Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-6bb9c88b65-xjl2h Total loading time: 0 Render date: 2025-07-22T04:40:08.607Z Has data issue: false hasContentIssue false

Bibliography

Published online by Cambridge University Press:  08 May 2025

Chenyang Xu
Chenyang Xu
Affiliation:
Princeton University, New Jersey
Get access

Information

Type
Chapter
Information
K-stability of Fano Varieties , pp. 399 - 408
Publisher: Cambridge University Press
Print publication year: 2025

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

Abban, Hamid, and Zhuang, Ziquan. 2022. K-stability of Fano varieties via admissible flags. Forum Math. Pi, 10, Paper No. e15, 43.CrossRefGoogle Scholar
Abban, Hamid, and Zhuang, Ziquan. 2023. Seshadri constants and K-stability of Fano manifolds. Duke Math. J., 172(6), 1109–1144.CrossRefGoogle Scholar
Abban, Hamid, Cheltsov, Ivan, Kasprzyk, Alexander, Liu, Yuchen, and Petracci, Andrea. 2022. On K-moduli of quartic threefolds. Algebr. Geom.Google Scholar
Abban, Hamid, Cheltsov, Ivan, Denisova, Elena, Etxabarri-Alberdi, Erroxe, Kaloghiros, Anne-Sophie, Jiao, Dongchen, Martinez-Garcia, Jesus, and Papazachariou, Theodoros. 2023. One-dimensional components in the K-moduli of smooth Fano 3-folds. J. Algebraic Geom.Google Scholar
Abhyankar, Shreeram. 1956. On the valuations centered in a local domain. Am. J. Math., 78(2), 321–348.CrossRefGoogle Scholar
Alper, Jarod. 2013. Good moduli spaces for Artin stacks. Ann. Inst. Fourier (Grenoble), 63(6), 2349–2402.CrossRefGoogle Scholar
Alper, Jarod, Hall, Jack, and Rydh, David. 2020a. A Luna étale slice theorem for algebraic stacks. Ann. Math., 191(3), 675–738.CrossRefGoogle Scholar
Alper, Jarod, Blum, Harold, Halpern-Leistner, Daniel, and Xu, Chenyang. 2020b. Reductivity of the automorphism group of K-polystable Fano varieties. Invent. Math., 222(3), 995–1032.CrossRefGoogle Scholar
Alper, Jarod, Halpern-Leistner, Daniel, and Heinloth, Jochen. 2023. Existence of moduli spaces for algebraic stacks. Invent. Math., 234(3), 949–1038.CrossRefGoogle Scholar
Araujo, Carolina, Castravet, Ana-Maria, Cheltsov, Ivan, Fujita, Kento, Kaloghiros, Anne-Sophie, Martinez-Garcia, Jesus, Shramov, Constantin, Süß, Hendrik, and Viswanathann, Nivedita. 2023. The Calabi problem for Fano threefolds. London Mathematical Society Lecture Note Series, vol. 485. Cambridge University Press, Cambridge.Google Scholar
Arezzo, Claudio, Ghigi, Alessandro, and Pirola, Gian Pietro. 2006. Symmetries, quotients and Kähler-Einstein metrics. J. Reine Angew. Math., 2006(591), 177–200.CrossRefGoogle Scholar
Ascher, Kenneth, Devleming, Kristin, and Liu, Yuchen. 2019. Wall crossing for K-moduli spaces of plane curves. Proc. Lond. Math. Soc., 128(6), e12615.Google Scholar
Ascher, Kenneth, Devleming, Kristin, and Liu, Yuchen. 2023a. K-moduli of curves on a quadric surface and K3 surfaces. J. Inst. Math. Jussieu, 22(3), 1251–1291.CrossRefGoogle Scholar
Ascher, Kenneth, DeVleming, Kristin, and Liu, Yuchen. 2023b. K-stability and birational models of moduli of quartic K3 surfaces. Invent. Math., 232(2), 471–552.CrossRefGoogle Scholar
Aubin, Thierry. 1978. Équations du type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math., 102(1), 63–95.Google Scholar
Bando, Shigetoshi, and Mabuchi, Toshiki. 1987. Uniqueness of Einstein Kähler metrics modulo connected group actions. Pages 11–40 of: Algebraic geometry, Sendai, 1985. Advanced Studies in Pure Mathematics, vol. 10. North Holland Publishing, Amsterdam.Google Scholar
Barth, Wolf P., Hulek, Klaus, Peters, Chris A. M., and Van de Ven, Antonius. 2004. Compact complex surfaces. Second ed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Berman, Robert J. 2016. K-polystability of ℚ-Fano varieties admitting Kähler-Einstein metrics. Invent. Math., 203(3), 973–1025.CrossRefGoogle Scholar
Berman, Robert, Boucksom, Sébastien, and Jonsson, Mattias. 2021. A variational approach to the Yau–Tian–Donaldson conjecture. J. Am. Math. Soc., 34(3), 605–652.CrossRefGoogle Scholar
Białynicki-Birula, A. 1973. Some theorems on actions of algebraic groups. Ann. Math., 98(3), 480–497.CrossRefGoogle Scholar
Birkar, Caucher. 2019. Anti-pluricanonical systems on Fano varieties. Ann. Math., 190(2), 345–463.CrossRefGoogle Scholar
Birkar, Caucher. 2021. Singularities of linear systems and boundedness of Fano varieties. Ann. Math., 193(2), 347–405.CrossRefGoogle Scholar
Birkar, Caucher, Cascini, Paolo, Hacon, Christopher D., and McKernan, James. 2010. Existence of minimal models for varieties of log general type. J. Am. Math. Soc., 23(2), 405–468.Google Scholar
Blum, Harold, and Jonsson, Mattias. 2020. Thresholds, valuations, and K-stability. Adv. Math., 365, 107062.CrossRefGoogle Scholar
Blum, Harold, and Liu, Yuchen. 2021. The normalized volume of a singularity is lower semicontinuous. J. Eur. Math. Soc., 23(4), 1225–1256.Google Scholar
Blum, Harold, and Liu, Yuchen. 2022. Openness of uniform K-stability in families of ℚ-Fano varieties. Ann. Sci. Éc. Norm. Supér., 55(1), 1–41.Google Scholar
Blum, Harold, and Xu, Chenyang. 2019. Uniqueness of K-polystable degenerations of Fano varieties. Ann. Math., 190(2), 609–656.CrossRefGoogle Scholar
Blum, Harold, Halpern-Leistner, Daniel, Liu, Yuchen, and Xu, Chenyang. 2021. On properness of K-moduli spaces and optimal degenerations of Fano varieties. Selecta Math. (N.S.), 27(4), 73.CrossRefGoogle Scholar
Blum, Harold, Liu, Yuchen, and Xu, Chenyang. 2022a. Openness of K-semistability for Fano varieties. Duke Math. J., 171(13), 2753–2797.Google Scholar
Blum, Harold, Liu, Yuchen, and Zhou, Chuyu. 2022b. Optimal destabilization of K–unstable Fano varieties via stability thresholds. Geom. Topol., 26(6), 2507–2564.CrossRefGoogle Scholar
Blum, Harold, Liu, Yuchen, Xu, Chenyang, and Zhuang, Ziquan. 2023. The existence of the Kähler-Ricci soliton degeneration. Forum Math. Pi, 11, e9.CrossRefGoogle Scholar
Boucksom, Sébastien. 2014. Corps d’Okounkov (d’après Okounkov, Lazarsfeld-Mustaţǎ et Kaveh-Khovanskii). Astérisque, Exp. No. 1059, vii, 1–41.Google Scholar
Boucksom, Sébastien, and Chen, Huayi. 2011. Okounkov bodies of filtered linear series. Compos. Math., 147(4), 1205–1229.CrossRefGoogle Scholar
Boucksom, Sébastien, and Jonsson, Mattias. 2023. A non-Archimedean approach to K-stability, II: Divisorial stability and openness. J. Reine Angew. Math., 805, 1–53.Google Scholar
Boucksom, Sébastien, and Jonsson, Mattias. 2024. A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations. Ann. Inst. Fourier (Grenoble). In press.Google Scholar
Boucksom, Sébastien, Favre, Charles, and Jonsson, Mattias. 2009. Differentiability of volumes of divisors and a problem of Teissier. J. Algebraic Geom., 18(2), 279–308.Google Scholar
Boucksom, Sébastien, Demailly, Jean-Pierre, Păun, Mihai, and Peternell, Thomas. 2013. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebraic Geom., 22(2), 201–248.Google Scholar
Boucksom, Sébastien, Favre, Charles, and Jonsson, Mattias. 2014. A refinement of Izumi’s theorem. Pages 55–81 of: Valuation theory in interaction. EMS Series of Congress of the European Mathematical Society, Zürich.CrossRefGoogle Scholar
Boucksom, Sébastien, Küronya, Alex, Maclean, Catriona, and Szemberg, Tomasz. 2015. Vanishing sequences and Okounkov bodies. Math. Ann., 361(3–4), 811–834.CrossRefGoogle Scholar
Boucksom, Sébastien, Hisamoto, Tomoyuki, and Jonsson, Mattias. 2017. Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble), 67(2), 743–841.CrossRefGoogle Scholar
Cheltsov, Ivan A., and Shramov, K. A. 2008. Log-canonical thresholds for nonsingular Fano threefolds. Uspekhi Mat. Nauk, 63(5(383)), 73–180.Google Scholar
Cheltsov, Ivan A., 2001. Log canonical thresholds on hypersurfaces. Mat. Sb., 192(8), 155–172.Google Scholar
Cheltsov, Ivan A., 2008. Log canonical thresholds of del Pezzo surfaces. Geom. Funct. Anal., 18(4), 1118–1144.CrossRefGoogle Scholar
Cheltsov, Ivan A., and Park, Jihun. 2002. Global log-canonical thresholds and generalized Eckardt points. Mat. Sb., 193(5), 149–160.Google Scholar
Cheltsov, Ivan A., Fujita, Kento, Kishimoto, Takashi, and Okada, Takuzo. 2023. K-stable divisors in ℙ1 × ℙ1 × ℙ2 of degree (1, 1, 2). Nagoya Math. J., 251, 686–714.CrossRefGoogle Scholar
Cheltsov, Ivan A., Denisova, Elena, and Fujita, Kento. 2024. K-stable smooth Fano threefolds of Picard rank two. Forum Math. Sigma, 12, e41.CrossRefGoogle Scholar
Cheltsov, Ivan A., Rubinstein, Yanir A., and Zhang, Kewei. 2019. Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces. Sel. Math. New Ser., 25(2), 34.CrossRefGoogle Scholar
Chen, Xiuxiong, Donaldson, Simon, and Sun, Song. 2015a. Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Am. Math. Soc., 28(1), 183–197.Google Scholar
Chen, Xiuxiong, Donaldson, Simon, and Sun, Song. 2015b. Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π. J. Am. Math. Soc., 28(1), 199–234.Google Scholar
Chen, Xiuxiong, Donaldson, Simon, and Sun, Song. 2015c. Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof. J. Am. Math. Soc., 28(1), 235–278.Google Scholar
Codogni, Giulio, and Patakfalvi, Zsolt. 2021. Positivity of the CM line bundle for families of K-stable klt Fano varieties. Invent. Math., 223(3), 811–894.CrossRefGoogle Scholar
Conrad, Brian. 2014. Reductive group schemes. Pages 93–444 of: Autour des schémas en groupes. Vol. I. Panoramas et Synthèses, vol. 42/43. Société mathématique de France. France, Paris.Google Scholar
Cutkosky, Steven Dale. 2013. Multiplicities associated to graded families of ideals. Algebra Number Theory, 7(9), 2059–2083.CrossRefGoogle Scholar
Demailly, Jean-Pierre, Ein, Lawrence, and Lazarsfeld, Robert. 2000. A subadditivity property of multiplier ideals. Dedicated to William Fulton on the occasion of his 60th birthday, Michigan Math. J. 48(1), 137–156.Google Scholar
Dervan, Ruadhaı́. 2016a. On K-stability of finite covers. Bull. Lond. Math. Soc., 48(4), 717–728.CrossRefGoogle Scholar
Dervan, Ruadhaı́. 2016b. Uniform stability of twisted constant scalar curvature Kähler metrics. Int. Math. Res. Not., 2016(15), 4728–4783.CrossRefGoogle Scholar
Ding, Wei Yue. 1988. Remarks on the existence problem of positive Kähler-Einstein metrics. Math. Ann., 282(3), 463–471.CrossRefGoogle Scholar
Ding, Wei Yue, and Tian, Gang. 1992. Kähler-Einstein metrics and the generalized Futaki invariant. Invent. Math., 110(2), 315–335.CrossRefGoogle Scholar
Dolgachev, Igor. 2003. Lectures on invariant theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge.Google Scholar
Donaldson, S. K. 1985. Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc., S3–S50(1), 1–26.Google Scholar
Donaldson, S. K. 1987. Infinite determinants, stable bundles and curvature. Duke Math. J., 54(1), 231–247.CrossRefGoogle Scholar
Donaldson, S. K. 2002. Scalar curvature and stability of toric varieties. J. Differential Geom., 62(2), 289–349.CrossRefGoogle Scholar
Donaldson, S. K. 2012. Kähler metrics with cone singularities along a divisor. Pages 49–79 of: Essays in mathematics and its applications. Springer, Heidelberg.Google Scholar
Ein, Lawrence, Küchle, Oliver, and Lazarsfeld, Robert. 1995. Local positivity of ample line bundles. J. Differential Geom., 42(2), 193–219.CrossRefGoogle Scholar
Ein, Lawrence, Lazarsfeld, Robert, and Smith, Karen E. 2003. Uniform approximation of Abhyankar valuation ideals in smooth function fields. Am. J. Math., 125(2), 409–440.CrossRefGoogle Scholar
Ein, Lawrence, Lazarsfeld, Robert, Mustaţă, Mircea, Nakamaye, Michael, and Popa, Mihnea. 2006. Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble), 56(6), 1701–1734.CrossRefGoogle Scholar
Ein, Lawrence, Lazarsfeld, Robert, Mustaţă, Mircea, Nakamaye, Michael, and Popa, Mihnea. 2009. Restricted volumes and base loci of linear series. Am. J. Math., 131(3), 607–651.CrossRefGoogle Scholar
Eisenbud, David. 1995. Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer-Verlag, New York.Google Scholar
Eyssidieux, Philippe, Guedj, Vincent, and Zeriahi, Ahmed. 2009. Singular Kähler-Einstein metrics. J. Am. Math. Soc., 22(3), 607–639.CrossRefGoogle Scholar
Fujino, Osamu. 2017. Notes on the weak positivity theorems. Pages 73–118 of: Algebraic varieties and automorphism groups. Advanced Studies in Pure Mathematics, vol. 75. The Mathematical Society of Japan. Tokyo.Google Scholar
Fujino, Osamu. 2018. Semipositivity theorems for moduli problems. Ann. of Math., 187(3), 639–665.CrossRefGoogle Scholar
Fujino, Osamu, and Mori, Shigefumi. 2000. A canonical bundle formula. J. Differential Geom., 56(1), 167–188.CrossRefGoogle Scholar
Fujita, Kento. 2016. On K-stability and the volume functions of ℚ-Fano varieties. Proc. Lond. Math. Soc., 113(5), 541–582.CrossRefGoogle Scholar
Fujita, Kento. 2018. Optimal bounds for the volumes of Kähler-Einstein Fano manifolds. Am. J. Math., 140(2), 391–414.CrossRefGoogle Scholar
Fujita, Kento. 2019a. K-stability of Fano manifolds with not small alpha invariants. J. Inst. Math. Jussieu, 18(3), 519–530.CrossRefGoogle Scholar
Fujita, Kento. 2019b. A valuative criterion for uniform K-stability of ℚ -Fano varieties. J. Reine Angew. Math., 751, 309–338.Google Scholar
Fujita, Kento. 2023. On K-stability for Fano threefolds of rank 3 and degree 28. Int. Math. Res. Not., 2023(15), 12601–12784.CrossRefGoogle Scholar
Fujita, Kento, and Odaka, Yuji. 2018. On the K-stability of Fano varieties and anticanonical divisors. Tohoku Math. J., 70(4), 511–521.CrossRefGoogle Scholar
Fujita, Takao. 1978. On Kähler fiber spaces over curves. J. Math. Soc. Japan, 30(4), 779–794.CrossRefGoogle Scholar
Fujita, Takao. 1979. On Zariski problem. Proc. Japan Acad. Ser. A Math. Sci., 55(3), 106–110.CrossRefGoogle Scholar
Futaki, A. 1983. An obstruction to the existence of Einstein Kähler metrics. Invent. Math., 73(3), 437–443.CrossRefGoogle Scholar
Gallardo, Patricio, Martinez-Garcia, Jesus, and Spotti, Cristiano. 2021. Applications of the moduli continuity method to log K-stable pairs. J. Lond. Math. Soc., 103(2), 729–759.CrossRefGoogle Scholar
Guedj, Vincent, and Zeriahi, Ahmed. 2017. Degenerate complex Monge-Ampère equations. EMS Tracts in Mathematics, vol. 26. European Mathematical Society, Zürich.Google Scholar
Hacon, Christopher D., McKernan, James, and Xu, Chenyang. 2013. On the birational automorphisms of varieties of general type. Ann. Math., 177(3), 1077–1111.CrossRefGoogle Scholar
Hacon, Christopher D., McKernan, James, and Xu, Chenyang. 2014. ACC for log canonical thresholds. Ann. Math., 180(2), 523–571.CrossRefGoogle Scholar
Halpern-Leistner, Daniel. 2022. On the structure of instability in moduli theory. arXiv:1411.0627Google Scholar
Hartshorne, Robin. 1977. Algebraic geometry. Graduate Texts in Mathematics, vol. 52. Springer-Verlag, New York/Heidelberg.Google Scholar
Hisamoto, Tomoyuki. 2016. Stability and coercivity for toric polarizations. arXiv:1610.07998Google Scholar
Jiang, Chen. 2020. Boundedness of ℚ-Fano varieties with degrees and alpha-invariants bounded from below. Ann. Sci. Éc. Norm. Supér., 53(5), 1235–1248.Google Scholar
Jonsson, Mattias, and Mustaţă, Mircea. 2012. Valuations and asymptotic invariants for sequences of ideals. Ann. Inst. Fourier (Grenoble), 62(6), 2145–2209.CrossRefGoogle Scholar
Jow, Shin-Yao, and Miller, Ezra. 2008. Multiplier ideals of sums via cellular resolutions. Math. Res. Lett., 15(2), 359–373.CrossRefGoogle Scholar
Kaloghiros, Anne-Sophie, Küronya, Alex, and Lazić, Vladimir. 2016. Finite generation and geography of models. Pages 215–245 of: Minimal models and extremal rays (Kyoto, 2011). Advanced Studies in Pure Mathematics, vol. 70. Mathematical Society of Japan, Tokyo.Google Scholar
Kaveh, Kiumars, and Khovanskii, A. G. 2012. Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory. Ann. Math., 176(2), 925–978.CrossRefGoogle Scholar
Kawamata, Yujiro. 1981. Characterization of abelian varieties. Compos Math., 43(2), 253–276.Google Scholar
Kawamata, Yujiro. 1998. Subadjunction of log canonical divisors. II. Am. J. Math., 120(5), 893–899.Google Scholar
Keel, Seán, and Mori, Shigefumi. 1997. Quotients by groupoids. Ann. Math., 145(1), 193–213.CrossRefGoogle Scholar
Kempf, George R. 1978. Instability in invariant theory. Ann. Math., 108(2), 299–316.CrossRefGoogle Scholar
Kleiman, Steven L. 1966. Toward a numerical theory of ampleness. Ann. Math., 84, 293–344.CrossRefGoogle Scholar
Knaf, Hagen, and Kuhlmann, Franz-Viktor. 2005. Abhyankar places admit local uniformization in any characteristic. Ann. Sci. École Norm. Super., 38(6), 833–846.Google Scholar
Knudsen, Finn Faye, and Mumford, David. 1976. The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div.” Math. Scand., 39(1), 19–55.Google Scholar
Kollár, János. 1990. Projectivity of complete moduli. J. Diff. Geom., 32(1), 235–268.Google Scholar
Kollár, János. 1996. Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32. Springer-Verlag, Berlin.Google Scholar
Kollár, János. 2007. Kodaira’s canonical bundle formula and adjunction. Pages 134–162 of: Flips for 3-folds and 4-folds. Oxford Lecture Series in Mathematics and Its Applications, vol. 35. Oxford University Press, Oxford.Google Scholar
Kollár, János. 2013. Singularities of the minimal model program. With a collaboration of Sándor Kovács. Cambridge Tracts in Mathematics, vol. 200. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Kollár, János. 2016. Maps between local Picard groups. Algebr. Geom., 3(4), 461–495.Google Scholar
Kollár, János. 2023. Families of varieties of general type. Cambridge Tracts in Mathematics, vol. 231. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Kollár, János, and Mori, Shigefumi. 1998. Birational geometry of algebraic varieties. Cambridge Tracts in Mathematics, vol. 134. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Kovács, Sándor J., and Patakfalvi, Zsolt. 2017. Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension. J. Am. Math. Soc., 30(4), 959–1021.Google Scholar
Kuipers, L., and Niederreiter, H. 1974. Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney.Google Scholar
Langton, Stacy G. 1975. Valuative criteria for families of vector bundles on algebraic varieties. Ann. Math., 101(1), 88–110.CrossRefGoogle Scholar
Lazarsfeld, Robert. 2004a. Positivity in algebraic geometry. I. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48. Classical setting: Line bundles and linear series. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Lazarsfeld, Robert. 2004b. Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 49. Springer-Verlag, Berlin.Google Scholar
Lazarsfeld, Robert, and Mustaţă, Mircea. 2009. Convex bodies associated to linear series. Ann. Sci. Éc. Norm. Supér., 42(5), 783–835.Google Scholar
Li, Chi. 2017. K-semistability is equivariant volume minimization. Duke Math. J., 166(16), 3147–3218.CrossRefGoogle Scholar
Li, Chi. 2018. Minimizing normalized volumes of valuations. Math. Z., 289(1–2), 491–513.CrossRefGoogle Scholar
Li, Chi. 2022. G-uniform stability and Kähler-Einstein metrics on Fano varieties. Invent. Math., 227(2), 661–744.CrossRefGoogle Scholar
Li, Chi, and Xu, Chenyang. 2014. Special test configuration and K-stability of Fano varieties. Ann. Math., 180(1), 197–232.CrossRefGoogle Scholar
Li, Chi, and Xu, Chenyang. 2018. Stability of valuations: Higher rational rank. Peking Math. J., 1(1), 1–79.CrossRefGoogle Scholar
Li, Chi, and Xu, Chenyang. 2020. Stability of valuations and Kollár components. J. Eur. Math. Soc., 22(8), 2573–2627.CrossRefGoogle Scholar
Li, Chi, Wang, Xiaowei, and Xu, Chenyang. 2018. Quasi-projectivity of the moduli space of smooth Kähler-Einstein Fano manifolds. Ann. Sci. Éc. Norm. Supér., 51(3), 739–772.Google Scholar
Li, Chi, Wang, Xiaowei, and Xu, Chenyang. 2019. On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties. Duke Math. J., 168(8), 1387–1459.CrossRefGoogle Scholar
Li, Chi, Liu, Yuchen, and Xu, Chenyang. 2020. A guided tour to normalized volume. Pages 167–219 of: Geometric analysis—in honor of Gang Tian’s 60th birthday. Progr. Math., vol. 333. Birkhäuser/Springer, Cham.Google Scholar
Li, Chi, Wang, Xiaowei, and Xu, Chenyang. 2021. Algebraicity of the metric tangent cones and equivariant K-stability. J. Am. Math. Soc., 34(4), 1175–1214.CrossRefGoogle Scholar
Li, Chi, Tian, Gang, and Wang, Feng. 2022. The uniform version of Yau-Tian-Donaldson conjecture for singular Fano varieties. Peking Math. J., 5(2), 383–426.CrossRefGoogle Scholar
Liu, Yuchen. 2018. The volume of singular Kähler-Einstein Fano varieties. Compos. Math., 154(6), 1131–1158.CrossRefGoogle Scholar
Liu, Yuchen. 2022. K-stability of cubic fourfolds. J. Reine Angew. Math., 786, 55–77.Google Scholar
Liu, Yuchen, and Xu, Chenyang. 2019. K-stability of cubic threefolds. Duke Math. J., 168(11), 2029–2073.CrossRefGoogle Scholar
Liu, Yuchen, Xu, Chenyang, and Zhuang, Ziquan. 2022. Finite generation for valuations computing stability thresholds and applications to K-stability. Ann. Math., 196(2), 507–566.CrossRefGoogle Scholar
Lyu, Shiji, and Murayama, Takumi. 2022. The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero. arXiv:2209.08732.Google Scholar
Mabuchi, Toshiki. 1986. K-energy maps integrating Futaki invariants. Tohoku Math. J., 38(4), 575–593.CrossRefGoogle Scholar
Mabuchi, Toshiki, and Mukai, Shigeru. 1993. Stability and Einstein-Kähler metric of a quartic del Pezzo surface. Pages 133–160 of: Einstein metrics and Yang-Mills connections (Sanda, 1990). Lecture Notes in Pure and Applied Mathematics, vol. 145. Dekker, New York.Google Scholar
Matsumura, Hideyuki. 1989. Commutative ring theory. Second ed. Translated from the Japanese by Reid, M.. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge.Google Scholar
Matsushima, Yozô. 1957. Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne. Nagoya Math. J., 11, 145–150.CrossRefGoogle Scholar
Mumford, D., Fogarty, J., and Kirwan, F. 1994. Geometric invariant theory. Third ed. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Nakayama, Noboru. 2004. Zariski-decomposition and abundance. MSJ Memoirs, vol. 14. Mathematical Society of Japan, Tokyo.Google Scholar
Narasimhan, M. S., and Seshadri, C. S. 1965. Stable and unitary vector bundles on a compact Riemann surface. Ann. Math., 82, 540–567.CrossRefGoogle Scholar
Odaka, Yuji. 2013a. A generalization of the Ross-Thomas slope theory. Osaka J. Math., 50(1), 171–185.Google Scholar
Odaka, Yuji. 2013b. The GIT stability of polarized varieties via discrepancy. Ann. of Math., 177(2), 645–661.CrossRefGoogle Scholar
Odaka, Yuji. 2015. Compact moduli spaces of Kähler-Einstein Fano varieties. Publ. Res. Inst. Math. Sci., 51(3), 549–565.CrossRefGoogle Scholar
Odaka, Yuji, Spotti, Cristiano, and Sun, Song. 2016. Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics. J. Diff. Geom., 102(1), 127–172.Google Scholar
Okounkov, Andrei. 1996. Brunn-Minkowski inequality for multiplicities. Invent. Math., 125(3), 405–411.CrossRefGoogle Scholar
Okounkov, Andrei. 2003. Why would multiplicities be log-concave? Pages 329–347 of: The orbit method in geometry and physics (Marseille, 2000). Progress in Mathematics, vol. 213. Birkhäuser, Boston, MA.Google Scholar
Patakfalvi, Zsolt, and Xu, Chenyang. 2017. Ampleness of the CM line bundle on the moduli space of canonically polarized varieties. Algebr. Geom., 4(1), 29–39.Google Scholar
Paul, Sean, and Tian, Gang. 2006. CM Stability and the Generalized Futaki Invariant I. arXiv:math/0605278.Google Scholar
Posva, Quentin. 2022. Positivity of the CM line bundle for K-stable log Fanos. Trans. Amer. Math. Soc., 375(7), 4943–4978.CrossRefGoogle Scholar
Pukhlikov, A. V. 2002. Birational geometry of higher-dimensional Fano hypersurfaces. Dokl. Akad. Nauk, 385(1), 28–30.Google Scholar
Reboulet, Rémi. 2022. Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy. J. Reine Angew. Math., 793, 59–103.Google Scholar
Ross, Julius, and Thomas, Richard. 2007. A study of the Hilbert-Mumford criterion for the stability of projective varieties. J. Algebraic Geom., 16(2), 201–255.Google Scholar
Rubinstein, Yanir A. 2008. Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kähler metrics. Adv. Math., 218(5), 1526–1565.CrossRefGoogle Scholar
Seshadri, C. S. 1972. Quotient spaces modulo reductive algebraic groups. Ann. Math., 95(2), 511–556; errata, ibid. (2) 96 (1972), 599.CrossRefGoogle Scholar
Shokurov, V. V. 1992. Three-dimensional log perestroikas. Izv. Ross. Akad. Nauk Ser. Mat., 56(1), 105–203.Google Scholar
Springer, T. A. 1998. Linear algebraic groups. 2nd ed. Progress in Mathematics, vol. 9. Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Székelyhidi, Gábor. 2011. Greatest lower bounds on the Ricci curvature of Fano manifolds. Compos. Math., 147(1), 319–331.CrossRefGoogle Scholar
Székelyhidi, Gábor. 2013. A remark on conical Kähler-Einstein metrics. Math. Res. Lett., 20(3), 581–590.CrossRefGoogle Scholar
Székelyhidi, Gábor. 2014. An introduction to extremal Kähler metrics. Graduate Studies in Mathematics, vol. 152. American Mathematical Society, Providence, RI.Google Scholar
Székelyhidi, Gábor. 2015. Filtrations and test-configurations. Math. Ann., 362(1–2), 451–484. With an appendix by Sebastien Boucksom.CrossRefGoogle Scholar
Székelyhidi, Gábor. 2016. The partial C0-estimate along the continuity method. J. Amer. Math. Soc., 29(2), 537–560.Google Scholar
Takagi, Shunsuke. 2006. Formulas for multiplier ideals on singular varieties. Am. J. Math., 128(6), 1345–1362.CrossRefGoogle Scholar
Takagi, Shunsuke. 2013. A subadditivity formula for multiplier ideals associated to log pairs. Proc. Am. Math. Soc., 141(1), 93–102.Google Scholar
Tian, Gang. 1987. On Kähler-Einstein metrics on certain Kähler manifolds with C1 (M) > 0. Invent. Math., 89(2), 225–246.CrossRefGoogle Scholar
Tian, Gang. 1990. On Calabi’s conjecture for complex surfaces with positive first Chern class. Invent. Math., 101(1), 101–172.CrossRefGoogle Scholar
Tian, Gang. 1992. On stability of the tangent bundles of Fano varieties. Int. J. Math., 3(3), 401–413.CrossRefGoogle Scholar
Tian, Gang. 1997. Kähler-Einstein metrics with positive scalar curvature. Invent. Math., 130(1), 1–37.CrossRefGoogle Scholar
Tian, Gang. 2015. K-stability and Kähler-Einstein metrics. Comm. Pure Appl. Math., 68(7), 1085–1156.Google Scholar
Uhlenbeck, K., and Yau, S.-T. 1986. On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Commun. Pure Appl. Math., 39(S1), S257–S293.CrossRefGoogle Scholar
Viehweg, Eckart. 1989. Weak positivity and the stability of certain Hilbert points. Invent. Math., 96(3), 639–667.CrossRefGoogle Scholar
Wang, Xiaowei. 2012. Height and GIT weight. Math. Res. Lett., 19(4), 909–926.CrossRefGoogle Scholar
Wang, Xiaowei, and Xu, Chenyang. 2014. Nonexistence of asymptotic GIT compactification. Duke Math. J., 163(12), 2217–2241.CrossRefGoogle Scholar
Nyström, Witt David. 2012. Test configurations and Okounkov bodies. Compos. Math., 148(6), 1736–1756.Google Scholar
Xu, Chenyang. 2020. A minimizing valuation is quasi-monomial. Ann. Math., 191(3), 1003–1030.CrossRefGoogle Scholar
Xu, Chenyang. 2021. Towards finite generation of higher rational rank valuations. Mat. Sb., 212(3), 157–174.Google Scholar
Xu, Chenyang. 2023. K-stability for varieties with a big anticanonical class. Épijournal Géom. Algébrique, Special volume in honour of C. Voisin, Art. 7, 9.Google Scholar
Xu, Chenyang, and Zhuang, Ziquan. 2020. On positivity of the CM line bundle on K-moduli spaces. Ann. Math., 192(3), 1005–1068.CrossRefGoogle Scholar
Xu, Chenyang, and Zhuang, Ziquan. 2021. Uniqueness of the minimizer of the normalized volume function. Camb. J. Math., 9(1), 149–176.CrossRefGoogle Scholar
Xu, Chenyang, and Zhuang, Ziquan. 2023. Stable degenerations of singularities. arXiv:2205.10915Google Scholar
Yau, Shing Tung. 1978. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Commun. Pure Appl. Math., 31(3), 339–411.Google Scholar
Zhuang, Ziquan. 2020. Birational superrigidity and K-stability of Fano complete intersections of index 1. Duke Math. J., 169(12), 2205–2229. With an appendix by Zhuang and Charlie Stibitz.CrossRefGoogle Scholar
Zhuang, Ziquan. 2021. Optimal destabilizing centers and equivariant K-stability. Invent. Math., 226(1), 195–223.CrossRefGoogle Scholar

Accessibility standard: Unknown

Accessibility compliance for the PDF of this book is currently unknown and may be updated in the future.

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge-org.demo.remotlog.com is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Bibliography
  • Chenyang Xu, Princeton University, New Jersey
  • Book: K-stability of Fano Varieties
  • Online publication: 08 May 2025
  • Chapter DOI: https://doi.org/10.1017/9781009538763.013
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Bibliography
  • Chenyang Xu, Princeton University, New Jersey
  • Book: K-stability of Fano Varieties
  • Online publication: 08 May 2025
  • Chapter DOI: https://doi.org/10.1017/9781009538763.013
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Bibliography
  • Chenyang Xu, Princeton University, New Jersey
  • Book: K-stability of Fano Varieties
  • Online publication: 08 May 2025
  • Chapter DOI: https://doi.org/10.1017/9781009538763.013
Available formats
×