Troncoso–Urzúa Chern-slope density conjecture

The conjecture concerns how densely Chern slopes can occur for surfaces with a prescribed fundamental group. Earlier work established density throughout $[1,3]$; the latest report claims an extension to $[1/2,3]$.

Known results

• The 2015 paper Chern slopes of simply connected complex surfaces of general type established fixed-genus fundamental-group examples with slopes approaching every $r \in [1,3]$.
• For simply connected surfaces, it obtained spin examples for $r \in [2,3]$ and non-spin minimal examples for $r \in [1,3]$.
• Savage Surfaces states density in $[1,3]$ for every topological fundamental group of a nonsingular complex projective surface, including finite groups.
• The corresponding freedom question below $1$ was recorded as open, with a specific restriction below $1/3$ for finite fundamental groups.

August 2026 claimed extension

On August 26, 2026, a report citing The geography of Chern slopes with prescribed fundamental group announced that density extends from $[1,3]$ to $[1/2,3]$. The interval below $1/2$ remains unresolved; the claim has not been independently verified in the retrieved material.

Current status (as of August 2026): Density is established on $[1,3]$, while the extension to $[1/2,3]$ is claimed but unverified and the range below $1/2$ remains open.