A Note on Donaldson’s“Tamed to Compatible”Question

(School of Mathematics,Yangzhou University,Yangzhou,Jiangsu,225002)

A Note on Donaldson’s“Tamed to Compatible”Question

TAN QIANG AND XU HAI-FENG

(School of Mathematics,Yangzhou University,Yangzhou,Jiangsu,225002)

Communicated by Lei Feng-chun

Recently,Tedi Draghici and Weiyi Zhang studied Donaldson’s“tamed to compatible”question(Draghici T,Zhang W.A note on exact forms on almost complex manifolds.arXiv:1111.7287v1[math.SG].Submitted on 30 Nov.2011). That is,for a compact almost complex 4-manifold whose almost complex structure is tamed by a symplectic form,is there a symplectic form compatible with this almost complex structure?They got several equivalent forms of this problem by studying the space of exact forms on such a manifold.With these equivalent forms,they proved a result which can be thought as a further partial answer to Donaldson’s question in dimension4.In this note,we give another simpler proof of their result.

compact almost complex 4-manifold,ω-tame almost complex structure,ω-compatible almost complex structure

1 Introduction

Donaldson[1]asked the following question:

Question 1.1For a compact almost complex4-manifold(M4,J),if J is tamed by a symplectic form,is there a symplectic form compatible with J?

An almost complex structureJon a manifoldM2nis tamed by a symplectic formω,ifωisJ-positive,i.e.,

An almost complex structureJis said to be compatible withω,ifωisJ-positive andJ-invariant,i.e.,

Taubes[2]showed that this question has a positive answer on all compact almost complex 4-manifolds(M4,J)withb+=1 for generic almost complex structures.This problem is related to an almost-K¨ahler analogue of Yau’s theorem(see[3]).

Draghici and Zhang[4]obtained the following result,which can be thought as a further partial answer to the Donaldson’s question in dimension 4.

Theorem 1.1[4]Let(M4,J)be a compact almost complex manifold.The following statements are equivalent:

(i)J admits a compatible symplectic form;

(ii)For any J-anti-invariant form α,there exists a J-tamed symplectic form whose J-anti-invariant part is α.

Draghici and Zhang[4]proposed several equivalent statements of Donaldson’s question. If we use some notations as introduced in[4](also see the next section for explanation),then Question 1.1 can be rewritten as

Question 1.2≠≠∅as well?

We denote by≠∅and≠∅the set of symplectic forms,which areJ-compatible andJ-tamed,respectively.

Question 1.1 has the following two equivalent forms:

Question 1.3Is it true that eitherdΩJ,−∩dΩJ,⊕=∅ordΩJ,⊕=dΩJ,+?

Question 1.4If α∈ΩJ,−satis fi esdα∈dΩJ,⊕,is it true thatd(−α)∈dΩJ,⊕as well?

For a detailed proof of equivalent statements,we refer to[4].

Let(M4,J)be a compact almost complex manifold.∧2(M)is the vector bundle of (real)2-forms onM.Ω2(M)denotes the space of realC∞2-forms,i.e.,theC∞sections of the bundle∧2(M).

Jacts onΩ2(M)as an involution via

whereαJ(·,·)=α(J·,J·).Thus,we can de fi neJ-invariant forms andJ-anti-invariant forms by

It is easy to see thatΩJ,+(M)andΩJ,−(M)are vector spaces and we have

Anyψ∈Ω2(M)admits an orthogonal splitting

whereψJ,+is theJ-invariant part andψJ,−is theJ-anti-invariant part,which are given by

respectively.Zk(M)is the space of closedk-forms onM.Let

They are subspaces ofZ2(M).Draghiciet al.[5]introduced cohomology subgroupsHJ,+(M) andHJ,−(M)ofH2(M;R)as follows:

Tanet al.[6]proved that the dimensionofHJ,−(M)is equal to 0 for generic almost complex structuresJonM.

2 Proof of Theorem 1.1

Lejmi[7]considered the following second order linear di ff erential operator on the smooth sectionsΩJ,−(M)of the bundle ofJ-anti-invariant 2-forms:

soPis de fi ned by

Lemma 2.1P is a self-adjoint strongly elliptic linear operator with kernel the g-harmonic

J-anti-invariant2-forms.Hence,

2-forms.

We give another proof of Theorem 1.1 by using the operatorP.

Proof of Theorem 1.1.(ii)⇒(i).Letα=0∈ΩJ,−(M).By the assumption,there exists a symplectic 2-formωtamingJsuch thatω=ω++ω−,where theJ-anti-invariant partω−=α=0.Soω=ω+inΩJ,+(M)and it is a compatible symplectic form.

(i)⇒(ii).Letω0be a symplectic 2-form compatible withJ.For anyα∈ΩJ,−,by Lemma 2.1,there existα1andα2=dθsuch thatα=α1+α2,whereθ∈Ω1.

Becauseω0is positive,we can choose a positive real numberλlarge enough so that

is positive.Soω1isJ-tamed.Set

and

For anyX∈TM,X≠0,we have

which implies

So for anyX∈TM,X≠0,we have

In addition,

Therefore,ωis a tamed symplectic form as wanted.This completes the proof of Theorem 1.1.

Acknowledgements The authors are very grateful to Professor Wang Hong-yu for his insightful discussions and the reviewers for having pointed out several improvements in writting this paper.

[1]Donaldson S K.Two-forms on Four-manifolds and Elliptic Equations.in:Inspired by Chern S S,Nankai Tracts Math.11.Hackensack,NJ:World Sci.Publ.,2006,153–172.

[2]Taubes C.Tamed to compatible:symplectic forms via moduli space integration.J.Symplectic Geom.,2011,9:161–250.

[3]Weinkove B.The Calabi-Yau equation on almost-K¨ahler four-manifolds.J.Di ff erential Geom., 2007,76(2):317–349.

[4]Draghici T,Zhang W.A note on exact forms on almost complex manifolds.arXiv:1111.7287v1 [math.SG].Submitted on 30 Nov.2011.

[5]Draghici T,Li T J,Zhang W.Symplectic forms and cohomology decomposition of almost complex 4-manifolds.Int.Math.Res.Not.IMRN,2010,(1):1–17.

[6]Tan Q,Wang H Y,Zhang Y,Zhu P.On cohomology of almost complex 4-manifolds.arXiv: 1112.0768v1[math.SG].Submitted on 4 Dec.2011.

[7]Lejmi M.Stability under deformations of extremal almost-K¨ahler metrics in dimension 4.Math. Res.Lett.,2010,(4):601–612.

tion:53D35,53D45,58D29

A

1674-5647(2014)02-0179-04

10.13447/j.1674-5647.2014.02.08

Received date:Nov.19,2012.

Foundation item:The NSF(11071208 and 11126046)of China and the Postgraduate Innovation Project (CXZZ130888)of Jiangsu Province.

E-mail address:tanqiang1986@hotmail.com.