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Celebratio Mathematica

Walter D. Neumann

Contributions on Lipschitz geometry of complex singularities

by Lev Birbrair and Anne Pichon

Anne

I first met Wal­ter in 2002 in New York at a con­fer­ence on sin­gu­lar­ity the­ory he or­gan­ized with Ágnes Sz­ilárd. He then vis­ited me sev­er­al times in Mar­seille, and we wrote two short pa­pers on ana­lyt­ic real­iz­a­tions of mul­tilinks in 3-di­men­sion­al man­i­folds, one with An­drás Némethi. A couple of years later, in June 2009, Wal­ter vis­ited Mar­seille with his col­lab­or­at­or Lev Birbrair. The first day, they star­ted to ex­plain their sem­in­al res­ults on in­ner Lipschitz geo­metry of com­plex sin­gu­lar germs on the black­board of the cafet­er­ia, and offered me one of the greatest days — if not the greatest — of my math­em­at­ic­al life. It turned out after a couple of hours that our view­points were per­fectly com­ple­ment­ary and would lead to a glob­al and com­plete pic­ture of the in­ner Lipschitz geo­metry of nor­mal sur­face germs. At the end of the day, we had a three-line state­ment that a sur­face sin­gu­lar­ity has a nat­ur­al thick-thin de­com­pos­i­tion, sim­il­ar to the one de­scribed by Gregori Mar­gulis for neg­at­ive curvature spaces. Our col­lab­or­a­tion really took off there, and I am very grate­ful to both of them to have brought me onto the team, and par­tic­u­larly to Wal­ter, who had the idea of this vis­it in Mar­seille.

Lev

I learned who Wal­ter Neu­mann is when I was a stu­dent. His clas­sic the­or­em about plumb­ing cal­cu­lus and res­ol­u­tion of com­plex sur­face sin­gu­lar­it­ies was already fam­ous. We met at sev­er­al con­fer­ences in sin­gu­lar­ity the­ory. His talks were al­ways very in­ter­est­ing, but be­fore the Trieste con­fer­ence in 2005 we nev­er dis­cussed math­em­at­ics. His lec­ture in Trieste stim­u­lated Al­ex­an­dre Fernandes and me to think about Lipschitz geo­metry of com­plex sur­faces. When I vis­ited Columbia Uni­versity that same year, Al­ex­an­dre and I had the first ex­amples of non­con­ic­al sin­gu­lar­it­ies of com­plex sur­faces with re­spect to the in­ner met­ric. Wal­ter said to me: “Ex­cel­lent. The Lipschitz geo­metry is nonempty. We must see what we can do.” This was the point were the col­lab­or­a­tion star­ted. When Anne joined the group, the the­ory be­came much deep­er. All Wal­ter’s ex­per­i­ence and ca­pa­city turned out to be ex­tremely use­ful for the cre­ation of the mod­ern Lipschitz geo­metry. Wal­ter is a won­der­ful col­lab­or­at­or and I am very happy that I had the great priv­ilege of work­ing with him.

We will present an over­view of the res­ults we proved with Wal­ter on the in­ner Lipschitz clas­si­fic­a­tion of com­plex sur­faces, and then, of some of the res­ults he proved with Anne on their out­er Lipschitz geo­metry and on Lipschitz nor­mal em­bed­dings. We refer to the ori­gin­al pa­pers for pre­cise defin­i­tions, state­ments and proofs, as well as to the re­cent lec­ture notes volume on Lipschitz geo­metry of sin­gu­lar­it­ies ed­ited by Wal­ter and Anne [11].

1. Seminal works

Let (X,0) be a germ of real (or com­plex) ana­lyt­ic space em­bed­ded in some (Rn,0) (or (Cn,0)). A nat­ur­al ques­tion is:

How does X look in a neigh­bor­hood of the ori­gin?

There are mul­tiple an­swers to this vague ques­tion, de­pend­ing on the cat­egory we work in. The fam­ous Con­ic­al Struc­ture The­or­em gives a com­plete an­swer in the to­po­lo­gic­al cat­egory: let Bε be the n-ball with ra­di­us ε centered at the ori­gin of Rn, let Sε be its bound­ary, and let X(ε)=SεX be the link at dis­tance ε of (X,0); for ε>0 suf­fi­ciently small, we have an homeo­morph­ism of pairs (Bε,XBε)(Bε,Cone(X(ε))), where Cone(X(ε)) de­notes the uni­on of seg­ments join­ing the ori­gin to a point xX(ε). But this state­ment com­pletely ig­nores the geo­met­ric prop­er­ties of the set (X,0), and a nat­ur­al ques­tion is then:

How does the link X(ε) evolve met­ric­ally as ε tends to zero?

In oth­er words, is XBε met­ric­ally con­ic­al, i.e., bi-Lipschitz equi­val­ent to the straight cone over X(ε)? Or are there some parts of X(ε) which shrink faster than lin­early when ε tends to zero?

These ques­tions can be ap­proached from two dif­fer­ent view­points de­pend­ing on the choice of the met­ric. Giv­en an ana­lyt­ic germ (X,0), and an em­bed­ding (X,0)(Rn,0), there are two nat­ur­al choices for the met­ric on (X,0): the out­er met­ric do in­duced by the Eu­c­lidean met­ric of the am­bi­ent space Rn, i.e., do(x,y)=xyRn, and the in­ner met­ric di defined as the in­duced arc-length met­ric on (X,0). We call out­er (resp. in­ner) Lipschitz geo­metry of (X,0) its equi­val­ence class up to loc­al out­er (resp. in­ner) bi-Lipschitz homeo­morph­ism.

The pi­on­eer­ing work on the Lipschitz geo­metry of com­plex spaces is the 1969 pa­per [e1] by Frédéric Pham and Bern­ard Teis­si­er (for an Eng­lish trans­la­tion, see the fi­nal chapter of [11]), where they present a com­plete clas­si­fic­a­tion of the Lipschitz geo­metry of com­plex curves for the out­er met­ric. The high­er di­men­sions re­mained un­ex­plored for dec­ades. In par­tic­u­lar, the in­ner Lipschitz geo­metry of com­plex sin­gu­lar­it­ies was com­pletely ig­nored. Ac­tu­ally, it was known for a long time that com­plex curve germs are met­ric­ally con­ic­al, i.e., in­ner bi-Lipschitz equi­val­ent to straight cones over their links, and it was be­lieved by ex­perts that the same was true in high­er di­men­sions. They were wrong! The first ex­ample of a non­con­ic­al com­plex sur­face germ was giv­en by Lev and Al­ex­an­dre Fernandes in [e13]: for k2, the sur­face sin­gu­lar­ity Ak:x2+y2+zk+1=0 is not met­ric­ally con­ic­al. Short after this pi­on­eer­ing work, Wal­ter joined the team, and they wrote a series of pa­pers giv­ing a lot of new ex­amples ([2], [3], [4]) sug­gest­ing that fail­ure of met­ric con­ic­alness is a com­mon phe­nomen­on. For ex­ample, they proved that if the two low­est weights of a weighted ho­mo­gen­eous sin­gu­lar­ity (X,0) are un­equal, then (X,0) is not met­ric­ally con­ic­al. In par­tic­u­lar, among ADE sin­gu­lar­it­ies, only A1 and D4 are met­ric­ally con­ic­al. This work brought about the re­sur­gence of the field. Lipschitz geo­metry of germs be­came since then one of the most at­tract­ive and flour­ish­ing fields in sin­gu­lar­ity the­ory.

Be­fore present­ing Wal­ter’s main con­tri­bu­tions to the field, let us say a few words on the reas­ons that make the Lipschitz geo­metry of sin­gu­lar spaces so in­ter­est­ing and pop­u­lar.

First, while the out­er and in­ner met­rics on (X,0) defined above de­pend on the choice of an em­bed­ding (X,0)(Rn,0), their out­er and in­ner Lipschitz geo­met­ries do not de­pend on this choice (see, e.g., Pro­pos­i­tion 7.2.13 of [e20]). In oth­er words, the in­ner and out­er Lipschitz geo­met­ries of (X,0) are de­term­ined by the ana­lyt­ic type of (X,0). Moreover, the out­er Lipschitz geo­metry of (X,0) de­term­ines the in­ner Lipschitz geo­metry, and the lat­ter ob­vi­ously de­term­ines its to­po­lo­gic­al type, i.e., its class up to loc­al homeo­morph­ism. In ad­di­tion, through the pi­on­eer­ing ex­amples men­tioned above, Wal­ter, Al­ex­an­dre Fernandes and Lev dis­covered that the in­ner Lipschitz geo­metry of a com­plex ana­lyt­ic germ may not be de­term­ined by its to­po­lo­gic­al type. There­fore, the in­ner and out­er Lipschitz geo­met­ries in­deed give two in­ter­est­ing in­ter­me­di­ate clas­si­fic­a­tions between the ana­lyt­ic type and the to­po­lo­gic­al type.

An­oth­er mo­tiv­a­tion comes from the “tame­ness” of Lipschitz clas­si­fic­a­tion of space germs. Ana­lyt­ic types of sin­gu­lar com­plex space germs con­tain con­tinu­ous mod­uli, and this is why it is dif­fi­cult to de­scribe a com­plete ana­lyt­ic clas­si­fic­a­tion. For ex­ample, con­sider the fam­ily of com­plex plane curves germs (Xt,0)tC with equa­tions xy(xy)(xty)=0. For every pair (t,t) with tt, (Xt,0) is not ana­lyt­ic­ally equi­val­ent to (Xt,0). On the con­trary, in [e3], Laurent Sieben­mann and Den­nis Sul­li­van con­jec­tured that the set of Lipschitz struc­tures is tame, i.e., the set of equi­val­ence classes of com­plex al­geb­ra­ic sets in Cn, defined by poly­no­mi­als of de­gree less than or equal to k is fi­nite. One of the most im­port­ant res­ults on Lipschitz geo­metry of com­plex al­geb­ra­ic sets is the proof of this con­jec­ture by Tadeusz Mostowski in [e4], which was gen­er­al­ized to the real set­ting a couple of years later by Adam Parusiński ([e6] and [e8]) and more re­cently to the o-min­im­al set­ting by Nguy­en Nhan and Guil­laume Valette ([e14]). Then a com­plete clas­si­fic­a­tion of the Lipschitz geo­metry of sin­gu­lar spaces seemed to be a more reach­able goal. But so far, there did not ex­ist ex­pli­cit de­scrip­tion of the equi­val­ence classes, ex­cept in the case of com­plex plane curves stud­ied by Frédéric Pham and Bern­ard Teis­si­er.

2. The thick-thin decomposition of surfaces and the complete inner Lipschitz classification

When we star­ted to work to­geth­er with Wal­ter in June 2009 in Mar­seille, it turned out after a couple of hours that our view­points were per­fectly com­ple­ment­ary. They would even­tu­ally lead to a glob­al and com­plete pic­ture of the in­ner Lipschitz geo­metry of nor­mal sur­face germs. At the end of the day, we had a three-line state­ment say­ing that a sur­face sin­gu­lar­ity has a nat­ur­al thick-thin de­com­pos­i­tion, sim­il­ar to the one de­scribed by Mar­gulis for neg­at­ive curvature spaces. From this start­ing point, it took us more than two years to write a full proof of the the­or­em and based on it, to give a com­plete clas­si­fic­a­tion of in­ner geo­metry of nor­mal com­plex sur­face germs, which even­tu­ally ap­peared in [7].

In the first ex­amples of non­con­ic­al nor­mal sur­face germs presen­ted by Lev, Al­ex­an­dre Fernandes and Wal­ter, the ob­struc­tion to met­ric­al con­ic­alness of nor­mal is giv­en by the ex­ist­ence of fast loops. A fast loop in (X,0) is a con­tinu­ous fam­ily of loops γεX(ε) para­met­rized by the ra­di­us ε>0 such that γε is ho­mo­top­ic­ally non­trivi­al in the link X(ε) and such that the length of γε shrinks faster than lin­early, i.e., there ex­ists a real num­ber q>1 such that limε0length(γε)εq=0.

The key res­ult of [7] is the Thick-Thin De­com­pos­i­tion The­or­em (The­or­em 1.6 in that art­icle) which states the ex­ist­ence and uni­city (up to a very nat­ur­al equi­val­ence re­la­tion) of what we called a (min­im­al) thick-thin de­com­pos­i­tion of every nor­mal sur­face germ (X,0): (X,0)=(Xthick,0)(Xthin,0). Roughly speak­ing, it re­flects the fact that all fast loops of min­im­al length in­side their ho­mo­topy class are con­cen­trated in a min­im­al 4-di­men­sion­al semi­al­geb­ra­ic sub­germ (Xthin,0) of (X,0), while the 4-di­men­sion­al semi­al­geb­ra­ic sub­germ (Xthick,0), defined as the clos­ure of the com­ple­ment of (Xthin,0) in (X,0) is es­sen­tially met­ric­ally con­ic­al.

As a first ap­plic­a­tion, the thick-thin de­com­pos­i­tion en­abled us to char­ac­ter­ize the nor­mal sur­face germs which are met­ric­ally con­ic­al:

The­or­em 2.1: ([7], Theorem 7.5, Corollary 1.8) A nor­mal sur­face germ (X,0) is met­ric­ally con­ic­al if and only if its thin part is empty, so that its thick part is the whole (X,0).

The thick-thin de­com­pos­i­tion is re­mark­ably re­lated to the to­po­logy of the link X(ε):

The­or­em 2.2: ([7], Theorem 1.7) The thick and thin parts can be de­com­posed as uni­ons of 4-di­men­sion­al semi­al­geb­ra­ic sets pair­wise dis­joint out­side the ori­gin: (Xthick,0)=i=1r(Yi,0) and (Xthin,0)=j=1s(Zj,0) in such a way that it in­duces a de­com­pos­i­tion of the link: X(ε)=i=1rYi,(ε)j=1sZj(ε) such that for ε>0 suf­fi­ciently small,
  1. each Yi(ε) is a Seifert fibered man­i­fold;
  2. each Zj(ε) is a graph man­i­fold (uni­on of Seifert man­i­folds glued along bound­ary com­pon­ents) and not a sol­id tor­us;
  3. there ex­ist ra­tion­al num­bers qj>1 and fibra­tions ζj(ε):Zj(ε)S1 de­pend­ing smoothly on εε0 whose fibers have dia­met­er of or­der εqj.

One of the re­mark­able prop­er­ties of the thick-thin de­com­pos­i­tion is that it can be ex­pli­citely de­scribed in a very easy way through a suit­able res­ol­u­tion of the sin­gu­lar­ity by us­ing the plumb­ing cal­cu­lus de­veloped by Wal­ter in [1] thirty years be­fore (see Jonath­an Wahl’s con­tri­bu­tion in the present volume). Let us de­scribe this more pre­cisely. Con­sider the min­im­al good res­ol­u­tion π:(X~,E)(X,0) which factors through the nor­mal­ized blowup e0 of the ori­gin. Call L-curve any ir­re­du­cible com­pon­ent of the ex­cep­tion­al di­visor π1(0) which maps sur­ject­ively onto an ir­re­du­cible com­pon­ent of e01(0). By blow­ing-up once each in­ter­sec­tion point (if any) between L-curves, we can as­sume that no L-curves in­ter­sect. De­note by Γ the dual graph of the ex­cep­tion­al di­visor π1(0), i.e., the graph whose ver­tices ν are in bijec­tion with the ex­cep­tion­al com­pon­ents Eν of π1(0) and the edges between ν and ν with the points of EνEν, and let V(Γ) be the set of ver­tices of Γ. Call L-node of Γ any νV(Γ) cor­res­pond­ing to an L-curve. A node of Γ is a ver­tex ν which is an L-node or has 3 in­cid­ent edges or such that the cor­res­pond­ing curve Eν has genus >0. For each ver­tex vV(Γ), let N(Ev) be a small tu­bu­lar neigh­bor­hood of Ev in X~. For each L-node ν, let Γν be the max­im­al con­nec­ted sub­graph of Γ con­tain­ing ν and no oth­er node, and set N(Γν)=vΓνN(Eν). Then Xthick=νL(Γ)π(N(Γν)), where L(Γ) de­notes the set of L-nodes.

Figure 1. The two thick-thin decompositions in the Briançon–Speder family
x5+z15+y7z+txy6=0.
Left: t0. Right: t=0.

Many ex­pli­cit ex­amples are de­scribed in [7]. Let us give just one. Con­sider the fam­ily of sur­face sin­gu­lar­it­ies (Xt,0)(C3,0) with equa­tions x5+z15+y7z+txy6=0. It is the μ-con­stant fam­ily in­tro­duced by Bri­ançon and Speder in [e2]. It has con­stant to­po­lo­gic­al type, while the thick-thin de­com­pos­i­tion changes rad­ic­ally when t be­comes 0. The two graphs of Fig­ure 1 are the res­ol­u­tion graphs of π for t=0 and for t0; the L-nodes are the circled ver­tices, the sub­trees Γν cor­res­pond­ing to the com­pon­ents of the thick part have black ver­tices, while the max­im­al sub­graphs with white ver­tices cor­res­pond to the com­pon­ents of the thin part. The neg­at­ive num­ber weight­ing each ver­tex is the self-in­ter­sec­tion of the cor­res­pond­ing ex­cep­tion­al curve in X~. The weight [8] means that the cor­res­pond­ing ex­cep­tion­al curve has genus 8, while the oth­ers have genus zero. For t0, (Xt,0) has three thick com­pon­ents and a single thin one. For t=0, it has one com­pon­ent of each type.

The second main res­ult of [7], built on a re­fine­ment of the thick-thin de­com­pos­i­tion, gives the com­plete in­ner Lipschitz clas­si­fic­a­tion of nor­mal sur­face germs:

The­or­em 2.3: ([7], Theorem 1.9) The bi-Lipschitz geo­metry of (X,0) de­term­ines and is uniquely de­term­ined by the fol­low­ing data:
  1. the graph de­com­pos­i­tion of X(ε) defined by the thick-thin de­com­pos­i­tion;
  2. in the nota­tion of The­or­em 2.2, for each thin zone Zj(ε), the ho­mo­topy class of the fo­li­ation by fibers of the fibra­tion ζj(ε):Zj(ε)S1, and for each Seifert piece Sν(ε) of Zj(ε), a ra­tion­al num­ber qνqj with qν=1 such that the fibers of the re­stric­tion ζj(ε) to Sν(ε) have dia­met­er of or­der εqν.

The high­er-di­men­sion­al case re­mains up to now al­most un­ex­plored.

3. Outer Lipschitz geometry

As already men­tioned, the com­plete Lipschitz clas­si­fic­a­tion of com­plex curve germs for the out­er met­ric was es­tab­lished by Frédéric Pham and Bern­ard Teis­si­er in [e1] us­ing the concept of Lipschitz sat­ur­a­tions. In fact, any curve germ (X,0)(CN,0) is bi-Lipschitz equi­val­ent to its im­age ((C),0) by a gen­er­ic lin­ear plane pro­jec­tion :CnC2. There­fore, it is suf­fi­cient to clas­si­fy plane curve germs. The main res­ult of [e1] says that the out­er Lipschitz type of a com­plex curve germ (C,0)(C2,0) de­term­ines and is de­term­ined by its em­bed­ded to­po­lo­gic­al type, i.e., by the homeo­morph­ism class of the pair (Sε3,C(ε)). This res­ult was re­vis­ited and com­pleted by Wal­ter and Anne with a geo­met­ric view­point:

The­or­em 3.1: ([8], Theorem 1.1) Let (C1,0)(C2,0) and (C2,0)(C2,0) be two germs of com­plex curves. The fol­low­ing are equi­val­ent:
  1. (C1,0) and (C2,0) have same Lipschitz geo­metry, i.e., there is a homeo­morph­ism of germs φ:(C1,0)(C2,0) which is bi-Lipschitz for the out­er met­ric;
  2. there is a homeo­morph­ism of germs φ:(C1,0)(C2,0), holo­morph­ic ex­cept at 0, which is bi-Lipschitz for the out­er met­ric;
  3. (C1,0) and (C2,0) have the same em­bed­ded to­po­logy, i.e., there is a homeo­morph­ism of germs h:(C2,0)(C2,0) such that h(C1)=C2;
  4. there is a bi-Lipschitz homeo­morph­ism of germs h:(C2,0)(C2,0) with h(C1)=C2.

The equi­val­ence of (ii) and (iii) is the res­ult of Pham and Teis­si­er. The equi­val­ence of (i), (iii) and (iv) is the new con­tri­bu­tion. The proof of (ii)(iii) based on what we call a bubble trick ar­gu­ment, which can be con­sidered as a pro­to­type for ex­plor­ing Lipschitz geo­metry of sin­gu­lar spaces in vari­ous set­tings. An ex­ten­ded ver­sion of the proof is giv­en in [e20]. The first oc­curence of a bubble trick ar­gu­ment we found in the lit­ter­at­ure is in the pa­per of Jean-Pierre Henry and Adam Parusiński [e11]. This im­port­ant tool is also used in re­cent works such has the con­struc­tion of the mod­er­ately dis­con­tinu­ous ho­mo­logy by Javi­er Fernández de Boba­dilla, Sonja Hein­ze, Maria Pe Pereira and José Ed­son Sam­paio in [e18].

The ques­tion of the out­er Lipschitz clas­si­fic­a­tion of high­er-di­men­sion­al germs is still open. Ac­tu­ally, a very small amount of ana­lyt­ic in­vari­ants are de­term­ined by the to­po­lo­gic­al type of an ana­lyt­ic germ (even if one con­siders the em­bed­ded to­po­lo­gic­al type), and a first nat­ur­al ques­tion to ask is wheth­er the Lipschitz clas­si­fic­a­tion is suf­fi­ciently ri­gid to catch ana­lyt­ic in­vari­ants:

Which ana­lyt­ic­al in­vari­ants are in fact Lipschitz in­vari­ants?

Dur­ing the last dec­ade, it was shown that a large amount of ana­lyt­ic in­vari­ants are de­term­ined by the out­er Lipschitz geo­metry in the case of a sur­face germ. The main res­ult of [8] is the pi­on­eer­ing res­ult in this dir­ec­tion. It shows that the out­er Lipschitz class con­tains a lot of in­form­a­tion on the sin­gu­lar­ity and con­firms that out­er Lipschitz geo­metry of sin­gu­lar­it­ies is a very prom­ising area to ex­plore. It can be seen as a first key­stone to a com­plete clas­si­fic­a­tion of the out­er Lipschitz geo­metry of com­plex sur­faces. The high­er-di­men­sion­al set­ting re­mains al­most un­ex­plored.

The­or­em 3.2: ([6], Theorem 1.2) If (X,0) is a nor­mal com­plex sur­face sin­gu­lar­ity then the out­er Lipschitz geo­metry on X de­term­ines:
  1. the dec­or­ated res­ol­u­tion graph of the min­im­al good res­ol­u­tion of (X,0) which re­solves the basepoints of a gen­er­al lin­ear sys­tem of hy­per­plane sec­tions, i.e., the min­im­al good res­ol­u­tion which factors through the blow-up of the max­im­al ideal;
  2. the mul­ti­pli­city of (X,0);
  3. the max­im­al ideal cycle in its res­ol­u­tion;
  4. for a gen­er­ic hy­per­plane H, the out­er Lipschitz geo­metry of the curve (XH,0);
  5. the dec­or­ated res­ol­u­tion graph of the min­im­al good res­ol­u­tion of (X,0) which re­solves the basepoints of the fam­ily of po­lar curves of plane pro­jec­tions, i.e., the min­im­al good res­ol­u­tion which factors through the Nash modi­fic­a­tion;
  6. the to­po­logy of the dis­crim­in­ant curve of a gen­er­ic plane pro­jec­tion;
  7. the out­er Lipschitz geo­metry of the po­lar curve of a gen­er­ic plane pro­jec­tion.

“Dec­or­ated res­ol­u­tion graph” means the res­ol­u­tion graph dec­or­ated with ar­rows cor­res­pond­ing to the com­pon­ents of the strict trans­form of the re­solved curve.

The proof of this res­ult is based on a soph­ist­ic­ated ver­sion of the bubble trick which leads to the graph de­com­pos­i­tion of the link X(ε) de­scribed by the res­ol­u­tion graph of point (i).

The Lipschitz in­vari­ance of the mul­ti­pli­city (point (ii) in the above the­or­em) was then ex­ten­ded bey­ond the nor­mal case by Al­ex­an­dre Fernandes and José Ed­son Sam­paio ([e15]) for a hy­per­sur­face in C3 and by Javi­er Fernández de Boba­dilla, Al­ex­an­dre Fernandes and José Ed­son Sam­paio ([e17]) for the gen­er­al sur­face case. However it is now known that the mul­ti­pli­city is not a Lipschitz in­vari­ant in high­er di­men­sions ([e19]).

4. Lipschitz normal embeddings

An­oth­er im­port­ant con­tri­bu­tion of Wal­ter in the area of Lipschitz geo­metry of sin­gu­lar­it­ies is his series of works on Lipschitz nor­mal em­bed­ding. A sub­set (or a germ) in Rn is Lipschitz Nor­mally Em­bed­ded (LNE for short) if the iden­tity map is a bi-Lipschitz homeo­morph­ism (loc­al in case of a germ) between the in­ner and out­er met­rics. Of course, every com­pact smooth sub­man­i­fold in Rn is LNE. But this prop­erty turns out to be fairly rare among sin­gu­lar spaces, and the res­ults ob­tained by Wal­ter with his coau­thors sug­gest that it plays an im­port­ant (quasi un­ex­plored) role in sev­er­al areas of sin­gu­lar­ity the­ory such as res­ol­u­tion of sin­gu­lar­it­ies or the struc­ture of arc and jet spaces.

The pi­on­eer­ing res­ult on LNE is the fol­low­ing the­or­em due to Lev and Tadeusz Mostowski. Its ori­gin­al ver­sion is proved in the semi­al­geb­ra­ic set­ting, but this res­ult re­mains true with the same proof in the more gen­er­al sub­ana­lyt­ic cat­egory and even, in the o-min­im­al cat­egory. We state it here in the sub­ana­lyt­ic set­ting.

The­or­em 4.1: [e9] Let XRm be a con­nec­ted sub­ana­lyt­ic set. Then there ex­ist a LNE sub­ana­lyt­ic set X~Rq and a sub­ana­lyt­ic in­ner bi-Lipschitz homeo­morph­ism p:XX~.

The map p:XX~ is called a nor­mal em­bed­ding of X.

As a con­sequence of this the­or­em, one ob­tains the fol­low­ing al­tern­at­ive defin­i­tion of the no­tion of Lipschitz nor­mal em­bed­ding. The proof was nev­er writ­ten be­fore. That is why we in­clude it in this manuscript.

Pro­pos­i­tion 4.2: A con­nec­ted sub­ana­lyt­ic sub­space X of Rn is LNE if and only if there ex­ist a sub­ana­lyt­ic bi-Lipschitz homeo­morph­ism f:(X,di)(X,do), that is, there ex­ists and a real num­ber K1 such that for all x,y in X, 1Kdi(f(x),f(y))do(x,y)Kdi(f(x),f(y)).

Proof. As­sume there ex­ist a sub­ana­lyt­ic bi-Lipschitz homeo­morph­ism f:(X,di)(X,do) and a real num­ber K1 as in the state­ment. Let p:(X,di)(X~,di) be a nor­mal em­bed­ding of X as in The­or­em 4.1. We de­duce that pf1 is bi-Lipschitz with re­spect to the out­er met­rics. There­fore pf1 is also bi-Lipschitz with re­spect to the in­ner met­rics. This im­plies that IdX=ff1=fp1pf1 is bi-Lipschitz from (X,di) to (X,do). ☐

Com­plex and real al­geb­ra­ic sets of Cn or Rn are sub­ana­lyt­ic sets. By The­or­em 4.1, they ad­mit a glob­al sub­ana­lyt­ic Lipschitz nor­mal em­bed­ding. Tadeusz Mostowski asked if there ex­ists a com­plex al­geb­ra­ic Lipschitz nor­mal em­bed­ding when X is a com­plex al­geb­ra­ic set, i.e., a (sub­ana­lyt­ic) Lipschitz nor­mal em­bed­ding for which the im­age set X~Cn is a com­plex al­geb­ra­ic set.

The an­swer is pos­it­ive in com­plex di­men­sion 1: every com­plex al­geb­ra­ic curve ad­mits a com­plex al­geb­ra­ic Lipschitz nor­mal em­bed­ding. This fol­lows from the fact that an ir­re­du­cible germ of com­plex al­geb­ra­ic curve is sub­ana­lyt­ic­ally bi-Lipschitz homeo­morph­ic with re­spect to the in­ner met­ric to the germ of C at the ori­gin (see for ex­ample Pro­pos­i­tion 2.1 of [8]). Then a nor­mal em­bed­ding can be con­struc­ted by us­ing the so-called tent pro­ced­ure in­tro­duced in [e9].

Wal­ter, Lev and Al­ex­an­dre Fernandes answered neg­at­ively to Mostowski’s ques­tion in high­er di­men­sion by prov­ing the fol­low­ing res­ult:

The­or­em 4.3: ([5], Theorem 2.3) If 1<a<b and a is not a di­visor of b then no neigh­bor­hood of 0 in the Brieskorn sur­face in C3 X={(x,y,z)C3xb+yb+za=0} ad­mits a com­plex al­geb­ra­ic nor­mal em­bed­ding.

The proof of this res­ult is based on a beau­ti­ful to­po­lo­gic­al ar­gu­ment due to Wal­ter that we re­pro­duce here:

Proof. As­sume there ex­ists a com­plex al­geb­ra­ic Lipschitz nor­mal em­bed­ding p:XX~. By The­or­em 1.3 of [2], the germ (X,0) is met­ric­ally con­ic­al for the in­ner met­ric by a sub­ana­lyt­ic homeo­morph­ism, and so is (X~,0). Since (X~,0) is LNE, it is then sub­ana­lyt­ic­ally met­ric­ally con­ic­al for the out­er met­ric. By a res­ult of Bernig and Lytchak (see Re­mark 2, The­or­em 1.2 of [e12]), this im­plies that (X~,0) is sub­ana­lyt­ic­ally bi-Lipschitz homeo­morph­ic to its tan­gent cone C0X~ at 0 for the out­er met­ric, which is the com­plex cone over a pro­ject­ive al­geb­ra­ic curve C. Since (X,0) has isol­ated sin­gu­lar­ity, then its link is a to­po­lo­gic­al man­i­fold, which im­plies that C is ir­re­du­cible and loc­ally ir­re­du­cible. Now, if C had a sin­gu­lar point, this would con­tra­dict the fact that C0X~ is LNE. There­fore, C0X~ is a com­plex al­geb­ra­ic cone over a smooth ir­re­du­cible pro­ject­ive curve, thus its link is a S1-bundle. On the oth­er hand, the link of X at 0 is a Seifert fibered man­i­fold with b sin­gu­lar fibers of de­gree a/gcd(a,b). This is a con­tra­dic­tion be­cause the Seifert fibra­tion of a Seifert fibered man­i­fold (oth­er than a lens space) is unique up to dif­feo­morph­ism. ☐

An­oth­er im­port­ant con­tri­bu­tion of Wal­ter is a char­ac­ter­iz­a­tion of nor­mal sur­face sin­gu­lar­it­ies which are LNE. In [e16], Lev proved with Rodrigo Mendes that a semi­al­geb­ra­ic germ (X,0) is LNE if and only if if all pairs of real arcs p1 and p2 in (X,0) have equal in­ner and out­er con­tacts. This cri­terion is dif­fi­cult to use ef­fect­ively in prac­tice to prove LNE-ness of a germ since it re­quires to com­pute the in­ner and out­er con­tact or­ders of an im­mense amount of pairs of arcs. Wal­ter, with Anne and Helge Ped­er­sen, es­tab­lished in [9] an ana­log­ous cri­terion for com­plex sur­face germs where the num­ber of pairs of arcs to be tested is re­duced drastic­ally to just fi­nitely many types of pairs.

This ef­fi­cient LNE cri­terion led to the dis­cov­ery of sev­er­al non­trivi­al in­fin­ite fam­ily of LNE nor­mal sur­face germs. For ex­ample:

The­or­em 4.4: [10] Let (X,0) be a nor­mal com­plex sur­face germ as­sume that it is ra­tion­al. Then (X,0) is LNE if and only if it is a min­im­al sin­gu­lar­ity.

Min­im­al sin­gu­lar­it­ies were in­tro­duced by Janos Kollár in [e5]. In di­men­sion 2, they are ra­tion­al sur­face sin­gu­lar­it­ies which play a ma­jor role in res­ol­u­tion the­ory (see [e7], [e10]).

Sev­er­al oth­er fam­il­ies were dis­covered later us­ing the cri­terion of [9], for ex­ample among su­per-isol­ated sin­gu­lar­it­ies [e21]. But LNE-ness of com­plex sur­faces is far from be­ing com­pletely un­der­stood, and the high­er-di­men­sion­al case is still in its in­fancy.

5. Dissemination

In June 2018, Wal­ter or­gan­ized with Anne an in­ter­na­tion­al school on Lipschitz geo­metry of sin­gu­lar­it­ies in Cuerna­vaca, Mex­ico. About fifty PhD stu­dents and young re­search­ers at­ten­ded the event, and the the­ory and prom­ising ex­ten­sions were in­tro­duced a huge vari­ety of view­points, by Wal­ter him­self and also Patrick Popes­cu-Pam­pu, Maria Apare­cida Soares Ru­as, Bern­ard Teis­si­er, Dav­id Trot­man, among oth­ers.

The lec­ture notes were com­pleted and col­lec­ted in a Lec­ture Notes in Math­em­at­ics volume [11], ed­ited by Wal­ter and Anne, which is the first in­tro­duct­ory book on Lipschitz geo­metry of sin­gu­lar­it­ies, and which is now in the hands of many PhD stu­dents and young re­search­ers. The volume also con­tains an eng­lish ver­sion of the his­tor­ic­al pa­per of Pham and Teis­si­er [e1], which was a pre­print in French of the École Poly­tech­nique dur­ing 50 years.

Works

[1] W. D. Neu­mann: “A cal­cu­lus for plumb­ing ap­plied to the to­po­logy of com­plex sur­face sin­gu­lar­it­ies and de­gen­er­at­ing com­plex curves,” Trans. Am. Math. Soc. 268 : 2 (1981), pp. 299–​344. MR 632532 Zbl 0546.​57002 article

[2] L. Birbrair, A. Fernandes, and W. D. Neu­mann: “Bi-Lipschitz geo­metry of weighted ho­mo­gen­eous sur­face sin­gu­lar­it­ies,” Math. Ann. 342 : 1 (2008), pp. 139–​144. MR 2415318 Zbl 1153.​14003 ArXiv 0704.​2041 article

[3] L. Birbrair, A. Fernandes, and W. D. Neu­mann: “Bi-Lipschitz geo­metry of com­plex sur­face sin­gu­lar­it­ies,” Geom. Ded­icata 139 (2009), pp. 259–​267. MR 2481850 Zbl 1164.​32005 ArXiv 0804.​0194 article

[4] L. Birbrair, A. Fernandes, and W. D. Neu­mann: “Sep­ar­at­ing sets, met­ric tan­gent cone and ap­plic­a­tions for com­plex al­geb­ra­ic germs,” Se­lecta Math. (N.S.) 16 : 3 (2010), pp. 377–​391. MR 2734336 Zbl 1200.​14010 ArXiv 0905.​4312 article

[5] L. Birbrair, A. Fernandes, and W. D. Neu­mann: “On nor­mal em­bed­ding of com­plex al­geb­ra­ic sur­faces,” pp. 17–​22 in Real and com­plex sin­gu­lar­it­ies (São Car­los, Brazil, 27 Ju­ly–2 Au­gust 2008). Edi­ted by M. Manoel, M. C. Romero Fuster, and C. T. C. Wall. Lon­don Math­em­at­ic­al So­ci­ety Lec­ture Note Series 380. Cam­bridge Uni­versity Press, 2010. Ded­ic­ated to our friends Maria (Cid­inha) Ru­as and Terry Gaffney in con­nec­tion to their 60th birth­days. MR 2759086 Zbl 1215.​14057 ArXiv 0901.​0030 incollection

[6] W. D. Neu­mann and A. Pichon: Lipschitz geo­metry of com­plex sur­faces: Ana­lyt­ic in­vari­ants and equisin­gu­lar­ity. Technical report, November 2012. ArXiv 1211.​4897 techreport

[7] L. Birbrair, W. D. Neu­mann, and A. Pichon: “The thick-thin de­com­pos­i­tion and the bilipschitz clas­si­fic­a­tion of nor­mal sur­face sin­gu­lar­it­ies,” Acta Math. 212 : 2 (2014), pp. 199–​256. MR 3207758 Zbl 1303.​14016 ArXiv 1105.​3327 article

[8] W. D. Neu­mann and A. Pichon: “Lipschitz geo­metry of com­plex curves,” J. Sin­gul. 10 (2014), pp. 225–​234. MR 3300297 Zbl 1323.​14003 ArXiv 1302.​1138 article

[9] W. D. Neu­mann, H. M. Ped­er­sen, and A. Pichon: “A char­ac­ter­iz­a­tion of Lipschitz nor­mally em­bed­ded sur­face sin­gu­lar­it­ies,” J. Lond. Math. Soc. (2) 101 : 2 (2020), pp. 612–​640. MR 4093968 Zbl 1441.​14015 ArXiv 1806.​11240 article

[10] W. D. Neu­mann, H. M. Ped­er­sen, and A. Pichon: “Min­im­al sur­face sin­gu­lar­it­ies are Lipschitz nor­mally em­bed­ded,” J. Lond. Math. Soc. (2) 101 : 2 (2020), pp. 641–​658. MR 4093969 Zbl 1441.​14016 ArXiv 1503.​03301 article

[11] In­tro­duc­tion to Lipschitz geo­metry of sin­gu­lar­it­ies (Cuerna­vaca, Mex­ico, 11–22 June 2018). Edi­ted by W. Neu­mann and A. Pichon. Lec­ture Notes in Math­em­at­ics 2280. Spring­er (Cham, Switzer­land), 2020. MR 4200092 Zbl 1456.​58002 book