Complex quotients by nonclosed groups and their stratifications

TL;DR

This paper introduces complex stratifications by quasifolds as a generalization of toric varieties, achieved through complex quotients by specific nonclosed subgroups of tori linked to convex polytopes, extending the rational case to nonrational scenarios.

Contribution

It defines complex stratifications by quasifolds and demonstrates their realization as complex quotients by certain nonclosed torus subgroups, broadening the scope of toric geometry.

Findings

Spaces obtained are complex quotients by nonclosed subgroups of tori.

These spaces generalize toric varieties to nonrational convex polytopes.

Provides a framework for complex stratifications in nonrational settings.

Abstract

We define the notion of complex stratification by quasifolds and show that such spaces occur as complex quotients by certain nonclosed subgroups of tori associated to convex polytopes. The spaces thus obtained provide a natural generalization to the nonrational case of the notion of toric variety associated with a rational convex polytope.