Positive scalar curvature Kähler cones on compact complex surfaces

@misc{sha2026positive,
  title = {{Positive scalar curvature Kähler cones on compact complex surfaces}},
  author = {Zehao Sha},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.27870},
  eprint = {2609.27870},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27870},
}

TY  - UNPB
TI  - Positive scalar curvature Kähler cones on compact complex surfaces
AU  - Zehao Sha
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.27870
UR  - https://arxiv.org/abs/2609.27870
AB  - We determine the positive scalar curvature Kähler cone of every minimal compact Kähler surface of Kodaira dimension −∞. By the Enriques--Kodaira classification, every such surface is either ℙ^2 or a geometrically ruled surface ℙ(E)→Σ_g, where E is a rank-two holomorphic vector bundle over a compact Riemann surface of genus g. On ℙ^2, every Kähler class admits a Kähler metric of positive scalar curvature, whereas on ℙ(E), every Kähler class of positive total scalar curvature admits a metric of positive scalar curvature if and only if g≤1 or E is slope-semistable. We further prove that if a smooth compact Kähler surface admits a birational morphism onto a ruled surface ℙ(E)→Σ_g, where g≥2 and E is slope-unstable, then it contains a Kähler class of positive total scalar curvature admitting no positive scalar curvature Kähler metric. Finally, for every smooth compact Kähler surface X birational to Y=ℙ^2 or Y=ℙ(E), we show that Kähler classes admitting positive scalar curvature metrics on Y are exactly the pushforwards of classes admitting such metrics on X.
ER  -