On canonical metrics of complex surfaces with split tangent and related geometric PDEs

TL;DR

This paper investigates canonical metrics on complex surfaces with split tangent bundles, introduces new geometric PDEs, and constructs solutions leading to canonical metrics on specific complex surfaces.

Contribution

It introduces a new non-linear PDE framework for such surfaces and solves the prescribed Bismut Ricci problem, providing canonical metrics on Hopf and Inoue surfaces.

Findings

Constructed 2 types of metric cones.

Established smooth solutions to the new PDE.

Obtained canonical metrics on primary Hopf and Inoue surfaces.

Abstract

In this paper, we study bi-Hermitian metrics on complex surfaces with split holomorphic tangent bundle and construct 2 types of metric cones. We introduce a new type of fully non-linear geometric PDE on such surfaces and establish smooth solutions. As a geometric application, we solve the prescribed Bismut Ricci problem. In various settings, we obtain canonical metrics on 2 important classes of complex surfaces: primary Hopf surfaces and Inoue surfaces of type $\mathcal{S}_{M}$.