homogeneous hypersurface singularities Derivations of the moduli algebras of weighted Journal of Algebra

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Journal of Algebra

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Derivations of the moduli algebras of weighted homogeneous hypersurface singularities

StephenS.-T. Yau^a,∗,1, Huaiqing Zuo^b,∗,2

a DepartmentofMathematicalSciences,TsinghuaUniversity,Beijing,100084, PR China

b YauMathematicalSciencesCenter,TsinghuaUniversity,Beijing,100084, PR China

a r t i c l e i n f o a bs t r a c t

Article history:

Received29December2014 Availableonlinexxxx CommunicatedbyStevenDale Cutkosky

Keywords:

Derivations Modulialgebra Isolatedsingularity

Let R = C[x1,x2,· · ·,zn]/(f) where f is a weighted ho-

mogeneous polynomial defining an isolated singularity at the origin. ThenR,andDer(R),theLie algebra ofderiva- tions on R, are graded. It is well-known that Der(R) has no negatively graded component [10]. J. Wahl conjectured that theabovefactisstilltrueinhighercodimensionalcase provided that R = C[x1,x2,· · ·,xn]/(f1,f2,· · ·,fm) is an isolated, normal and complete intersection singularity and

f1,f2,· · ·,fm are weighted homogeneous polynomials with

the sameweight type (w1,w2,· · ·,wn).On the other hand the first author Yau conjectured that the moduli algebra A(V)=C[x1,x2,· · ·,xn]/(∂f /∂x1,· · ·,∂f /∂xn) hasnoneg- atively weightedderivationswheref isaweighted homoge- neouspolynomialdefininganisolatedsingularityattheorigin.

Assuming this conjecture has a positive answer, he gave a characterization of weightedhomogeneous hypersurfacesin- gularitiesonlyusingtheLiealgebraDer(A(V)) ofderivations onA(V).TheconjectureofYaucanbethoughtasanArtinian analogueofJ.Wahl’sconjecture.Forthelowembeddingdi- mension, the Yau conjecturehas a positive answer. In this paperweprovethisconjectureforanyhigh-dimensionalsin-

* Correspondingauthors.

E-mailaddresses:yau@uic.edu(S.S.-T. Yau),hqzuo@math.tsinghua.edu.cn(H. Zuo).

1 ThisworkissupportedbyNSFC(grantno.11531007)andthestart-upfundfromTsinghuaUniversity.

2 ThisworkissupportedbytheNSFC(grantnos.11401335,11531007),TsinghuaUniversityInitiative ScientificResearchProgramandthestart-upfundfromTsinghuaUniversity.

http://dx.doi.org/10.1016/j.jalgebra.2016.03.003 0021-8693/© 2016ElsevierInc.All rights reserved.

gularitiesundertheconditionthatthelowestweightisbigger thanorequaltohalfofthehighestweight.

© 2016ElsevierInc.All rights reserved.

1. Introduction

LetA be aweighted zerodimensionalcomplete intersection, i.e., acommutativeal- gebraoftheform

A=C[x₁, x₂,· · ·, x_n]/I

wheretheidealIisgeneratedbyaregularsequenceoflengthn,(f₁,f₂,· · ·,f_n).Herethe variableshavestrictlypositiveintegralweights,denotedbywt(x_i)=w_i,1≤i≤n,and theequationsareweightedhomogeneouswithrespecttotheseweights.Theyarearranged forfutureconvenienceinthedecreasingorderofthedegrees:pi:=degfi,i= 1,2,· · ·,n and p1 ≥ p2 ≥ · · · ≥ pn. Consequently the algebra A is graded and one may speak aboutitshomogeneousdegreek derivations(k isaninteger).Alinearmap D:A→A is a derivation if D(ab) = D(a)b+aD(b), for any a,b ∈ A. D belongs to Der^k(A) if D:A^∗→A^∗+k.Oneofthemostimportantopenproblemsinrationalhomotopytheory isrelatedtothevanishingof theabovederivationsinstrictlynegativedegrees:

HalperinConjecture.(See[5].) IfA isasabove, thenDer^<0(A)= 0.

Assuming thatall the weights w_i are even, this has the following topological inter- pretation.IfaspaceX hasH^∗(X,C)=Aasgradedalgebras,thenitisknownthatthe vanishing of Der^<0(A)= 0 implies the collapsing at theE2-term of theSerre spectral sequencewith C-coefficientsof anyorientablefibration havingX as fiber.Actually the abovecollapsing propertiesalso implies vanishing propertieswhen C is replaced by Q andX arationalspace,seee.g.[5].TheHalperinConjecturehasbeenverifiedinseveral particularcases[10]:

1) equal weights(w1=w2=· · ·=wn),see [14];

2) n= 2,3,see[9,3];

3) “fibered”algebras see[4];

4) assumingC[x1,x2,· · ·,xn]/(f1,f2,· · ·,f_n−1) isreduced,see [6].

5) homogeneousspaces ofequalrankcompactconnectedLie groups(A=H^∗(G/K)), see [8].

OntheotherhandS.S.-T. Yaudiscoveredindependentlythefollowing conjectureon the nonexistence of the negative weight derivation from his work on Lie algebras of

derivations ofthe moduli algebras of isolatedhypersurface singularities,and especially hisworkonmicro-localcharacterizationofquasi-homogeneoushypersurfacesingularities ([1,11–13]).

Yau Conjecture. Let (V,0) ={(x₁,x₂,. . . ,x_n) ∈Cⁿ:f(x₁,x₂,. . . ,x_n) = 0}be an iso- latedsingularitydefinedbytheweightedhomogeneouspolynomialf(x1,x2,. . . ,xn).Then thereisnonon-zeronegativeweightderivationonthemodulialgebra(=Milnoralgebra) A(V)=C[x1,x2,. . . ,xn]/(∂f /∂x1,∂f /∂x2,. . . ,∂f /∂xn).

In case f is a homogeneous polynomial, then it was shown in [11] that L(V) is a graded Liealgebrawithoutnegative weight.Infacttheyprovedthefollowingtheorem.

Theorem 1.1. (See[11].)Let A=⊕^ti=0Ai bea commutative Artinian local algebra with A0 = C. Suppose that maximal ideal of A is generated by Aj for some j > 0. Then Der(A)isanonnegativelygradedLie Algebra ⊕^tk=0Der^k(A).

This conjecture was proved in the low-dimensional case n ≤ 4 ([1,2]) by explicit calculations.

Theorem 1.2. (See[1].)Letf(x1,x2,x3) be aweightedhomogeneous polynomial oftype (w₁,w₂,w₃;d)withisolatedsingularityattheorigin.Assumethatd≥2w₁≥2w₂≥2w₃. LetD beaderivationof themodulialgebra

C[x1, x2, x3]/(∂f /∂x1, ∂f /∂x2, ∂f /∂x3).

Then D≡0if D isnegativelyweighted.

Theorem 1.3. (See [2].) Let f(x1,x2,x3,x4) be a weighted homogeneous polynomial of type (w₁,w₂,w₃,w₄;d) with isolated singularity at the origin. Assume that d≥ 2w₁ ≥ 2w2≥2w3≥2w4.LetD be aderivationof themodulialgebra

C[x1, x2, x3, x4]/(∂f /∂x1, ∂f /∂x2, ∂f /∂x3, ∂f /∂x4).

Then D≡0if D isnegativelyweighted.

Let (V,0) = {(x1,x2,. . . ,xn) ∈ Cⁿ: f(x1,x2,. . . ,xn) = 0} be an isolated singu- larity defined by the weighted homogeneous polynomial f(x₁,x₂,. . . ,x_n) of type (d : w1,w2,· · ·,wn).ThenbyaresultofSaito(seeTheorem 2.1),wecanalwaysassumewith- outlossofgeneralitythat2wi≤dforall1≤i≤n.Ifwegivethevariablexi weightwi

for 1≤i≤n, themoduli algebraA(V)=C[x₁,· · ·,x_n]/(∂f /∂x₁,∂f /∂x₂,. . . ,∂f /∂x_n) is a graded algebra _∞

i=0A_i(V) and the Lie algebra of derivations Der(A(V)) is also graded.

Inthispaper,wedealwithYauconjectureforhigh-dimensionalsingularities.Weprove thefollowingresult whichanswersYauconjecturepositivelyforsomehigh-dimensional singularities.

MainTheorem.Let(V,0)={(x1,x2,. . . ,xn)∈Cⁿ:f(x1,x2,. . . ,xn)= 0}beanisolated singularitydefinedby theweightedhomogeneouspolynomialf(x1,x2,. . . ,xn)oftype(d: w₁,w₂,· · ·,w_n).Assumethatd≥2w₁≥2w₂≥ · · · ≥2w_nwithoutlossofgenerality.Let Der(A(V))betheLiealgebra of derivationsof themodulialgebra

A(V) =C[x₁,· · ·, x_n]/(∂f /∂x₁, ∂f /∂x₂, . . . , ∂f /∂x_n).

If wn ≥w1/2,thenDer^<0(A(V))= 0.

In§2, we recall some definitionsand results which are necessary to prove themain theorems.In§3,weshallgivetheproofofourmaintheorem.

2. Preliminary

In this section, we recall some known results which are needed to prove the Main Theorem.

LetA=_∞

i=0A_i beafinitelygeneratedintegraldomainoverC,withA₀=C.LetA havehomogeneousgeneratorsx_i∈A_ni,i= 1,· · ·,s;thenAisgraded quotient

C[x₁, x₂,· · ·, x_n]/I,wt(x_i) =n_i.

AderivationDofA(i.e.D∈Der(A))canbeviewedasaderivationofC[x1,x2,· · ·,xn] whichpreserveI.So,byabuseofnotation,weshallwrite

D=

a_i∂/∂x_i (a_i∈A).

ThemoduleDer(A) ofderivationsisgradedbysayingDasabovehasweightkifai are weightedhomogeneouswithwt(a_i)=n_i+k,orequivalently, D(A_i)⊂A_i+k foralli.In particular,theEulerderivation

Δ =

nixi∂/∂xi

(whichisaderivation ofAsinceIis graded)hasweight0.Wewrite Der(A) =

k∈Z

Der^k(A).

(ClearlyDer^−k(A)= 0 fork >max{ni,i= 1,· · ·,s}.)

Definition 2.1. A polynomial f(x1,x2,· · ·,xn) is weighted homogeneous of type (d : w1,w2,· · ·,wn) whered andw1,w2,· · ·,wn are fixed positiveintegers,ifit canbe ex- pressedasalinearcombinationofmonomialsxⁱ₁¹xⁱ₂²· · ·xⁱ_nⁿ forwhichw1i1+w2i2+· · ·+ w_ni_n=d.Here,discalled thedegreeoff.

Let f be a weightedhomogeneous polynomial of type (d : w1,w2,· · ·,wn) with an isolated singularityattheorigin.Thenthemodulialgebra

A(V) =C[x1,· · ·, xn]/(∂f /∂x1, ∂f /∂x2, . . . , ∂f /∂xn) is a graded algebra _∞

i=0Ai(V) and the Lie algebra of derivations Der(A(V)) is also graded.

Lemma 2.1. (See [13].) Let (A,m) be a commutative Artinian local algebra and D ∈ Der(A) be any derivation of A. Then D preserve the m-adic filtration of A, i.e., D(m)⊂m.

Definition 2.2.Letf,g: (Cⁿ,0)→(C,0) begermsofholomorphicfunctionsdefiningan isolatedhypersurfacesingularitiesattheoriginrespectively.Letφ: (Cⁿ,0)→(Cⁿ,0) be agermofbiholomorphicmap.f is calledrightequivalent tog,ifg=f ◦φ.

Theorem 2.1. (See [7].) Let f : (Cⁿ,0) → (C,0) be the germ of a complex analytic function withan isolatedcriticalpointattheorigin.

(a) f is rightequivalent toaweighted homogeneouspolynomialif andonlyif μ=τ or f ∈Jf := (∂f /∂x1, ∂f /∂x2, . . . , ∂f /∂xn)

(b) If f is weighted homogeneous with normalized weight system (d :w₁,· · ·,w_n) with 0 < wn ≤ · · · ≤ w1 < d and if f ∈ m³_Cn,0, then the weight system is unique and 0< wn ≤ · · · ≤w1<^d₂

(c) If f ∈ J_f then f is right equivalent to a weighted homogeneous polynomial z²₁ +

· · ·+z_k²+g(z_k+1,· · ·,z_n)with g∈m³_Cn,0.Especially, itsnormalizedweight system satisfies0< wn≤ · · · ≤wk+1< wk =· · ·=w1= ^d₂

(d) If f andf¯∈ OCⁿ,0 are right equivalent andweighted homogeneouswith normalized weight systems (d:w₁,· · ·,w_n)and (d: ¯w₁,· · ·,w¯_n)with 0< w_n ≤ · · · ≤w₁ ≤ ^d₂ and0<w¯_n ≤ · · · ≤w¯₁≤^d₂ thenw_i= ¯w_i.

3. Proofofthe MainTheorem

Proof. SincetheHalperinconjectureistrueforn= 2,byTheorem 1.2andTheorem 1.3, we onlyneed toprovethemain Theoremforn≥5.

Sincef isaweightedhomogeneouspolynomial,

A(V) =C[x1,· · ·, xn]/(∂f /∂x1, ∂f /∂x2, . . . , ∂f /∂xn) isgraded, so Der(A(V))=

kDer^k(A(V)) isalso graded. Any D∈Der^k(A(V)) is of the form n

i=1p_i(x₁,x₂,· · ·,x_n)∂/∂x_i therefore k ≥ −w₁. If Der(A(V)) has negative graded part we can take Der^q(A(V)) to be the most negatively graded part, that is Der(A(V))=

k≥qDer^k(A(V)) and Der^q(A(V))= 0. Whatwe wantto proveisthat anyD∈Der^q(A(V)) hasto be0.

Lemma3.1. Letf ∈C[x₁,· · ·,x_n]be aweightedpolynomial of type(d:w₁,w₂,· · ·,w_n) withisolated singularityattheorigin.Assume thatd≥2w₁≥2w₂≥ · · · ≥2w_n without lossof generality.Let D∈Der^q(A(V))be a negativeweight derivation asabove. Then fori≥2,

∂(Df)/∂xi∈(∂f /∂x1, ∂f /∂x2,· · ·, ∂f /∂xi−1).

Proof. Bycomparisonoftheweighteddegreesitisclearthatanegativeweightderivation D hasto bethefollowingform

D=p1(x2, x3,· · ·, xn)∂/∂x1+p2(x3, x4,· · ·, xn)∂/∂x2+· · ·+c1x^k_n∂/∂x_n−1+c2∂/∂xn

where degpi =wi+q < wi and c1,c2 are constants.ByLemma 2.1 c2 = 0 and k≥1.

Thereforethecommutator[∂/∂x1,D]= 0 and[∂/∂xi,D],i= 2,3· · ·,nisofthefollow- ingformbyadirectcomputation.

[∂/∂x_i, D] =∂p₁/∂x_i·∂/∂x₁+∂p₂/∂x_i·∂/∂x₂+· · ·+∂p_i−1/∂x_i·∂/∂x_i−1. (3.1) LetJ_f = (∂f /∂x₁,∂f /∂x₂,· · ·,∂f /∂x_n) betheJacobianideal.SinceDisaderivation onA(V) it has to preserveJ_f,thatis, D(J_f)⊂J_f.By therelation d≥2w₁ ≥2w₂ ≥

· · · ≥2wn andacomparison ofweighteddegreeswehave

D(∂f /∂x1) = 0 and fori≥2, D(∂f /∂xi)∈(∂f /∂x1, ∂f /∂x2,· · ·, ∂f /∂xi−1). (3.2) By(3.1)wehave

∂(Df)/∂xi =D(∂f /∂xi) +∂p1/∂xi·∂f /∂x1+∂p2/∂xi·∂f /∂x2+· · ·

+∂p_i−1/∂x_i·∂f /∂x_i−1. (3.3)

Theright-handsideof(3.3)is in(∂f /∂x₁,∂f /∂x₂,· · ·,∂f /∂x_i−1) by(3.2). 2

ThefollowingLemmaisanimmediateconsequencefromtheproofoftheLemma 3.1.

Lemma3.2. Under thesame conditionswith Lemma 3.1,thenwehave ∂(Df)/∂x1= 0.

Proof. Itiseasytosee that∂(Df)/∂x1=D(∂f /∂x1)= 0. 2

Now we continue to prove our main Theorem. By Lemma 3.2, ∂(Df)/∂x₁ = 0, it is clear that Df does not depend on x₁. Then by Lemma 3.1, ∂(Df)/∂x₂ = q(x2,x3,· · ·,xn)∂f /∂x1. If q(x2,x3,· · ·,xn) is not 0, then degq(x2,x3,· · ·,xn) ≥ wt(xn) = wn. It follows that wt(q(x2,x3,· · ·,xn)∂f /∂x1) ≥ wn + degf − w1. Since wt(∂(Df)/∂x₂) = wt(D)+degf −w₂. By assumption w_n ≥ w₁/2, we have wt(∂(Df)/∂x₂)=wt(D)+degf−w₂≤wt(D)+degf−w_n<degf−w_n≤w_n+degf−w₁≤ wt(q(x2,x3,· · ·,xn)∂f /∂x1), a contradiction. Therefore q(x2,x3,· · ·,xn) ≡ 0 and

∂(Df)/∂x2= 0 whichimpliesDf doesnotdependonx2.For3≤i≤n.ByLemma 3.1,

∂(Df)/∂x_i=q₁∂f /∂x₁+q₂∂f /∂x₂+· · ·+q_i−1∂f /∂x_i−1.Observethatq_j,1≤j≤i−1 dependsonlyonx_j+1,x_j+2,· · ·,x_n variablesbyweightconsiderationsinceD hasnega- tive weight.Ifthere existsaqj,j= 1,· · ·,i−1 whichisnot0,then

wt(q1(x2, x3,· · ·, xn)∂f /∂x1+q2(x3, x4,· · ·, xn)∂f /∂x2+· · · +qi−1(xi, xi+1,· · ·, xn)∂f /∂xi−1)≥wn+degf −w1.

On the other hand, wt(∂(Df)/∂xi) = wt(D)+degf −wi ≤ wt(D)+degf −wn <

degf−wn ≤wn+degf−w1≤wt(q1(x2,x3,· · ·,xn)∂f /∂x1+q2(x3,x4,· · ·,xn)∂f /∂x2+

· · ·+q_i−1(x_i,x_i+1,· · ·,x_n)∂f /∂x_i−1), a contradiction. Therefore, q_j ≡ 0 for all j = 1,· · ·,i−1. Thus we have ∂(Df)/∂xi = 0 for i = 1,· · ·,n. It follows that Df is a constant and wt(D)+d= 0 whered is thedegree of f. Since wt(D) ≥ −w1, we have that d ≤ w1 which contradicts with our assumption d ≥ 2w1. The main Theorem is proved. 2

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