Foliation divisorial contraction by the Sasaki-Ricci flow on Sasakian 5-manifolds

TL;DR

This paper develops a minimal model program for compact quasi-regular Sasakian 5-manifolds with cyclic quotient foliation singularities, using the Sasaki-Ricci flow to perform canonical surgical contractions and extremal ray contractions.

Contribution

It introduces a foliation minimal model program for Sasakian manifolds and proves the existence of canonical contractions via the Sasaki-Ricci flow, extending the analytic minimal model program to this setting.

Findings

Derived the foliation minimal model program for Sasakian 5-manifolds.

Proved existence of foliation canonical surgical contractions under Sasaki-Ricci flow.

Established a Sasaki analogue of the analytic minimal model program.

Abstract

Let (M,{\eta},{\xi},{\Phi},g) be a compact quasi-regular Sasakian 5-manifold with finite cyclic quotient foliation singularities of type (1/r)(1,a). First, we derive the foliation minimal model program by applying the resolution of cyclic quotient foliation singularities. Secondly, based on the study of local model of resolution of foliation singularities, we prove the foliation canonical surgical contraction or the foliation extremal ray contraction under the Sasaki-Ricci flow. As a consequence, we prove a Sasaki analogue of analytic minimal model program with the Keahler-Ricci flow due to Song-Tian and Song-Weinkove.