Generalized K\"ahler-Ricci flow on toric Fano varieties

TL;DR

This paper investigates the behavior of the generalized K"ahler-Ricci flow on toric Fano varieties, demonstrating long-term existence, convergence properties, and extending key functionals to this geometric setting.

Contribution

It extends the theory of K"ahler-Ricci flow to toric Fano varieties with symplectic initial data, establishing global existence and convergence results, and generalizing Perelman's entropy and Mabuchi's K-energy.

Findings

Global existence of the normalized flow on toric Fano varieties.

Extension of Perelman's entropy functional to this setting.

Weak convergence of the flow via an extended Mabuchi's K-energy.

Abstract

We study the generalized K\"ahler-Ricci flow with initial data of symplectic type, and show that this condition is preserved. In the case of a Fano background with toric symmetry, we establish global existence of the normalized flow. We derive an extension of Perelman's entropy functional to this setting, which yields convergence of nonsingular solutions at infinity. Furthermore, we derive an extension of Mabuchi's $K$-energy to this setting, which yields weak convergence of the flow.