Spatial degeneracy along weak geodesics in the space of Kähler potentials

@misc{xiao2026spatial,
  title = {{Spatial degeneracy along weak geodesics in the space of Kähler potentials}},
  author = {Xia Xiao and Qi Yao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00744},
  eprint = {2611.00744},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00744},
}

TY  - UNPB
TI  - Spatial degeneracy along weak geodesics in the space of Kähler potentials
AU  - Xia Xiao
AU  - Qi Yao
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00744
UR  - https://arxiv.org/abs/2611.00744
AB  - We study spatial Hessian degeneracy along weak Kähler geodesics. We construct examples in three settings: (1) On every compact Kähler manifold, the spatial form degenerates on a nonempty open set during an interior time interval, although both endpoints are smooth and strictly Kähler. (2) On ALE Kähler manifolds with a holomorphic quotient end and a decaying potential to the Euclidean metric, spatial degeneracy occurs on open regions arbitrarily far along an unbounded horn. (3) For cone and Poincaré endpoint metrics along a smooth divisor, normal spatial degeneracy occurs on open sets approaching the divisor. Both cone endpoints retain the prescribed angle. The first construction answers negatively the spatial-positivity question recorded in Darvas's survey. The ALE and divisor constructions show that weak geodesics need not preserve the asymptotic behavior of the endpoint metrics.
ER  -