Chern holonomy under the first and second Chern--Ricci flows

@misc{liao2026chern,
  title = {{Chern holonomy under the first and second Chern--Ricci flows}},
  author = {Yuan Liao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34665},
  eprint = {2610.34665},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34665},
}

TY  - UNPB
TI  - Chern holonomy under the first and second Chern--Ricci flows
AU  - Yuan Liao
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34665
UR  - https://arxiv.org/abs/2610.34665
AB  - We compare the Chern holonomy of the first and second Chern-Ricci flows on compact complex manifolds. For the first flow, we construct an explicit immortal pluriclosed solution on a complex two-torus with full and restricted holonomy U(2), except at one finite time when both groups are U(1)^2; its smooth limit has holonomy SU(2). The solution is uniformly nondegenerate and has bounds of every fixed differential order for all time. In contrast, the second Chern-Ricci flow preserves full and restricted Chern holonomy under the natural unitary identification, including the initial time and nonclosed groups. The proof combines preservation of a connection reduction, positive-time analyticity, and a path-transport comparison. As an application, the optimal minimum mean curvature on a holomorphic bundle is unchanged when metrics are restricted to any realizable full and restricted holonomy class. An explicit projected-curvature source explains the failure of the corresponding Ricci-flow holonomy argument.
ER  -