Isotopy of nodal symplectic spheres in rational manifolds

Abstract

In 1985, M. Gromov proved that any symplectic sphere of degree 1 in CP2 is isotopic to an algebraic line. J. Barraud extended Gromov's work to show that any symplectic sphere of degree d in CP2 with only positive ordinary double point singularities is symplectically isotopic to an algebraic curve. In this paper, We imitate Barraud's approach and further extend the result to the nodal symplectic spheres in rational manifolds. We prove that if (M, w) is a rational symplectic 4-manifold, and A a homology class in H2(M, Z) with Kw(A) < 0, then the space of nodal symplectic spheres in the homology class A has only finitely many isotopy classes.