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**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.

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 |