Circle bundles with PSC over large manifolds
Aditya Kumar, Balarka Sen
TL;DR
The paper constructs infinite families of $2n$-dimensional manifolds $Z$ as Donaldson divisors in integral symplectic $(2n+2)$-manifolds, with $[Z]=\mathrm{PD}(k[\omega])$ for large $k$, such that $Z$ is non‑PSC and $\dim_{\mathrm{MC}}(\widetilde{Z})=2n$. It then builds twisted circle bundles $S^1\to E_Z\to Z$ whose total spaces $E_Z^{2n+1}$ admit metrics of positive scalar curvature while satisfying $\dim_{\mathrm{MC}}(\widetilde{E_Z})\le n+1$, effectively showing a macroscopic dimension drop under the circle bundle. The results address Gromov’s question on circle bundles over enlargeable bases in all dimensions, with a robust symplectic-geometry framework based on Donaldson divisors and Gromov–Lawson–Schoen–Yau surgery to preserve PSC. The work also analyzes the 3‑manifold case, showing obstructions to PSC total spaces in that setting, and provides numerous examples and constructions illustrating how PSC, enlargeability, and macroscopic dimension interact in twisted product manifolds.
Abstract
We construct infinitely many examples of macroscopically large manifolds of dimension $m \geq 4$ equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most $\lceil m/2 \rceil + 1$. In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.