## A minimizing valuation is quasi-monomial.(English)Zbl 1469.14033

In this important paper, the author proves a number of fundamental results for valuations, singularities and K-stability.
The first main theorem in the paper under review confirms the weak version of Conjecture B in [M. Jonsson and M. Mustaţă, Ann. Inst. Fourier 62, No. 6, 2145–2209 (2012; Zbl 1272.14016)]. Let $$(X,\Delta)$$ be a klt pair and $$\mathfrak{a}_{\bullet}:=\{\mathfrak{a}_k\}_{k\in \mathbb{N}}$$ is a graded sequence of ideals such that the log canonical threshold $$\text{lct}(\mathfrak{a}_{\bullet})<\infty$$. Then there exists a quasi-monomial valuation calculating the log canonical threshold of $$\mathfrak{a}_{\bullet}$$.
There are several interesting applications of the theorem above. The first application confirms a quasi-monomialness part of the Stable Degenerate Conjecture proposed by C. Li [Math. Z. 289, No. 1–2, 491–513 (2018; Zbl 1423.14025)]. Moreover precisely, for a klt singularity $$x\in(X,\Delta)$$, C. Li has defined the normalized volue function for real valuations centred at $$x$$. The Stable Degenerate Conjecture asserts that the minimizer of the normalized volume function exists, is unique up to scaling, is quasi-monomial with a finitely generated associated graded ring, and degenerates the singularity to a K-semistable log Fano cone. The existence is proved by H. Blum [Compos. Math. 154, No. 4, 820–849 (2018; Zbl 1396.14007)]. The uniqueness was shown by the author of the paper under review in his joint work with Z. Zhuang [Camb. J. Math. 9, No. 1, 149–176 (2021; Zbl 1483.14005)]. As an application of the theorem above, the author in the paper under review proves the quasi-monomialness part of the conjecture above.
As a second application, the author finishes the algebraic approach for solving the Demailly-Kollár Openness Conjcture initiated in [M. Jonsson and M. Mustaţă, J. Inst. Math. Jussieu 13, No. 1, 119–144 (2014; Zbl 1314.32047)]. Here we recall that the Openness Conjecture was solved by By B. Berndtsson [Abel Symp. 10, 29–44 (2015; Zbl 1337.32001)] using analytic tools, see also [Q. Guan and X. Zhou, Ann. Math. (2) 182, No. 2, 605–616 (2015; Zbl 1329.32016)] for a more stronger version.
The second main theorem of the paper under review is the constructibility of local volume of $$\mathbb{Q}$$-Gorenstein family of klt singularities (Theorem 1.3). We recall that in H. Blume and C. Liu proved the lower semi-continuity of the local volume in [H. Blum and Y. Liu, J. Eur. Math. Soc. (JEMS) 23, No. 4, 1225–1256 (2021; Zbl 1470.14008)].
This theorem above has important applications to study the moduli space of $$K$$-semistable Fano varieties. Indeed, combining the theorem above with the cone construction, the author showes in Theorem 1.4 that for a $$\mathbb{Q}$$-Gorenstein family of $$\mathbb{Q}$$-Fano varieties, the locus where the fibre is $$K$$-semistable is an open set, see also [H. Blum, “Openness of K-semistability for Fano varieties”, arXiv:1907.02408] for another proof given by Blum, Liu and Xu. This together with a series of significant recent works [C. Jiang, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 5, 1235–1248 (2020; Zbl 1473.14079); H. Blum and C. Xu, Ann. Math. (2) 190, No. 2, 609–656 (2019; Zbl 1427.14084); J. Alper et al., Invent. Math. 222, No. 3, 995–1032 (2020; Zbl 1465.14043)] leads to the existence of K-moduli spaces of Fano varieties as a separated good moduli space. In the recent remarkable works of [C. Xu and Z. Zhuang, Ann. Math. (2) 192, No. 3, 1005–1068 (2020; Zbl 1465.14047)], and Y. Liu et al. [“Finite generation for valuations computing stability thresholds and applications to K-stability”, arXiv:2102.09405], it is shown that this good moduli space is even projective.
There are two key ingredients of the proof of the main theorems. The first one is approximating a valuation computing the log canonical threshod by a sequence of rescalings of Kollár components $$S_i$$ using some ideas developed in [C. Li and C. Xu, J. Eur. Math. Soc. (JEMS) 22, No. 8, 2573–2627 (2020; Zbl 1471.14076)]. The second one is the boundedness of complements established recently by C. Birkar [Ann. Math. (2) 190, No. 2, 345–463 (2019; Zbl 1470.14078)]. In particular, the author shows that all the Kollár components $$S_i$$ can be obtained as log canonical places of a bounded family of $$\mathbb{Q}$$-Cartier divisors.
Reviewer: Jie Liu (Nice)

### MSC:

 14E30 Minimal model program (Mori theory, extremal rays) 14J17 Singularities of surfaces or higher-dimensional varieties 14J45 Fano varieties
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