國立清華大學 數學系

The isotopy problem for symplecticsubmanifoldsin a symplecticmanifold is always a very interesting topic in symplectictopology. One can use various techniques to study the isopotyproblems for different categories of symplecticsubmanifoldsin different symplecitcmanifolds. For the compact symplectic4-manifold case, one of the approaches to attack the isotopy problem for the symplecticsurfaces in a compact symplectic4-manifold (M,w) is to study the isotopies for J-holomorphic curves in (M,w) where J is an w-tamed almost complex structure on M. This approach is first introduced by M. Gromov. He (1985) proved that any symplecticsphere of degree 1 in CP2 is symplecticallyisotopic to an algebraic line. J. Barraud (1999) extended Gromov'swork to show that any symplecticsphere of degree d in CP2 with only positive ordinary double point singularities is symplecticallyisotopic to an algebraic curve. In fact, We can imitate Barraud's work and further extend the result to the nodal symplecticspheres in rational manifolds by studying the nodal J-holomorphic spheres in them. In this talk, we will briefly introduce the idea and take a walk in the moduli space of J-holomorphic curves.