Bi-Lipschitz geometry of weighted homogeneous surface singularities

BI-LIPSCHITZ GEOMETR Y OF WEIGHTED HOMOGENEOUS SURF A CE SINGULARITIES LEV BIRBRAIR, ALEXANDRE FERNANDES, AND W AL TER D. NEUMANN Abstract. W e show that a weigh ted homogeneous complex surface singular- ity is metrically conical (i.e., bi- Lipsch itz equiv alen t to a metric cone) only if its t w o lo west we ights are equal. W e also give an example of a pair of weigh ted homogeneou s complex sur face singularities that are topol ogicall y equiv alen t but not bi- Li psc hitz equiv alen t. 1. Introduction and main resul ts A natural question of metric theory of singula rities is the ex istence of a metr ically conical s tr ucture ne a r a singula r po int of an alge braic set. F or example, co mplex algebraic curves, equipp ed with the inner metric induced fro m an embedding in C N , a lwa ys have metrically co nical singular ities . It w as discovered recently (see [1]) that w eighted homogeneous complex surface singularities are not necessar ily metrically co nical. In this pap er we show that they are rar ely metrically co nical. Let ( V , p ) be a normal complex surface singularity germ. An y set z 1 , . . . , z N of g enerator s for O ( V ,p ) induces an embedding of ge r ms ( V , p ) → ( C N , 0). The Riemannian metric on V − { p } induced by the standard metric on C N then gives a metric space str ucture o n the germ ( V , p ). This metric space structure, in which distance is given b y ar clength within V , is called the inner metric (as opp osed to outer metric in which distance b etw een points of V is dis tance in C N ). It is easy to see that, up to bi-Lipschitz equiv alence, this inner metric is indep en- dent of choices. It dep ends str ongly o n the a nalytic structure, howev er, and may not b e what one first expects. F or ex ample, we shall see that if ( V , p ) is a quotient singularity ( V , p ) = ( C 2 /G, 0), with G ⊂ U (2) finite a cting freely , then this metric is usually not bi-Lipschitz equiv alent to the conical metric induced by the standard metric on C 2 . If M is a s mo o th co mpact manifold then a c one on M will mean the co ne on M with a s ta ndard Riemannian metric off the cone p oint. This is the metric completion of the Riemannian manifold R + × M with metric g iven (in terms of an elemen t o f arc length) by ds 2 = dt 2 + t 2 ds 2 M where t is the co or dina te on R + and ds M is given by an y Riemannia n metric o n M . It is easy to see that this metric completion simply a dds a single point at t = 0 and, up to bi-Lipschitz equiv alence, the metric on the cone is indep endent of choice of metric on M . If M is the link of an isolated c omplex singula r ity ( V , p ) then the germ ( V , p ) is homeomorphic to the germ of the cone p oint in a cone C M . If this ho meomorphism Key wor ds and phr a ses. bi-Lipschitz, complex sur face si ngularity . Researc h supp orted under CNPq grant no 300985/93-2. Researc h supp orted under CNPq grant no 300393/200 5-9. Researc h supp orted under NSA grant H98230-06-1-011 and NSF gran t no. DMS-0206464. 1 2 LEV BIRBRAIR, ALEXANDRE FERNANDES, AND W AL TER D. NEUMAN N can b e chosen to b e bi-Lipschitz we say , following [3 ], that the germ ( V , p ) is metri- c al ly c o nic al . In [3] the appro ach taken is to consider a semialgebr aic triangula tion of V and consider the s tar of p according to this triangulatio n. The p o int p is metrically c o nical if the intersection V ∩ B ǫ [ p ] is bi- L ips chit z ho meomorphic to the star of p , considered with the s tandard metric of the simplicial complex. Suppo se no w that ( V , p ) is w eighted ho mo geneous. That is, V admits a go o d C ∗ –action (a holo morphic a ction with p ositive weigh ts: each orbit { λx | λ ∈ C ∗ } approaches zero as λ → 0). The w eights v 1 , . . . , v r of a minimal set of homogeneo us generator s of the graded ring of V are ca lle d the weights of V . W e shall order them by size, v 1 ≥ · · · ≥ v r , so v r − 1 and v r are the tw o low est weights. If ( V , p ) is a cyclic quotien t singularity V = C 2 /µ n (where µ n denotes the n –th ro ots of unity) then it do es not hav e a unique C ∗ –action. In this cas e we use the C ∗ –action induced b y the diago na l a ction on C 2 . If ( V , p ) is homo geneous, that is, the weigh ts v 1 , . . . , v r are a ll equa l, then it is easy to see that ( V , p ) is metrically conical. Theorem 1. If the two lowest weights of V ar e une qual t hen ( V , p ) is not metric al ly c onic al. F or example, the Kleinian singularities A k , k ≥ 1, D k , k ≥ 4, E 6 , E 7 , E 8 are the quotient s ing ularities C 2 /G with G ⊂ S U (2) finite. The dia gonal action of C ∗ on C 2 induces a n a c tion on C 2 /G , so they a r e weigh ted homogeneous . They are the weigh ted homogeneous hypersur face singular ities: equation w eigh ts A k : x 2 + y 2 + z k +1 = 0 ( k + 1 , k + 1 , 2) or ( k +1 2 , k +1 2 , 1) D k : x 2 + y 2 z + z k − 1 = 0 ( k − 1 , k − 2 , 2), k ≥ 4 E 6 : x 2 + y 3 + z 4 = 0 (6 , 4 , 3) E 7 : x 2 + y 3 + y z 3 = 0 (9 , 6 , 4) E 8 : x 2 + y 3 + z 5 = 0 (15 , 10 , 6 ) By the theorem, none of them is metrically co nical except for the quadric A 1 and po ssibly 1 the quater nion gro up quo tient D 4 . The general cyclic quotient singular ity is of the form V = C 2 /µ n where the n –th ro ots of unit y act on C 2 by ξ ( u 1 , u 2 ) = ( ξ q u 1 , ξ u 2 ) for so me q prime to n with 0 < q < n ; the link of this singularity is the lens space L ( n, q ). It is ho mogeneous if and o nly if q = 1. Theorem 2 . A cyclic quotient singularity is metric al ly c onic al if and only if it is homo g ene ous. Many non-homogeneous cyclic quotient sing ularities hav e their t wo lo west w eigh ts equal, so the conv erse to Theorem 1 is not g enerally true. W e c a n a lso so metimes distinguish weigh ted homo g eneous singular ities with the same top olo g y from each other. Theorem 3. L et ( V , p ) and ( W, q ) b e two weighte d ho mo gene ous normal surfac e singularities, with weights v 1 ≥ v 2 ≥ · · · ≥ v r and w 1 ≥ w 2 ≥ · · · ≥ w s r esp e c- tively. If either v r − 1 v r > w 1 w s or w s − 1 w s > v 1 v r then ( V , p ) and ( W , q ) ar e not bi-Lipschitz home o morphic. 1 W e hav e a tenta tive pro of that the quaternion quotien t is metrically conical, see [2]. BI-LIPSCHITZ GEOMETR Y OF WEIGHTED HOMOGENEOUS SURF ACE SINGULARITIES 3 Corollary 4. L et V , W ⊂ C 3 b e define d by V = { ( z 1 , z 2 , z 3 ) ∈ C 3 : z 2 1 + z 51 2 + z 102 3 = 0 } and W = { ( z 1 , z 2 , z 3 ) ∈ C 3 : z 12 1 + z 15 2 + z 20 3 = 0 } . Then, t he germs ( V , 0) and ( W , 0) ar e home o morphic, but they ar e not bi-Lipschitz home omorphic. The co rollary fo llows b ecause in bo th cas es the link o f the s ing ularity is an S 1 bundle of Euler class − 1 o ver a curve of genus 2 6; the w eights are (51 , 2 , 1) and (5 , 4 , 3) resp ectively and Theor em 2 applies since 2 1 > 5 3 . The idea o f the pro of of Theorem 1 is to find a n ess e ntial clo sed cur ve in V − { p } with the prop erty that as w e shrink it tow ards p using the R ∗ action, its diameter shrinks fa s ter than it could if V w ere bi-Lipsc hitz equiv alent to a cone. An y ess ent ial closed curve in V − { p } that lies in the h yp erplane se c tio n z r = 1 will hav e this prop erty , so w e must show that the hyperplane sectio n co n tains such cur ves. The pro ofs of Theorems 2 and 3 ar e similar. 2. Proofs Let z 1 , . . . , z r be a minimal set of ho mo geneous generator s of the graded r ing of V , with z i of weigh t v i and v 1 ≥ v 2 ≥ . . . v r − 1 ≥ v r . Then x 7→ ( z 1 ( x ) , . . . , z r ( x )) embeds V in C r . This is a C ∗ –equiv ariant embedding for the C ∗ –action on C r given by z ( z 1 , . . . , z r ) = ( z v 1 z 1 , . . . , z v r z r ) Consider the subset V 0 := { x ∈ V | z r ( x ) = 1 } of V . This is a nonsingula r complex curve. Lemma 2. 1. Supp ose ( V , p ) is not a homo gene ous cyclic quotient singularity. Then for any c omp onent V ′ 0 of V 0 the map π 1 ( V ′ 0 ) → π 1 ( V − { p } ) is non-trivial. Pr o of. Denote v = l cm ( v 1 , . . . , v r ). A conv enient v ersio n o f the link of the singu- larity is given by M = S ∩ V with S = { z ∈ C r | | z 1 | 2 v/v 1 + · · · + | z r | 2 v/v r = 1 } . The ac tion of S 1 ⊂ C ∗ restricts to a fixed-p oint free action on M . If we denote the quotient M /S 1 = ( V − { p } ) / C ∗ by P then the orbit map M → P is a Seifer t fibration, so P has the structur e of an orbifo ld. The orbit map induces a surjectio n of π 1 ( V − { p } ) = π 1 ( M ) to the orbifold fundament al gro up π or b 1 ( P ) (see eg [5, 6]) so the lemma will follow if we show the imag e of π 1 ( V ′ 0 ) in π or b 1 ( P ) is nontrivial. Denote V r := { z ∈ V | z r 6 = 0 } a nd P r := { [ z ] ∈ P | z r 6 = 0 } a nd π : V → P the pro jection. Ea ch generic orbit of the C ∗ –action on V r meets V 0 in v r po int s; in fact the C ∗ –action on V r restricts to an action of µ v r (the v r –th r o ots of unity) on V 0 , and V 0 /µ v r = V r / C ∗ = P r . Thus V 0 → P r is a cyclic cov er of orbifolds, so the same is true for any comp onent V ′ 0 of V 0 . Thus π 1 ( V ′ 0 ) → π or b 1 ( P r ) maps π 1 ( V ′ 0 ) injectively to a norma l subgro up with cyclic quo tient. On the o ther hand π or b 1 ( P r ) → π or b 1 ( P ) is surjective, since P r is the complement of a finite set of p oints in P . Hence, the image of π 1 ( V ′ 0 ) in π or b 1 ( P ) is a normal subgroup with cyclic quotient. Thus the lemma follows if π or b 1 ( P ) is not c y clic. If π or b 1 ( P ) is cyclic then P is a 2–sphere with at mos t t wo or bifold points, so the link M must b e a lens space, so ( V , p ) is a cyclic quotient s ingularity , say V = C 2 /µ n . Here µ n acts on C 2 by ξ ( u 1 , u 2 ) = ( ξ q u 1 , ξ u 2 ) with ξ = e 2 π i/n , for some 0 < q < n with q prime to n . Recall that we are using the diagonal C ∗ –action. The base orbifold is then ( C 2 /µ n ) / C ∗ = ( C 2 / C ∗ ) /µ n = P 1 C /µ n . Note that µ n may not act effectiv ely o n 4 LEV BIRBRAIR, ALEXANDRE FERNANDES, AND W AL TER D. NEUM ANN P 1 C ; the kernel of the a c tion is µ n ∩ C ∗ = { ( ξ qa , ξ a ) | ξ qa = ξ a } = { ( ξ qa , ξ a ) | ξ ( q − 1) a = 1 } = µ d with d = g cd( q − 1 , n ) . So the actual a ction is by a cyclic gro up of or de r n ′ := n /d and the o rbifold P is P 1 C / ( Z /n ′ ), which is a 2–spher e with tw o degr ee n ′ cone p oints. The ring of functions o n V is the ring of inv a riants for the action of µ n on C 2 , which is gene r ated by functions of the form u a 1 u b 2 with q a + b ≡ 0 (mo d n ). The minimal set of generato r s is included in the set consisting o f u n 1 , u n 2 , a nd all u a 1 u b 2 with q a + b ≡ 0 (mo d n ) a nd 0 < a, b < n . If q = 1 these are the elements u a 1 u n − a 2 which all hav e e q ual weigh t, a nd ( V , p ) is homoge neo us and a co ne; this case is excluded by our ass umptions . If q 6 = 1 then a gener ator of least weigh t will b e so me u a 1 u b 2 with a + b < n . Then V 0 is the subset of V given b y the quo tien t of the set ¯ V 0 = { ( u 1 , u 2 ) ∈ C 2 | u a 1 u b 2 = 1 } b y the µ n –action. Each fib er of the C ∗ –action on C 2 int ersec ts ¯ V 0 in exactly a + b p oints, so the comp osition ¯ V 0 → C 2 − { 0 } → P 1 C induces an ( a + b )– fo ld covering ¯ V 0 → P 1 C − { 0 , ∞} . Note that d = gcd( q − 1 , n ) divides a + b s ince a + b = ( q a + b ) − ( q − 1) a = nc − ( q − 1) a for some c . Hence the subgr oup µ d = µ n ∩ C ∗ is in the covering transfor mation g roup of the a bove cov ering, so the cov ering V 0 → P 0 obtained b y quotien ting by the µ n –action ha s degree at most ( a + b ) /d . Restricting to a comp onent V ′ 0 of V 0 gives us po ssibly smaller degree. Since ( a + b ) /d < n/ d = n ′ , the image of π 1 ( V ′ 0 ) in π 1 ( P ) = Z /n ′ is non-trivia l, completing the pro of.  Pr o of of The or em 1. Assume v r − 1 /v r > 1. By Lemma 2.1 we can find a clos ed curve γ in V 0 which represents a non-trivial element o f π 1 ( V − { p } ). Suppo se we have a bi-Lipschitz homeomo rphism h from a neig hborho o d of p in V to a neighborho o d in the co ne C M . Using the R ∗ + –action on V , cho ose ǫ > 0 small enough that tγ is in the given neighborho o d of p for 0 < t ≤ ǫ . Consider the map H of [0 , 1] × (0 , ǫ ] to V given by H ( s, t ) = t − v r h ( tγ ( s )). Her e tγ ( s ) r efers to the R ∗ + –action on V , and t − v r h ( v ) refers to the R ∗ + –action o n C M . Note that the co or dinate z r is constant equal to t v r on ea ch tγ and the o ther co ordinates have b een m ultiplied by at most t v r − 1 . Hence, for e a ch t the curve tγ is a c lo sed curve of length of o rder b ounded by t v r − 1 , so h ( tγ ) has length of the same o rder, so t − v r h ( tγ ) has length of o rder t v r − 1 − v r . T his leng th approaches zero as t → 0, so H extends to a contin uous map H ′ : [0 , 1 ] × [0 , ǫ ] → V for w hich H ([0 , 1] × { 0 } ) is a point. Note that t − v r h ( tγ ) is nev er closer to p than distance 1 /K , where K is the bi-Lips chitz constant o f h , so the same is true for the ima ge of H ′ . Thus H ′ is a n ull-homoto p y o f ǫγ in V − { p } , contradicting the fact that γ was homotopica lly nontrivial.  Pr o of of The or em 2. Supp ose ( V , p ) is a non-homo g eneous cyclic quotient singular - it y , as in the pro o f of Lemma 2.1 and supp ose Theor em 1 do es no t apply , so the tw o low es t w eights a r e equal (in the notation of that pro of this happ ens, for example, if n = 4 k a nd q = 2 k + 1 for some k > 1 : the generators of the ring of functions of low es t w eight are u 1 u 2 k − 1 2 , u 3 1 u 2 k − 3 2 , . . . , o f weigh t 2 k ). L e t u a 1 u b 2 be the genera tor of low est weight that has smallest u 1 –exp onent a nd choos e this one to b e the co or - dinate z r in the notation of Lemma 2 .1. Cons ider now the C ∗ –action induced b y the ac tio n t ( u 1 , u 2 ) = ( t α u 1 , t β u 2 ) on C 2 for some pairwise prime pair of p ositive BI-LIPSCHITZ GEOMETR Y OF WEIGHTED HOMOGENEOUS SURF ACE SINGULARITIES 5 int eger s α > β . With r e spe ct to this C ∗ –action the w eight αa ′ + β b ′ of any generator u a ′ 1 u b ′ 2 with a ′ > a will b e grea ter that the w eight αa + β b of z r (since a ′ + b ′ ≥ a + b , which implies αa ′ + β b ′ = αa + α ( a ′ − a ) + β b ′ > αa + β ( a ′ − a ) + β b ′ ≥ αa + β b ). On the other hand, any generator u a ′ 1 u b ′ 2 with a ′ < a had a ′ + b ′ > a + b by our choice of z r , and if α/ β is chosen close eno ugh to 1 we will still hav e αa ′ + β b ′ > αa + β b , so it will still hav e la r ger weigh t than z r . Thus z r is then the unique genera tor of low es t weight , so we can c a rry out the pro of of Theorem 1 us ing this C ∗ –action to prov e non-conicalnes s of the singula rity .  Pr o of The or em 3. Let h : ( V , p ) → ( W, q ) be a K –bi-L ips chit z homeomor phism. Let us suppose tha t v r − 1 v r > w 1 w s . Let γ b e a lo o p in V 0 representing a non-trivial element of π 1 ( V − { p } ) (see Lemma 2.1). W e choose ǫ a s in the previous pro o f. F or t ∈ (0 , ǫ ] co nsider the curve tγ , wher e tγ refers to R ∗ + -action on V . Its length l ( tγ ), considered as a function of t , has the or der b ounded by t v r − 1 . The distance of the curve tγ from p is of orde r t v r . Since h is a bi-Lipschitz map, we obtain the same estimates for h ( tγ ). Since the smallest weight for W is w s , the cur ve t − v r /w s h ( tγ ) will b e distance at least 1 /K from p . Moreov er its length will b e o f order at most t − w 1 v r /w s l ( tγ ) which is o f order t v r − 1 − w 1 v r /w s . This approa ches zero as t → 0 so, a s in the previous pro of, we ge t a contradiction to the non-trivia lit y of [ γ ] ∈ π 1 ( V − { p } ) = π 1 ( W − { q } ). By exchanging the roles of V and W we see that w s − 1 w s > v 1 v r also leads to a c ontradiction.  References [1] Lev Birbrair and Al exandre F ernandes, Metric geometry of complex algebraic surf aces with isolated singularities, Preprint 2006. [2] Lev Birbrair, Alexandre F ernandes, and W alter N eumann, C onical quasihomogeneous complex sur f ace si ngulari ties, in preparation [3] JP Brasselet, M Goresky , R MacPherson, Si mplicial differential forms with p oles. Amer. J. Math. 113 (1991), 1019–1052. [4] W alter Neumann and Mark Jankins, L ectur e s on Seifert manifolds . Brandeis Lecture Notes, 2. Brandeis Unive rsity , W altham, MA , 1983. [5] W alter D Neumann, F rank Ra ymond, Seifert manifolds, plum bing, -inv ari an t and orienta- tion r ev ersing maps. Al gebr aic and ge ometric top olo gy (Pr o c. Symp os., Univ. California, Santa Barb ar a, Calif., 1977) Lecture Notes in Math., 6 64 (Springer, Ber l in, 1978), 163– 196 [6] Pet er Scott, The geometries of 3-manifolds. Bull . London Math. So c. 15 (1983), 401–487. Dep ar t am ento de M atem ´ atica, Universidade Federal do Cear ´ a (UF C), Campus do Picici, Bloco 91 4, Cep. 6 0455-760 . For t aleza-Ce, Brasil E-mail addr ess : birb@ufc. br Dep ar t am ento de M atem ´ atica, Universidade Federal do Cear ´ a (UF C), Campus do Picici, Bloco 91 4, Cep. 6 0455-760 . For t aleza-Ce, Brasil E-mail addr ess : alex@mat. ufc.br Dep ar tment of Ma them a tics, Barnard College, Columb ia University, New York, NY 10027 E-mail addr ess : neumann@m ath.columbia .edu