The pseudo-effective cone of a non-K\"ahlerian surface and applications

TL;DR

This paper characterizes the positive and pseudo-effective cones of non-Kählerian surfaces and explores their implications for Ricci scalars and stability of canonical extensions, aiding in the proof of the GSS conjecture.

Contribution

It provides a detailed description of cones on non-Kählerian surfaces and applies these results to Ricci scalar sets and stability of canonical extensions, advancing understanding in complex surface theory.

Findings

Characterization of positive and pseudo-effective cones on non-Kählerian surfaces

Determination of the deformation invariance of Ricci scalar sets

Analysis of stability of canonical extensions in class VII surfaces

Abstract

We describe the positive cone and the pseudo-effective cone of a non-K\"ahlerian surface. We use these results for two types of applications: - Describe the set $\sigma(X)$ of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-K\"ahhlerian surface, and decide whether the assignment $X\mapsto\sigma(X)$ is a deformation invariant. - Study the stability of the canonical extension $$0\to {\cal K}_X\to {\cal A}\to{\cal O}_X\to 0$$ of a class VII surface $X$ with positive $b_2$. This extension plays an important role in our strategy to prove the GSS conjecture using gauge theoretical methods.