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

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 $\mathcal{S}_{M}$.

💡 Research Summary

The paper investigates bi‑Hermitian metrics on compact complex surfaces whose holomorphic tangent bundle splits as a direct sum of two non‑trivial line sub‑bundles, a condition that appears in the classification of class VI surfaces (primary Hopf surfaces and Inoue surfaces of type S_M). The authors introduce a “split‑type” framework: a (1,1)‑form is called split if it decomposes into a sum of a (+)‑part and a (–)‑part, each living in the exterior algebra of the corresponding sub‑bundle. An involution (\iota) changes the sign of the (–)‑part.

A key analytic tool is the operator
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