A polyptych of multi-centered deformation spaces

TL;DR

This paper introduces a new framework for studying deformation spaces via multi-centered dilatations, establishing isomorphisms that relate complex deformation spaces to simpler ones, and enabling detailed stratification analysis.

Contribution

It develops a novel class of asymmetric deformation spaces linked to chains of immersions and proves panelization isomorphisms that connect deformation spaces of different lengths.

Findings

Established panelization isomorphisms between deformation spaces of different lengths

Constructed a polyptych of deformation spaces for arbitrary chain lengths

Enabled computation of strata within deformation spaces

Abstract

We study deformation spaces using multi-centered dilatations. Interpolating Fulton simple deformation space and Rost asymmetric double deformation space, we introduce (asymmetric) deformation spaces attached to chains of immersions of arbitrary lengths. One of the main results of this paper is the so-called panelization isomorphism, producing several isomorphisms between the deformation space of length $n$ and deformation spaces of smaller lengths. Combining these isomorphisms, we get a polyptych $\mathscr{P}(n)$ of deformation spaces. Having these panelization isomorphisms allows to compute the strata -- certain restrictions of special interests -- of deformation spaces.