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Convergence of Generalized Kähler--Ricci Flow on Toric Fano Manifolds
Abstract
We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold (M,J,𝕋), and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial 𝕋-invariant generalized Kähler metric with its symplectic form in 2πc1(M,J), we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.
Related
- Shrinking Kähler-Ricci solitons on toric fano fibrations
- Extremal Kähler--Ricci solitons on Fano manifolds are Kähler--Einstein
- Finite-Time Singularities of the Kähler--Ricci Flow on a ℂPm-Bundle over a Product of Kähler--Einstein Manifolds
- Complete Kähler Manifolds with pinched Ricci curvature
- An L1 bound for Ricci curvature on Kähler manifolds
BibTeX
@misc{huang2026convergence,
title = {{Convergence of Generalized Kähler--Ricci Flow on Toric Fano Manifolds}},
author = {Liding Huang and Xiaohua Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05201},
eprint = {2610.05201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05201},
}
RIS
TY - UNPB
TI - Convergence of Generalized Kähler--Ricci Flow on Toric Fano Manifolds
AU - Liding Huang
AU - Xiaohua Zhu
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05201
UR - https://arxiv.org/abs/2610.05201
AB - We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold (M,J,𝕋), and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial 𝕋-invariant generalized Kähler metric with its symplectic form in 2πc_1(M,J), we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.
ER -