets in a real analytic space are defined in Section 1. 2 A subset B of Rn is a quadrant if there exists a partition {1,...,n}= I ∪I ∪I such 0 1 − that B is the set of x ∈ Rn such that x = 0 for all i∈ I , x > 0 for all j ∈ I and x < 0 i 0 j + k for all k ∈ I where x ,...,x are the natural coordinates of x in Rn. − 1 n We thank Jan Denef for suggesting rectilinearization as an application of local mono- mialization of analytic morphisms, and for discussion and encouragement. We also thank Bernard Teissier for discussions on this and related problems. 1. Semi analytic and sub analytic sets We review the definitions of semi analytic and sub analytic sets from Chapter 6 [20] (also see Chapter 1 [8] or [7]). Let X be a set and ∆ be a family of subsets of X. The elementary closure ∆˜ of ∆ is the smallest family of subsets of X containing ∆ which is stable under finite intersection, finite union and complement. Suppose that U is an open subset of a real analytic space X. Let ∆ (U) be the set of subsets A of U of the form A = {x ∈ U | f(x) > 0} for some + real analytic function f on U. A subset A of X is said to be semi analytic at x ∈ X if 0 thereexists an open neighborhoodU of x in X such that A∩U belongs to theelementary 0 closure of ∆ (U). A is said to be semi analytic in X if it is semi analytic at every point + of X. LetΓ(U)betheset ofthoseclosed subsetsofU whichareimages ofproperrealanalytic maps g : Y → U. A subset A of X is said to be sub analytic at x ∈ X if there exists 0 an open neighborhood U of x in X such that A∩U belongs to the elementary closure of 0 Γ(U). A is said to be sub analytic in X if it is sub analytic at every point of X. 2. Preliminaries on ´etoiles and local blow ups We require that an analytic space be Hausdorff. An ´etoile is defined in Definition 2.1 [21]. An ´etoile e over a complex analytic space X is defined as a subcategory of the category of sequences of local blow ups E(X) over X. If π : X′ → X ∈ e then a point eX′ ∈ X′ is associated to e. We will call eX′ the center of e on X′. The ´etoile associates a point e ∈ X to X and if π : X → U is a local X 1 1 blow up of X such that e ∈ U then π ∈ e and e ∈ X satisfies π (e ) = e . If X 1 X1 1 1 X1 X π : X → U is a local blow up of X such that e ∈ U then π π ∈ e and e ∈ X 2 2 1 1 X1 1 1 2 X2 2 satisfies π (e ) = e . Continuing in this way, we can construct sequences of local blow 2 X2 X1 ups X →πn X → ··· → X →π1 X n n−1 1 such that π ···π ∈ e, with associated points e ∈ X such that π (e ) = e for all i. 1 i Xi i i Xi Xi−1 Let X be a complex analytic space. Let E be the set of all ´etoiles over X and for X π : X → X a product of local blow ups, let 1 E = {e ∈ E |π ∈e}. π X Then the E form a basis for a topology on E . The space E with this topology is π X X called the vouˆte ´etoil´ee over X (Definition 3.1 [21]). The vouˆte ´etoil´ee is a generalization to complex analytic spaces of the Zariski Riemann manifold of a variety Z in algebraic geometry (Section 17, Chapter VI [28]). We have a canonical map P : E → X defined by P (e) = e which is continuous, X X X X surjective and proper (Theorem 3.4 [21]). It is shown in Section 2 of [21] that given a 3 product of local blow ups π : X → X, there is a natural homeomorphism j : E → E 1 π X1 π giving a commutative diagram E ∼= E ⊂ E X1 π X P ↓ ↓ P X1 X π X → X. 1 The join of π ,π ∈ E(Y) is defined in Proposition 2.9 [21]. The join is a morphism 1 2 J(π ,π ) : Y → Y. It has the following universal property: Suppose that f : Z → Y 1 2 J is a strict morphism (Definition 2.1 [21]). Then there exists a Y-morphism Z → Y if J and only if there exist Y-morphisms Z → Y and Z → Y . It follows from 2.9.2 [21] that 1 2 if π ,π ,∈ e ∈ E , then J(π ,π ) ∈ e. We describe the construction of Proposition 2.9 1 2 Y 1 2 [21]. In the case when π and π are each local blowups, which are described by the data 1 2 (U ,E ,π ), J(π ,π ) is the blow up i i i 1 2 J(π ,π ) :Y = B(I I O |U ∩U ) → Y. 1 2 J E1 E2 Y 1 2 Now suppose that π is a product α α ···α where α : Y → Y are local blow ups 1 0 1 r i i+1 i defined by the data (U ,E ,α ), and π is a product α′α′ ···α′ where α′ : Y′ → Y′ i i i 2 0 1 r i i+1 i are local blow ups defined by the data (U′,E′,α′). We may assume (by composing with i i i identity maps) that the length of each sequence is a common value r. We define J(π ,π ) 1 2 byinductiononr. AssumethatJ = J(α α ···α ,α′α′ ···α′ )hasbeenconstructed, r 0 1 r−1 0 1 r−1 with projections γ : Y → Y and δ : Y → Y′. Then we define J(π ,π ) to be the blow Jr r Jr r 1 2 up J(π1,π2) :YJ = B(IErIEr′OJr|γ−1(Ur)∩δ−1(Ur′)) → Y. Suppose that ϕ : X → Y is a morphism of complex analytic spaces, and π : Y′ → Y ∈ E(Y). The morphism ϕ−1[π] : ϕ−1[Y′] → X will denote the strict transform of ϕ by π (Section 2 of [22]). Inthecaseofasinglelocalblowup(U,E,π)ofY,ϕ−1[Y′]istheblowupB(I O |ϕ−1(U)). E X In the case when π = π π ···π with π : Y → Y given by local blow ups (U ,E ,π ), 0 1 r i i+1 i i i i we inductively define ϕ−1[π]. Assume that π−1[π ···π ] has been constructed. Let 0 r−1 h = π ···π , so that π = hπ . Let ϕ′ : ϕ−1[Y ] → Y be the natural morphism. Then 0 r−1 r r r define ϕ−1[Yr+1] to be the blow up B(IErOϕ−1[Yr]|(ϕ′)−1(Ur)). 3. Rectilinearization In this section we prove rectilinearization, Theorem 3.5. We use the method of com- plexification of a real analytic morphism (Section 1, [20]). Theorem 3.1. Suppose that ϕ : Y → X is a morphism of reduced complex analytic spaces, K is a compact neighborhood in Y and f is an ´etoile over X. Then there exist local monomializations Y →ϕi X i i (1) δ ↓ ↓ γ i i ϕ Y → X 4 for 1 ≤ i ≤ r and π : X → X ∈ f such that there are commutative diagrams for f 1 ≤ i≤ t ≤ r Y →τi X i f β ↓ ↓ α i i (2) Y →ϕi X i i δ ↓ ↓ γ i i ϕ Y → X where Y = ϕ−1[X ]. Let ψ = δ β and π = γ α . There exists a closed analytic subspace i i f i i i i i G of X which is nowhere dense in X such that X \G → X is an open embedding, f f f f f π−1(π(G )) = G , the vertical arrows are products of a finite number of local blow ups of f f smooth subspaces and ∪t α−1(ϕ (δ−1(K)))\G = ∪t τ (ψ−1(K))\G = π−1(ϕ(K))\G . i=1 i i i f i=1 i i f f There exists a compact neighborhood L of f in X , a morphism of reduced complex Xf f analytic spaces u :Z → X and a compact neighborhood M in Z such that dimZ < dimY, u(Z) ⊂ ϕ(Y) and ϕ(K)∩π(G ∩L) = u(M). f Proof. By Theorem 0.2, we may construct local monomializations (1), with relatively compact open neighborhoods C in Y with closures K such that {E ,...,E } is an i i i C1 Cr open cover of the compact set ρ−1(K) (ρ is proper by Theorem 3.16 [20]) and γ are Y Y i sequences of local blow ups of smooth subspaces. We have that (3) K ⊂ ∪r δ (C ). i=1 i i Further, thereexist closed analytic subspacesG ofX whicharenowheredenseinX such i i i that X \G → X is an open embedding and ϕ−1(G ) is nowhere dense in Y . Reindex the i i i i i diagrams (1) so that f ∈ E if 1 ≤ i ≤ t and f 6∈ E if t < i ≤ r. Suppose that i is an Xi Xi index such that f 6∈E (that is, t < i ≤ r). The morphism X → X has a factorization Xi i X = V →σn ··· → V →σ2 V →σ1 V = X i n 2 1 0 where each σ : V → V is a local blowup (U ,E ,σ ). There exists a smallest j such j j j−1 j j j that f 6∈ U . Vj−1 j Let X∗ be an open neighborhood of f in V which is disjoint from U . Then i Vj−1 j−1 j fof∈thEeXji∗oiannodfEtXhie∗ ∩XEXwih=ich∅.saLtiestfyπf: ∈XfE→ aXndbeofath(geloXba∗l)surcehsoltuhtaitonfo6∈f sEingulaanrditiseos i Xi i Xi that π : X → X ∈ f is a sequence of local blowups whose centers are nonsingular and f such that α−1(ϕ (Y )) is nowhere dense in X for all i. Then, E ⊂ E if f ∈ E and i i i f Xf Xi Xi E ∩E = ∅ if f 6∈ E . We have factorizations Xf Xi Xi X →αi X →γi X f i of π if f ∈ E . For 1 ≤ i≤ t let Xi Y →τi X i f β ↓ ↓ α i i Y →ϕi X i i be the natural commutative diagram of morphisms, with Y = ϕ−1[X ]. Let ψ = δ β . i i f i i i Let G be the union of the preimages of the subspaces blown up in a factorization of π f by local blowups. Then G is a nowhere dense closed analytic subset of X such that f f X \G → X is an open embedding and π−1(π(G )) = G . Further, τ−1(G ) is nowhere f f f f i f 5 dense in Y for all i. Let U = X \ G . Suppose that q ∈ ϕ(K) ∩ π(U). There there i f f exists p ∈ K such that ϕ(p) = q and there exists i and p′ ∈ C such that δ (p′) = p by i i (3). Let q′ = ϕ (p′) ∈ X . There exists λ ∈ E such that λ = q′ and thus λ = q. i i X Xi X Since q ∈ π(U), we can regard q as an element of X with λ = q. Thus λ ∈ E f Xf Xf so that λ ∈ E ∩E , and so f ∈ E as this intersection is nonempty. We have that Xf Xi Xi ϕ (p′) = q′ and α : X → X is an open embedding in a neighborhood of q, so β is an i i f i i open embedding in a neighborhood of p′. Thus q = λ ∈ τ (ψ−1(K)). Whence Xf i i π−1(ϕ(K))∩U ⊂ ∪ τ (β−1(C ∩δ−1(K)))∩U ⊂ ∪ τ (ψ−1(K))∩U. i i i i i i i i We have ∪ τ (ψ−1(K)) ⊂ π−1(ϕ(K)) i i i since πτ (ψ−1(K)) = ϕψ (ψ−1(K)) ⊂ ϕ(K) i i i i for 1 ≤ i≤ t. Thus (4) ∪ τ (ψ−1(K))∩U = π−1(ϕ(K))∩U. i i i Let V bearelatively compact open neighborhoodof f in X . LetL bethe closureof f Xf f V in X . Then β−1(C )∩τ−1(V ) are relatively compact open subsets of Y with closures f f i i i f i L = β−1(K )∩τ−1(L) for 1 ≤ i ≤ t. Further, i i i i ∪ ϕψ (ψ−1(K)∩L )⊂ π(L)∩ϕ(K) i i i i and so ∪ ϕψ (ψ−1(K)∩L )\π(G ) = (π(L)\π(G ))∩ϕ(K) i i i i f f by (4). For all i, the compact set π(G )∩ [ϕψ (ψ−1(K)∩ L )] is nowhere dense in the f i i i compactsetϕψ (ψ−1(K)∩L )sinceτ−1(G )∩ψ−1(K)∩L isnowheredenseinthecompact i i i i f i i neighborhood ψ−1(K)∩L in Y . i i i Thus the compact set ∪ ϕψ (ψ−1(K) ∩ L ) is everywhere dense in the compact set i i i i π(L)∩ϕ(K). Thus ∪ ϕψ (ψ−1(K)∩L )= π(L)∩ϕ(K) i i i i and so ∪ ϕϕ (ψ−1(K)∩L ∩τ−1(G ))= π(G ∩L)∩ϕ(K). i i i i i f f Let Z = τ−1(G ) be the disjoint union of the analytic spaces τ−1(G ) with `1≤i≤t i f i f associated morphism u = ϕψ : Z → X and compact subset M = ψ−1(K)∩L ∩ `i i `i i i τ−1(G ) of Z. i f Then dimZ < dimY , u(Z) ⊂ ϕ(Y) and ϕ(K)∩π(G ∩L)= u(M). (cid:3) f Suppose ϕ : Y → X is a morphism of reduced real analytic spaces such that X is smooth. Let Y˜ → X˜ be a complexification of ϕ : Y → X such that X˜ is smooth and Y˜ is reduced. Suppose that K˜ is a compact neighborhood in Y˜ which is invariant under the auto conjugation of Y˜. Let K be the real part of K˜, which is a compact neighborhood in Y. Let Y˜ = Y˜(n) ⊃ Y˜(n−1) ⊃ ··· ⊃ Y˜(0) = ∅ be the stratification of Y˜ where Y˜(i−1) = sing(Y˜(i)) is the singular locus of Y˜(i), and let Y = Y(n) ⊃Y(n−1) ⊃ ··· ⊃ Y(0) 6 be the induced smooth real analytic stratification of Y. We have induced compact neigh- borhoods K˜ ∩ Y(i) in Y(i), with K = K˜ ∩ Y(n). There exist global resolutions of sin- gularities λ˜ : (Y˜(i))∗ → Y˜(i) which have an auto conjugation such that the real part of i λ˜ :(Y˜(i))∗ → Y˜(i) is λ : (Y(i))∗ → Y(i) where (Y(i))∗ is smooth (Desingularization I, 5.10 i i [20]). The morphism λ˜ is proper, so K˜ = λ˜−1(K ∩Y˜(i)) is a compact neighborhood in i i i Y˜(i) with compact real neighborhood K = λ−1(K ∩Y(i)) in Y(i). i i Let Y′ = (Y(i))∗. We have that the induced morphism ϕ∗ : Y′ → Y is proper and `i surjective. Let K′ = (λ∗)−1(K), a compact neighborhood in Y′. Let Y˜′ = (Y˜(i))∗ with `i induced complex analytic morphism λ˜∗ :Y˜′ → Y˜. Then λ˜∗ :Y˜′ → Y˜ is a complexification of λ∗ : Y′ → Y. Let K˜′ = (λ∗)−1(K˜), which is a compact neighborhood in Y˜′ with (K˜′)∩Y′ = K′. By Theorem3.1, applied tothecomplex analytic morphismϕ˜λ˜∗ :Y˜′ → X˜, thecompact neighborhood K˜′ in Y˜′ and an ´etoile f over X˜, there exist commutative diagrams Y˜ →τ˜i X˜ i f β˜ ↓ ↓ α˜ i i Y˜ →ϕ˜i X˜ i i δ˜ ↓ ↓ γ˜ i i Y˜′ ϕ˜→λ˜∗ X˜ ց ր Y˜ and a closed analytic subspace G˜ of X˜ such that f f (5) ∪t τ˜(ψ˜−1(K˜′))\G˜ = π˜−1(ϕ˜λ˜∗(K˜′))\G˜ i=1 i i f f and there exists a compact neighborhood L˜ of f in X˜ , a morphism of reduced complex X˜f f analytic spaces u˜ :Z˜ → X˜ and a compact neighborhood M˜ in Z˜ such that dimZ˜ < dimY˜ and ϕ˜λ˜∗(K˜′)∩π˜(G˜ ∩L˜)= u˜(M˜). f We can construct the above complex analytic spaces and morphisms so that there are compatible auto conjugations which preserve G˜ , Z˜, L˜ and M˜ and so that the real part f X of X˜ is nonempty if and only if f is a real point (by Theorems 8.4 and 8.5 [14]). f f X˜f Taking the invariants of the auto conjugations, we thus have whenever X 6= ∅, induced f commutative diagrams of real analytic spaces and morphisms Y →τi X i f β ↓ ↓ α i i Y →ϕi X i i δ ↓ ↓ γ i i Y′ ϕ→λ∗ X ց ր Y with a closed real analytic subspace G = G˜ ∩X of X . We have that G is nowhere f f f f f dense in X since X is smooth (for instance by Lemma 8.2 [14]), and thus dimG < f f f dimX = dimY. f 7 Also, taking the real part of u˜ : Z˜ → Y˜′, we have a morphism of reduced real analytic spaces u : Z → Y′ and a compact subset M of Z such that dimZ < dimY. The analog of (3) in Theorem 3.1, K′ ⊂ ∪r δ (C ) where C is the real part of C˜ , is true as Y′ is i=1 i i i i smooth (by Theorem 8.7 [14] and its proof). Then the argument following (3) in Theorem 3.1 shows that ∪t τ (ψ−1(K′))\G = π−1(ϕλ∗(K′))\G = π−1(ϕ(K))\G i=1 i i f f f and π(K)∩π(G ∩L) = ϕλ∗(K′)∩π(G ∩L)= u(M). f f Theorem 3.2. Suppose that ϕ :Y → X is a morphism of reduced complex analytic spaces, K ⊂ Y is a compact neighborhood in Y and h ∈ E . Then there exists d : X → X ∈ h, X h h morphisms of reduced complex analytic spaces ϕ : Y → X for 0 ≤ i ≤ t with compact i i neighborhoods K in Y such that ϕ = ϕ, Y = Y, K = K, ϕ (Y ) ⊂ ϕ (Y ), i i 0 0 0 i+1 i+1 i i dimY < dimY for all i and Y = ∅. There exist commutative diagrams for 0 ≤ i≤ t i+1 i t Yˆ σ→ij X ij h b ↓ ↓a ij i Y →τij X ij i (6) β ↓ ↓α ij ij Y ϕ→ij X ij ij δ ↓ ↓γ ij ij Y →ϕi X i where ϕ : Y → X are monomial morphisms, Y = ϕ−1[X ] and Yˆ = τ−1[X ], ij ij ij ij ij i ij ij h ψ = δ β , c =ψ b , π = γ α , ε = α a and d = π a such that ij ij ij ij ij ij i ij ij ij ij i h i i (7) d−1(ϕ(K)) = ∪ a−1[∪ τ (ψ−1(K ))] = ∪ ε−1(ϕ (δ−1(K))) h i i j ij ij i i,j ij ij ij Proof. We construct commutative diagrams Y → X ij i ↓ ↓ Y → X ij ij ↓ ↓ Y → X i satisfyingtheconclusions ofthetheorem byinductiononi,usingTheorem3.1. Inparticu- lar, there exist nowheredenseclosed analytic subsets G of X such that π−1(π (G )) = G i i i i i i for all i and X \G → X is an open embedding and there exist compact neighborhoods i i K in Y and L of f in X such that X = ∅, and for all i, i i i Xi i t (8) ∪ τ (ψ−1(K ))\G = π−1(ϕ (K ))\G j ij ij i i i i i i and (9) ϕ (K )∩π (G ∩L ) = ϕ (K ). i i i i i i+1 i+1 We then have (by the definition of an ´etoile) that there exists X → X ∈ h such that we h have a commutative diagram (6) such that a (X ) ⊂ L for all i. For all i, (8) implies i h i (10) a−1[∪ τ (ψ−1(K ))]∩(X \a−1(G )) = d−1(ϕ (K ))∩(X \a−1(G )). i j ij ij i f i i h i i f i i Now, (9) implies π−1 (ϕ (K ))∩G ∩π−1(π (L ))= π−1 (ϕ (K )) i−1 i−1 i−1 i−1 i−1 i−1 i−1 i−1 i i 8 for all i, and so since a (X ) ⊂ L , i−1 h i−1 d−1(ϕ (K ))∩a−1 (G ) = d−1(ϕ (K )) h i−1 i−1 i−1 i−1 h i i for all i. Thus (11) d−1(ϕ(K))∩a−1(G )∩···∩a−1 (G ) = d−1(ϕ (K )). h 0 0 i−1 i−1 h i i Since G = ∅, and we certainly have t ∪ a−1[∪ τ (ψ−1(K ))] ⊂ d−1(ϕ(K)), i i j ij ij i h (7) follows from induction on i, using (10) and (11) and since ∪t [a−1(G )∩···∩a−1 (G )]∩(X \a−1(G )) = X . i=0 0 0 i−1 i−1 f i i f (cid:3) From the discussion after Theorem 3.1 and Theorem 3.2, we obtain the following state- ment. Suppose ϕ : Y → X is a morphism of reduced real analytic spaces such that X is smooth. Let Y˜ → X˜ be a complexification of ϕ : Y → X such that X˜ is smooth and Y˜ is reduced. Suppose that K˜ is a compact neighborhood in Y˜ which is invariant under the auto conjugation of Y˜. Let K be the real part of K˜ which is a compact neighborhood in Y. Suppose that h ∈ E . Then there exists d˜ : X˜ → X˜ ∈ h, morphisms of reduced X h h complex analytic spaces ϕ˜ : Y˜ → X˜ for 0 ≤ i ≤ t with compact neighborhoods K˜ in Y˜ i i i i suchthatϕ˜ (K˜ ) =ϕ˜(K˜),dimY˜ <dimY˜ foralliandY˜ =∅. Thereexistcommutative 0 0 i+1 i t diagrams for 0 ≤ i≤ t Y˜ˆ σ→˜ij X˜ ij h ˜b ↓ ↓a˜ ij i Y˜ →τ˜ij X˜ ij i (12) β ↓ ↓α ij ij Y˜ ϕ→˜ij X˜ ij ij δ˜ ↓ ↓γ˜ ij ij Y˜ →ϕ˜i X˜ i where ϕ : Y˜ → X˜ are monomial morphisms, Y˜ = ϕ˜−1[X ] and Y˜ˆ = τ˜−1[X˜ ]. Let ij ij ij ij ij i ij ij h ψ˜ = δ˜ β˜ , c˜ = ψ˜ ˜b , π˜ = γ˜ α˜ and d˜ =π˜ a˜ such that ij ij ij ij ij ij i ij ij h i i (13) d˜−1(ϕ˜(K˜)) = ∪ ε˜−1(ϕ˜ (δ˜−1(K˜))) =∪ a˜−1[∪ τ˜ (ψ˜−1(K˜ ))] h i,j ij ij ij i i j ij ij i Further, there are compatible auto conjugations of these analytic spaces and morphisms such that the real parts are d : X → X, morphisms of reduced real analytic spaces h h ϕ : Y → X for 0 ≤ i ≤ t with compact neighborhoods K in Y with K = K˜ ∩ Y i i i i i i i such that ϕ (K ) = ϕ(K), dimY < dimY for all i and Y = ∅. We may assume that 0 0 i+1 i t X 6= ∅ if and only if h is a real point of X˜ . Suppose that X 6= ∅. Then there exist h X˜h h h 9 commutative diagrams for 0 ≤i ≤ t Yˆ σ→ij X ij h b ↓ ↓ a ij i Y →τij X ij i (14) β ↓ ↓ a ij ij Y ϕ→ij X ij ij δ ↓ ↓ γ ij ij Y →ϕi X i where ϕ : Y → X are monomial morphisms, Y = ϕ−1[X ] and Yˆ = τ−1[X ], ij ij ij ij ij i ij ij h ψ = δ β , c = ψ b , π = γ α and d =π a such that ij ij ij ij ij ij i ij ij h i i (15) d−1(ϕ(K)) = ∪ ε−1(ϕ (δ−1(K))) =∪ a−1[∪ τ (ψ−1(K ))] h i,j ij ij ij i i j ij ij i Theorem 3.3. Suppose that X and Y are real analytic spaces such that X is smooth and ϕ :Y → X is a proper real analytic map. Let p ∈ X. Then there exists a finite number of real analytic maps π :V → X such that: α α 1) Each V is smooth and each π is a composition of local blowups of nonsingular α α sub varieties, 2) There exist compact neighborhoods N in V for all α such that ∪ π (N ) is a α α α α α compact neighborhood of p in X, 3) For all α, π−1(ϕ(Y)) is a semi analytic subset of V . α α Proof. Let ϕ˜ : Y˜ → X˜ be a complexication of ϕ : Y → X so that X˜ is smooth and Y˜ is reduced. Let U˜ be a relatively compact open neighborhood of p in X˜ which is invariant under the auto conjugation of X˜ and let L˜ be the closure of U˜ in X˜. Let L =L˜∩X, a compact neighborhood of p in X. Let K′ = ϕ˜−1(L˜). The real part of K′ is K = ϕ−1(L) which is compact since ϕis proper. LetN bea compact neighborhoodof K in Y˜ which contains K and is preserved by the auto conjugation of Y˜. Let K˜ = K′∩N. The set K˜ is a compact neighborhood in Y˜ which is preserved by the auto conjugation of Y˜ such that the real part of K˜ is K. Let U be the real part of U˜ which is an open neighborhood of p in X with closure L in X. Let V = ϕ−1(U), whose closure is K = ϕ−1(L). We have that (16) ϕ(V) = ϕ(K)∩U. For each h ∈ E such that X 6= ∅, we have associated complex analytic morphisms X˜ h d˜ : X˜ → X˜ with real part d : X → X, and associated diagrams (12) with real part h h h h (14). For all i,j, we have that d−1(ϕ(K)) = d−1(ϕ(K))∩[∪ ε−1(ϕ (Y ))] h h i,j ij ij ij by (15). Thus d−1(ϕ(V)) = d−1(ϕ(K)∩U) =d−1(ϕ(K))∩d−1(U) h h h h (17) = d−h1(ϕ(K))∩[∪i,jε−ij1(ϕij(Yij))]∩d−h1U) = d−1(U)∩[∪ ε−1(ϕ (Y ))]. h i,j ij ij ij We now establish that d−1(ϕ(V)) is a semi analytic subset of X . For all i,j, ϕ (Y ) is h h ij ij semi analytic in X since ϕ is a monomial morphism(by the TarskiSeidenbergtheorem, ij ij c.f. Theorem1.5 [8]. Thus d−1(ϕ(V)) is semianalytic in X by (17). h h 10