K-stability is an algebro-geometric stability theory that characterizes the existence of Kähler-Einstein metrics on Fano varieties. It is usually a challenging problem to check K-stability for a given Fano variety. In this paper, the authors develop a general approach, known as the Abban-Zhuang approach in the literature, to prove K-stability of Fano varieties. This approach has become one of the most powerful techniques in showing K-stability.
We first summarize the main applications. In this paper, the authors prove K-stability of all smooth Fano hypersurfaces of Fano index \(2\). Note that K-stability of index \(1\) smooth Fano hypersurfaces was proved by K. Fujita [J. Inst. Math. Jussieu 18, No. 3, 519–530 (2019; Zbl 1523.14074)] based on Tian’s criterion of alpha-invariants, which does not work as soon as the Fano index is at least \(2\). In particular, the authors give a new proof of K-stability of smooth cubic threefolds, which were obtained earlier by Y. Liu and C. Xu [Duke Math. J. 168, No. 11, 2029–2073 (2019; Zbl 1436.14085)] using moduli methods. There have been many applications of the Abban-Zhuang approach since the appearance of this paper. In a subsequent paper [H. Abban and Z. Zhuang, “Seshadri constants and K-stability of Fano manifolds”, Preprint, arXiv:2101.09246] by the same authors, it was proved that any smooth Fano hypersurface of dimension \(n\) and Fano index \(r\) is K-stable whenever \(n\geq r^3\). In the book [The Calabi problem for Fano threefolds. Cambridge University Press (to appear)], the Abban-Zhuang approach in the equivariant setting was extensively used to obtain an exhaustive list of families of smooth Fano threefolds whose general member is K-polystable.
We briefly explain the approach to prove K-stability from this paper. For a Fano variety \(X\), an \(m\)-basis type divisor \(D\) is given by \[ D = \frac{1}{mN_m}\sum_{i=1}^{N_m} \{s_i = 0\}, \] where \(s_1,\dots, s_{N_m}\) form a basis of the vector space \(V_m=H^0(X, -mK_X)\). Then \(\delta_m(X):=\inf\mathrm{lct}(X; D_m)\) where the infimum is taken over all \(m\)-basis type divisor. The stability threshold of \(X\) is defined as \[ \delta(X) = \lim_{m\to \infty} \delta_m(X). \] From Fujita-Li’s valuative criteria of K-stability, we know that \(X\) is K-stable if and only if \(\delta(X)>1\). Thus to show K-stability of \(X\) it suffices to estimate its stability threshold \(\delta(X)\). The first observation of the authors is that we can take \(D\) to be compatible with any filtration \(\mathcal{F}\) on \(V_m\), that is, every subspace \(\mathcal{F}^j V_m\) is spanned by some of the \(s_i\)’s. For instance, if the filtration \(\mathcal{F}\) is induced by a prime divisor \(E\) over \(X\), we can write a compatible \(m\)-basis type divisor \(D = S_m(E) E + D_0\) where \(D_0\) does not contain \(E\). The second key observation of the authors is that by inversion of adjunction, to show \((X, D)\) is klt near \(E\), it suffices to show two things: \(A_X(E) > S(E)\), and the pair \((E, D_0|_E)\) is klt. Now \(D_0|_E\) becomes a basis type divisor of a bi-graded linear system on \(E\) as restrictions of \(\mathcal{F}^j V_m\). Repeating this process for \((E, D_0|_E)\), or more generally for an admissible flag of subvarieties coming from the construction of Okounkov bodies, the authors established a strong estimate for local stability thresholds in terms of lower dimensional stability thresholds (Theorem 3.4).