On the Nash modification of a germ of complex analytic singularity
aa r X i v : . [ m a t h . AG ] A ug On the Nash Modification of a Germ of ComplexAnalytic Singularity
Arturo Giles Flores ∗ September 24, 2018
Abstract
For a germ ( X, ⊂ ( C n ,
0) of reduced, equidimensional complex analyticsingularity its Nash modification can be constructed as an analytic subvariety Z ⊂ C n × G ( k, n ). We give a characterization of the subvarieties of C n × G ( k, n )that are the Nash modification of its image under the projection to C n . This resultgeneralizes the characterization of conormal varieties as Legendrian subvarieties of C n × ˇ P n − with its canonical contact structure. As a by-product we define the d -conormal space of ( X,
0) for any d ∈ { k, . . . , n − } which is a generalization ofboth the Nash modification and the conormal variety of ( X, For a germ of analytic singularity ( X, ⊂ ( C n ,
0) the set of limits of tangentspaces plays a big role in the study of equisingularity. If ( X,
0) is a reducedand irreducible germ of analytic singularity of pure dimension d , this set isobtained as the preimage ν − (0) of the Nash modification ν : N X → X . Itis then a subvariety of the Grasmannian G ( d, n ) of d -planes of C n and sohas the structure of a projective algebraic variety.When X is a hypersurface the Grassmannian G ( d, n ) is a projectivespace ˇ P n − and the set ν − (0) can be described via projective duality bya finite family of subcones of the tangent cone C X, , which includes all ofthe irreducible components, known as the aur´eole of the singularity. [LT88,Thm 2.1.1 & Coro 2.1.3]The generalization of this result to germs of arbitrary codimension needsto replace the Nash modification N X by the conormal space C ( X ). Recallthat the conormal space of X in C n is an analytic space C ( X ) ⊂ X × ˇ P n − ,together with a proper analytic map κ : C ( X ) → X , where the fiber overa smooth point x ∈ X is the set of tangent hyperplanes to X at x , that ∗ Research partially supported by CONACYT (Mexico) grant 221635
1s the hyperplanes H ∈ ˇ P n − containing the direction of the tangent space T x X . We are then able to once again describe the set of limits of tangenthyperplanes via the aur´eole and projective duality. See proposition [Tei82,pg. 378-381]What is so useful about this change from tangent spaces to tangenthyperplanes is that eventhough the space C ( X ) depends on the embeddingthere is a “numerical” characterization ( in terms of the dimension of a fiber)of the Whitney conditions via the normal/conormal diagram [Tei82, Chapter5, Thm 1.2] and in theory it is possible to recover the fiber of the Nash modi-fication (which does not depend on the embedding) from the conormal fiber.The idea is that every limit of tangent hyperplanes H ∈ κ − (0) containsa limit of tangent spaces T ∈ ν − (0), and so to each such T there corre-sponds, via projective duality, a linear subspace ˇ P n − d − ⊂ κ − (0) ⊂ ˇ P n − .This means we have to look for linear subspaces of the right dimension con-tained in the conormal fiber and take their projective duals.The problem is that not every T obtained this way is a limit of tangentspaces, and it is a simple dimensionality question. Take for instance a germof surface ( S, ⊂ ( C ,
0) with an exceptional tangent. According to whatwe just said each limit of tangent planes T corresponds to a ˇ P ⊂ κ − (0) ⊂ ˇ P . But the existence of the exceptional tangent tells us that the projectivedual of this point of P is contained in κ − (0). Its projective dual is a ˇ P ,and so inside it we have a G (2 ,
3) (dimension 2) of possible limits of tangentspaces. But they can’t all be limits of tangent spaces because we know thatthe dimesion of ν − (0) is at most 1!!!!!!! And even in a simple case like thiswe do not know how to distinguish the ones that are limits of tangent spacesfrom the ones that are not. More generally we do not know the size of thecontribution of an exceptional cone to the Nash fiber.
One of the key results that made working with the conormal easier thanwith the Nash modification is that conormal varieties can be character-ized as Legendrian subvarieties of projectivized cotangent spaces with theircanonical contact structure. In this spirit we try to characterize analyticsubvarieties Z of C n × G ( d, n ) such that:1. Z has dimension d .2. Its image (by the projection) X in C n has dimension d .3. Z is the Nash modification of X In order to do this we define an analytic k -plane distribution on C n × G ( d, n ) locally defined by a system of analytic forms and look at the corre-2ponding integral subvarieties. Even though we want to find subvarieties Z of dimension d , there are subvarieties of dimension greater than d that arecompatible with the distribution in the sense that for every smooth point p ∈ Z we have that the tangent space T p Z is contained in the corresponding k -plane H p determined by the distribution.However, if X ⊂ C n is of dimension k ≤ d then we can define an analyticsubvariety of C n × G ( d, n ), that generalizes both the Nash modification N X and the conormal space C ( X ) via the limits of tangent d -planes. Zak workswith this kind of spaces in his book [Zak93] but only in the case of projectivevarieties and calls them higher order Gauss maps. k -plane distribution on C n × G ( d, n ) Let us first recall that one of the ways of defining analytic charts for theGrassmannian G ( d, n ) is to view its points as graphs of linear maps definedon a fixed d -dimensional subspace of C n and taking values in another fixed( n − d )-subspace of C n , where these two fixed subspaces are transversal.This is done as follows.Fix a point W ∈ G ( d, n ) and a n − d linear subspace W ⊂ C n such that C n = W ⊕ W For every linear map L ∈ Hom C ( W , W ) we have that its graph in W × W = C n is a linear subspace W of dimension d , that is, a point in G ( d, n ).Moreover, we have that W ∈ G ( d, n ) is the graph of one such linear map L if and only if W is transversal to W .Consider the open subset of the Grasmannian G d ( n, W ) := { W ∈ G ( d, n ) | W ⋔ W } and note that it contains W . If we denote by π j the linear projection from C n to W j then we have a bijectionΦ W ,W : G d ( n, W ) −→ Hom C ( W , W ) W L := π ◦ ( π | W ) − : W → W Indeed, for every W ∈ G d ( n, W ) the restriction map π | W : W → W isa linear isomorphism and the L thus defined has W as its graph. The col-lection of the charts Φ W ,W , when ( W , W ) runs over the set of all directsum decompositions of C n , with W of dimension d, is an analytic atlasfor G ( d, n ). Note that to cover G ( d, n ) it is enough to consider the charts3orresponding to all the coordinate d − planes with their corresponding com-plementary coordinate ( n − d ) − planes. (See [PT08])To better understand the construction of the k -plane distribution on C n × G ( d, n ) let us first recall the canonical contact structure on the pro-jectivized cotangent bundle P T ∗ C n = C n × ˇ P n − with coordinate system( x , . . . , x n ) , [ a : · · · : a n ]. If we look at the chart ϕ : U → C n − where a = 0 ( x , . . . , x n ) , [ a : · · · : a n ] (cid:18) x , . . . , x n , a a , . . . , a n a (cid:19) then the hyperplane of the tangent space T ~x, [ a ] P T ∗ C n chosen by this distri-bution is given by the kernel of the 1-form dx + a a dx + · · · + a n a dx n ( ∗ )But if we identify the tangent space T ~x, [ a ] P T ∗ C n with the product of tangentspaces T ~x C n × T [ a ] ˇ P n − then the kernel H ~x, [ a ] of ( ∗ ) is identified with e H × T [ a ] ˇ P n − where e H ⊂ C n is the hyperplane determined by the point [ a ] ∈ ˇ P n − . Definition 2.1.
On the n + d ( n − d ) dimensional analytic manifold C n × G ( d, n ) we define a d + d ( n − d ) -plane distribution as follows. Let ( ~z, W ) be a point C n × G ( d, n ) and identify its tangent space with the product oftangent spaces T ~z C n × T W G ( d, n ) = C n × T W G ( d, n ) . Then the plane givenby the distribution at this point is: H ( ~z, W ) := W × T W G ( d, n ) Proposition 2.2.
The distribution H is locally defined by the kernel of asystem of analytic 1-forms of C n × G ( d, n ) .Proof. Recall that it is enough to consider charts of the form C n × G d ( n, W )where W is a coordinate (n-d)-plane of C n , and W the corresponding“complementary” coordinate d − plane . To simplify notation and withoutloss of generality we will assume W = h ~e , . . . , ~e d i and W = h ~e d +1 , . . . , ~e n i .Now from the Grassmannian chartΦ W ,W : G d ( n, W ) −→ Hom C ( W , W ) W L := π ◦ ( π | W ) − : W → W and after identifying each linear map L ∈ Hom C ( W , W ) with the corres-ponding ( n − d ) × d matrix with respect to the basis previously establishedwe obtain the chart of C n × G ( d, n ) given by:Ψ W ,W : C n × G d ( n, W ) −→ C n × C d ( n − d ) ( z , . . . , z n ) , W ( z , . . . , z n , a ij ) , i = 1 , . . . , n − d ; j = 1 , . . . , d W = h ~e + L ( ~e ) , . . . , ~e d + L ( ~e d ) i is the graph of the correspondinglinear map L = Φ W ,W ( W ) ∈ Hom C ( W , W ).In this chart we can define the following system of analytic 1-forms: dz d +1 dz d +2 ... dz n = a · · · a d a · · · a d ... ... ... a ( n − d )1 · · · a ( n − d ) d dz dz ... dz d whose kernel at a point ( z , . . . , z n ) , W ∈ C n × G ( d, n ) is H ( ~z, W ) = W × T W G ( d, n ) ⊂ C n × T W G ( d, n ) = T z,W ( C n × G ( d, n )) Once we defined the k − plane distribution the next step is to characterize,or find the corresponding integral subvarieties. Definition 3.1.
The analytic subvariety Z ⊂ C n × G ( d, n ) is an integralsubvariety of ( C n × G ( d, n ) , H ) if for every smooth point ( ~z, W ) ∈ Z we havethat T ~z,W Z ⊂ H ( ~z, W ) . The definition of the distribution puts a restriction on both the dimensionof the integral subvariety Z and the dimension of its projection on C n . Proposition 3.2.
Let π : C n × G ( d, n ) → C n be the projection onto C n .If Z ⊂ C n × G ( d, n ) is an integral subvariety of ( C n × G ( d, n ) , H ) then t := dim π ( Z ) ≤ d and dim Z ≤ t + ( d − t )( n − d ) .Proof. Just by looking at the definition of integral subvariety we have that T p,W Z ⊂ H ( p, W ) and this implies that dim Z ≤ d + d ( n − d ). Since π is a proper map π ( Z ) ⊂ C n is an analytic subvariety, and the restriction π : Z → π ( Z ) is generically submersive. Then, for any (sufficiently general)point ( p, W ) ∈ Z with smooth image p ∈ π ( Z ) we have that T p π ( Z ) ⊂ D p π ( H ( p, W )) = W therefore t := dim π ( Z ) ≤ d .In order to bound the dimension of Z we are going to calculate a boundfor the dimension of the fiber π − ( p ) for a generic point p ∈ π ( Z ). For asufficiently general smooth point p ∈ π ( Z ) we have that π − ( p ) ⊂ { p } × { W ∈ G ( d, n ) | W ⊃ T p π ( Z ) } π ( Z ) is of dimension t then by choosing any (linear) direct sum decom-position of C n = E n − t L T p π ( Z ) we get a 1 to 1 correspondence betweenthe set { W ∈ G ( d, n ) | W ⊃ T p π ( Z ) } and the set of d − t linear subspacesof E n − t , i.e. a Grassmanian G ( d − t, n − t ) of dimension ( d − t )( n − d ).Therefore dim Z ≤ t + ( d − t )( n − d ).In the proof of this result we have seen that the fiber over a non-singularpoint p ∈ π ( Z ) is contained in the set of tangent d − planes to π ( Z ) at p , that is d − dimensional linear subspaces W of C n such that W ⊃ T p π ( Z ).This means, we are looking at a natural generalization of both the Nash mo-dification and the conormal space of a germ of singularity ( X, ⊂ ( C n , d in { dim X, . . . , n − } . Zak considers these spaces in [Zak93] in the case ofprojective varieties and subvarieties of complex tori. C d ( X ) inside C n × G ( d, n ) Definition 4.1.
Let ( X, ⊂ ( C n , be a germ of analytic,reduced andirreducible analytic singularity of dimension k . For any d ∈ { k, k +1 , . . . , n − } define the d − conormal of X by C d ( X ) := { ( z, W ) ∈ X × G ( d, n ) | T z X ⊂ W } where X denotes the smooth part of X , G ( d, n ) is the Grassmann varietyof d − dimensional linear subspaces of C n and the bar denotes closure in X × G ( d, n ) . We will denote by ν d : C d ( X ) → X the restriction of theprojection to the first coordinate. Note that for d = k we have that C k ( X ) is the Nash modification of X and for d = n − X . Lemma 4.2.
In the setting of definition 4.1 we have that C d ( X ) is ananalytic space of dimension k + ( d − k )( n − d ) and ν d : C d ( X ) → X is aproper map. Moreover it is an integral subvariety of ( C n × G ( d, n ) , H ) .Proof. That C d ( X ) is analytic follows from the fact that X is analytic andthe incidence condition T z X ⊂ W defining the fiber over a smooth pointis algebraic. Moreover the map ν d is proper because G ( d, n ) is compact.Regarding its dimension, it is the same calculation we did in proposition3.2. That is, for any smooth point z ∈ X we have that ν − d ( z ) = { z } × { W ∈ G ( d, n ) | W ⊃ T z X } and the set in the second factor is a Grassmannian G ( d − k, n − k ). Thisimplies that for a smooth germ ( C k , ⊂ ( C n ,
0) we have that C d ( C k ) isisomorphic to C k × G ( d − k, n − k ) and so if z is a smooth point of X then6ny point ( z, W ) ∈ ν − d ( z ) is smooth in C d ( X ).Finally, recall that by definition, for any point ( z, W ) ∈ C n × G ( d, n ) wehave H ( z, W ) = W × T W G ( d, n )Now, since the map ν d is just the restriction of the projection onto thefirst factor, then the tangent map D ( z,W ) ν d is also a projection and for anytangent vector ( ~u, ~v ) ∈ T ( z,W ) C k ( X ) ⊂ T z C n × T W G ( k, n ) we have that D ( z,W ) ν k ( ~u, ~v ) = ~u ∈ T z X ⊂ W that is ( ~u, ~v ) ∈ H ( z, W ) and so C d ( X ) is an integral subvariety of ( C n × G ( d, n ) , H ). Theorem 4.3.
Let Z ⊂ C n × G ( d, n ) be a reduced, analytic and irreduciblesubvariety and X = π ( Z ) where π : C n × G ( d, n ) → C n denotes the projectionto C n . If the dimension of X is equal to t , then the following statements areequivalent:i) Z is the d-conormal space of X ⊂ C n .ii) Z is an integral subvariety of ( C n × G ( d, n ) , H ) of dimension t + ( d − t )( n − d ) Proof. i ) ⇒ ii ) was proved in lemma 4.2.First note that since X is of dimension t and Z is of dimension t +( d − t )( n − d )then the generic fiber of π : Z → X is of dimension ( d − t )( n − d ). Now, let z be a smooth point of X , then for any sufficiently general smooth point ofits fiber ( z, W ) ∈ Z we have that D ( z,W ) π ( T ( z,W ) Z ) = T z X Since Z is an integral subvariety we have that T ( z,W ) Z ⊂ W × T W G ( d, n )and so T z X ⊂ W . This implies that the ( d − t )( n − d ) dimensional fiber π − ( Z ) is contained in the ( d − t )( n − d ) dimensional variety { z } × { W ∈ G ( d, n ) | T z X ⊂ W } , and so they must be equal. But this is precisely thedefinition of the d − conormal variety C d ( X ).Note that when d = n − t +( d − t )( n − d ) = n − C d ( X ) ⊂ C n × ˇ P n − is the usual conormal space of X . Moreover, this theorem recovers thecharacterization of conormal varieties as legendrian subvarieties of C n × ˇ P n − with its canonical contact structure. (See [Pha79, Section 10.1, pg 91-92]) Corollary 4.4.
Let Z be an integral subvariety of ( C n × G ( d, n ) , H ) of di-mension d . Then Z is the Nash modifcation of its image in C n if and only iffor every smooth point ( z, W ) ∈ Z the tangent space T ( z,W ) Z is transverseto the subspace T W G ( d, n ) of T ( z,W ) ( C n × G ( d, n )) . roof. ⇒ ] Note that for any point ( z, W ) in C n × G ( d, n ) the kernel ofthe differential Dπ : T z C n × T W G ( d, n ) → T z C n is T W G ( d, n ). On theother hand, the Nash modification ν : N X → X is an isomorphism overthe smooth part of X so for any smooth point z ∈ X we have that thedifferential D ( z ,T z X ) ν : T ( z ,T z X ) N X → T z X is an isomorphism. Since the map ν can be realized as the restriction to N X of the projection π : C n × G ( d, n ) → C n this implies that T ( z ,T z X ) N X is transverse to T W G ( d, n ). ⇐ ] We know that the projection π : Z → X is generically a submersionwith the kernel of the differential D ( z,W ) π : T ( z,W ) Z → T z X being equal tothe intersection of T ( z,W ) Z and T W G ( d, n ) , but the transversality conditionmeans that this this intersection is of dimension zero which implies that T z X and therefore X is of dimension d . By theorem 4.3 this is equivalentto Z being the Nash modification of X . Example 4.5.
For a germ of surface ( S, ⊂ ( C ,
0) we have the followingspaces: Nash modification ν : N S → S dimension 23 − conormal ν : C ( S ) → S dimension 4Conormal κ : C ( S ) → S dimension 4Since N S ⊂ S × G (2 ,
5) and C ( S ) ⊂ S × G (3 ,
5) it would be interesting to tryto use that these two Grassmannians are isomorphic to define a morphism N S → C ( S ) making the following diagram commute: N S / / ν ❇❇❇❇❇❇❇❇❇ C ( S ) ν | | ②②②②②②②② S This could be a first step to work out a way from the conormal fiber κ − (0)to the Nash fiber ν − (0).As a first application of how this d-conormal spaces can be used, wewill characterize Whitney conditions in the Nash modification of X in ananalogous way to the characterization in the conormal space C ( X ) given in[LT88, Proposition 1.3.8].Consider a germ of analytic, reduced and irreducible singularity ( X, ⊂ ( C n ,
0) of dimension d such that its singular locus ( Y,
0) is smooth of dimen-sion t . We will fix a coordinate system ( y . . . , y n , z t +1 , . . . , z n ) in C n andwe can assume that Y is equal to C t × { } .8ote that the d-conormal of C d ( Y ) ⊂ C n × G ( d, n ) of Y in C n is equalto Y × { W ∈ G ( d, n ) | W ⊃ Y } and so it is enough to consider the charts C n × G d ( n, W ) of C n × G ( d, n ) where W is a coordinate n − d linear sub-space such that W ∩ Y = { } .Moreover, after identifying G d ( n, W ) with Hom C ( W , W ), we can take W = C · (cid:10) e , . . . , e t , e i t +1 , . . . , e i d (cid:11) and in this chart the W ’s that contain Y correspond to linear morphisms L : W → W such that Y ⊂ Ker( L ).We will use the fact that in complex analytic geometry Whitney’s con-dition b) is equivalent ([Tei82, Chap. 5]) to condition w) which we nowrecall. The couple ( X , Y ) satisfies condition w) at the origin if there existsan open neighborhood of the origin U ⊂ X and a real positive constant C such that for every y ∈ U ∩ Y and x ∈ U ∩ X we have that δ ( T y Y, T x X ) ≤ Cd ( x, Y )where d ( x, Y ) is the euclidean distance in C n , δ is defined for linear subspaces A, B ⊂ C n by: δ ( A, B ) := sup ~u ∈ B ⊥ \{ } ,~v ∈ A \{ } |h ~u, ~v i||| ~u || || ~v || and h ~u, ~v i denotes the usual hermitian product in C n . Proposition 4.6.
Let I denote the ideal of O N X that defines the intersec-tion C d ( Y ) ∩ N X and J the ideal defining ν − ( Y ) .1. The couple ( X \ Y, Y ) satisfies Whitney’s condition a ) at the origin ifand only if at every point (0 , T ) ∈ ν − (0) we have that √I = √ J in O N X, (0 ,T ) .2. The couple ( X \ Y, Y ) satisfies condition w ) at the origin if and onlyif at every point (0 , T ) ∈ ν − (0) the ideals I and J have the sameintegral closure in O N X, (0 ,T ) .Proof. Note that we always have the inclusion C d ( Y ) ∩ N X ⊂ ν − ( Y ), orequivalently I ⊃ J .For 1), recall that Whitney’s condition a) demands that every limit of tan-gent spaces T to X at 0 contains the tangent space to Y at 0, which we canidentify with Y since it is linear. This is exactly what the set-theoreticalequality C d ( Y ) ∩ ν − (0) = ν − (0) means which is equivalent to √I = √ J in O N X, (0 ,T ) for every point (0 , T ) ∈ ν − (0).2) ⇐ ]Now suppose that at every point (0 , T ) ∈ ν − (0) the ideals I and J areequal in O N X, (0 ,T ) , in particular they have the same radical, and so by 1)9e have that Y ⊂ T and by the discussion prior to the proposition we cansee it in a chart of C n × G ( d, n ) of the form C n × Hom C ( W , W ), where W is an n − d linear coordinate subspace transversal to Y and the d linearsubspace W can be taken of the form C · (cid:10) e , . . . , e t , e i t +1 , . . . , e i d (cid:11) .In this chart we have a coordinate system( y , . . . , y t , z t +1 , . . . , z n , a ij ) i = 1 , . . . , n − d, j = 1 , . . . , d where J = h z t +1 , . . . , z n i O N X and since W ∈ G ( d, n ) contains Y if and onlyif Y is in the kernel of the corresponding linear map L W ∈ Hom C ( W , W ),that is L W ( e i ) = ~ i = 1 , . . . , t we have that I = h z t +1 , . . . , z n , a ij ; i = 1 , . . . , n − d ; j = 1 , . . . t i O N X J = h z t +1 , . . . , z n i The equality of integral closures I = J implies that the coordinate func-tions a ij ∈ J O N X, (0 ,T ) and by [LJT08, Thm 2.1] this is equivalent to the existence of an open set V ′ ⊂ N X and a real positive constant C V ′ such that (0 , T ) ∈ V ′ and forevery ( p, W ) ∈ V ′ we have that | a ij | ≤ C V ′ sup {| z t +1 | , . . . , | z n |} ≃ C V ′ d ( p, Y )Doing this for every point (0 , T ) ∈ ν − (0) we obtain an open cover of thefiber and since it is compact we can obtain a finite subcover ν − (0) ⊂ ( V , C ) ∪ · · · ∪ ( V r , C r )Note that U := ν ( V ∪ V ∪ · · · ∪ V r ) is an open neighborhood of the origin in X , and define C := max { C , . . . , C r } . Now for any smooth point p ∈ U ∩ X we have that the point ( p, T p X ) | a ij | ≤ C j sup {| z t +1 | , . . . , | z n |} ≤ C sup {| z t +1 | , . . . , | z n |} ≃ Cd ( p, Y )Now to finish the proof we will show that δ ( T y Y, T p X ) ≤ (cid:16) Ct √ n − d ) (cid:17) d ( p, Y )Using the local coordinates of the chosen chart it is enough to prove thatfor any point ( x, W ) in this chart we have that δ ( Y, W ) ≤ t √ n − d sup {| a ij | , i = 1 , . . . , n, j = 1 , . . . , t }
10y definition we have δ ( Y, W ) := sup ~u ∈ W ⊥ \{ } ,~v ∈ Y \{ } |h ~u, ~v i||| ~u || || ~v || Now Y = C · h ˆ e , . . . , ˆ e t i and W = C · h (ˆ e , a i ) , . . . , (ˆ e d , a id ) i and using theHermitian product we get the following relations for ~u ∈ W ⊥ :0 = h (ˆ e , a i ) , ~u i = u + a u d +1 + a u d +2 + · · · + a ( n − d )1 u n h (ˆ e , a i ) , ~u i = u + a u d +1 + a u d +2 + · · · + a ( n − d )2 u n ...0 = h (ˆ e d , a id ) , ~u i = u d + a d u d +1 + a d u d +2 + · · · + a ( n − d ) d u n And so we have: |h ~u, ~v i||| ~u || || ~v || = (cid:12)(cid:12)(cid:10) ~u, P ti =1 λ i ˆ e i (cid:11)(cid:12)(cid:12) || ~u || || P ti =1 λ i ˆ e i || = (cid:12)(cid:12)P ti =1 λ i u i (cid:12)(cid:12) || ~u || || P ti =1 λ i ˆ e i ||≤ P ti =1 (cid:12)(cid:12) λ i u i (cid:12)(cid:12) || ~u || || P ti =1 λ i ˆ e i || ≤ (cid:12)(cid:12) λ u (cid:12)(cid:12) || ~u || || λ ˆ e || + · · · + (cid:12)(cid:12) λ t u t (cid:12)(cid:12) || ~u || || λ t ˆ e t || = t X i =1 u i || ~u || = | P n − dj =1 a j u d + j ||| ~u || + · · · + | P n − dj =1 a jt u d + j ||| ~u ||≤ || (0 , a , . . . , a ( n − d )1 || + · · · + || (0 , a t , . . . , a ( n − d ) t ||≤ √ n − d sup {| a | , . . . , | a ( n − d )1 |} + · · · + √ n − d sup {| a t | , . . . , | a ( n − d ) t |}≤ t √ n − d sup {| a ij | , i = 1 , . . . , n − d ; j = 1 , . . . , t } ⇒ ]By hypothesis the couple ( X \ Y, Y ) satisfies condition w) at the origin,and since in complex analytic geometry this condition is equivalent to Whit-ney conditions, then for every point (0 , T ) ∈ ν − (0) we have that Y ⊂ T and so we can restrict ourselves to look at the charts we have been workingon. Without loss of generality we will look at the chart C n × Hom C ( W , W )with coordinate system( y , . . . , y t , z t +1 , . . . , z n , a ij ) ; i = 1 . . . , n − d, j = 1 . . . , d where W = C · h e , . . . , e d i and W = C · h e d +1 , . . . , e n i . In this coordinatesystem we have the ideals J = h z t +1 , . . . , z n i O N X I = h z t +1 , . . . , z n , a ij ; i = 1 , . . . , n − d ; j = 1 , . . . t i O N X I = J in O N X, (0 ,T ) for every point (0 , T ) ∈ ν − (0).Again by hypothesis we have an open neighborhood of the origin U ⊂ X and a real positive constant C such that for every smooth point p ∈ U ∩ X C sup {| z t +1 | , . . . , | z n |} ≥ δ ( Y, T p X ) := sup ~u ∈ ( T p X ) ⊥ \{ } ,~v ∈ Y \{ } |h ~u, ~v i||| ~u || || ~v || Note that for any W ∈ Hom C ( W , W ) with coordinates ( b ij ) in this chart,using the relations previously obtained, we have that ~u ∈ W ⊥ if and only ifit is of the form: u u ... u d u d +1 ... · u n = λ − b − b ... − b d + λ − b − b ... − b d + · · · + λ n − d − b ( n − d )1 − b ( n − d )2 ... − b ( n − d ) d with λ i ∈ C .Fix a point (0 , T ) in the Nash fiber and consider an open neighbourhood V := { ( a ij ) ∈ C d ( n − d ) | | a ij | < M } where M is a sufficiently big real positiveconstant. Now for any point ( p, W ) ∈ U × V we have C sup {| z t +1 | , . . . , | z n |} ≥ δ ( Y, W ) := sup ~u ∈ W ⊥ \{ } ,~v ∈ Y \{ } |h ~u, ~v i||| ~u || || ~v || in particular, by setting ~v = ˆ e j and ~u = ( − b k , . . . , − b kd , , . . . , , , , . . . , j ∈ { , . . . , t } and k ∈ { , . . . , n − d } we get the inequality C sup {| z t +1 | , . . . , | z n |} ≥ |h ~u, ˆ e j i||| ~u || || ˆ e j || = | b kj ||| ~u || > | b kj | M ′ the last inequality coming from the fact that the b ij ’s are bounded since W is in V . This implies that for every j ∈ { , . . . , t } and i ∈ { , . . . , n − d } wehave that a ij ∈ J which finishes the proof.As a final comment we would like to point out that the classic construc-tion of the local polar ( P k ( X ) ,
0) varieties using the Nash modification, orthe conormal space ([Tei82, Chap. 4, Coro 1.3.2 & Prop 4.1.1]) carries overpractically word for word to the d-conormal.Recall that for a germ of reduced and equidimensional complex analyticsingularity ( X, ⊂ ( C n ,
0) of dimension d and a sufficiently general linear12pace D of dimension n − d + k − k ∈ { , . . . , d − } ) the polar variety P k ( X ; D ) ⊂ X is the closure in X of the critical locus of the linear projectionwith kernel D Π D : X → C d − k +1 It is a reduced analytic variety of dimension d − k , with the property thatthe multiplicity of ( P k ( X ; D ) ,
0) is an analytic invariant of the germ ( X, ℓ ∈ { d, . . . , n − } and k ∈ { , . . . , d − } take the Schubertvariety c k ( D ) := { W ∈ G ( ℓ, n ) | dim W ∩ D ≥ k + ℓ − d } and consider the diagram C ℓ ( X ) ⊂ X × G ( ℓ, n ) γ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ ν ℓ w w ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ X G ( ℓ, n )then1. P k ( X ; D ) = ν ℓ (cid:0) γ − ( c k ( D )) (cid:1)
2. The equalitydim (cid:0) ν − ℓ (0) ∩ γ − ( c k ( D )) (cid:1) = dim ν − ℓ (0) − ( ℓ − d )( n − ℓ ) − k is true if the intersection is not empty.where ( ℓ − d )( n − ℓ ) is the dimension of the fiber ν − ℓ ( p ) for any smoothpoint p ∈ X . References [LJT08] M. Lejeune-Jalabert and B. Teissier. Clˆoture int´egrale des id´eauxet ´equisingularit´e.
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