J-Holomorphic Curves in a Nef Class

Abstract

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 ...

The curve cone of almost complex 4-manifolds

“…In other words, is ‐nef when is ‐ample. Then by [, Theorem 1.5], we know any irreducible component of a subvariety in class is a rational curve with . Then the conclusion follows since there is a subvariety in class passing through any given point.…”

“…Hence by [, Proposition 4.5], for any given point of , we have a unique possibly reducible ‐holomorphic rational curve in each of these new square 0 nef classes passing through any given point. By [, Theorem and Corollary 4.11], reducible curves happen only when all components are in . Hence, for any square 0 nef class and for any point on outside the negative curve locus, we have a smooth curve in class .…”

“…The conclusion follows from non‐negativity of intersection of different classes in and since classes in are of type with . Moreover, any reducible representative of such a curve class will have only negative square components by [, Theorem and Corollary 4.11]. Such a representative cannot have as a component since otherwise by nefness of and .…”

“…Another interesting new feature of this reducible curve is one of its component is of genus 1 although the original class is of genus 0. Recall that this cannot happen if the original class is ‐nef by . However, the following question still makes sense.…”

“…Usually, to prove a geometric result for a general (non‐generic) tamed almost complex structure is far more delicate than the generic case, since we have to deal with very general subvarieties. The techniques developed in enable us to have a fairly clear structural picture for subvarieties in a ‐nef class. In this paper, we would develop more techniques to work with a general tamed almost complex structure.…”