Abstract

We study the moduli space of J-holomorphic subvarieties in a 4-dimensional symplectic manifold. For an arbitrary tamed almost complex structure, we show that the moduli space of a sphere class is formed by a family of linear system structures as in algebraic geometry. Among the applications, we show various uniqueness results of J-holomorphic subvarieties, e.g. for the fiber and exceptional classes in irrational ruled surfaces. On the other hand, non-uniqueness and other exotic phenomena of subvarieties in complex rational surfaces are explored. In particular, connected subvarieties in an exceptional class with higher genus components are constructed. The moduli space of tori is also discussed, and leads to an extension of the elliptic curve theory.

Highlights

  • We study the moduli space of J -holomorphic subvarieties where the almost complex structure J is tamed by a symplectic form

  • J -holomorphic subvarieties are the analogues of one dimensional subvarieties in algebraic geometry

  • In Proposition 3.2, we have shown that all curves having the homology class aU + bT − i ci Ei must have a ≥ 0

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Summary

Introduction

We study the moduli space of J -holomorphic subvarieties where the almost complex structure J is tamed by a symplectic form. There are classes of exceptional curves, such that the moduli space are of complex dimension 1 and some representatives have an elliptic curve component The first statement follows from the fact that the positive fiber class of an irrational ruled surface is J -nef for any tamed J (Proposition 3.2). This is an important ingredient for almost all the results in this paper It follows directly from the second statement of Theorem 1.1 that the J holomorphic subvariety in class E is connected and has no cycle in its underlying graph for any tamed J by Gromov compactness, since these properties hold for the Gromov limit of smooth pseudoholomorphic rational curves

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J-holomorphic subvarieties
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J-nef classes
Intersection of subvarieties
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Seiberg–Witten invariants and subvarieties
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Almost Kähler Hodge conjecture
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Irrational ruled surfaces
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Rational surfaces
Connectedness of moduli spaces of subvarieties
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Part 4: Me is path connected when e is a big J -nef class
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J-holomorphic tori
Non-associative addition on elliptic curve
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Moduli space of tori: a case study
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More applications
Spaces of tamed almost complex structures
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Symplectic isotopy of spheres
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