Jieming YANG

Abstract The author shows that if a locally conformal Khler metric is Hermitian Yang-Mills with respect to itself with Einstein constant c≤0,then it is a Khler-Einstein metric.In the case of c>0,some identities on torsions and an inequality on the second Chern number are derived.

Keywords Hermitian Yang-Mills metric,Locally conformal Khler metric,Torsion,Chern number inequality

1 Introduction

Let(X,g)be a compact Hermitian manifold of complex dimension n≥2.Let ω=iΣgijdzi∧dzjbe the associated positive definite(1,1)-form,which is also called a Hermitian metric.

Let Rωbe the curvature of the Chern connection of ω.A Hermitian metric ω is a Hermitian Yang-Mills(HYM for short)metric with respect to itself if

In this paper we consider how to generalize Gauduchon and Ivanov’s result to the higher dimensional case.We need some definitions.

A Hermitian metric ω is called a Gauduchon metric if=0.A well-known result in[3]says that there exists a unique Gauduchon metric,up to a constant conformal factor,in the conformal class of a Hermitian metric.

A Hermitian metric ω is called a locally conformal Khler(l.c.K for short)metric if for any point x∈X,there exist an open neighbourhood U of x and a smooth function ϕ∈such that ω′=eϕω is a Khler metric on U.

Denote torsions of the Chern connection of a Hermitian metric ω to be

Then τ=ΣTidziis the torsion 1-form of ω.A Hermitian metric ω is l.c.K if and only if equations

and

hold.Note that when n=2,equation(1.2)always holds for any Hermitian metric ω and when n≥3,(1.2)implies(1.3).These results can be consulted in[2].

As we will see,the natural metric ω on the Hopf manifold of complex dimension n≥2 is a Gauduchon,l.c.K and HYM metric.Our main result is as follows.

Theorem 1.1Let ω be a l.c.K and HYM metric on a compact complex manifold X of dimension n≥2.If c≤0,then ω is a Khler-Einstein metric;If c>0 and ω is also a(non-Khler)Gauduchon metric,then|τ|2=(n−1)c and

Theorem 1.2Let ω be a Gauduchon,l.c.K and HYM metric on a compact complex manifold X of dimension n≥2.Then

The equality holds if and only if ω is either a flat Khler metric or the natural metric on the Hopf surface.

from which we can easily get

where the equality holds if and only if ω is projectively flat.Under the assumption in Theorem 1.2,we will show thathence the inequality(1.5)follows.

This paper is arranged as follows.In Section 2,the geometry of the natural metric on the Hopf manifold of dimension n is studied.In Section 3,some identities on torsion of a Gauduchon and HYM metric are derived and in particular identity(1.4)in Theorem 1.1 is proved.In Section 4,we finish the proof of Theorem 1.1 and in Section 5 we prove Theorem 1.2.

We follow the notations in[5].For a Hermitian metric ω,we denote Rωto be the curvature of the Chern connection of ω.Locally,its components are

2 Hopf Manifolds

Let Hn=S2n−1×S1with n≥2 be the standard Hopf manifold(see[6,Section 6]),equipped with the natural metric

It is direct to check that ω is both Gauduchon and l.c.K.

The torsions of the Chern connection of ω are

which imply D′τ=0 and

The curvature Rωis

and the mean curvature Kωis

Hence ω satisfies the HYM equation(1.1)with

By(2.2),the Ricci curvature of ω is

and hence

i.e.,ω is projectively flat.So the equality in the Bogomolov-Lbke inequality(1.7)holds.

Now we assume n>2.Since

and ω is projectively flat,by the formula in[5,p.42],we have

Moreover,we calculate

Hence the natural metric ω on Hndoes not satisfy the Miyaoka-Yau inequality(1.6),but satisfies the inequality(1.5).

3 Some Identities on Torsion

The start point of Theorem 1.1 is the following identities.Let ω be a HYM metric.Denote

We have

where

Let ω be a Gauduchon metric.Integrating(3.1)over X yields

where the left hand side,after integration by parts,is equal to

Thus we obtain the following result.

Proposition 3.1If a Gauduchon metric ω satisfies the HYM equation(1.1),then

For any Hermitian metric ω,we obtain from the calculation(3.1)that

From the HYM equation(1.1)and the calculation(3.2),we obtain the following result.

Proposition 3.2If a Gauduchon metric ω satisfies the HYM equation(1.1),then

The curvature Rωof the Chern connection of a Hermitian metric ω satisfies the following Bianchi identity

which imples

Combining the Bianchi identity

with the HYM equation(1.1),we obtain

Inserting it into(3.5)yields

Moreover,we have

Let ω be a l.c.K metric.By(1.2),we have

Notice that inserting(3.7)into Proposition 3.1 recovers(3.3).Inserting(3.7)and the HYM equation(1.1)into(3.6),we obtain

Moreover,we have

Integrating it over X and using integration by parts as in(3.2)to the left hand side yields

which implies

Comparing it with(3.3),we obtain the following result.

Proposition 3.3Let n>2 and ω be a Gauduchon and l.c.K metric.If ω satisfies the HYM equation(1.1),then identity(1.4)in Theorem 1.1 holds.

For n=2,by[4]the Hopf surface(H2,ω)is the only non-Khler HYM metric with respect to itself.By(2.1),we have|D′τ|=0 and c|τ|2=|D′′τ|2.Hence,the identity(1.4)also holds for n=2.

4 Proof of Theorem 1.1

Let us recall a well-known result in[3].

Let ω be a l.c.K and HYM metric on a compact complex manifold X of dimension n≥2.We follow the idea in[1]to prove Theorem 1.1.

ProofThere are two scalar curvatures of any Hermitian metric ω:

Since ω satisfies the HYM equation(1.1),and so s=nc.By(3.4),we have

Inserting these and the HYM equation(1.1)into Proposition 3.2 in[1]yields

which implies c=0 and τ=0.Hence ω is a Khler metric due to(3.7).

Indeed,let ω′=efω be a Khler metric for some function f∈(X).By(1.2),

Since

we obtain from(4.1)that

In this case,c<0.Notice that

Inserting γ into the right hand side above,we have

By the Cauchy-Schwarz inequality,we obtain

Hence,the above inequalities hold if and only if f is a constant.Combining the above arguments,we obtain the first part of Theorem 1.1.

As to the second part,we obtain from Lemma 4.1 and(4.2)that−c is a constant.If−c is not identically zero,then ω is Khler.Hence−c≡0,and

where the first identity holds for any Gauduchon metric.

In the case c>0,we obtain from(4.1)that

which implies

By these facts,it seems that the Hopf manifold(Hn,ω)is the only(non-Khler)l.c.K metric satisfying the HYM equation(1.1)with positive Einstein constant.

5 Proof of Theorem 1.2

Let ω be a Gauduchon,l.c.K and HYM metric on a compact complex manifold X of dimension n≥2.We are ready to prove Theorem 1.2.

ProofBy the Bogomolov-Lbke inequality(1.7),the inequality(1.5)holds if

where the second equality follows from the formula[5,(4.1)]and the last one follows from Kω=ρω.

If c>0,then we use again the formula[5,(4.1)]to calculate

From(4.4),(4.3)and(1.4),we obtain

which implies the inequality(5.1),and hence the inequality(1.5).For the equality,we obtain from the Bogomolov-L¨ubke inequality(1.7)that

which contradicts(5.2)unless n=2.By the result in[4],ω is the natural metric on the Hopf surface.

AcknowledgementThe author thanks Professor Jixiang Fu for everything.