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A Yau-Tian-Donaldson correspondence for degenerations

  • Gabriel Frey
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Abstract

We prove an equivalence between the existence of a unique Kähler-Einstein metric on a degeneration of Fano varieties and uniform CM minimization. This proves a discretely valued analogue of the (uniform) Yau-Tian-Donaldson conjecture, in the sense of non-Archimedean geometry. As a main tool, we prove that Ding stability of a degeneration implies uniform Ding minimization, proving a converse to work of Blum-Xu. We also prove an analogous result in geometric invariant theory (GIT), stating that uniform GIT height minimization of a family of points is equivalent to GIT stability, proving a converse to work of Wang, Wang-Xu and the author.

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Cite

BibTeXRIS
BibTeX
@misc{frey2026yautiandonaldson,
  title = {{A Yau-Tian-Donaldson correspondence for degenerations}},
  author = {Gabriel Frey},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39996},
  eprint = {2610.39996},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39996},
}
RIS
TY  - UNPB
TI  - A Yau-Tian-Donaldson correspondence for degenerations
AU  - Gabriel Frey
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39996
UR  - https://arxiv.org/abs/2610.39996
AB  - We prove an equivalence between the existence of a unique Kähler-Einstein metric on a degeneration of Fano varieties and uniform CM minimization. This proves a discretely valued analogue of the (uniform) Yau-Tian-Donaldson conjecture, in the sense of non-Archimedean geometry. As a main tool, we prove that Ding stability of a degeneration implies uniform Ding minimization, proving a converse to work of Blum-Xu. We also prove an analogous result in geometric invariant theory (GIT), stating that uniform GIT height minimization of a family of points is equivalent to GIT stability, proving a converse to work of Wang, Wang-Xu and the author.
ER  -