RC-positivity and scalar-flat metrics on ruled manifolds

TL;DR

This paper characterizes when ruled surfaces over a curve admit scalar-flat Hermitian metrics, linking geometric properties to the genus and an intrinsic complex-structure-dependent number.

Contribution

It provides a precise criterion involving genus and an intrinsic number for the existence of scalar-flat Hermitian metrics on ruled surfaces.

Findings

Scalar-flat Hermitian metrics exist if and only if g ≥ 2 and m(X) > 2 - 2g.

The existence depends on the genus and an intrinsic complex-structure-dependent number.

The paper establishes a complete characterization for ruled surfaces over curves.

Abstract

Let $X$ be a ruled surface over a curve of genus $g$. We prove that $X$ has a scalar-flat Hermitian metric if and only if $g\geq 2$ and $m(X)>2-2g$ where $m(X)$ is an intrinsic number depends on the complex structure of $X$.