# Regularizing properties of Complex Monge-Amp\`ere flows II: Hermitian manifolds

## TL;DR

This paper proves that complex Monge-Ampère flows on Hermitian manifolds can start from arbitrary initial conditions and confirms a conjecture that the Chern-Ricci flow performs canonical surgical contractions, also exploring a twisted version.

## Contribution

It establishes the ability to run Monge-Ampère flows from any initial condition and confirms a key conjecture about the Chern-Ricci flow's behavior on Hermitian manifolds.

## Key findings

- Flow can start from arbitrary initial conditions with zero Lelong number.
- Confirmed that Chern-Ricci flow performs canonical surgical contraction.
- Studied a generalized twisted Chern-Ricci flow.

## Abstract

We prove that a general complex Monge-Amp\`ere flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow.