Rational curves on toroidal compactifications of Picard modular varieties

@misc{memariansorkhabi2026rational,
  title = {{Rational curves on toroidal compactifications of Picard modular varieties}},
  author = {Soheil Memariansorkhabi and Sai-Kee Yeung},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12071},
  eprint = {2610.12071},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12071},
}

TY  - UNPB
TI  - Rational curves on toroidal compactifications of Picard modular varieties
AU  - Soheil Memariansorkhabi
AU  - Sai-Kee Yeung
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.12071
UR  - https://arxiv.org/abs/2610.12071
AB  - It was known that the canonical bundle of a smooth toroidal compactification of a noncompact finite-volume ball quotient is nef in dimension n≥3 and ample in dimension n≥6. We prove that the canonical bundle is ample in dimension n≥4 and show that this dimension bound is sharp. We construct infinitely many pairwise noncommensurable noncompact Picard modular threefolds whose smooth projective toroidal compactifications have non-ample canonical bundle. Each compactification contains a smooth rational curve meeting the interior and having canonical degree zero. More generally, in every dimension n≥2, we construct infinitely many pairwise noncommensurable noncompact Picard modular varieties whose smooth toroidal compactifications contain smooth rational curves meeting the interior. In particular, such curves persist in arbitrarily high dimension, even when the canonical bundle is ample.
ER  -