Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor
Authors:
Daniel Greb, Stefan Kebekus and Thomas Peternell
Journal:
J. Algebraic Geom. 31 (2022), 467-496
DOI:
https://doi.org/10.1090/jag/785
Published electronically:
March 8, 2022
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by $\mathbb {Q}$-Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.
References
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- M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207. MR 86359, DOI 10.1090/S0002-9947-1957-0086359-5
- Indranil Biswas, Semistability and restrictions of tangent bundle to curves, Geom. Dedicata 142 (2009), 37–46, DOI 10.1007/s10711-009-9356-3. Preprint, arXiv:0901.4161.
- Lukas Braun, The local fundamental group of a Kawamata log terminal singularity is finite, Invent. Math. 226 (2021), no. 3, 845–896. DOI 10.1007/s00222-021-01062-0. Preprint, arXiv:2004.00522, 2020.
- Bang-yen Chen and Koichi Ogiue, Some characterizations of complex space forms in terms of Chern classes, Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 104, 459–464. MR 405303, DOI 10.1093/qmath/26.1.459
- Stéphane Druel, Henri Guenancia, and Mihai Păun, A decomposition theorem for $\mathbb {Q}$-Fano Kähler-Einstein varieties, Preprint, arXiv:2008.05352, August 2020.
- Igor Dolgachev, Weighted projective varieties, In Group actions and vector fields (Vancouver, B.C., 1981), Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71, DOI 10.1007/BFb0101508.
- Simon K. Donaldson, Holomorphic discs and the complex Monge-Ampère equation, J. Symplectic Geom. 1 (2002), no. 2, 171–196, euclid.jsg/1092316649.
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345, Available from the author’s web site https://www-fourier.ujf-grenoble.fr/$\sim $demailly/manuscripts/dps1.pdf.
- Stéphane Druel, The Zariski-Lipman conjecture for log canonical spaces, Bull. Lond. Math. Soc. 46 (2014), no. 4, 827–835, DOI 10.1112/blms/bdu040, Preprint, arXiv:1301.5910.
- David Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995, DOI 10.1007/978-1-4612-5350-1.
- William Fulton and Robert Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 26–92. MR 644817
- Hubert Flenner, Divisorenklassengruppen quasihomogener Singularitäten, J. Reine Angew. Math. 328 (1981), 128–160 (German). MR 636200, DOI 10.1515/crll.1981.328.128
- Hubert Flenner, Restrictions of semistable bundles on projective varieties, Comment. Math. Helv. 59 (1984), no. 4, 635–650, DOI 10.1007/BF02566370.
- Takao Fujita, On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan 32 (1980), no. 1, 153–169, DOI 10.2969/jmsj/03210153.
- Takao Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990, DOI 10.1017/CBO9780511662638
- Kento Fujita, Examples of K-unstable Fano manifolds with the Picard number 1, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 4, 881–891, DOI 10.1017/S0013091516000432, Preprint, arXiv:1508.04290.
- Daniel Greb, Stefan Kebekus, Sándor J. Kovács, and Thomas Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. 114 (2011), no. 1, 87–169, DOI 10.1007/s10240-011-0036-0. An extended version with additional graphics is available as arXiv:1003.2913.
- Daniel Greb, Stefan Kebekus, and Thomas Peternell, Movable curves and semistable sheaves, Int. Math. Res. Not. IMRN 2016 (2016), no. 2, 536–570, DOI 10.1093/imrn/rnv126, Preprint, arXiv:1408.4308.
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- Daniel Greb, Stefan Kebekus, and Thomas Peternell, Projectively flat KLT varieties, Journal de l’École polytechnique—Mathématiques (JEP) Tome 8 (2021), 1005–1036. DOI 10.5802/jep.164. Preprint, arXiv:2010.06878, October 2020.
- Daniel Greb, Stefan Kebekus, Thomas Peternell, and Behrouz Taji, The Miyaoka-Yau inequality and uniformisation of canonical models, Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 6, 1487–1535, DOI 10.24033/asens.2414, Preprint, arXiv:1511.08822.
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
- Alexander Grothendieck, Représentations linéaires et compactification profinie des groupes discrets, Manuscripta Math. 2 (1970), 375–396, DOI 10.1007/BF01719593.
- Daniel Greb and Michael Lennox Wong, Canonical complex extensions of Kähler manifolds, J. Lond. Math. Soc. 101 2020, no. 2, 786–827. DOI 10.1112/jlms.12287. Preprint, arXiv:1807.01223.
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977, DOI 10.1007/978-1-4757-3849-0.
- Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, second edition, Cambridge University Press, Cambridge, 2010, DOI 10.1017/CBO9780511711985.
- V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR 1668579
- Akihiro Kanemitsu, Fano manifolds and stability of tangent bundles, J. Reine Angew. Math. 774 (2021), 163–183. DOI 10.1515/crelle-2020-0043. Preprint, arXiv:1912.12617.
- Stefan Kebekus, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, Complex geometry (Göttingen, 2000) Springer, Berlin, 2002, pp. 147–155. MR 1922103
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original, DOI 10.1017/CBO9780511662560.
- Shoshichi Kobayashi and Takushiro Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47, DOI 10.1215/kjm/1250523432.
- Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. MR 909698, DOI 10.1515/9781400858682
- Robert Lazarsfeld, Positivity in algebraic geometry. II, 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. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
- Chi Li, Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds, J. Reine Angew. Math. 733 (2017), 55–85, DOI 10.1515/crelle-2014-0156, Preprint, arXiv:1302.6681.
- Gustav I. Lehrer and Donald E. Taylor, Unitary reflection groups, Australian Mathematical Society Lecture Series, vol. 20, Cambridge University Press, Cambridge, 2009. MR 2542964
- Steven Lu and Behrouz Taji, A characterization of finite quotients of abelian varieties, Int. Math. Res. Not. IMRN (2018), no. 1, 292–319, DOI 10.1093/imrn/rnw251, Preprint, arXiv:1410.0063.
- Shigefumi Mori and Shigeru Mukai, On Fano $3$-folds with $B_{2}\geq 2$, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 101–129. MR 715648, DOI 10.2969/aspm/00110101
- Shigefumi Mori and Shigeru Mukai. Erratum: “Classification of Fano 3-folds with $B_2\geq 2$”, Manuscripta Math. 36 (1981/82), no. 2, 147–162, DOI 10.1007/s00229-002-0336-2. MR0641971 (83f:14032).
- Shigefumi Mori and Shigeru Mukai, Classification of Fano $3$-folds with $B_{2}\geq 2$, Manuscripta Math. 36 (1981/82), 147–162, DOI 10.1007/BF01170131.
- Shigefumi Mori, On a generalization of complete intersections, J. Math. Kyoto Univ. 15 (1975), no. 3, 619–646, DOI 10.1215/kjm/1250523007.
- Noboru Nakayama. Zariski-decomposition and abundance, volume 14 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2004. DOI:10.2969/msjmemoirs/014010000.
- Yoshinori Namikawa, Smoothing Fano $3$-folds, J. Algebraic Geom. 6 (1997), no. 2, 307–324. MR 1489117
- Yuji Odaka, The GIT stability of polarized varieties via discrepancy, Ann. of Math. (2) 177 (2013), no. 2, 645–661. MR 3010808, DOI 10.4007/annals.2013.177.2.6
- David Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375–386. MR 210944
- Yuri G. Prokhorov, The degree of Fano threefolds with canonical Gorenstein singularities, Mat. Sb. 196 (2005), no. 1, 81–122, DOI 10.1070/SM2005v196n01ABEH000873.
- Thomas Peternell and Jarosław A. Wiśniewski, On stability of tangent bundles of Fano manifolds with $b_2=1$, J. Algebraic Geom. 4 (1995), no. 2, 363–384. MR 1311356
- Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963
- Stephen Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), no. 3, 495–550, DOI 10.2307/2374768.
- Jason Starr, Volume of $-K_X$ for a weighted projective variety, MathOverflow, 2017, mathoverflow.net/q/285155 (version: 2017-11-03).
- Shigeharu Takayama, Simple connectedness of weak Fano varieties, J. Algebraic Geom. 9 (2000), no. 2, 403–407. MR 1735807
- Gang Tian, On stability of the tangent bundles of Fano varieties, Internat. J. Math. 3 (1992), no. 3, 401–413, DOI 10.1142/S0129167X92000175.
- Gang Tian, Kähler-Einstein metrics on algebraic manifolds, Transcendental methods in algebraic geometry (Cetraro, 1994) Lecture Notes in Math., vol. 1646, Springer, Berlin, 1996, pp. 143–185. MR 1603624, DOI 10.1007/BFb0094304
- J. M. Wahl, A cohomological characterization of $\textbf {P}^{n}$, Invent. Math. 72 (1983), no. 2, 315–322. MR 700774, DOI 10.1007/BF01389326
- B. A. F. Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 76, Springer-Verlag, New York-Heidelberg, 1973. MR 0335656
- Chenyang Xu, K-stability of Fano varieties: an algebro-geometric approach, EMS Surv. Math. Sci. 8 (2021), no. 1-2, 265–354. DOI 10.4171.EMSS/51. Preprint, arXiv:2011.10477, November 2020.
- JianMing Yu and GuangFeng Jiang, Reducibility of finite reflection groups, Sci. China Math. 55 (2012), no. 5, 947–948, DOI 10.1007/s11425-011-4341-3.
References
- Mark A. Armstrong, On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965), 639–646, DOI 10.1017/s0305004100038974.
- Michael F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207. MR 86359, DOI 10.2307/1992969
- Indranil Biswas, Semistability and restrictions of tangent bundle to curves, Geom. Dedicata 142 (2009), 37–46, DOI 10.1007/s10711-009-9356-3. Preprint, arXiv:0901.4161.
- Lukas Braun, The local fundamental group of a Kawamata log terminal singularity is finite, Invent. Math. 226 (2021), no. 3, 845–896. DOI 10.1007/s00222-021-01062-0. Preprint, arXiv:2004.00522, 2020.
- Bang-Yen Chen and Koichi Ogiue, Some characterizations of complex space forms in terms of Chern classes, Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 104, 459–464. MR 405303, DOI 10.1093/qmath/26.1.459
- Stéphane Druel, Henri Guenancia, and Mihai Păun, A decomposition theorem for $\mathbb {Q}$-Fano Kähler-Einstein varieties, Preprint, arXiv:2008.05352, August 2020.
- Igor Dolgachev, Weighted projective varieties, In Group actions and vector fields (Vancouver, B.C., 1981), Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34–71, DOI 10.1007/BFb0101508.
- Simon K. Donaldson, Holomorphic discs and the complex Monge-Ampère equation, J. Symplectic Geom. 1 (2002), no. 2, 171–196, euclid.jsg/1092316649.
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345, Available from the author’s web site https://www-fourier.ujf-grenoble.fr/$\sim $demailly/manuscripts/dps1.pdf.
- Stéphane Druel, The Zariski-Lipman conjecture for log canonical spaces, Bull. Lond. Math. Soc. 46 (2014), no. 4, 827–835, DOI 10.1112/blms/bdu040, Preprint, arXiv:1301.5910.
- David Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995, DOI 10.1007/978-1-4612-5350-1.
- William Fulton and Robert Lazarsfeld, Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 26–92. MR 644817
- Hubert Flenner, Divisorenklassengruppen quasihomogener Singularitäten, J. Reine Angew. Math. 328 (1981), 128–160 (German). MR 636200, DOI 10.1515/crll.1981.328.128
- Hubert Flenner, Restrictions of semistable bundles on projective varieties, Comment. Math. Helv. 59 (1984), no. 4, 635–650, DOI 10.1007/BF02566370.
- Takao Fujita, On the hyperplane section principle of Lefschetz, J. Math. Soc. Japan 32 (1980), no. 1, 153–169, DOI 10.2969/jmsj/03210153.
- Takao Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990, DOI 10.1017/CBO9780511662638
- Kento Fujita, Examples of K-unstable Fano manifolds with the Picard number 1, Proc. Edinb. Math. Soc. (2) 60 (2017), no. 4, 881–891, DOI 10.1017/S0013091516000432, Preprint, arXiv:1508.04290.
- Daniel Greb, Stefan Kebekus, Sándor J. Kovács, and Thomas Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. 114 (2011), no. 1, 87–169, DOI 10.1007/s10240-011-0036-0. An extended version with additional graphics is available as arXiv:1003.2913.
- Daniel Greb, Stefan Kebekus, and Thomas Peternell, Movable curves and semistable sheaves, Int. Math. Res. Not. IMRN 2016 (2016), no. 2, 536–570, DOI 10.1093/imrn/rnv126, Preprint, arXiv:1408.4308.
- Daniel Greb, Stefan Kebekus, and Thomas Peternell, Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J. 165 (2016), no. 10, 1965–2004, DOI 10.1215/00127094-3450859 Preprint, arXiv:1307.5718.
- Daniel Greb, Stefan Kebekus, and Thomas Peternell, Projectively flat KLT varieties, Journal de l’École polytechnique—Mathématiques (JEP) Tome 8 (2021), 1005–1036. DOI 10.5802/jep.164. Preprint, arXiv:2010.06878, October 2020.
- Daniel Greb, Stefan Kebekus, Thomas Peternell, and Behrouz Taji, The Miyaoka-Yau inequality and uniformisation of canonical models, Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 6, 1487–1535, DOI 10.24033/asens.2414, Preprint, arXiv:1511.08822.
- Alexander Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
- Alexander Grothendieck, Représentations linéaires et compactification profinie des groupes discrets, Manuscripta Math. 2 (1970), 375–396, DOI 10.1007/BF01719593.
- Daniel Greb and Michael Lennox Wong, Canonical complex extensions of Kähler manifolds, J. Lond. Math. Soc. 101 2020, no. 2, 786–827. DOI 10.1112/jlms.12287. Preprint, arXiv:1807.01223.
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977, DOI 10.1007/978-1-4757-3849-0.
- Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, second edition, Cambridge University Press, Cambridge, 2010, DOI 10.1017/CBO9780511711985.
- Vasily A. Iskovskikh and Yuri G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR 1668579
- Akihiro Kanemitsu, Fano manifolds and stability of tangent bundles, J. Reine Angew. Math. 774 (2021), 163–183. DOI 10.1515/crelle-2020-0043. Preprint, arXiv:1912.12617.
- Stefan Kebekus, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, Complex geometry (Göttingen, 2000) Springer, Berlin, 2002, pp. 147–155. MR 1922103
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original, DOI 10.1017/CBO9780511662560.
- Shoshichi Kobayashi and Takushiro Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47, DOI 10.1215/kjm/1250523432.
- Shoshichi Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5. MR 909698, DOI 10.1515/9781400858682
- Robert Lazarsfeld, Positivity in algebraic geometry. II, 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. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
- Chi Li, Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds, J. Reine Angew. Math. 733 (2017), 55–85, DOI 10.1515/crelle-2014-0156, Preprint, arXiv:1302.6681.
- Gustav I. Lehrer and Donald E. Taylor, Unitary reflection groups, Australian Mathematical Society Lecture Series, vol. 20, Cambridge University Press, Cambridge, 2009. MR 2542964
- Steven Lu and Behrouz Taji, A characterization of finite quotients of abelian varieties, Int. Math. Res. Not. IMRN (2018), no. 1, 292–319, DOI 10.1093/imrn/rnw251, Preprint, arXiv:1410.0063.
- Shigefumi Mori and Shigeru Mukai, On Fano $3$-folds with $B_{2}\geq 2$, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 101–129. MR 715648, DOI 10.2969/aspm/00110101
- Shigefumi Mori and Shigeru Mukai. Erratum: “Classification of Fano 3-folds with $B_2\geq 2$”, Manuscripta Math. 36 (1981/82), no. 2, 147–162, DOI 10.1007/s00229-002-0336-2. MR0641971 (83f:14032).
- Shigefumi Mori and Shigeru Mukai, Classification of Fano $3$-folds with $B_{2}\geq 2$, Manuscripta Math. 36 (1981/82), 147–162, DOI 10.1007/BF01170131.
- Shigefumi Mori, On a generalization of complete intersections, J. Math. Kyoto Univ. 15 (1975), no. 3, 619–646, DOI 10.1215/kjm/1250523007.
- Noboru Nakayama. Zariski-decomposition and abundance, volume 14 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2004. DOI:10.2969/msjmemoirs/014010000.
- Yoshinori Namikawa, Smoothing Fano $3$-folds, J. Algebraic Geom. 6 (1997), no. 2, 307–324. MR 1489117
- Yuji Odaka, The GIT stability of polarized varieties via discrepancy, Ann. of Math. (2) 177 (2013), no. 2, 645–661. MR 3010808, DOI 10.4007/annals.2013.177.2.6
- David Prill, Local classification of quotients of complex manifolds by discontinuous groups, Duke Math. J. 34 (1967), 375–386. MR 210944
- Yuri G. Prokhorov, The degree of Fano threefolds with canonical Gorenstein singularities, Mat. Sb. 196 (2005), no. 1, 81–122, DOI 10.1070/SM2005v196n01ABEH000873.
- Thomas Peternell and Jarosław A. Wiśniewski, On stability of tangent bundles of Fano manifolds with $b_2=1$, J. Algebraic Geom. 4 (1995), no. 2, 363–384. MR 1311356
- Miles Reid, Young person’s guide to canonical singularities, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 345–414. MR 927963
- Stephen Semmes, Complex Monge-Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), no. 3, 495–550, DOI 10.2307/2374768.
- Jason Starr, Volume of $-K_X$ for a weighted projective variety, MathOverflow, 2017, mathoverflow.net/q/285155 (version: 2017-11-03).
- Shigeharu Takayama, Simple connectedness of weak Fano varieties, J. Algebraic Geom. 9 (2000), no. 2, 403–407. MR 1735807
- Gang Tian, On stability of the tangent bundles of Fano varieties, Internat. J. Math. 3 (1992), no. 3, 401–413, DOI 10.1142/S0129167X92000175.
- Gang Tian, Kähler-Einstein metrics on algebraic manifolds, Transcendental methods in algebraic geometry (Cetraro, 1994) Lecture Notes in Math., vol. 1646, Springer, Berlin, 1996, pp. 143–185. MR 1603624, DOI 10.1007/BFb0094304
- Jonathan Wahl, A cohomological characterization of ${\mathbf {P}}^{n}$, Invent. Math. 72 (1983), no. 2, 315–322. MR 700774, DOI 10.1007/BF01389326
- Bertram A. F. Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Matematik und ihrer Grenzgebiete, Band 76. MR 0335656
- Chenyang Xu, K-stability of Fano varieties: an algebro-geometric approach, EMS Surv. Math. Sci. 8 (2021), no. 1-2, 265–354. DOI 10.4171.EMSS/51. Preprint, arXiv:2011.10477, November 2020.
- JianMing Yu and GuangFeng Jiang, Reducibility of finite reflection groups, Sci. China Math. 55 (2012), no. 5, 947–948, DOI 10.1007/s11425-011-4341-3.
Additional Information
Daniel Greb
Affiliation:
Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg–Essen, 45117 Essen, Germany
MR Author ID:
888778
Email:
daniel.greb@uni-due.de
Stefan Kebekus
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany; and Freiburg Institute for Advanced Studies (FRIAS), Freiburg im Breisgau, Germany
MR Author ID:
637173
Email:
stefan.kebekus@math.uni-freiburg.de
Thomas Peternell
Affiliation:
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
MR Author ID:
138450
Email:
thomas.peternell@uni-bayreuth.de
Received by editor(s):
June 18, 2020
Published electronically:
March 8, 2022
Additional Notes:
The second author gratefully acknowledges partial support through a fellowship of the Freiburg Institute of Advanced Studies (FRIAS)
Article copyright:
© Copyright 2022
University Press, Inc.
