Dual complexes of log Calabi–Yau pairs with big boundary components
Follow the latest progress by email.
For a surface qdlt log Calabi–Yau pair with three big boundary components, must the dual complex be the boundary of a two-dimensional simplex? The scoped counterexample target has a square dual complex.
Why it matters and what counts as progress
Why it matters: It tests a proposed combinatorial rigidity of log Calabi–Yau boundaries under bigness assumptions.
What counts as progress: Construct a qualifying log Calabi–Yau pair whose dual complex is not the proposed simplex boundary, or prove a corrected classification.
How the problem developed
- 29 Jul 2025 · Open problem [1]
Paolo Mauri and Joaquín Moraga Asked whether a log Calabi–Yau pair with n+1 big boundary divisors must be qdlt with dual complex the boundary of an n-simplex.
- 24 Aug 2026 · Counterexample claim [2]
Junyan Liu The preprint gives a three-dimensional example that is not qdlt and a qdlt surface example whose dual complex is a square, disproving the proposed conclusions in the stated cases.
AI contribution
On a question of Mauri and Moraga
A surface qdlt log Calabi–Yau pair with three big boundary components has a square dual complex, rather than the boundary of a two-dimensional standard simplex. This answers part (2) of Mauri and Moraga’s Question 9.9 negatively.
Sources
Website timeline
Publication dates and editorial changes.