@article{Zhou2026, 
author = {Chuyu Zhou},
title = {On the Shape of the K-semistable Domain and Wall Crossing for K-stability},
year = {2026},
journal = {Peking Mathematical Journal},
volume = {9},
number = {2},
pages = {401-431},
keywords = {Log Fano pair, K-stability, K-semistable domain, K-moduli, Wall crossing},
url = {https://www.sciopen.com/article/10.1007/s42543-024-00094-8},
doi = {10.1007/s42543-024-00094-8},
abstract = {Fixing two positive integers d and k, a positive number v, and a positive integer I, we prove that the K-semistable domain of the log pair    (  X  ,      ∑          j      =      1        k        D    j    ) is a rational polytope lying in the k-dimensional simplex              Δ      k        ¯  , where X is a Fano variety of dimension d,        D    j        ∼                  Q              −      K    X  ,    (  −      K    X        )    d    =  v,    I  (      K    X    +      D    j    )  ∼  0, and    (  X  ,      ∑          j      =      1        k        c    j        D    j    ) is a K-semistable log Fano pair for some        c    j    ∈  [  0  ,  1  )  ∩            Q      . Moreover, we show that there are only finitely many polytopes that may appear as the K-semistable domains for such log pairs. Based on this, we establish a wall crossing theory for K-moduli with multiple boundaries.}
}