Skip to content
DraconiticResearch
Search Sources API
194,161 studies31 sourcesUpdated 1 min ago
PreprintCC BY 4.0arXiv

K-unstable Toric Varieties and Secondary Polytopes

  • Chi Li
  • Li Sheng
  • Yi Yao
Open at sourcePDFDOICite

Abstract

For K-unstable toric varieties, Székelyhidi's optimal test-function ΘP is a mysterious concave function over the moment polytope P, which gives the maximal destabilizer for K-stability, and encodes the limiting behavior of the divergent Calabi flow. As balanced norms quantize cscK metrics, we show that ΘP can be quantized by the maximal destabilizers for Chow-stability, which are given by the shortest GKZ vectors, i.e., the least norm point on the secondary polytope for the set P∩ k−1ℤn. Properties and algorithm for general sGKZ vectors are given. Our result may provide a new method to detect the K-unstability of toric varieties.

Related

  1. K2-regularity is equivalent to smoothness for toric varieties
    Christian Haesemeyer, George Henderson-Walshe
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Holomorphic polyvector fields on toric varieties via Klyachko filtrations
    Wei Hong, Maosong Xiang
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. G0 of complete split toric varieties
    Zeyu Shen
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. There are no unexpected toric positroid varieties
    Avery McNeill
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  5. On everywhere locally soluble varieties in families over toric bases
    Vladi Mitankin, Leo Schäfer, Damaris Schindler +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{li2026kunstable,
  title = {{K-unstable Toric Varieties and Secondary Polytopes}},
  author = {Chi Li and Li Sheng and Yi Yao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05179},
  eprint = {2610.05179},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05179},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - K-unstable Toric Varieties and Secondary Polytopes
AU  - Chi Li
AU  - Li Sheng
AU  - Yi Yao
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05179
UR  - https://arxiv.org/abs/2610.05179
AB  - For K-unstable toric varieties, Székelyhidi's optimal test-function Θ_P is a mysterious concave function over the moment polytope P, which gives the maximal destabilizer for K-stability, and encodes the limiting behavior of the divergent Calabi flow. As balanced norms quantize cscK metrics, we show that Θ_P can be quantized by the maximal destabilizers for Chow-stability, which are given by the shortest GKZ vectors, i.e., the least norm point on the secondary polytope for the set P∩ k^−1ℤ^n. Properties and algorithm for general sGKZ vectors are given. Our result may provide a new method to detect the K-unstability of toric varieties.
ER  -