# Simpson-Mochizuki Correspondence for $\lambda$-Flat Bundles

## TL;DR

This paper extends Mochizuki's Kobayashi--Hitchin correspondence to stable $
lambda$-flat bundles on compact balanced manifolds and explores its applications to moduli space homeomorphisms and dynamical systems.

## Contribution

It generalizes Mochizuki's correspondence to a broader class of manifolds and studies its implications for moduli space topology and dynamics.

## Key findings

- Existence of harmonic metrics on stable $
lambda$-flat bundles over balanced manifolds.
- Homeomorphism between moduli spaces of $
lambda$-flat bundles and Dolbeault moduli space.
- Development and analysis of two-parameter dynamical systems on moduli spaces.

## Abstract

The notion of flat $\lambda$-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi--Hitchin-type theorem for $\lambda$-flat bundles ($\lambda\neq 0$), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable $\lambda$-flat bundles ($\lambda\neq 0$). On the other hand, we study two applications of the Simpson--Mochizuki correspondence to moduli spaces. More concretely, we show this correspondence provides a homeomorphism between the moduli space of (semi)stable $\lambda$-flat bundles over a complex projective manifold and the Dolbeault moduli space, and also provides dynamical systems with two parameters on the latter moduli space. We investigate such dynamical systems, in particular, we calculate the first variation, the fixed points and discuss the asymptotic behaviour.