Compact Complex Surfaces of Locally Conformally Flat Type

TL;DR

This paper classifies compact complex surfaces that admit locally conformally flat metrics, showing they cannot contain certain rational curves and must be minimal, providing a specific list of such surfaces.

Contribution

It establishes new restrictions on the structure of complex surfaces with locally conformally flat metrics and provides a classification list.

Findings

Surfaces with such metrics cannot contain smooth rational curves of odd self-intersection.

Such surfaces must be minimal.

A list of possible surfaces admitting these metrics is provided.

Abstract

We show that if a compact complex surface admits a locally conformally flat metric, then it cannot contain a smooth rational curve of odd self-intersection. In particular, the surface has to be minimal. Then we give a list of possibilities of such surfaces.