## Kyoto Journal of Mathematics

### Uniform K-stability and plt blowups of log Fano pairs

Kento Fujita

#### Abstract

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test the uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant finite covers. Moreover, we give another proof of the K-semistability of the projective plane.

#### Article information

Source
Kyoto J. Math., Volume 59, Number 2 (2019), 399-418.

Dates
Revised: 14 February 2017
Accepted: 9 March 2017
First available in Project Euclid: 19 April 2019

https://projecteuclid.org/euclid.kjm/1555660964

Digital Object Identifier
doi:10.1215/21562261-2019-0012

Mathematical Reviews number (MathSciNet)
MR3960299

Zentralblatt MATH identifier
07080110

Subjects
Primary: 14J45: Fano varieties
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)

#### Citation

Fujita, Kento. Uniform K-stability and plt blowups of log Fano pairs. Kyoto J. Math. 59 (2019), no. 2, 399--418. doi:10.1215/21562261-2019-0012. https://projecteuclid.org/euclid.kjm/1555660964

#### References

• [1] R. Berman, S. Boucksom, and M. Jonsson, A variational approach to the Yau–Tian–Donaldson conjecture, preprint, arXiv:1509.04561v2 [math.DG].
• [2] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468.
• [3] H. Blum, Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018), no. 4, 820–849.
• [4] H. Blum, On divisors computing MLD’s and LCT’s, preprint, arXiv:1605.09662v3 [math.AG].
• [5] S. Boucksom, C. Favre, and M. Jonsson, Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom. 18 (2009), no. 2, 279–308.
• [6] S. Boucksom, T. Hisamoto, and M. Jonsson, Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, 743–841.
• [7] R. Dervan, On K-stability of finite covers, Bull. Lond. Math. Soc. 48 (2016), no. 4, 717–728.
• [8] R. Dervan, Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. IMRN 2016, no. 15, 4728–4783.
• [9] S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289–349.
• [10] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye, and M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1701–1734.
• [11] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye, and M. Popa, Restricted volumes and base loci of linear series, Amer. J. Math. 131 (2009), no. 3, 607–651.
• [12] K. Fujita, On K-stability and the volume functions of $\mathbb{Q}$-Fano varieties, Proc. Lond. Math. Soc. (3) 113 (2016), no. 5, 541–582.
• [13] K. Fujita, A valuative criterion for uniform K-stability of $\mathbb{Q}$-Fano varieties, J. Reine Angew. Math., published electronically 25 October 2016.
• [14] K. Fujita, K-stability of Fano manifolds with not small alpha invariants, J. Inst. Math. Jussieu, published electronically 30 March 2017.
• [15] K. Fujita and Y. Odaka, On the K-stability of Fano varieties and anticanonical divisors, Tohoku Math. J. (2) 70 (2018), no. 4, 511–521.
• [16] T. Fujita, Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17 (1994), no. 1, 1–3.
• [17] Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
• [18] S. Ishii, Extremal functions and prime blow-ups, Comm. Algebra 32 (2004), no. 3, 819–827.
• [19] G. R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316.
• [20] J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
• [21] R. Lazarsfeld, Positivity in Algebraic Geometry, I: Classical Setting: Line Bundles and Linear Series, Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin, 2004.
• [22] R. Lazarsfeld, Positivity in Algebraic Geometry, II: Positivity for Vector Bundles, and Multiplier Ideals, Ergeb. Math. Grenzgeb. (3) 49, Springer, Berlin, 2004.
• [23] R. Lazarsfeld and M. Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783–835.
• [24] C. Li, K-semistability is equivariant volume minimization, Duke Math. J. 166 (2017), no. 16, 3147–3218.
• [25] C. Li and C. Xu, Stability of valuations and Kollár components, to appear in J. Eur. Math. Soc. (JEMS), preprint, arXiv:1604.05398v5 [math.AG].
• [26] J. Park and J. Won, K-stability of smooth del Pezzo surfaces, Math. Ann. 372 (2018), no. 3–4, 1239–1276.
• [27] Y. G. Prokhorov, “Blow-ups of canonical singularities” in Algebra (Moscow, 1998), de Gruyter, Berlin, 2000, 301–317.
• [28] Y. G. Prokhorov, Lectures on Complements on Log Surfaces, MSJ Mem. 10, Math. Soc. Japan, Tokyo, 2001.
• [29] J. Ross and R. Thomas, A study of the Hilbert–Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), no. 2, 201–255.
• [30] V. V. Shokurov, $3$-fold log models, J. Math. Sci. 81 (1996), no. 3, 2667–2699.
• [31] G. Székelyhidi, Extremal metrics and K-stability, Bull. Lond. Math. Soc. 39 (2007), no. 1, 76–84.
• [32] G. Székelyhidi, Filtrations and test-configurations, with an appendix by S. Boucksom, Math. Ann. 362 (2015), no. 1–2, 451–484.
• [33] D. Testa, A. Várilly-Alvarado, and M. Velasco, Big rational surfaces, Math. Ann. 351 (2011), no. 1, 95–107.
• [34] G. Tian, Kähler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1–37.
• [35] C. Xu, Finiteness of algebraic fundamental groups, Compos. Math. 150 (2014), no. 3, 409–414.