Seminar

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Abouzaid-McLean-Smith have constructed global charts for moduli spaces of (pseudo)-holomorphic curves via methods of differential geometry. This means that, unlike the Kuranishi approach which gives a local description of the moduli spaces as (finite) quotients of zero-loci of maps from G-manifolds to G-representations, this work produces a single manifold, equipped with an action of a compact Lie group, and an equivariant vector bundle, and a section thereof, so that the moduli spaces of genus-0 curves are described as the quotient by the group of the zero locus. This geometric construction has the outcome that the resulting G-manifold submerses over (an open subset) of the moduli space of curves in projective space, with fibres having canonical smooth structures. Since the moduli space of curves in projective space has an algebro-geometric construction, this determines a fibrewise smooth structure over a smooth manifold, which, through methods of smoothing theory, yields a smooth structure on the total space. The conclusion thus is that the moduli spaces of genus 0 curves have smooth global charts.

Floer Homotopy Foundations Seminar: Global Charts In Genus 0 Gromov-Witten Theory