arXiv subjects
Publications and source records attributed to Matthew Gibson.
We introduce natural deformation classes of generalized K\"ahler structures using the Courant symmetry group. We show that these yield natural extensions of the notions of K\"ahler class and K\"ahler cone to generalized K\"ahler geometry. Lastly we show that the generalized K\"ahler-Ricci flow preserves this generalized K\"ahler cone, and the underlying real Poisson tensor.
We analyze four-dimensional symplectic manifolds of type $X=S^1 \times M^3$ where $M^3$ is an open $3$-manifold admitting inequivalent fibrations leading to inequivalent symplectic structures on $X$. For the case where $M^3 \subset S^3$ is the complement of a $4$-component link constructed by McMullen-Taubes, we provide a general algorithm for computing the monodromy of the fibrations explicitly. We use this algorithm to show that certain inequivalent symplectic structures are distinguished by the dimensions of the primitive cohomologies of differential forms on $X$. We also calculate the primitive cohomologies on $X$ for a class of open $3$-manifolds that are complements of a family of fibered graph links in $S^3$. In this case, we show that there exist pairs of symplectic forms on $X$, arising from either equivalent or inequivalent pairs of fibrations on the link complement, that have different dimensions of the primitive cohomologies.