Fibrations of highly singular map germs
Maico F. Ribeiro, Raimundo N. Araújo dos Santos, Mihai Tibăr
aa r X i v : . [ m a t h . AG ] M a y FIBRATIONS OF HIGHLY SINGULAR MAP GERMS
RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂRAbstract.
We define local fibration structures for real map germs with strictly positivedimensional discriminant: a local fibration structure over the complement of the discrim-inant, and a complete local fibration structure which includes the stratified discriminantinto the picture. We provide new classes of map germs endowed with such local fibrationstructures. Introduction
The existence of a local fibration structure defined by a map germ G : ( K m , → ( K p , , m ≥ p > , K = R or C , is fundamental for the study of the topological properties ofthe map. This has been first shown by Milnor [Mi] in case of holomorphic functions f : ( C n , → ( C , , and had tremendous developments over several decades. In [Mi] it is proved that the Milnor fibration is independent of the local data. Moreprecisely, the local fibration existence statement includes the essential requirement thatthe locally trivial C ∞ -fibrations(1) f | : B nε ∩ f − ( D ∗ η ) → D ∗ η are independent, up to isotopy, of the small enough ε ≫ η > , where B nε is the openball at ∈ C n of radius ε , and D ∗ η denotes the pointed disk of radius η at ∈ C .For holomorphic maps, Hamm [Ha] showed that if the map germ G : ( C n , → ( C k , , n > k > , is an isolated complete intersection singularity (abbreviated ICIS in thefollowing), then G defines a local fibration over the complement of its hypersurface dis-criminant:(2) G | : B nε ∩ G − ( B kη \ Disc G ) → B kη \ Disc G In particular this includes the independency of the choices of the small enough neighbour-hoods, similar to Milnor’s case evoked just before.
Mathematics Subject Classification.
Key words and phrases. singularities of real analytic maps, Milnor fibrations, mixed functions.The authors acknowledge the support of USP-COFECUB Uc Ma 163-17. M. Tibar acknowledgesthe support of Labex CEMPI (ANR-11-LABX-0007-01), and R.N. Araújo dos Santos acknowledges thesupport of Fapesp grant 2017/20455-3 and CNPq research-grant 313780/2017-0. See also [Le] for holomorphic function germs on a singular space germ X , a different stream whichwe do not discuss here. The ICIS condition amounts to the condition
Sing G ∩ G − (0) = { } . One of the richest sourcesof information on ICIS is Looijenga’s book [Lo2, Lo3] recently re-edited, and we refer to it for all thestatements in the ICIS context. RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂR
In case of real map germs, as well as beyond the ICIS case in the holomorphic setting,the existence of local fibrations is no more insured. Milnor showed it in [Mi, §11] thatanalytic maps G : ( R m , → ( R p , define local tube fibrations in case they have isolatedsingularity Sing G = { } , but this is the only setting which needs no further conditions. More recently there have been considered classes of singular maps G for which Disc G is the point { } only, thus zero dimensional, and having non-isolated singularities in thefibre over { } , cf [RSV, AT1, Mas, AT2, ACT1, DA, MS, PT1] and many more others,but we do not intend to discuss more about this particular setting here.Beyond the ICIS setting, the positive dimensional discriminant Disc G in local fibrationsoccurred before in some very special cases, e.g. [ACT1, Theorem 1.3], [CGS, Prop. 2.1].We address here the possibility of defining locally trivial fibrations in case of real orcomplex map germs with positive dimensional discriminant Disc G . In this new settingthe following problems have to be taken into account such that the fibration problem canbe well posed, and that the fibration becomes a local invariant:(a) the local fibration must be independent of the small enough neighbourhood data,like in (1) and (2). This does not come automatically for map germs outside theICIS case;(b) the image of the map germ G may not be a neighbourhood of in R p . Moreover, itmay not be independent of the radius ε of the ball B mε ⊂ R m , and thus the imageof G may not be well defined as a set germ in ( R p , , see Definition 2.2;(c) the discriminant of G may not be well defined as a set germ. In case the image G (Sing G ) of the singular locus is a set germ, and when the image Im G is a setgerm too and has a boundary which contains the origin 0, then the discriminant Disc G should contain this boundary.Here we show how all the above issues can be treated in the same time.We consider first the existence of a local fibration outside the discriminant. Since weneed to rely on the image Im G of G as a well-defined set germ at , as well as on thediscriminant as a well-defined germ of the image of the singular locus, we start in §2 thestudy of the delicate question whether the image of an analytic map is well-defined asgerm at the origin in the target . We single out classes of map germs G such that Im G and Disc G are well-defined as germs at and we call them “nice” (Definition 2.2). Thiscategory includes of course the complex ICIS. Within this framework, we then give anappropriate definition of the discriminant Disc G as the locus where the topology of thefibres may change , Definition 2.6.In §3 we prove the existence result, Lemma 3.3, under the most general conditions. Thelocally trivial fibration provided by this lemma in case of positive dimensional Disc G will In [Mi] Milnor also defines a local sphere fibration for holomorphic function germs, and subsequentstudy in the real setting has been concentrated to maps with isolated singularities (whenever such mapsexist, see e.g. [Lo1, AHSS]). We do not address this type of fibration here, we only refer to [ART] forthe newest developments. Recent progress [JT] was made in classifying map germs with respect to this problem.
IBRATIONS OF HIGHLY SINGULAR MAP GERMS 3 be called
Milnor-Hamm fibration in reference to the fibration structure defined in the1970’s by Hamm [Ha] in case of an ICIS.Comparing our local invariant to the existing literature, let us remark that [CGS, Prop.2.1] states a fibration result in case of a positive dimensional discriminant, and anotherdefinition of a fibration in case of a positive dimensional discriminant was given in thevery beginning of [MS, Section 2]. However these differ radically from our approach tolocal fibrations since they do not address the above displayed key issues (a), (b) and (c).We next prove that a weaker Thom condition, the ∂ -Thom regularity , is sufficient forthe existence of the Milnor-Hamm fibration. We find classes of singularities which satisfyour conditions, for instance certain maps of type f ¯ g (see Theorem 4.3, Proposition 4.5).We construct in §5 examples where the map germ G has Milnor-Hamm fibration withoutbeing ∂ -Thom regular, which is another new feature.The second aim of this paper is to introduce the singular Milnor tube fibration (cfDefinition 6.3) which includes the fibres over the discriminant. We give in §6 an existencecondition which is more general than the Thom regularity condition, and we find examplesof singular Milnor tube fibrations either having Thom regularity or not having it. Thelater possibility has been recently discovered and rises interesting questions, see e.g. [PT1,§5.1].The following very simple class shows several aspects of the new fibrations that weintroduce in this paper: Example . Let h : ( C n , → ( C , be a holomorphic function germ. Consider thereal valued function germ H : ( C n , → ( R , given by H := h ¯ h ; this is clearly a nicemap germ (Definition 2.2). One has Sing H = V H and one can show that the map H isThom regular at V H . By Proposition 4.2, H has a Milnor-Hamm fibration over R \ { } with two types of fibres, one empty and one diffeomorphic to B ε ∩ h − ( S η ) , for some < η ≪ ε .If we consider H as a mixed map f : C n → C with f := h ¯ h , then Sing f = C n and Disc f = Im f = R ≥ , thus its Milnor-Hamm fibration over C \ Disc G has empty fibre.Nevertheless, its singular tube fibration (Definition 6.3) exists and has precisely the twotypes of fibres of the map H .As final remark, let us point out that with the same methods based on stratifications,particularly those in Section 6, one can extend the framework to maps G : ( X, → ( R p , where ( X, ⊂ ( R N , is singular analytic. More generally, one can work inside someo-minimal structure. We leave to the reader the technical details.2. Images, discriminants and nice map germs
While working with polynomial or analytic map germs, and with semialgebraic or sub-analytic sets, respectively, the first obstruction to define a fibration is that the image ofa set germ at 0 in the source must be a set germ at 0 in the target, and moreover, that We prefer not to call this a “tube fibration” since a large part from B pδ may be missing. We reserve“singular tube fibration” for our second type of fibration presented in §6. As introduced in the 1990’s, see Definition 4.1 and the discussion before it.
RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂR the discriminant of the map should be well-defined as a set germ. This holds for functiongerms and also for the complex ICIS maps, but it is far from being true in general, neitherin the real setting, nor in the complex one. Such a negative phenomenon can be observedeven for very simple polynomial maps, as follows:
Example . Let G : R → R , G ( x, z ) = ( x, xz ) . For the 2-disks D t := {| x | < t, | z | < t } as basis of open neighbourhoods of 0 for t > , we get that the image A t := G ( D t ) is thefull angle with vertex at 0, having the horizontal axis as bisector, and of slope < t . Sincethe relations defining A t depend of t , it means that the image of G is not well-defined asa germ. A similar behaviour happens over C instead of R .Let A, A ′ ⊂ R p be subsets containing the origin and let ( A, and ( A ′ , denote theirgerms at 0. We recall that one has the equality of set germs ( A,
0) = ( A ′ , if andonly if there exists some open ball B ε ⊂ R p centred at 0 and of radius ε > such that A ∩ B ε = A ′ ∩ B ε . Definition 2.2.
Let G : ( R m , → ( R p , , m ≥ p > , be a continuous map germ. Wesay that the image G ( K ) of a set K ⊂ R m containing is a well-defined set germ at ∈ R p if for any open balls B ε , B ε ′ centred at 0, with ε, ε ′ > , we have the equality ofgerms ( G ( B ε ∩ K ) ,
0) = ( G ( B ε ′ ∩ K ) , .Whenever the images Im G and G (Sing G ) are well-defined as set germs at 0, we saythat G is a nice map germ . Remark . In support of the above definition, let us point out that even if the image Im G of a map G is well-defined as a germ, the restriction of G to some subset mightbe not. This behaviour can be seen in the following example related to the above one,namely let G : C → C , G ( x, y, z ) = ( x, z ) and K := { ( x, y, z ) | z = xy } ⊂ C . Then theimage G ( K ) is not well-defined as a set germ.We have first the following general result: Lemma 2.4.
Let G : ( R m , → ( R p , , m ≥ p > be an analytic map germ. (a) If Sing G ∩ G − (0) ( G − (0) then Im G is well-defined as a set germ. (b) If Sing G ∩ G − (0) = { } ( G − (0) then G is nice.Proof. (a). Let q ∈ G − (0) \ Sing G = ∅ by hypothesis. Then G is a submersion onsome small open neighbourhood N q of q , thus the restriction G | N q is an open map, andtherefore Im G contains some open neighbourhood of the origin of the target. This showsthe equality of set germs (Im G,
0) = ( R p , .(b). Let us show that G (Sing G ) is well-defined as set germ at ∈ R p . By contradiction, ifthis is not the case, then there exist ε > ε ′ > and a sequence of points p n ∈ R p , p n → , p n ∈ G ( B ε ∩ Sing G ) , p n G ( B ε ′ ∩ Sing G ) for all integers n ≫ . Let then x n ∈ B ε ∩ Sing G with G ( x n ) = p n and there is a subsequence ( x n k ) k ∈ N which tends to some point x inthe closure ¯ B ε ∩ Sing G . We have G ( x ) = lim k →∞ G ( x nk ) = lim k →∞ p n k = 0 . Since Sing G ∩ G − (0) = 0 , the point x must be 0. But then we must have x n k ∈ B ε ′ ∩ Sing G for some k ≫ , which implies that p n k ∈ G ( B ε ′ ∩ Sing G ) which is a contradiction to theassumptions about the sequence ( p n ) n . IBRATIONS OF HIGHLY SINGULAR MAP GERMS 5
Since the hypothesis in (b) is more particular than that in (a), it then follows in additionthat G is nice. (cid:3) Remark . It is known that for analytic map germs G : ( R m , → ( R p , , m > p ,the condition Sing G ∩ V G = { } is equivalent to the finite C - K -determinacy of G ,see [Wa, CN] for details. Observe also that the condition in Lemma 2.4(a) cannot beweakened, like for instance in the example f ( x, y, z ) = ( x + y , ( x + y ) z ) taken from[Han].We still assume that G : ( R m , → ( R p , , m ≥ p > is an analytic map germ.Whenever Im G is well-defined as a set germ, its boundary ∂ Im G := Im G \ int(Im G ) is a closed subanalytic proper subset of R p , where int A := ˚ A denotes the p -dimensionalinterior of a subanalytic set A ⊂ R p (hence it is empty whenever dim A < p ), and A denotes the closure of it. We consider here ∂ Im G as a set germ at ∈ R p ; this is ofcourse empty if (and only if) the equality (Im G,
0) = ( R p , holds. Definition 2.6.
We call discriminant of a nice map germ G the following set:(3) Disc G := G (Sing G ) ∪ ∂ Im G which is a closed subanalytic germ of dimension strictly less than p .All the above subsets are considered as germs at the respective origins, and they are well-defined since G is nice. In particular, the complementary of Disc G in a neighbourhoodof the origin is the disjoint union of finitely many open connected subanalytic sets, well-defined as germs at the origin.We shall denote by gcd( f, g ) = 1 the greatest common divisor of two holomorphicfunction germs f, g : ( C n , → ( C , , thus well-defined only modulo units in the algebraof such function germs. Theorem 2.7.
Let f, g : ( C n , → ( C , , n > , be holomorphic germs such that gcd( f, g ) = 1 . Then f ¯ g : ( C n , → ( C , is a nice map germ.Proof. By our assumption we have ord f > and ord g > . Let us consider somerepresentatives f, g : B ε → C of the holomorphic germs, well-defined on some small ball B ε . For any α ∈ S ⊂ C the set A α := { f − αg = 0 } ⊂ B ε is a complex hypersurfacegerm at the origin ∈ C n , hence of dimension n − . The image of the restriction of f ¯ g to A α is Im α | g | A α | , and we claim that the restriction g | A α is not constant = 0 .If not the case, then g | A α ≡ implies that there is some holomorphic germ v of positiveorder such that v divides both g and f − αg = 0 , so that its zero locus contains A α . Thisfurther implies that v divides f , contradicting the irreducibility of the fraction f /g .We have thus shown that Im g | A α contains some small open disk D r ( α ) centred at 0 ofpositive radius r ( α ) , for any α ∈ S . The radius r ( α ) is a lower semi-continuos functionof α in the following sense: let α ∈ S be fixed; for any < d < r ( α ) , there is a smallneighbourhood N α ⊂ S of α such that if α ′ ∈ N α then Im g | A α ′ contains some opendisc D d centred at 0 of radius d . Indeed, for d < d ′ < r ( α ) , there exists some point x α ∈ B ε ∩ ( g | A α ) − ( ∂ ¯ D d ′ ) . Then the continuous function g maps some small enoughneighbourhood M x α ⊂ B ε of x α to the complement of the disk D d . On the other hand, RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂR M x α is intersected by all hypersurfaces A α ′ for α ′ in some small enough neighbourhood of α . Now since S is compact, it follows that there is ρ > such that r ( α ) > ρ for any α ∈ S . Altogether this shows that Im ( f ¯ g ) | B contains an open neighbourhood of ∈ C ,and this implies that the image of f ¯ g is well-defined as germ.Let us show that the discriminant Disc f ¯ g is also well defined as a set germ at . Firstwe remark that the discriminant Disc ( f, g ) of the map ( f, g ) is well defined as a set germat the origin of C , namely it is either or a the germ of a complex curve which may haveseveral irreducible components. This follows from the fact that the restriction of ( f, g ) to Sing ( f, g ) is either constant or locally a complex curve (i.e. at points where the rank ofthis restriction is 1).Next, by considering the decomposition of the map f ¯ g into ( f, g ) followed by u ¯ v : C → C (see [PT1]), one finds that the discriminant of f ¯ g is the image by u ¯ v of the critical locusof all the restrictions u ¯ v | C : ( C, → ( C , to irreducible components ( C, ⊂ Disc ( f, g ) .By considering the Puiseux parametrisation p C : ( C , → ( C, , the critical locus of u ¯ v | C ◦ p C is defined as the solution of real analytic equations, and therefore it is a finitecollection of real analytic curves. The images by u ¯ v | C ◦ p C of these real analytic curves isa finite union of real semi-analytic curves (by Łojasiewicz’ result in dimension 1) whichare obviously well defined as set germs at 0. It then follows that Disc f ¯ g is well definedas a set germ, and it is either or a finite union of real semi-analytic curve germs.This completes the proof that f ¯ g is a nice map germ. (cid:3) Milnor-Hamm fibration
We still work with a non-constant nice analytic map germ G : ( R m , → ( R p , , m ≥ p > . We denote V G := G − (0) and B kη signifies an open ball of radius η centred at ∈ R k , for some positive integer k .After (3), the open subset B pη \ Disc G is the disjoint union ⊔ i C i of finitely many opensubanalytic sets considered as germs at the origin, and we have by definition:(4) Im G ∩ C i = ∅ ⇔ Im G ⊃ C i . We would like to have a locally trivial C ∞ -fibration with well-defined fibre over eachsuch component C i . Definition 3.1.
Let G : ( R m , → ( R p , , m ≥ p > , be a non-constant nice analyticmap germ. We say that G has Milnor-Hamm fibration if, for any ε > small enough,there exists < η ≪ ε such that the restriction:(5) G | : B mε ∩ G − ( B pη \ Disc G ) → B pη \ Disc G is a locally trivial fibration over each connected component C i ⊂ B pη \ Disc G , such that itis independent of the choices of ε and η up to diffeomorphisms.It follows that, if it exists, this fibration has nonempty fibres only if dim Im G = p . Thecase dim Im G < p will be included in a more general treatment in §6.Let U ⊂ R m be an open set such that ∈ U , and let ρ : U → R ≥ be a non-negative proper function which defines the origin, for instance the Euclidean distance. IBRATIONS OF HIGHLY SINGULAR MAP GERMS 7
The transversality of the fibres of a map G to the levels of ρ is called ρ -regularity and it isa sufficient condition for the existence of locally trivial fibrations. It was used in the local(stratified) setting by Thom, Milnor, Mather, Looijenga, Bekka, e.g. [Th1, Th2, Mi, Lo2,Be] and more recently [ACT1, AT2], as well as at infinity, e.g. [NZ, Ti2, Ti3, ACT2, DRT],but also in many other recent papers and under different names. We consider the followingdefinition: Definition 3.2.
Let G : ( R m , → ( R p , be a non-constant analytic map germ, m ≥ p > . We call: M ( G ) := { x ∈ U | ρ ⋔ x G } the set of ρ -nonregular points of G , or the Milnor set of G , where ρ denotes here theEuclidean distance function.We then consider the following condition:(6) M ( G ) \ G − (Disc G ) ∩ V G ⊆ { } where the analytic closure of the subanalytic set M ( G ) \ G − (Disc G ) is considered asa set germ at the origin. Condition (6) is a direct extension of the condition used in[AT1, AT2, Mas, ACT1] for the case Disc G = { } , where it was shown that it insuresthe existence of a locally trivial fibration called Milnor tube fibration . The basic resulton the existence of Milnor-Hamm fibrations is the following: Lemma 3.3.
Let G : ( R m , → ( R p , be a non-constant nice analytic map germ, m > p > . If G satisfies condition (6) , then G has a Milnor-Hamm fibration (5) .Proof. The map G satisfies condition (6) if and only if there exists ε > such that, forany < ε < ε , there exists η , < η ≪ ε , such that the restriction map(7) G | : S m − ε ∩ G − ( B pη \ Disc G ) → B pη \ Disc G is a smooth submersion and moreover that the map G | : B mε ∩ G − ( B pη \ Disc G ) → B pη \ Disc G is a submersion on a manifold with boundary, and it is proper. We maytherefore apply Ehresmann’s Theorem (see for instance [Mat]) to conclude to the existenceof a locally trivial fibration (5) over every connected component of B pη \ Disc G (with emptyor non-empty fibre). (cid:3) Remark . A map germ G such that ( M ( G ) ,
0) = ( R m , can still have a Milnor-Hamm fibration, here is an example: G := ( x + y , z + w ) : R → R . The map G isnice, with Im G = R ≥ × R ≥ , Disc G = R ≥ × { } ∪ { } × R ≥ . We have V G = { } and ( M ( G ) ,
0) = ( R , , compare to Theorem 3.5 below.Nevertheless, G satisfies condition (6) and the Milnor-Hamm fibration exists. Its fibreover the positive quadrant is compact, diffeomorphic to a torus S × S which does notintersect the Milnor-Hamm sphere S m − ε in (7).The next statement represents a real counterpart of Hamm’s result [Ha] that a holo-morphic map which defines an ICIS has a locally trivial fibration over the complement ofthe discriminant: We send to §6 for the extension of this fibration.
RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂR
Theorem 3.5.
Let G : ( R m , → ( R p , be a analytic map germ. If V G ∩ Sing G = { } and dim V G > , then G is nice and has a Milnor-Hamm fibration (5) .Proof. Since V G ∩ Sing G = { } and dim V G > , we may apply Lemma 2.4(b) and concludethat G is nice.Next, since transversality is an open condition, it follows that for each ε > one canfind < η ≪ ε , such that for any < η ≤ η the restriction map(8) G | : S m − ε ∩ G − ( B pη ) → B pη is a smooth submersion and therefore Lemma 3.3 applies. (cid:3) Example . The map G : R n → R , G ( x , . . . , x n ) = ( x , x + · · · + x n − − x n ) is nice.One has V G = { x = 0 } ∩ { x + · · · + x n − − x n = 0 } and Sing G = { x = · · · = x n = 0 } isthe x -axis. The condition V G ∩ Sing G = { } is satisfied and the Milnor-Hamm fibration(5) exists by Theorem 3.5.4. Partial Thom regularity condition and Milnor-Hamm fibration
Given some stratification of a neighbourhood of ∈ R m we recall, after [GLPW, Mat],that a stratum A is Thom regular over a stratum B ⊂ ¯ A \ A at x ∈ B relative to G (or,equivalently, that the pair ( A, B ) satisfies the Thom (a G )-regularity condition at x ), if thefollowing condition holds: for any { x n } n ∈ N ⊂ A such that x n → x , if T x n ( G | A ) converges,when n → ∞ , to a limit H in the appropriate Grassmann bundle, then T x ( G | B ) ⊂ H .The notion of ∂ -Thom regularity has been introduced in [Ti1, Def. 2.1] for analyticfunction germs, and in [Ti3, A.1.1.1] and [DRT, §6] as a regularity condition in theneighbourhood of infinity. Here we give the map germ version. The difference to theclassical Thom regularity is that here we do not ask all possible conditions between couplesof strata ( A, B ) for A outside V G and B inside V G , but only those for A equal to the singlestratum B mε \ G − (Disc G ) . Definition 4.1.
Let G : ( R m , → ( R p , be a non-constant analytic map germ. We saythat G is ∂ -Thom regular at V G if there is a a ball B mε centred at ∈ R m and a Whitney(a) semi-analytic stratification H = { H α } of B mε ∩ V G such that, for any stratum H α , thepair ( B mε \ G − (Disc G ) , H α ) satisfies the Thom (a G )-regularity condition.We then also say that such a stratification is a ∂ -Thom ( a G ) -stratification of V G .Let us remark that Thom regularity at V G obviously implies ∂ -Thom regularity at V G .The following result shows that the weaker regularity condition is sufficient to insure theexistence of the Milnor-Hamm fibration. Proposition 4.2.
Let G : ( R m , → ( R p , be a non-constant nice analytic map germ.If G is ∂ -Thom regular at V G , then G has a Milnor-Hamm fibration (5) .Proof. The ∂ -Thom regularity implies that for any ε > small enough, there exists aneighborhood N ε of S m − ε ∩ V G in S m − ε such that for any x ∈ N ε \ G − (Disc G ) one has S m − ε ⋔ x G − ( G ( x )) . IBRATIONS OF HIGHLY SINGULAR MAP GERMS 9
It follows that there is η ε > such that S m − ε ∩ G − ( B mη ε \ Disc G ) ⊂ N ε . Then therestriction (7) is a submersion and consequently G satisfies condition (6) and one mayapply Lemma 3.3. (cid:3) Classes of maps with ∂ -Thom regularity. We single out several classes of maps whichhave Milnor-Hamm fibration because of being ∂ -Thom regular, thus to which Proposition4.2 applies. For that, let us formulate the definition of the ∂ -Thom regularity set , inanalogy to that of the Thom regularity set of [PT1, §3]. Referring to Definition 4.1, let(9) ∂ NT G := (cid:26) x ∈ V G (cid:12)(cid:12)(cid:12)(cid:12) there is no ∂ -Thom ( a G ) -stratification of V G suchthat x belongs to a positive dimensional stratum. (cid:27) The definition of the
Thom regularity set NT G of [PT1, §3] is obtained by just replacing“ ∂ -Thom” by “Thom” in the above definition.Let us first consider maps G = ( f, g ) , where f, g : ( C n , → ( C , , n > , are non-constant complex analytic function germs, and let f ¯ g : ( C n , → ( C , . The latter are aspecial class of mixed functions. Theorem 4.3.
Let f, g : ( C n , → ( C , , n > , be non-constant holomorphic functions.Then: (a) If ( f, g ) defines an ICIS then the map germ f ¯ g is nice, Thom regular and has aMilnor-Hamm fibration. (b) If gcd( f, g ) = 1 and the map ( f, g ) is Thom regular (i.e. NT ( f,g ) ⊂ { } ), then themap f ¯ g has a Milnor-Hamm fibration. (c) If gcd( f, g ) = 1 , if Disc ( f, g ) consists of only lines (for instance if f and g arehomogeneous of the same degree), and if ( f, g ) is ∂ -Thom regular, then the map f ¯ g is ∂ -Thom regular and has a Milnor-Hamm fibration. Remark . There are some well-known attempts to endowing f ¯ g with a Thom regularstratification. Pichon and Seade stated and proved a general theorem in n > variables.As this was pointed out to be false [Pa], [Ti4], in the subsequent erratum [PS2] the authorspresented Parusinski’s counterexample [Pa], [Ti4], [ACT1, p. 827], [Oka2]. Even in n = 2 variables, the proof of the statement published earlier [PS1, Proposition 1.4] has the sameproblem since it tacitly assumes that the normal space at a point of some fibre of f ¯ g isgenerated by a single vector.The status of the existence of a Thom regular stratification for f ¯ g was establishedrecently in [PT1, Theorem 3.1] (see [PT2] for the rectified statement of this theorem),based on different grounds, and for a large class of maps f ¯ g in n ≥ variables. Inparticular the proof of [PT1, Theorem 3.1] shows that the statement of the above discussedresult in 2 variables [PS1, Proposition 1.4] is nevertheless correct. Proof of Theorem 4.3.
In the following we use the proof of [PT1, Theorem 3.1]. Thatproof starts with the hypothesis
Disc f ¯ g ⊂ { } and therefore remains unchanged after therectification [PT2] of the statement of [PT1, Theorem 3.1]. The corrigendum [PT2] rectifies the erroneous criterion for the inclusion
Disc f ¯ g ⊂ { } given in[PT1], as follows: “ Disc f ¯ g ⊂ { } iff the discriminant Disc ( f, g ) of the map ( f, g ) contains only curvecomponents which are tangent to the coordinate axes ”. RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂR (a). If ( f, g ) defines an ICIS then there is the equality of germs [Im ( f, g )] = [ C ] , and Disc ( f, g ) is a well-defined curve germ in ( C , , see e.g. [Lo2, Lo3]. In particular itfollows that ( f, g ) is a nice map. It then follows from the proof of Theorem 2.7 that f ¯ g isa nice map germ.In order to prove the ∂ -Thom regularity of f ¯ g , we proceed as follows. By Step 1 of theproof of [PT1, Theorem 3.1], we have NT ( f,g ) ⊂ V ( f,g ) . Since Sing ( f, g ) ∩ V ( f,g ) = { } , itfollows that ( f, g ) is Thom regular. One remarks that the Thom ( a ( f,g ) ) -regularity of thepair of strata ( B ε \ Sing ( f, g ) , V ( f,g ) \ { } ) implies the Thom ( a ( f ¯ g ) ) -regularity of the samepair; one can find all details in Step 2 of the proof of [PT1, Theorem 3.1] where this claimis showed. In our simple situation, this just means that f ¯ g is ∂ -Thom regular.One may now apply Proposition 4.2 to conclude that the Milnor-Hamm fibration exists.(b). If gcd( f, g ) = 1 , then it follows from Theorem 2.7 that the maps f ¯ g is nice.In order to prove the ∂ -Thom regularity of f ¯ g it is only necessary to check the Thomregularity condition outside the inverse image of the discriminant set Disc f ¯ g . This avoidsthe line components of the discriminant of ( f, g ) but not the other components of thesame Disc ( f, g ) . Nevertheless, the Step 2 of the proof of [PT1, Theorem 3.1] takes intoaccount the components of the singular set Sing ( f, g ) which do not have line images.It therefore appears as a result of this proof that the Thom ( a ( f,g ) ) -regularity condition,including along the components of the singular set Sing ( f, g ) , implies the ∂ -Thom ( a ( f ¯ g ) ) -regularity.Lastly we may apply the same Proposition 4.2 for getting the Milnor-Hamm fibration.(c). Referring to the proof of Theorem 2.7, let us observe that if Disc f ¯ g contains a realhalf-line, then its inverse image by u ¯ v is a line component of Disc ( f, g ) . Therefore theproof of (c) can be extracted from the proof of (b) by remarking that since there areonly line components of Disc ( f, g ) we only deal with the Thom condition between strataoutside the singular locus of ( f, g ) , which means the we only need the ∂ -Thom regularityof ( f, g ) and not its full Thom regularity. (cid:3) While we do not know if the following statement is true for the stronger Thom regularity,we can prove it for the ∂ -Thom regularity, which is enough for our purpose, in view ofProposition 4.2. Proposition 4.5.
Let f : ( R m , → ( R p , and g : ( R n , → ( R k , be nice analyticmaps, in separate variables, and satisfying the ∂ -Thom regularity condition at V f = { } and V g = { } , respectively. Then the map G := ( f, g ) is nice and ∂ -Thom regular at V G .In particular, if f and g are holomorphic functions in separate variables, then f ¯ g is ∂ -Thom regular and thus has a Milnor-Hamm fibration.Proof. The niceness matter is an easy exercise which we may safely leave to the reader.Next, the fibre G − ( c , c ) ⊂ R m × R n of the map G := ( f, g ) : ( R m × R n , → ( R p × R k , is the product f − ( c ) × g − ( c ) . If S f and S g are ∂ -Thom regular stratifications of f and g , respectively, then the product stratification S f × S g on ( R m × R n , is a ∂ -Thomregular stratification of V G . The proof is also standard and we may safely skip it.Our last claim follows from Theorem 4.3(c) since the holomorphic functions f and g satisfy the hypotheses of that statement, as one can easily check. (cid:3) IBRATIONS OF HIGHLY SINGULAR MAP GERMS 11
Example . Let G : ( C , → ( C , given by G ( z , z , z , z ) = ( z ¯ z , z ¯ z ) . ByProposition 4.5 each component of G is a ∂ -Thom regular function. Since the map G is in separate variables, Proposition 4.5 yields that G is ∂ -Thom regular as well. ByLemma 3.3 the Milnor-Hamm fibration (5) exists for G . One has V G ∩ Sing G ) { } and Disc G = { ( z, ∈ C | z ∈ C } ∪ { (0 , w ) ∈ C | w ∈ C } . Example . We construct a map germ G := ( f, g ) : ( R × R , → ( R , as follows.Let f : ( R , → ( R , , f ( x, y, z ) = ( y − z x − x , xy ) . One has that V f = Sing f = { x = y = 0 } , and Im f = R , thus Disc f = { } , in particular the map is nice. It wasproved in [ACT1, Example 5.3] that f is Thom regular.Let g := ( g , g ) : ( R , → ( R , be defined by: ( g ( a, b, c, d ) = − d ab − d a c + 3 d b c + 6 dabc + a c − b c g ( a, b, c, d ) = d a − d b − d abc − da c + 3 db c + 2 abc . By computing the gradients, we get h∇ g , ∇ g i = 0 and k∇ g k = ( a + b )( d + c ) (4 d + 9 a + 9 b + 4 c ) = k∇ g k , which shows that g is a simple Ł-map after [Mas,Definition 3.5], and in particular has isolated critical value. Applying [Mas, Theorem2.12, Lemma 5.5], one obtains that g is Thom regular at V g . In particular its image coversan open subset containing the origin, hence we get that Disc g = { } , and that g is nice.It then follows by Proposition 4.5 that G is nice and is ∂ -Thom regular and hence ithas a Milnor-Hamm fibration. Since Disc G = { } × R ∪ R × { } , the complement of Disc G is connected, hence there is a unique fibre.5. Constructing examples
The first example with Milnor tube fibration but without Thom regularity occurred incase
Disc G = { } in [Ti4, ACT1]; see also [Oka2, PT1] and [Han]. We show here someexamples in our new setting Disc G ) { } , based on the following: Lemma 5.1.
Let f : ( R m , → ( R p , be an analytic map germ and let g : ( R n , → ( R n , be the identity function g ( y ) = y . Consider the analytic map germ G := ( f, g ) :( R m × R n , → ( R p × R n , . Then: (a) f is ∂ -Thom regular if and only if G is ∂ -Thom regular. (b) f satisfies condition (6) if and only if G satisfies condition (6) .Proof. (a) is a simple exercise. Let us prove (b). Assume that f satisfies condition(6). Let p = ( x , y ) ∈ M ( G ) \ G − (Disc G ) ∩ V G . One has a sequence of points p n =( x n , y n ) → p such that p n ∈ M ( G ) \ G − (Disc G ) = M ( f ) × R n \ f − (Disc f ) × R n and p ∈ V G = V f × { } . Thus, y = 0 , x ∈ V f and x n ∈ M ( f ) \ f − (Disc f ) . Consequently, x ∈ M ( f ) \ f − (Disc f ) ∩ V f ⊂ { } which implies p = (0 , and G satisfies the condition(6).Assume that M ( G ) \ G − (Disc G ) ∩ V G ⊆ { } . One has that (cid:0) M ( f ) \ f − (Disc f ) (cid:1) × { } ⊂ M ( G ) \ G − (Disc G ) . For any x ∈ M ( f ) \ f − (Disc f ) ∩ V f there exists a sequence x n ⊂ M ( f ) \ f − (Disc f ) such that x n → x . Hence, ( x n , ∈ M ( G ) \ G − (Disc G ) for all n ∈ N and the limit RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂR point ( x , ∈ M ( G ) \ G − (Disc G ) ∩ V G . We conclude that x = 0 , which implies that f satisfies the condition (6). (cid:3) Example . Let G = ( f, g ) = ( R , → ( R , , where f ( x, y, z ) = ( x, y ( x + y ) + xz ) and g ( w ) = w . After [Han, Example 1.4.9], the map f has isolated critical value andsatisfies the condition (6). We may easily check that f is an open map at 0, and that f is nice. By Lemma 5.1, G satisfies the condition (6), therefore G has a Milnor-Hammfibration. Note that Disc G = { (0 , } × R , and that G is nice.On the other hand, [Han, Example 2.3.9] shows that the map f is not Thom regular.Therefore, by Lemma 5.1(a), the map G cannot be ∂ -Thom regular. Example . Let F be one of the mixed functions:1) F ( x, y ) = xy ¯ x from [ACT1],2) F ( x, y, z ) = ( x + z k )¯ xy for a fixed k ≥ from [PT1],3) F ( w , . . . , w n ) = w (cid:16)P kj =1 | w j | a j − P nt = k +1 | w t | a t (cid:17) from [Oka2].They are all polar weighted-homogeneous and thus, by [ACT1, Theorem 1.4], one obtainsthat Disc F = { } and that F is nice and has Milnor tube fibration. It was also provedin the respective papers that F is not Thom regular.Let then G := ( F, g ) , where g ( v ) = v and note that Disc G = { } × C . By Lemma5.1 the map G satisfies condition (6) and therefore, by Lemma 3.3, G has Milnor-Hammfibration. However, G is not ∂ -Thom regular, cf Lemma 5.1(a). Example . Let G = ( f, g ) : ( R × R n , → ( R × R n , , where g : ( R n , → ( R n , is the identity map germ and f := ( f , f , f , f ) : ( R , → ( R , where f ( x, y, z, w, a, b ) = yw + xzf ( x, y, z, w, a, b ) = xw − yzf ( x, y, z, w, a, b ) = ax + byf ( x, y, z, w, a, b ) = ay − bx One has V f = { x + y = 0 } ∪ { a + b + z + w = 0 } , Sing f = { x + y = 0 } .One can show that f is a simple Ł-map , and therefore it is nice. The Milnor set is M ( f ) = { x + y = 0 } ∪ { a + b + z + w − x − y = 0 } . Note that M ( G ) = M ( f ) × R n .Since f has isolated critical value and satisfies condition (6), it follows by Lemma 5.1 that G satisfies condition (6) and therefore has a Milnor-Hamm fibration by Lemma 3.3. Itsdiscriminant set Disc G = { } × R n has positive dimension.6. The singular Milnor tube fibration
We extend here the definition of Milnor-Hamm fibration to that of a “tube fibration”by including the discriminant in the picture.
Definition 6.1.
Let G : ( R m , → ( R p , be a non-constant analytic map germ, m ≥ p > . Let G ε : B mε → Im G ε denote the restriction of G to a small ball.By the stratification theory, see e.g. [GLPW, GM], there exist locally finite subanalyticWhitney stratifications ( W , S ) of the source of G ε and of its target, respectively, such that IBRATIONS OF HIGHLY SINGULAR MAP GERMS 13 Im G ε is a union of strata, that Disc G ε is a union of strata, and that G ε is a stratifiedsubmersion. In particular every stratum is a nonsingular, open and connected subanalyticset at the respective origin, and morever:(i) The image by G ε of a stratum of W is a single stratum of S ,(ii) The restriction G | : W α → S β is a submersion, where W α ∈ W , and S β ∈ S .One calls ( W , S ) a regular stratification of the map germ G . We say that G is S-nice whenever all the above subsets of the target are well-definedas subanalytic germs, independent of the radius ε . Remark . By Theorem 2.7, the map germs of type f ¯ g such that f /g is an irreduciblefraction are S-nice. Definition 6.3.
Let G : ( R m , → ( R p , be a non-constant S-nice analytic map germ.We say that G has a singular Milnor tube fibration relative to some regular stratification ( W , S ) , which is well-defined as germ at the origin by our assumption, if for any smallenough ε > there exists < η ≪ ε such that the restriction:(10) G | : B mε ∩ G − ( B pη \ { } ) → B pη \ { } is a stratified locally trivial fibration which is independent, up to stratified homeomor-phisms, of the choices of ε and η .What means more precisely “independent, up to stratified homeomorphisms, of thechoices of ε and η ”: when replacing ε by some ε ′ < ε and η by some small enough η ′ < η ,then the fibration (10) and the analogous fibration for ε ′ and ǫ ′ have the same stratifiedimage in the smaller disk B pη ′ \ { } , and the fibrations are stratified diffeomorphic over thisdisk. This property is based on the fact that the image of G is well-defined as a stratifiedset germ, which amounts to our assumption of “S-nice”.By stratified locally trivial fibration we mean that for any stratum S β , the restriction G | G − ( S β ) is a locally trivial stratwise fibration. The fibres are of course empty over thecomplement of Im G , whereas the fibre over some connected stratum S β ⊂ Im G of S is asingular stratified set, namely the union of all fibres of the strata of W which map onto S β .We define the stratwise Milnor set M ( G ) as the union of the Milnor sets of the re-strictions of G to each stratum. Namely, let W α ∈ W be the germ at the origin of somestratum, and let M ( G | W α ) be the Milnor set as in Definition 3.2, namely: M ( G | W α ) := (cid:8) x ∈ W α | ρ | W α ⋔ x G | W α (cid:9) where ρ | W α denotes the restriction of the distance function ρ to the subset W α . Definition 6.4.
We call M ( G ) := ⊔ α M ( G | W α ) the set of stratwise ρ -nonregular points of G with respect to the stratifications W and S .Note that M ( G ) is closed, due to the fact that W is a Whitney (a) stratification. Wethen consider the following condition:(11) M ( G ) \ V G ∩ V G ⊂ { } . and remark that (11) restricted to M ( G ) \ G − (Disc G ) is condition (6). RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂR
In this new setting, the basic existence result is the following:
Theorem 6.5.
Let G : ( R m , → ( R p , be a non-constant S-nice analytic map germ. If G satisfies condition (11) , then G has a singular Milnor tube fibration (10) .Proof. Let us fix a regular stratification ( W , S ) , cf Definition 6.1. Condition (11) impliesthe existence of ε > such that, for any < ε < ε , there exists η , < η ≪ ε , such thatevery restriction map(12) G | : W α ∩ B mε ∩ G − ( B pη \ { } ) → S β ∩ B pη \ { } is a submersion on a manifold with boundary. Indeed, since the sphere S m − ε is transversalto all the finitely many strata of the Whitney stratification W at ∈ R n , it followsthat the intersection S m − ε ∩ W is a Whitney stratification W S,ε which refines W . Bycondition (11), the map G is not only transversal to the stratification W but also to thestratification W S,ε , for any < ε < ε . It then follows that the map (12) is a stratifiedsubmersion and it is proper, thus it is a stratified fibration by Thom-Mather IsotopyTheorem. Moreover, condition (11) tells that this fibration is independent of ε and η upto stratified homeomorphisms. (cid:3) By construction
Disc G and the complement Im G \ Disc G are unions of strata. Inparticular, if the singular tube fibration exists, then the Milnor-Hamm fibration existstoo. Thus, from the above proof one immediately derives: Corollary 6.6.
Under the hypotheses of Theorem 6.5, the map G has a Milnor-Hammfibration over B pη \ Disc G , with nonsingular Milnor fibre over each connected component. (cid:3) Example . Let G : ( R , → ( R , , G ( x, y, z ) = ( xy, z ) . One has: V G = { x = z = 0 } ∪ { y = z = 0 } Im G = R × R ≥ ( R Sing G = { x = y = 0 } ∪ { z = 0 } G (Sing G ) = { } × R ≥ ∪ R × { } Disc G = { (0 , β ) | β ≥ } ∪ { ( λ, | λ ∈ R } G − (Disc G ) = { x = 0 } ∪ { y = 0 } ∪ { z = 0 } M ( G ) = { x = ± y } ∪ { z = 0 } M ( G ) \ G − (Disc G ) = { x = ± y } . It follows that G is nice and satisfies the condition (6), thus G has a Milnor-Hamm fibrationby Lemma 3.3. The complement R \ Disc G consists of 3 connected components. We have:the fibre over R × R < is empty; the fibre over R > × R > and the fibre over R < × R > are two non-intersecting hyperbolas, with 4 connected components.It moreover follows that G is S-nice and satisfies the condition (11), thus it has asingular tube fibration by Theorem 6.5. The discriminant has 4 strata, with singularfibres over each of them, as follows: the positive vertical axis, the fibre over which has twodisconnected components each of which being two intersecting lines; the positive and thenegative horizontal axis, the fibres over which are both hyperbolas with two components;and the origin, the fibre over which is two intersecting lines. IBRATIONS OF HIGHLY SINGULAR MAP GERMS 15
Relation with the Thom regularity.
Let us explain shortly the relation to Thomregularity, extending our discussion in §4.
Definition 6.8.
Let G : ( R m , → ( R p , be a non-constant analytic map germ. Wesay that G is Thom regular at V G if there exists a Whitney stratification ( W , S ) like inDefinition 6.1 such that is a point stratum in S , that V G is a union of strata of W , andthat the Thom (a G )-regularity condition is satisfied at any stratum of V G .Note that we do not ask the full Thom regularity of the map G , i.e. we do not askthat W is a Thom regular stratification; we only ask that Thom regularity holds at thefibre over 0. This condition is enough to insure condition (11) and thus we derive thefollowing statement from Theorem 6.5 and from the proof of Theorem 3.5 adapted to ournew setting: Corollary 6.9.
Let G : ( R m , → ( R p , be a non-constant S-nice analytic map germ.If G is Thom regular at V G , dim V G > , then G has a singular Milnor tube fibration (10) .In particular, if V G ∩ Sing G = { } and dim V G > , then G has a Milnor-Hammfibration (5) . (cid:3) Theorem 4.3 already provides classes of maps f ¯ g which are S-nice, Thom regular at V f ¯ g , and thus have singular Milnor tube fibration. Here is an example with this property: Example . Let f, g : C → C given by f ( x, y ) = xy + x and g ( x, y ) = y . One has V ( f,g ) = { (0 , } and Sing ( f, g ) = { y = 0 } ∪ { y = − x } , thus ( f, g ) is obviously Thomregular. However Disc ( f, g ) = { ( x , | x ∈ C } ∪ { ( − x , x ) | x ∈ C } and therefore f ¯ g has non-isolated critical value. It then follows from Corollary 6.9 that f ¯ g is Thom regular,hence it has a Milnor-Hamm fibration, and also a singular Milnor tube fibration.6.2. An example without Thom regularity.
The singular Milnor tube fibration mayexist without the Thom regularity, as shown by the following example:
Example . Let us consider again the real map germ G from Example 5.2, where itwas shown that G is nice but not ∂ -Thom regular at V G , hence not Thom regular at V G .Let ( W , S ) be the following stratification of the source and of the target of G : W := B ε \ G − (Disc G ) , W := { (0 , , z, w ) ∈ R | w > } , W := { (0 , , z, w ) ∈ R | w < } ,W := { (0 , , z, ∈ R | z > } , W := { (0 , , z, ∈ R | z < } , W := { (0 , , , } ,and S := B η \ Disc ( G ) , S := { ((0 , , γ ) ∈ R | γ > } , S := { ((0 , , γ ) ∈ R | γ < } ,S = S = S := { (0 , , } .The restriction maps are G j := G | Wj : W j → S j , j = 1 , . . . , . We had shown inExample 5.2 that G satisfies condition (6). Each restriction map G j , ≤ j ≤ , is onto,in particular, it follows that G is also S-nice.The Milnor sets of the remaining maps are M ( G ) = M ( G ) = the Ow -axis, and M ( G i ) \ V G = ∅ , for i = 4 , , . It follows that condition (11) holds, and thus G has asingular Milnor tube fibration by Theorem 6.5.Since Disc G = { (0 , , γ ) ∈ R | γ ∈ R } does not disconnect the target space, the Milnorfibre outside the discriminant set (i.e. the Milnor-Hamm fibre) is unique and consists of3 disconnected lines. The fibre over each stratum of the discriminant is a single line. RAIMUNDO N. ARAÚJO DOS SANTOS, MAICO F. RIBEIRO, AND MIHAI TIBĂR
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