Ricci Positive Metrics on the Moment-Angle Manifolds?

Liman CHEN Feifei FAN

Abstract In this paper,the authors consider the problem of which(generalized)momentangle manifolds admit Ricci positive metrics.For a simple polytope P,the authors can cut o ffone vertex v of P to get another simple polytope Pv,and prove that if the generalized moment-angle manifold corresponding to P admits a Ricci positive metric,the generalized moment-angle manifold corresponding to Pvalso admits a Ricci positive metric.For a special class of polytope called Fano polytopes,the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics.Finally some conjectures on this problem are given.

Keywords Moment-Angle manifolds,Simple polytope,Cutting o ffface,Positive Ricci curvature,Fano polytope

1 Introduction

The moment-angle manifold Z comes from two different ways:

(1)The transverse intersections in Cnof real quadrics of the formwith the unit Euclidean sphere of Cn.

(2)An abstract construction from a simple polytope Pnwith m facets.

The study of the first one led to the discovery of a new special class of compact non-k¨ahler complex manifolds in the work of L′opez,Verjovsky and Meersseman[10–12],now known as the LV-M manifolds,which helps us understand the topology of non-k¨ahler complex manifolds.

The study of the second one is related to the quasitoric manifolds in the following way:For every quasitoric manifold π :M2n→ Pnthere is a principal Tm?n-bundle Z → M2nwhose composite map with π makes Z a Tm-manifold with orbit space Pn.The topology of the manifolds Z provides an e ff ective tool for understanding inter-relations between algebraic and combinatorial aspects such as the Stanley-Reisner rings,the subspace arrangements and the cubical complexes,etc.

Nowadays,studies of moment-angle manifolds are mainly focused on the following two aspects:

(1)The cohomology of moment-angle manifolds and topology of some special moment-angle manifolds.

(2)The geometry of moment-angle manifolds in its convex,complex-analytic,symplectic and Lagrangian aspects.

In this paper,we pay attention to the Riemannian metric property of moment-angle manifolds in aspect of Ricci curvature.In Riemannian geometry,one of the most important themes is to study the relationship between the curvature and globally topological or geometrical property of Riemannian manifolds.In three types of curvature,scalar curvature has the weakest relationship with the geometrical property of the manifolds,but it is the best understood one according to the work of Gromov-Lawson[8–9]and Schoen-Yau[13–14].Sectional curvature has the closest relationship with the topological and geometrical property.In some sense,sectional curvature controls nearly all aspects of Riemannian geometry.In order to get some geometric properties of manifolds,usually we should give some restriction of sectional curvature.Besides,one of the most important problems in geometry is the classification of the Riemannian manifolds with sectional positive(or non-negative)metrics and sectional negative(or non-positive)metrics,which as known is far from totally understanding.As a second order symmetric tensor,Ricci curvature is closely related to many elliptic,parabolic and nonlinear differential equations in geometry.Ricci curvature also plays an important role in the general relativity theory in physics and the existence problem of Ricci positive metric(or Einstein metric,K¨ahler-Einstein metric)on a given manifold is also important.However,we have few methods to determine whether a given manifold can admit a Ricci positive metric.Until now,we have not known many examples of manifolds with positive Ricci curvature(Biquotients,connected sums of Sni×Smi(see[15,17]),Fano varieties,some principal G bundles on Ricci positive manifolds(see[6],etc).

For the moment-angle manifolds,the existence of scalar positive metric can be easily proved(see[16]).As far as we know,[1]is the only paper concerned with the Ricci positive metrics on moment-angle manifolds.The authors constructed Riemannian metrics of positive Ricci curvature on 3 special moment-angle manifolds.In this paper,we also study the problem of which moment-angle manifolds admit Ricci positive metrics.We prove the following two theorems.

Theorem 1.1 Let P be a simple polytope,Pvbe a simple polytope which is obtained by cutting o ffone vertex v on P.If the generalized moment-angle manifold corresponding to P admits a Ricci positive metric,then the generalized moment-angle manifold corresponding to Pvalso admits a Ricci positive metric.

Theorem 1.2 If P is a Fano polytope,then the moment-angle manifold corresponding to P admits a Ricci positive metric.

In the next two sections,we respectively prove these two theorems and finally give two conjectures concerned with the existence of positive Ricci curvature on moment-angle manifolds.

2 Cutting o ffOne Vertex on a Simple Polytope

Definition 2.1 A convex polytope P is the convex hull of a finite set of points in some Rn.0-dimensional faces are called vertices,codimension one faces are called facets.If there are exactly n facets meeting at each vertex of n-dimensional convex polytope Pn,this polytope are called simple.

Given a simple polytope Pn,let F={F1,···,Fm}be the set of facets of P.For each facet Fi∈F,we use TFito denote the 1-dimensional coordinate subgroup of TF～=Tmcorresponding to Fi.Then assign to every face G the coordinate subtorus.For every point q∈P we denote by G(q)the unique face containing q in the relative interior.We can define the moment-angle manifold corresponding to P by the following way.

Definition 2.2 For a simple polytope P introduce the moment-angle manifold ZP=(TF×P)/～,where(t1,p)～ (t2,q)if and only if p=q and

Alternatively,we can define moment-angle manifold in another way:Since P is a simple polytope,the dual of the boundary of P is a simplicial(n?1)-sphere,which we denote by K.Let[m]={1,···,m}represent the vertices of K,σ be a simplex in the complex K.Define

and define the moment-angle complex ZKcorresponding to K as

From[2],ZKis homeomorphic to ZP.When we study the topology of moment-angle manifold corresponding to simple polytopes(or simplicial complexes),we consider the behavior of moment-angle manifolds under some surgeries on the simple polytopes(or simplicial complexes).One important surgery is cutting o fffaces of polytopes(or bistellar moves on simplicial complexes).

Definition 2.3 Let P be a simple polytope of dimension n with m facets,which is the convex hull of finitely many vertices in Rn.For any face G in P,we can find a hyperplanesatisfying that H(v)>b and H(w)

Let K be the dual simplicial complex of the boundary of P on the vertex set[m],and σ∈K be the simplex dual to the face G of P.Then the dual simplicial complex KGof the boundary of PGcan be expressed as

where linkKσ ={τ∈ K:τ?σ ∈ K}and{?}is an additional point.

We consider a simple case that linkKσ is the boundary of k-simplex τ.In this case,

By the definition,the moment-angle complex corresponding to KGis

Obviously,ZKGis diffeomorphic towhich can be interpreted as performing an“equivariant surgery operation”on ZK×S1.

It may be very complicated if we consider the topology of the moment-angle manifold corresponding to KGwhen G is a high dimensional face.However,we have known the change of topology of the moment-angle manifold after cutting o ffa vertex v on a polytope P.

After cutting o ffone vertex v of the simple polytope P,we obtain a new simple polytope Pv.Let KPand KPvbe the duals of the boundary of P and Pv,σ be the maximal simplex in KPdual to the vertex v of the simple polytope P.Then we have KPv=KP#σ?△n(△nis the standard n-dimensional simplex,and the choice of a maximal simplex in?△nis irrelevant).By the definition,the moment-angle complex corresponding to P(or KP)is

Then we can express the moment-angle complex corresponding to Pv(or KPv)as follows(see[2,6.4]):

In[7],Gitler and L′opez conjectured that ZPvis diffeomorphic to

In[3],we proved this conjecture by the following way.

First,we construct an isotopy ofin Z to move it to the regular embedding(see Remark 2.1)Tm?n? Dm?n+1? Dm+n? Z.The key lemma in the construction is as follows.

Lemma 2.1 We have two embeddings of Tkin Dk+2:

(1)Tk=Tk?1×S1? Dk×D2,where Tk?1is the regular embedding in Dk.

(2)Tk?Dk+1?Dk+1×D1,where Tkis the regular embedding in Dk+1.

These two embeddings are isotopic.

Proof The normal bundle of the regular embedding Tk?1in Dkis trivial,so we can choose a neighborhood of Tk?1which is diffeomorphic to Tk?1×R1.We can construct an isotopy of Tkin Dk+2:

An examination of this isotopy proves the lemma.

Using this lemma,we can inductively construct an isotopy ofin Z to move it to the regular embedding Tm?n? Dm?n+1? Dm+n? Z,thus prove the following proposition.

Proposition 2.1 ZPvis diffeomorphic to

where Tm?n×D2nis the regular embedding in Sm+n.

Remark 2.1 We construct the regular embedding of Tkinto Rk+1as follows:S1?D2?R2.Assume that we have constructed the embedding of Ti?1into Di? Ri.Represent(i+1)-sphere as Si+1=Di× S1∪ Si?1× D2.By the assumption,the torus Ti=Ti?1× S1can be embedded into Di×S1and therefore into Si+1.Since Tiis compact and Si+1is the one-point compactification of Ri+1,we have Ti?Ri+1.Inductively,we can construct the regular embedding of Tkinto Rk+1(or Dk+1).The regular embedding of Tkinto Rnis Tk? Rk+1×{0}? Rk+1×Rn?k?1,where Tk? Rk+1×{0}is the regular embedding of Tkinto Rk+1.

Similarly,we can construct the regular embedding of(Sn)kinto Rnk+1.

Then we prove the following by induction.

Proposition 2.2 ?[(Sm+n?Tm?n×D2n)×D2]is diffeomorphic to

where Tm?n×D2nis the regular embedding in Sm+n.

Combining these two propositions,the conjecture is proved.However,we can replace the pair(D2,S1)with(Dk+1,Sk)(k≥2)in the definition of moment-angle manifolds to obtain generalized moment-angle manifolds.We use ZP,kto denote the generalized moment-angle manifold corresponding to P,then the generalized moment-angle manifold ZPv,kcorresponding to Pvis diffeomorphic to ZPv,k～= ?[(Z ? (Sk)m?nbσ× (Dk+1)nσ)× Dk+1].In a way similar to the case of k=1,we construct an isotopy ofin ZP,kto move it to the regular embedding(Sk)m?n?Dk(m?n)+1?Dkm+n?ZP,kand we have a lemma similar to Lemma 2.1.

Lemma 2.2 There are two embedding of(Sn)kinto Dnk+2:

(1)(Sn)k?Dnk+1×{0}?Dnk+2,where(Sn)k?Dnk+1×{0}is the regular embedding.

(2)(Sn)k=(Sn)k?1× Sn? Dnk?n+1× Dn+1=Dnk+2,where(Sn)k?1? Dnk?n+1and Sn?Dn+1are regular embeddings.

These two embeddings are isotopic to each other in Dnk+2.

ProofThe normal bundle of the regular embedding(Sn)k?1in Dn(k?1)+1is trivial,so we can choose a neighborhood of(Sn)k?1which is diffeomorphic to(Sn)k?1× R1.We can construct an isotopy of(Sn)kin Dnk+2:

where we use(y1,y2,···,yn+1)to express the unit sphere Sn(1)in Rn+1.An examination of this isotopy proves the lemma.

By this lemma,we can construct an isotopy ofin ZP,kto move it to the regular embeddingthus prove the following proposition.

Proposition 2.3 ZPv,kis diffeomorphic to

where(Sk)m?n×D(k+1)nis the regular embedding in Skm+n.

Then using the same method of proving Proposition 1.2 in[3],we can prove the following by induction(see Section 4).

Proposition 2.4 ?[(Skm+n?(Sk)m?n×D(k+1)n)×Dk+1]is diffeomorphic to

where(Sk)m?n×D(k+1)nis the regular embedding in Skm+n.

Combining these two propositions,we can prove the following theorem.

Theorem 2.1 If P is a simple polytope,the generalized moment-angle manifold ZPv,kcorresponding to Pvis diffeomorphic to

In order to prove Theorem 1.1,we firstly recall some notations and theorems.

Suppose that we are given a Riemannian manifold Mp+dhaving positive Ricci curvature and an isometric embedding: ι:Sp(ρ)×Dd(R,N)→ M where Sp(ρ)is the p-sphere with the round metric of radius ρ,Dd(R,N)denotes a geodesic ball of radius R in the d-sphere with the round metric of radius N.We can regard ι as a trivialization of the normal bundle of ι(Sp× {0}).A corollary of the main Lemma 1 in[15]is the following result.

Theorem 2.2(see[17,Section 4,Theorem])Let～=(M ?Sp×Dd)∪Dp+1×Sd?1be the result of performing surgery on ι(Sp× {0})using the trivialization ι,and assume p ≥ 1,d ≥ 3.Then there exists κ(p,d,RN?1)>0 such that if<κ thencan be equipped with a Ricci positive curvature,the metric on a neighborhood Sd?1× Dp+1of Sd?1is the product of the metric on a round sphere Sd?1and the metric on a geodesic ball Dp+1in the(p+1)-sphere.

Remark 2.2 In[15],the authors used the warped product to construct a Ricci positive metric on Dd×Spsuch that the metric on a submanifold(Sd?1×I)×Sp? Dd×Sp(Sd?1×{0}×Spis the boundary?Dd×Sp)is Ricci positive satisfying that

(1)the metric on the submanifold Sd?1× [0,?]× Spis isomeric to a neighborhood of the boundary of the product of a geodesic ball Ddin the d-sphere and a round sphere Sp,

(2)the metric on the submanifold Sd?1×[1??,1]×Spis isometric to a neighborhood of the boundary of the product of a round sphere Sd?1and a geodesic ball Dp+1in the(p+1)-sphere.

So there exists a Ricci positive metric on

such that

(1)the metric on M?Sp×Ddinherits from the Ricci positive metric on M,

(2)the metric on Sd?1×I×Spis the Ricci positive metric constructed above,

(3)the metric on Dp+1× Sd?1is isometric to the product of the metric on a geodesic ball Dp+1in the(p+1)-sphere and the metric on a round sphere Sd?1.

The proof of Theorem A in[17]shows the following theorem.

Theorem 2.3 Let Sm×Dn(m>n≥3)be a neighborhood of an embedded sphere Smin M.If manifold M admits a Ricci positive metric such that the restricted metric on Sm×Dnis the product metric of the round metric of sphere Smand a geodesic ball Dnin the n-sphere,then any connected sum M#Sm1× Sn1#···#Smk× Snkadmits a metric of positive Ricci curvature for mi,ni≥3 and mi+ni=m+n for all i.

Remark 2.3 Consider the Ricci positive metrics on Dn×Sp+q+1,where n≥3,p≥2,q≥1.Let Dn+q+1×Sp=Dn×(Dq+1×Sp)?Dn×Sp+q+1be the product of embedding DnId→Dnand Dq+1×Sp?Sp+q+1.Then there is a Ricci nonnegative metric on Dn×Sp+q+1such that

(1)a neighborhood of?Dn×Sp+q+1is isomeric to a neighborhood of the boundary of the product of a geodesic ball Dnin the n-sphere and a round sphere Sp+q+1,

(2)the submanifold Dn+q+1×Spis isometric to the product metric of a geodesic ball Dn+q+1in the(n+q+1)-sphere and a round sphere Sp.

Without loss of generality,assume that mi≥ni,so m?ni≥1.M#Smi×Snican be expressed by(M ?Sni?1×Dmi+1)∪Dni×Smi,where Sni?1×Dmi+1? Sm×Dn? M is the product of embedding(Sni?1×Dm+1?ni)?Smand DnId→Dn.So there exists a Ricci nonnegative metric on(M ?Sni?1×Dmi+1)∪Dni×Smisuch that

(1)the metric on M?Sm×Dninherits from the Ricci positive metric on M,

(2)the metric on Sm×Dn?Sni?1×Dmi+1inherits from the Ricci nonnegative metric on Sm×Dnconstructed by the method above,

(3)the metric on Dni×Smiis isometric to the product of the metric on a geodesic ball Dniin the ni-sphere and the metric on a round sphere Smi.

By choosing several small geodesic sub-balls Dnof Dnand constructing a Ricci nonnegative metric on each Sm×Dnby the method above,we obtain a metric of nonnegative Ricci curvature on M#Sm1×Sn1#···#Smk×Snk.As the metric is Ricci positive at many points,by[5]this metric can be deformed to one with everywhere strictly positive Ricci curvature.

Now we come to the proof of Theorem 1.1.

Proof of Theorem 1.1 As noted in[15,p.134],if manifold Mmadmits a Ricci positive metric,then the metric can be deformed to be a Ricci positive one containing a geodesic ball Dmin the m-sphere.So we can always assume that the manifold M with a Ricci positive metric contains a geodesic ball Dmin the m-sphere.If the generalized moment-angle manifold ZP,kadmits a Ricci positive metric,the product of the metric on ZP,kand a round metric on Skis Ricci positive containing Dn+km×Skthe metric of which is the product of the metric on a geodesic ball Dn+kmin the(n+km)-sphere and the metric on a round sphere Sk.With Theorem 2.2,we can prove that

admits a Ricci positive metric,the restricted metric on a neighborhood Sn+km?1×Dk+1of Sn+km?1is the product of the metric on a round sphere Sn+km?1and the metric on a geodesic ball Dk+1in the(k+1)-sphere,when m,n,k ≥ 2,n+km?1>k+1.By Theorems 2.2–2.3,we can prove that the generalized moment-angle manifold

admits a Ricci positive metric if generalized moment-angle manifold ZP,kadmits a Ricci positive metric.

3 Fano Polytope

In this section,we will prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics.Now we come to the definition of Fano polytope.

Definition 3.1 Let Q be a simplicial convex polytope in Rnwhose vertices are primitive lattice vectors{li}(li∈ Zn),and which contains 0 in the interior.If a1,···,anare the vertices of a facet of Q,we suppose det(a1,···,an)= ±1 for every facet.Then we call Q a Fano polytope.

The boundary of Q is a simplicial sphere K,from which we can construct a moment-angle manifold ZK.Alternatively,we can define the moment-angle manifold in another way:The dual of Q:P={u ∈ Rn|hu,vi≤ 1,?v∈ Q}is a simple polytope.The normal vector of each facet can be chosen as one of the lattice vectors{li},we assume that the lattice vector corresponding to facet Fiis li.We can construct the moment-angle manifold ZPcorresponding to P which is homeomorphic to ZK.

In order to prove Theorem 1.2,we firstly recall a theorem in[6].

Theorem 3.1 Let Y be a compact connected Riemannian manifold with a metric of positive Ricci curvature.Let π:P →Y be a principal bundle over Y with compact connected structure group G.If the fundamental group of P is finite,then P admits a G invariant metric with positive Ricci curvature so that π is a Riemannian submersion.

Now we come to the proof of Theorem 1.2.

Proof of Theorem 1.2 Given a Fano polytope Q,we can define the complete fan Σ(Q)whose cones are generated by those sets of vertices li1,···,likwhich are in one face of Q.From this fan,we can construct a toric variety MP.This toric variety is smooth and Fano(see[4])(Fano means that the anticanonical divisor is ample).By Calabi-Yau’s theorem(see[18]),the Fano variety MPadmits a Ricci positive metric.

Topologically,toric Fano variety can be constructed from the polytope P and the lattice vectors{li}by the following way(see[2]):We identify the torus Tnwith the quotient Rn/Zn.For each point q∈P,define G(q)as the smallest face that contains q in its relative interior.The normal subspace to G(q)is spanned by the primitive vectors licorresponding to those facets Fiwhich contain G(q).Since N is a rational space,it projects to a subtorus of Tn,which we denote by T(q).Then as a topological space,the toric Fano variety

where(t1,p)～ (t2,q)if and only if p=q and t1t?12∈ T(q).

From[2],the moment-angle manifold ZPis a principal Tm?nbundle ZP→ MP.Since ZPis simply connected and MPadmits a Ricci positive metric,by Theorem 3.1,ZPadmits a Tm?ninvariant metric with positive Ricci curvature.

Now we give a conjecture.

Conjecture 3.1 P is a simple polytope.

(1)k≥1.If a generalized moment-angle manifold ZP,kadmits a Ricci positive metric,so does ZP,k+1.

(2)For k≥2,ZP,kadmits a Ricci positive metric for every simple polytope P.Momentangle manifold ZPadmits a Ricci positive metric for every irreducible simple polytope P.

If the conjecture(1)is true,by Theorem 1.2,the generalized moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics;by Theorem 1.1,we can prove that the generalized moment-angle manifolds corresponding to polytopes obtained by cutting o ffvertices of Fano polytopes admit Ricci positive metrics.So we can obtain a class of polytopes that the corresponding generalized moment-angle manifolds admit Ricci positive metrics.Besides,in[1],the authors constructed a Ricci positive metric on the moment-angle manifold corresponding to the polytope Pvobtained by cutting o ffone vertex v on the 3-cube P3.However,the dual of Pvis a Fano polytope.So by Theorem 1.2,we can prove that the corresponding moment-angle manifold admits a Ricci positive metric.

From[8],we know that any manifold obtained from a manifold which admits scalar positive curvature by performing surgeries in codimension≥3 also admits a scalar positive curvature.For the Ricci curvature,when we perform surgery on manifolds with Ricci positive metrics,whether the manifold(M ? Sp× Dn?p)∪ Dp+1× Sn?p?1obtained by surgery can admit Ricci positive curvature may depend on the restricted metric of Sp×Dn?pin M(see[15,17]).Similarly,suppose that the generalized moment-angle manifold ZP,kcorresponding to P admits a Ricci positive metric.After cutting o ffa face G of P,the dual simplicial complex KGof the boundary of PGcan be expressed as

where linkKσ ={τ∈ K:τ?σ ∈ K}and{?}is an additional point.We hope that the restricted metric of the submanifold in ZP,kcorresponding to σ ?linkKσ can be“good” enough that we can extend the Ricci positive metric to ZPG,k.

4 Appendix

In this appendix,we will prove Proposition 2.4 by induction.

While m?n=1,the manifold

Inductively suppose that we have proved thatis diffeomorphic to

where(Sk)i×D(k+1)n? D(k+1)n+kiis the regularembedding.So the manifold ?[(S(k+1)n+k(i+1)?(Sk)i+1×D(k+1)n)×Dk+1]is diffeomorphic to

By induction,it is diffeomorphic to

As

is diffeomorphic to

Recalling Lemmas 1–2 in[7],we can generalize these two lemmas as the following.

Lemma 4.1 Assume k≥2,

(1)Let M and N be connected and closed n-manifolds.Then?[(M#N?Dn)×Dk]is diffeomorphic to?[(M ?Dn)×Dk]#?[(N ?Dn)×Dk].

(2)Let M,N be connected n-manifolds.If M is closed but N has non-empty boundary,then?[(M#N)×Dk]is diffeomorphic to?[(M ?Dn)×Dk]#?(N ×Dk).

(3)?[(Sp×Sq?Dp+q)×Dk]=Sp×Sq+k?1#Sp+k?1×Sq.

The proof of the lemma is the same as that of Lemma 1 and Lemma 2 in[7].

By induction,we can prove Proposition 2.4.