Real and symmetric quasi-maps
Tsao-Hsien Chen, David Nadler

TL;DR
This paper constructs a stratified homeomorphism linking polynomial arc groups of real reductive groups and symmetric varieties, extending to moduli spaces of quasi-maps, revealing local homeomorphism of orbit closures to complex varieties.
Contribution
It introduces a multi-point homeomorphism between arc spaces and moduli spaces of quasi-maps, generalizing loop group factorizations and degeneration techniques.
Findings
Stratified homeomorphism between arc groups and symmetric varieties.
Extension to moduli spaces of quasi-maps from $P^1$.
Orbit closures have singularities locally homeomorphic to complex varieties.
Abstract
Let be a real reductive group and let be the corresponding complex symmetric variety under the Cartan bijection. We construct a stratified homeomorphism between the based polynomial arc group of and the based polynomial arc space of . We also prove a multi-point version where we replace arcs by moduli spaces of quasi-maps from the projective line to and . The key ingredients in the proof include: (i) a multi-point generalization of the ``Gram-Schmidt" factorization of loop groups, and (ii) a nodal degeneration of moduli spaces of quasi-maps. As an application, we show that for the closures of real spherical orbits in the real affine Grassmannian, their singularities near the base point are locally homeomorphic to complex algebraic varieties.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Nonlinear Waves and Solitons · Algebraic Geometry and Number Theory
