Moduli Spaces of Stable Polygons and Symplectic Structures on ${\bar{M}}_{0,n}$

In this paper, certain natural and elementary polygonal objects in Euclidean space, {\it the stable polygons}, are introduced, and the novel moduli spaces ${\text{bfmit}M}_{\mathbf{r},ϵ}$ of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space ${\overline{\text{cM}}}_{0,n}$ of the Deligne-Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data leading toward to a family of (classes of) symplectic (K"ahler) forms. To some degree, ${\bfmit M}{{\bf r}, \epsilon}$maybeconsideredassymplecticcounterpartsof$\bar{\cM}{0,n}$andKaprano{v}^{\mathrm{\prime }}sChowquotientconstructionof$\bar{\cM}_{0,n}$. All these together brings up a new tool to study the K"ahler topology of ${\overline{\text{cM}}}_{0,n}$.