Moduli Spaces of Stable Polygons and Symplectic Structures on $$\overline {\mathcal{M}} _{0,n} $$
Tóm tắt
In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces
$$\mathfrak{M}_{{\text{r,}}\varepsilon } $$
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
$$\overline {\mathcal{M}} _{0,n} $$
of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to
$$\overline {\mathcal{M}} _{0,n} $$
, brings up a new tool to study the Kähler topology of
$$\overline {\mathcal{M}} _{0,n} $$
. A wild but precise conjecture on the shape of the Kähler cone of
$$\overline {\mathcal{M}} _{0,n} $$
is given in the end.
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