Topics In Almost Complex4-manifolds

An almost complex structure J on M is an endomorphism of the tangent space TxM at every point x∈M, which squares to-1and varies smoothly on the manifold. Then (M, J) is an almost complex manifold.In Chapter1, we study almost Hermitian geometry. In Section1.1, we give a decomposition of the complexification of the cotangent bundle T*M according to the eigenvalues±i of the almost complex structure J, and then obtain the decom-position of the exterior differential operator d on an almost complex manifold. We also introduce two important geometric elements ηc and ηh. By the Newlander-Nirenberg theorem (cf.[72]), the almost complex structure is integrable iff ηc=0. An almost Hermitian manifold (M,g,J,F) is said to be Kahler if V9F=0, or equivalently if η=0. In Section1.2, we introduce the (almost) Hermitian connec-tion. A linear connectionâ–½on M acting on sections of the tangent bundle TM is an almost Hermitian connection ifâ–½J=â–½g=0. By Gauduchon’s work in [38], one can define the canonical connection. It is well known that there exists a unique almost Hermitian connection on(M,g, J, F) whose torsion has everywhere vanish-ing (1,1) part, that is, the second canonical connection. In Section1.3, the last section of this chapter, we study the curvature of the canonical almost Hermitian connection.In Chapter2, our main work is on the J-anti-invariant cohomology. In Section2.1, we study the symplectic cohomology on compact symplectic manifold (M,ω). We consider the symplectic star operator*ω:Ωk(M)â†'Ω2n-k(M) analogously to the Hodge star operator but with respect to the symplectic form ω. J. Brylinski introduced in [12] the notion of symplectic harmonic forms. Further he conjectured that on a compact symplectic manifold, every de Rham cohomology class contains a symplectic harmonic representative. Olivier Mathieu disproved Brylinski’s conjec-ture in [68]. In fact, Mathieu proved that every de Rham cohomology class contains a symplectic harmonic form if and only if the symplectic manifold satisfies the hard Lefschetz property. However, the hard Lefschetz property is too strong for symplec- tic manifolds. So Fernandez M., Munoz V. and Ugarte L. introduced s-Lefschetz property in [34] which is weaker than hard Lefschetz property. With Lin Y, Tseng and Yau’s work, we get that the Lefschetz decomposition for de Rham cohomology also holds for a compact symplectic manifold which is s-Lefschetz. In Section2.2, we study the J-invariant and J-anti-invariant cohomology subgroups HJ±(see [63]). For a closed almost Kahler2n-manifold(M,g, J,ω), we introduce the second order linear differential operator Pj on Ω02(cf.[83]), and get that if dim ker PJ=b2-1, then J is C∞pure and full. Additionally, we investigate the relationship between J-anti-invariant cohomology and new symplectic cohomologies introduced by L.-S. Tseng and S.-T. Yau on a closed symplectic4-manifold. In Section2.3, we compute the dimension of the J-anti-invariant cohomology subgroup HJ-. We denote by hJ-the dimension of HJ-, and get that the set of almost complex structures J on M with hJ-=0is an open dense subset of J in the C∞-topology on a closed4-manifold M admitting almost complex structures.In Chapter3, we study the tamed4-manifolds. In Section3.1, we re-describe Donaldson’s tame to compatible question:For a compact almost complex4-manifold (M4, J), if J is tamed by a symplectic form, is there a symplectic form compatible with J? For the above question, we know that it is trivial when J is integrable. Indeed, we know that on a closed complex surface (M, J), the following are equiv-alent:(i) b1is even;(ii) b+=hJ-+1;(iii) J is tamed. It is a well-known fact that any closed complex surface with b1even is Kahler, that is, the Kodaira conjecture. Note that if (M, J) is a tamed, closed almost complex4-manifold, then it is easy to see that0≤hJ-≤b+-1([81]). Hence, if b+=1, then hJ-=b+-1=0. Thus, Donaldson question (in particular, b+=1) can be regarded as a symplectic analogue of the Kodaira conjecture for tamed almost complex4-manifolds (cf.[26]). In Section3.2, we study the intersection paring on weakly dJ-closed (1,1)-forms (cf.[7]). Our approach is along the lines used by Buchdahl in [15] to give a unified proof of the Kodaira conjecture. We get a critical lemma, i.e., Lemma3.31which is similar to the Lemma7in [15]. Lemma3.31will paly a very important role in the following section, i.e., Section3.3. In Section3.3, we study the tamed almost complex4-manifolds with hJ=b+-1. Finally, we will give an affirmative answer to Donaldson question when hJ-=b+-1by using a very different approach.