We introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and Jcompatible symplectic cones over a large class of almost complex manifolds, including all Kähler manifolds, almost Kähler 4-manifolds and complex surfaces.

For any compact almost complex manifold (M, J), the last two authors [8] defined two subgroups H + J (M ), H − J (M ) of the degree 2 real de Rham cohomology group H 2 (M, R). These are the sets of cohomology classes which can be represented by J-invariant, respectively, Janti-invariant real 2−forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of H 2 (M, R). This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini [6]. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given. 1 TEDI DRAGHICI, TIAN-JUN LI, AND WEIYI ZHANG is not C ∞ -pure (the intersection of H + J (M ) and H − J (M ) is non-empty). 1 Taking products of this example with arbitrary almost complex manifolds, one obtains examples in all dimensions ≥ 6 of almost complex structures which are not C ∞ -pure. Also in section 2, for a compact 4-manifold with an integrable J, we show that subgroups H + J (M ) and H − J (M ) relate naturally with the (complex) Dolbeault cohomology groups. We also show that a complex type decomposition for cohomology does not hold for non-integrable almost complex structures (see Lemma 2.12 and Corollary 2.14).In section 3 we focus on almost complex structures J which admit compatible or tame symplectic forms and we give estimates for the dimensions h ± J in this case. If there are J-compatible symplectic forms, then the collection of cohomology classes of all such forms, the so-called J−compatible cone, K c J (M ), is a subcone of H 2 (M ; R). In fact,is the collection of cohomology classes of J−tamed symplectic forms. Thus it is also important to understand the group H − J (M ). Our investigation of almost complex structures which are tamed by symplectic forms is also motivated by the following question of Donaldson ([4]).Question 1.1. If J is an almost complex structure on a compact 4-manifold M which is tamed by a symplectic form ω, is there a symplectic form compatible with J?In [8] it was shown that the question has an affirmative answer when J is integrable. For progress on a related problem proposed by Donaldson, the symplectic Calabi-Yau equation, and its relation to Question 1.1, the reader is referred to [4], [13], [12], [11].We observe in Theorem 3.3 that an estimate on h + J which is immediate for compatible J's can be carried over to the case of tamed J's as well. Section 3 ends with an equivalent formulation of Donaldson's Question 1.1.In a later paper [5] we will further study the group H − J . We appreciate V. Apostolov for his very useful comments, R. Hind, T. Perutz for their interest, A. Fino and A. Tomassini for sending us their paper [6], and NSF for the partial support. We also thank the referees for their careful reading of the manuscript and useful remarks. 1 We learned of the preprint [6] while putting together the final form of our paper. There are further interes...

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Abstract. We study Nakai-Moishezon type question and Donaldson's "tamed to compatible" question for almost complex structures on rational four manifolds. By extending Taubes' subvarieties-current-form technique to J−nef genus 0 classes, we give affirmative answers of these two questions for all tamed almost complex structures on S 2 bundles over S 2 as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.

Abstract. Taubes established fundamental properties of J−holomorphic subvarieties in dimension 4 in [9]. In this paper, we further investigate properties of reducible J−holomorphic subvarieties. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is J−nef. For a spherical class, it has particularly strong consequences. It is shown that, for any tamed J, each irreducible component is a smooth rational curve. It might be even new when J is integrable. We also completely classify configurations of maximal dimension. To prove these results we treat subvarieties as weighted graphs and introduce several combinatorial moves. Let (M, J) be a closed, almost complex 4−manifold. In this paper we study properties of reducible J−holomorphic subvarieties in M . Here J is not always assumed to be tamed. A closed set C ⊂ M with finite, nonzero 2-dimensional Hausdorff measure is said to be an irreducible J−holomorphic subvariety if it has no isolated points, and if the complement of a finite set of points in C, called the singular points, is a connected smooth submanifold with J−invariant tangent space. ContentsA J−holomorphic subvariety Θ is a finite set of pairs {(C i , m i ), 1 ≤ i ≤ n}, where each C i is irreducible J−holomorphic subvariety and each m i is a non-negative integer. The set of pairs is further constrained so thatPseudo-holomorphic subvarieties are closely related to, but clearly different from pseudo-holomorphic maps. They are the real analogues of one dimensional subvarieties in algebraic geometry. When J is understood, we will simply call a J−holomorphic subvariety a subvariety. An irreducible subvariety is said to be smooth if it has no singular points. A subvariety Θ = {(C i , m i )} is said to be connected if ∪C i is connected.Taubes provides a systematic analysis of pseudo-holomorphic subvarieties in [9]. The knowledge of the structure of reducible J−holomorphic subvarieties is very important, in both the integrable case and the tamed case. Among others, two aspects are especially significant for applications. Firstly, under natural conditions, we need to know that the irreducible components are not too complicated. This point is used for example in the argument in [2] on the structure of rational curves. Secondly, we need to know the moduli space of the reducible subvarieties is not too large to ensure the existence of irreducible subvarieties. This is used in [6] for the study of Donaldons's tamed-to-compatible question and almost Kähler Nakai-Moishezon criterion. These aspects are the main focus of this paper.Suppose C is an irreducible subvariety. Then it is the image of a J−holomorphic map φ : Σ → M from a complex connected curve Σ, where φ is an embedding off a finite set. Σ is called the model curve and φ is called the tautological map. The map φ is uniquely determined up to automorphisms of Σ. This understood, the associated homology class e C is defined to be the push forward of the fundamental class of Σ via φ. And for a subvariety Θ, the associated class ...