We introduce new boundary conditions for differential forms on symplectic manifolds with boundary. These boundary conditions, dependent on the symplectic structure, allows us to write down elliptic boundary value problems for both second-order and fourth-order symplectic Laplacians and establish Hodge theories for the cohomologies of primitive forms on manifolds with boundary. We further use these boundary conditions to define a relative version of the primitive cohomologies and to relate primitive cohomologies with Lefschetz maps on manifolds with boundary. As we show, these cohomologies of primitive forms can distinguish certain Kahler structures of Kahler manifolds with boundary.

Let$C$and$C$′ be two smooth self-transverse immersions of$S$^{1}into ℝ^{2}. Both$C$and$C$′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes$C$to$C$′ induces a one-to-one correspondence between the disks of$C$and$C$′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking$C$to$C$′ is that there exists an isotopy for which the area of every disk of$C$equals that of the corresponding disk of$C$′. In this paper we show that this is also a sufficient condition.

We study Nakai-Moishezon type question and Donaldson’s “tamed to compatible” question for almost complex structures on rational four manifolds. By extending Taubes’ subvarieties-currentform technique to J-nef genus 0 classes, we give affirmative answers of these two questions for all tamed almost complex structures on S^2 bundles over S^2 as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.

We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using T-duality and generating functions of open Gromov--Witten invariants. The variety is singular in general. We construct a resolution of the variety using the wall and chamber structure on the base of the SYZ fibration.

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