We compute the genus zero family Gromov–Witten invariants for K3 surfaces using the topological recursion formula and the symplectic sum formula for a degeneration of elliptic K3 surfaces. In particular we verify the Yau–Zaslow formula for non-primitive classes of index two.

No abstract uploaded!

We compute the genus g = 1 family GW-invariants of K3 surfaces for non-primitive classes. These calculations verify the G¨ottsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two topological recursion formula and the symplectic sum formula to establish relationships among various generating functions.

Vector cross product structures on manifolds include symplectic, volume, G2- and Spin(7)-structures. We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi–Yau manifold defines a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure.We also relate the Calabi–Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.

In this article, we define a new class of middle dimensional submanifolds of a Hyperkahler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation, tt is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.