Special test configuration  and K-stability of Fano varieties

Chi Li; Chenyang Xu

doi:https://doi.org/10.4007/annals.2014.180.1.4

Abstract

For any flat projective family $(\mathcal{X},\mathcal{L})\rightarrow C$ such that the generic fibre $\mathcal{X}_\eta$ is a klt $\mathbb{Q}$-Fano variety and $\mathcal{L}|_{\mathcal{X}_\eta}\sim_{\mathbb{Q}}-K_{X_{\eta}}$, we use the techniques from the minimal model program (MMP) to modify the total family. The end product is a family such that every fiber is a klt $\mathbb{Q}$-Fano variety. Moreover, we can prove that the Donaldson-Futaki invariants of the appearing models decrease. When the family is a test configuration of a fixed Fano variety $(X,-K_X)$, this implies Tian’s conjecture: given $X$ a Fano manifold, to test its K-(semi, poly)stability, we only need to test on the special test configurations.

Note: To view the article, click on the URL link for the DOI number.

Show/hide bibliography for this article

[Ad] $Go to document$ M. Andreatta, "Actions of linear algebraic groups on projective manifolds and minimal model program," Osaka J. Math., vol. 38, iss. 1, pp. 151-166, 2001.

Show bibtex

@article {Ad, MRKEY = {1824904}, AUTHOR = {Andreatta, Marco}, TITLE = {Actions of linear algebraic groups on projective manifolds and minimal model program}, JOURNAL = {Osaka J. Math.}, FJOURNAL = {Osaka Journal of Mathematics}, VOLUME = {38}, YEAR = {2001}, NUMBER = {1}, PAGES = {151--166}, ISSN = {0030-6126}, CODEN = {OJMAA7}, MRCLASS = {14L30 (14E30 14M17)}, MRNUMBER = {1824904}, MRREVIEWER = {Jaros{\l}aw A. Wi{\'s}niewski}, ZBLNUMBER = {1054.14061}, URL = {http://projecteuclid.org/euclid.ojm/1153492317}, }
[Ale94] V. Alexeev, "Moduli spaces $M_{g,n}(W)$ for surfaces," in Higher-Dimensional Complex Varieties, Berlin: de Gruyter, 1996, pp. 1-22.

Show bibtex

@incollection {Ale94, MRKEY = {1463171}, AUTHOR = {Alexeev, Valery}, TITLE = {Moduli spaces {$M\sb {g,n}(W)$} for surfaces}, BOOKTITLE = {Higher-Dimensional Complex Varieties}, VENUE={{T}rento, 1994}, PAGES = {1--22}, PUBLISHER = {de Gruyter}, ADDRESS = {Berlin}, YEAR = {1996}, MRCLASS = {14D20 (14D22 14J10)}, MRNUMBER = {1463171}, MRREVIEWER = {Wolfgang K. Seiler}, ZBLNUMBER = {0896.14014}, }
[ALV] $Go to document$ C. Arezzo, A. Della Vedova, and G. La Nave, "Singularities and $K$-semistability," Int. Math. Res. Not., vol. 2012, iss. 4, p. no. 4, 849-869.

Show bibtex

@article {ALV, MRKEY = {2889160}, KEY={ADVLN12}, AUTHOR = {Arezzo, Claudio and Della Vedova, Alberto and La Nave, Gabriele}, TITLE = {Singularities and {$K$}-semistability}, JOURNAL = {Int. Math. Res. Not.}, FJOURNAL = {International Mathematics Research Notices. IMRN}, SORTYEAR = {2012}, NUMBER = {4}, PAGES = {no.~4, 849--869}, ISSN = {1073-7928}, MRCLASS = {32Q26}, MRNUMBER = {2889160}, MRREVIEWER = {Julius Ross}, ZBLNUMBER = {237.32003}, VOLUME = {2012}, DOI = {10.1093/imrn/rnr044}, }
[BCHM] $Go to document$ C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, "Existence of minimal models for varieties of log general type," J. Amer. Math. Soc., vol. 23, iss. 2, pp. 405-468, 2010.

Show bibtex

@article {BCHM, MRKEY = {2601039}, AUTHOR = {Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James}, TITLE = {Existence of minimal models for varieties of log general type}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {23}, YEAR = {2010}, NUMBER = {2}, PAGES = {405--468}, ISSN = {0894-0347}, MRCLASS = {14E30 (14E05)}, MRNUMBER = {2601039}, MRREVIEWER = {Mark Gross}, ZBLNUMBER = {1210.14019}, DOI = {10.1090/S0894-0347-09-00649-3}, }
[BHPV] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact Complex Surfaces, Second ed., New York: Springer-Verlag, 2004, vol. 4.

Show bibtex

@book {BHPV, MRKEY = {2030225}, AUTHOR = {Barth, Wolf P. and Hulek, Klaus and Peters, Chris A. M. and Van de Ven, Antonius}, TITLE = {Compact Complex Surfaces}, SERIES = {Ergeb. Math. Grenzgeb.}, VOLUME = {4}, EDITION = {Second}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {2004}, PAGES = {xii+436}, ISBN = {3-540-00832-2}, MRCLASS = {14Jxx (14-02 32-02 32J15 57R57)}, MRNUMBER = {2030225}, MRREVIEWER = {I. Dolgachev}, ZBLNUMBER = {1036.14016}, }
[CCT] $Go to document$ J. Cheeger, T. H. Colding, and G. Tian, "On the singularities of spaces with bounded Ricci curvature," Geom. Funct. Anal., vol. 12, iss. 5, pp. 873-914, 2002.

Show bibtex

@article {CCT, MRKEY = {1937830}, AUTHOR = {Cheeger, J. and Colding, T. H. and Tian, G.}, TITLE = {On the singularities of spaces with bounded {R}icci curvature}, JOURNAL = {Geom. Funct. Anal.}, FJOURNAL = {Geometric and Functional Analysis}, VOLUME = {12}, YEAR = {2002}, NUMBER = {5}, PAGES = {873--914}, ISSN = {1016-443X}, CODEN = {GFANFB}, MRCLASS = {53C21 (53C20)}, MRNUMBER = {1937830}, MRREVIEWER = {Zhongmin Shen}, ZBLNUMBER = {1030.53046}, DOI = {10.1007/PL00012649}, }
[CDSI] X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities.

Show bibtex

@misc{CDSI, author={Chen, X. and Donaldson, S. and Sun, S.}, TITLE={Kähler-{E}instein metrics on {F}ano manifolds, {I}: approximation of metrics with cone singularities}, ARXIV={1211.4566}, SORTYEAR={2011}, }
[CDSII] X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metrics on Fano manifolds, II: limits with cone angle less than $2\pi$.

Show bibtex

@misc{CDSII, author={Chen, X. and Donaldson, S. and Sun, S.}, TITLE={Kähler-{E}instein metrics on {F}ano manifolds, {II}: limits with cone angle less than $2\pi$}, ARXIV={1212.4714}, SORTYEAR={2012}, }
[CDSIII] X. Chen, S. Donaldson, and S. Sun, Kähler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches $2\pi$ and completion of the main proof.

Show bibtex

@misc{CDSIII, author={Chen, X. and Donaldson, S. and Sun, S.}, TITLE={Kähler-{E}instein metrics on {F}ano manifolds, {III}: limits as cone angle approaches $2\pi$ and completion of the main proof}, SORTYEAR={2013}, ARXIV={1302.0282}, }
[DM69] $Go to document$ P. Deligne and D. Mumford, "The irreducibility of the space of curves of given genus," Inst. Hautes Études Sci. Publ. Math., vol. 36, pp. 75-109, 1969.

Show bibtex

@article {DM69, MRKEY = {0262240}, AUTHOR = {Deligne, P. and Mumford, D.}, TITLE = {The irreducibility of the space of curves of given genus}, JOURNAL = {Inst. Hautes Études Sci. Publ. Math.}, FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques}, VOLUME = {36}, YEAR = {1969}, PAGES = {75--109}, ISSN = {0073-8301}, MRCLASS = {14.20}, MRNUMBER = {0262240}, MRREVIEWER = {Manfred Herrmann}, ZBLNUMBER = {0181.48803}, DOI = {10.1007/BF02684599}, }
[Dol] $Go to document$ S. K. Donaldson, "Scalar curvature and stability of toric varieties," J. Differential Geom., vol. 62, iss. 2, pp. 289-349, 2002.

Show bibtex

@article {Dol, MRKEY = {1988506}, AUTHOR = {Donaldson, S. K.}, TITLE = {Scalar curvature and stability of toric varieties}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {62}, YEAR = {2002}, NUMBER = {2}, PAGES = {289--349}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {32Q15 (14M25 53C21 53C55)}, MRNUMBER = {1988506}, ZBLNUMBER = {1074.53059}, URL = {http://projecteuclid.org/euclid.jdg/1090950195}, }
[DT] $Go to document$ W. Y. Ding and G. Tian, "Kähler-Einstein metrics and the generalized Futaki invariant," Invent. Math., vol. 110, iss. 2, pp. 315-335, 1992.

Show bibtex

@article {DT, MRKEY = {1185586}, AUTHOR = {Ding, Wei Yue and Tian, Gang}, TITLE = {Kähler-{E}instein metrics and the generalized {F}utaki invariant}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {110}, YEAR = {1992}, NUMBER = {2}, PAGES = {315--335}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {53C25 (32L07 53C55)}, MRNUMBER = {1185586}, MRREVIEWER = {Antonella Nannicini}, ZBLNUMBER = {0779.53044}, DOI = {10.1007/BF01231335}, }
[FS90] $Go to document$ A. Fujiki and G. Schumacher, "The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics," Publ. Res. Inst. Math. Sci., vol. 26, iss. 1, pp. 101-183, 1990.

Show bibtex

@article {FS90, MRKEY = {1053910}, AUTHOR = {Fujiki, Akira and Schumacher, Georg}, TITLE = {The moduli space of extremal compact {K}ähler manifolds and generalized {W}eil-{P}etersson metrics}, JOURNAL = {Publ. Res. Inst. Math. Sci.}, FJOURNAL = {Kyoto University. Research Institute for Mathematical Sciences. Publications}, VOLUME = {26}, YEAR = {1990}, NUMBER = {1}, PAGES = {101--183}, ISSN = {0034-5318}, CODEN = {KRMPBV}, MRCLASS = {32G13 (32L07 58G26)}, MRNUMBER = {1053910}, MRREVIEWER = {P. E. Newstead}, ZBLNUMBER = {0714.32007}, DOI = {10.2977/prims/1195171664}, }
[Fut] $Go to document$ A. Futaki, "An obstruction to the existence of Einstein Kähler metrics," Invent. Math., vol. 73, iss. 3, pp. 437-443, 1983.

Show bibtex

@article {Fut, MRKEY = {0718940}, AUTHOR = {Futaki, A.}, TITLE = {An obstruction to the existence of {E}instein {K}ähler metrics}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {73}, YEAR = {1983}, NUMBER = {3}, PAGES = {437--443}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {53C55 (32J99)}, MRNUMBER = {0718940}, MRREVIEWER = {J. L. Kazdan}, ZBLNUMBER = {0506.53030}, DOI = {10.1007/BF01388438}, }
[Ha] $Go to document$ R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.

Show bibtex

@book {Ha, MRKEY = {0463157}, AUTHOR = {Hartshorne, Robin}, TITLE = {Algebraic Geometry}, SERIES = {Grad. Texts in Math}, VOLUME={52}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1977}, PAGES = {xvi+496}, ISBN = {0-387-90244-9}, MRCLASS = {14-01}, MRNUMBER = {0463157}, MRREVIEWER = {Robert Speiser}, ZBLNUMBER = {0531.14001}, DOI = {10.1007/978-1-4757-3849-0}, }
[HM07] $Go to document$ C. D. Hacon and J. Mckernan, "On Shokurov’s rational connectedness conjecture," Duke Math. J., vol. 138, iss. 1, pp. 119-136, 2007.

Show bibtex

@article {HM07, MRKEY = {2309156}, AUTHOR = {Hacon, Christopher D. and Mckernan, James}, TITLE = {On {S}hokurov's rational connectedness conjecture}, JOURNAL = {Duke Math. J.}, FJOURNAL = {Duke Mathematical Journal}, VOLUME = {138}, YEAR = {2007}, NUMBER = {1}, PAGES = {119--136}, ISSN = {0012-7094}, CODEN = {DUMJAO}, MRCLASS = {14E30 (14E05 14J45)}, MRNUMBER = {2309156}, MRREVIEWER = {Mihnea Popa}, ZBLNUMBER = {1128.14028}, DOI = {10.1215/S0012-7094-07-13813-4}, }
[HoXu] $Go to document$ A. Hogadi and C. Xu, "Degenerations of rationally connected varieties," Trans. Amer. Math. Soc., vol. 361, iss. 7, pp. 3931-3949, 2009.

Show bibtex

@article {HoXu, MRKEY = {2491906}, AUTHOR = {Hogadi, Amit and Xu, Chenyang}, TITLE = {Degenerations of rationally connected varieties}, JOURNAL = {Trans. Amer. Math. Soc.}, FJOURNAL = {Transactions of the American Mathematical Society}, VOLUME = {361}, YEAR = {2009}, NUMBER = {7}, PAGES = {3931--3949}, ISSN = {0002-9947}, CODEN = {TAMTAM}, MRCLASS = {14M22 (14D06 14F45)}, MRNUMBER = {2491906}, MRREVIEWER = {Alexandr V. Pukhlikov}, ZBLNUMBER = {1173.14032}, DOI = {10.1090/S0002-9947-09-04715-1}, }
[KKMS] $Go to document$ G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal Embeddings. I, New York: Springer-Verlag, 1973, vol. 339.

Show bibtex

@book {KKMS, MRKEY = {0335518}, AUTHOR = {Kempf, G. and Knudsen, Finn Faye and Mumford, D. and Saint-Donat, B.}, TITLE = {Toroidal Embeddings. {I}}, SERIES = {Lecture Notes in Math.}, VOLUME={339}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1973}, PAGES = {viii+209}, MRCLASS = {14E15 (14D20 14E05 14M20 20G15)}, MRNUMBER = {0335518}, MRREVIEWER = {G. Harder}, ZBLNUMBER = {0271.14017}, DOI = {10.1007/BFb0070318}, }
[KM] $Go to document$ J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge: Cambridge Univ. Press, 1998, vol. 134.

Show bibtex

@book {KM, MRKEY = {1658959}, AUTHOR = {Koll{á}r, J{á}nos and Mori, Shigefumi}, TITLE = {Birational Geometry of Algebraic Varieties}, SERIES = {Cambridge Tracts in Math.}, VOLUME = {134}, PUBLISHER = {Cambridge Univ. Press}, ADDRESS = {Cambridge}, YEAR = {1998}, PAGES = {viii+254}, ISBN = {0-521-63277-3}, MRCLASS = {14E30}, MRNUMBER = {1658959}, MRREVIEWER = {Mark Gross}, ZBLNUMBER = {0926.14003}, DOI = {10.1017/CBO9780511662560}, }
[Kol07] $Go to document$ J. Kollár, "A conjecture of Ax and degenerations of Fano varieties," Israel J. Math., vol. 162, pp. 235-251, 2007.

Show bibtex

@article {Kol07, MRKEY = {2365862}, AUTHOR = {Koll{á}r, J{á}nos}, TITLE = {A conjecture of {A}x and degenerations of {F}ano varieties}, JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {162}, YEAR = {2007}, PAGES = {235--251}, ISSN = {0021-2172}, CODEN = {ISJMAP}, MRCLASS = {14D06 (12E30 14G05 14J45)}, MRNUMBER = {2365862}, MRREVIEWER = {Tam{á}s Szamuely}, ZBLNUMBER = {1140.14009}, DOI = {10.1007/s11856-007-0097-4}, }
[Ko] J. Kollár, Lectures on Resolution of Singularities, Princeton, NJ: Princeton Univ. Press, 2007, vol. 166.

Show bibtex

@book {Ko, MRKEY = {2289519}, AUTHOR = {Koll{á}r, J{á}nos}, TITLE = {Lectures on Resolution of Singularities}, SERIES = {Ann. of Math. Stud.}, VOLUME = {166}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, NJ}, YEAR = {2007}, PAGES = {vi+208}, ISBN = {978-0-691-12923-5; 0-691-12923-1}, MRCLASS = {14E15 (32S45)}, MRNUMBER = {2289519}, MRREVIEWER = {Dan Abramovich}, ZBLNUMBER = {1113.14013}, }
[Kol] J. Kollár, "Moduli of varieties of general type," in Handbook of Moduli: Volume II, Somerville, MA: Internat. Press, 2013, vol. 25, pp. 115-130.

Show bibtex

@incollection{Kol, author = {Koll{á}r, J{á}nos}, TITLE = {Moduli of varieties of general type}, BOOKTITLE={Handbook of Moduli: Volume II}, PAGES={115--130}, SERIES={Adv. Lect. Math.}, VOLUME={25}, PUBLISHER={Internat. Press}, ADDRESS={Somerville, MA}, YEAR={2013}, ZBLNUMBER = {1260.14002}, }
[KSB] $Go to document$ J. Kollár and N. I. Shepherd-Barron, "Threefolds and deformations of surface singularities," Invent. Math., vol. 91, iss. 2, pp. 299-338, 1988.

Show bibtex

@article {KSB, MRKEY = {0922803}, AUTHOR = {Koll{á}r, J{á}nos and Shepherd-Barron, N. I.}, TITLE = {Threefolds and deformations of surface singularities}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {91}, YEAR = {1988}, NUMBER = {2}, PAGES = {299--338}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {14J10 (14D20 14J30 32G10 32G13)}, MRNUMBER = {0922803}, MRREVIEWER = {Yujiro Kawamata}, ZBLNUMBER = {0642.14008}, DOI = {10.1007/BF01389370}, }
[Lai] $Go to document$ C. Lai, "Varieties fibered by good minimal models," Math. Ann., vol. 350, iss. 3, pp. 533-547, 2011.

Show bibtex

@article {Lai, MRKEY = {2805635}, AUTHOR = {Lai, Ching-Jui}, TITLE = {Varieties fibered by good minimal models}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {350}, YEAR = {2011}, NUMBER = {3}, PAGES = {533--547}, ISSN = {0025-5831}, CODEN = {MAANA}, MRCLASS = {14E30}, MRNUMBER = {2805635}, MRREVIEWER = {Daniel Greb}, ZBLNUMBER = {1221.14018}, DOI = {10.1007/s00208-010-0574-7}, }
[Mab] $Go to document$ T. Mabuchi, "$K$-energy maps integrating Futaki invariants," Tohoku Math. J., vol. 38, iss. 4, pp. 575-593, 1986.

Show bibtex

@article {Mab, MRKEY = {0867064}, AUTHOR = {Mabuchi, Toshiki}, TITLE = {{$K$}-energy maps integrating {F}utaki invariants}, JOURNAL = {Tohoku Math. J.}, FJOURNAL = {The Tohoku Mathematical Journal. Second Series}, VOLUME = {38}, YEAR = {1986}, NUMBER = {4}, PAGES = {575--593}, ISSN = {0040-8735}, CODEN = {TOMJAM}, MRCLASS = {53C25 (58E11)}, MRNUMBER = {0867064}, MRREVIEWER = {John C. Wood}, ZBLNUMBER = {0619.53040}, DOI = {10.2748/tmj/1178228410}, }
[Mum] $Go to document$ D. Mumford, "The topology of normal singularities of an algebraic surface and a criterion for simplicity," Inst. Hautes Études Sci. Publ. Math., vol. 9, pp. 5-22, 1961.

Show bibtex

@article {Mum, MRKEY = {0153682}, AUTHOR = {Mumford, David}, TITLE = {The topology of normal singularities of an algebraic surface and a criterion for simplicity}, JOURNAL = {Inst. Hautes Études Sci. Publ. Math.}, FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques}, VOLUME = {9}, YEAR = {1961}, PAGES = {5--22}, ISSN = {0073-8301}, MRCLASS = {14.18 (14.55)}, MRNUMBER = {0153682}, MRREVIEWER = {T. Matsusaka}, ZBLNUMBER = {0108.16801}, DOI = {10.1007/BF02698717}, }
[Oda2] $Go to document$ Y. Odaka, "The Calabi conjecture and K-stability," Int. Math. Res. Not., vol. 2012, iss. 10, pp. 2272-2288, 2012.

Show bibtex

@article {Oda2, MRKEY = {2923166}, AUTHOR = {Odaka, Yuji}, TITLE = {The {C}alabi conjecture and {K}-stability}, JOURNAL = {Int. Math. Res. Not.}, FJOURNAL = {International Mathematics Research Notices. IMRN}, YEAR = {2012}, NUMBER = {10}, PAGES = {2272--2288}, ISSN = {1073-7928}, MRCLASS = {32Q26 (32Q20)}, MRNUMBER = {2923166}, MRREVIEWER = {Jacopo Stoppa}, ZBLNUMBER = {06156617}, DOI = {10.1093/imrn/rnr107}, VOLUME = {2012}, }
[Oda1] $Go to document$ Y. Odaka, "The GIT stability of polarized varieties via discrepancy," Ann. of Math., vol. 177, iss. 2, pp. 645-661, 2013.

Show bibtex

@article {Oda1, MRKEY = {3010808}, AUTHOR = {Odaka, Yuji}, TITLE = {The {GIT} stability of polarized varieties via discrepancy}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {177}, YEAR = {2013}, NUMBER = {2}, PAGES = {645--661}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {14L24}, MRNUMBER = {3010808}, ZBLNUMBER = {1271.14067}, DOI = {10.4007/annals.2013.177.2.6}, }
[Oda] $Go to document$ Y. Odaka, "A generalization of the Ross-Thomas slope theory," Osaka J. Math., vol. 50, iss. 1, pp. 171-185, 2013.

Show bibtex

@article {Oda, MRKEY = {3080636}, AUTHOR = {Odaka, Yuji}, TITLE = {A generalization of the {R}oss-{T}homas slope theory}, JOURNAL = {Osaka J. Math.}, FJOURNAL = {Osaka Journal of Mathematics}, VOLUME = {50}, YEAR = {2013}, NUMBER = {1}, PAGES = {171--185}, ISSN = {0030-6126}, MRCLASS = {14L24 (32Q15 32Q26)}, MRNUMBER = {3080636}, ZBLNUMBER = {1271.14067}, URL = {http://projecteuclid.org/euclid.ojm/1364390425}, }
[OX] $Go to document$ Y. Odaka and C. Xu, "Log-canonical models of singular pairs and its applications," Math. Res. Lett., vol. 19, iss. 2, pp. 325-334, 2012.

Show bibtex

@article {OX, MRKEY = {2955764}, AUTHOR = {Odaka, Yuji and Xu, Chenyang}, TITLE = {Log-canonical models of singular pairs and its applications}, JOURNAL = {Math. Res. Lett.}, FJOURNAL = {Mathematical Research Letters}, VOLUME = {19}, YEAR = {2012}, NUMBER = {2}, PAGES = {325--334}, ISSN = {1073-2780}, MRCLASS = {14E30}, MRNUMBER = {2955764}, MRREVIEWER = {James McKernan}, ZBLNUMBER = {06194653}, DOI = {10.4310/MRL.2012.v19.n2.a5}, }
[PP10] $Go to document$ B. Pasquier and N. Perrin, "Local rigidity of quasi-regular varieties," Math. Z., vol. 265, iss. 3, pp. 589-600, 2010.

Show bibtex

@article {PP10, MRKEY = {2644311}, AUTHOR = {Pasquier, Boris and Perrin, Nicolas}, TITLE = {Local rigidity of quasi-regular varieties}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {265}, YEAR = {2010}, NUMBER = {3}, PAGES = {589--600}, ISSN = {0025-5874}, CODEN = {MAZEAX}, MRCLASS = {14L30 (14B12 14F10 14M27)}, MRNUMBER = {2644311}, MRREVIEWER = {Paolo Bravi}, ZBLNUMBER = {1200.14097}, DOI = {10.1007/s00209-009-0531-x}, }
[PauTi] S. T. Paul and G. Tian, CM stability and the generalized Futaki invariant I.

Show bibtex

@misc{PauTi, MRKEY = {2674882}, AUTHOR = {Paul, Sean Timothy and Tian, Gang}, TITLE = {C{M} stability and the generalized {F}utaki invariant {I}}, ARXIV={math/0605278}, SORTYEAR={2014}, }
[PauTi2] S. T. Paul and G. Tian, "CM stability and the generalized Futaki invariant II," Astérisque, vol. 328, pp. 339-354, 2009.

Show bibtex

@article {PauTi2, MRKEY = {2674882}, AUTHOR = {Paul, Sean Timothy and Tian, Gang}, TITLE = {C{M} stability and the generalized {F}utaki invariant {II}}, JOURNAL = {Astérisque}, FJOURNAL = {Astérisque}, VOLUME = {328}, YEAR = {2009}, PAGES = {339--354}, ISSN = {0303-1179}, ISBN = {978-2-85629-289-1}, MRCLASS = {32Q26 (14D06 32Q20)}, MRNUMBER = {2674882}, MRREVIEWER = {Julien Keller}, ZBLNUMBER = {1204.53061}, }
[RT] $Go to document$ J. Ross and R. Thomas, "A study of the Hilbert-Mumford criterion for the stability of projective varieties," J. Algebraic Geom., vol. 16, iss. 2, pp. 201-255, 2007.

Show bibtex

@article {RT, MRKEY = {2274514}, AUTHOR = {Ross, Julius and Thomas, Richard}, TITLE = {A study of the {H}ilbert-{M}umford criterion for the stability of projective varieties}, JOURNAL = {J. Algebraic Geom.}, FJOURNAL = {Journal of Algebraic Geometry}, VOLUME = {16}, YEAR = {2007}, NUMBER = {2}, PAGES = {201--255}, ISSN = {1056-3911}, MRCLASS = {14L24 (14L30 32L10 32Q20 53C25)}, MRNUMBER = {2274514}, MRREVIEWER = {Julien Keller}, ZBLNUMBER = {1200.14095}, DOI = {10.1090/S1056-3911-06-00461-9}, }
[St] $Go to document$ J. Stoppa, "K-stability of constant scalar curvature Kähler manifolds," Adv. Math., vol. 221, iss. 4, pp. 1397-1408, 2009.

Show bibtex

@article {St, MRKEY = {2518643}, AUTHOR = {Stoppa, Jacopo}, TITLE = {K-stability of constant scalar curvature {K}ähler manifolds}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {221}, YEAR = {2009}, NUMBER = {4}, PAGES = {1397--1408}, ISSN = {0001-8708}, CODEN = {ADMTA4}, MRCLASS = {32Q26 (14L24 32Q15)}, MRNUMBER = {2518643}, MRREVIEWER = {Julius Ross}, ZBLNUMBER = {1181.53060}, DOI = {10.1016/j.aim.2009.02.013}, }
[Tian1] $Go to document$ G. Tian, "On a set of polarized Kähler metrics on algebraic manifolds," J. Differential Geom., vol. 32, iss. 1, pp. 99-130, 1990.

Show bibtex

@article {Tian1, MRKEY = {1064867}, AUTHOR = {Tian, Gang}, TITLE = {On a set of polarized {K}ähler metrics on algebraic manifolds}, JOURNAL = {J. Differential Geom.}, FJOURNAL = {Journal of Differential Geometry}, VOLUME = {32}, YEAR = {1990}, NUMBER = {1}, PAGES = {99--130}, ISSN = {0022-040X}, CODEN = {JDGEAS}, MRCLASS = {32L07 (32C17 53C55)}, MRNUMBER = {1064867}, MRREVIEWER = {John M. Lee}, ZBLNUMBER = {0706.53036}, URL = {http://projecteuclid.org/euclid.jdg/1214445039}, }
[Tian97] $Go to document$ G. Tian, "Kähler-Einstein metrics with positive scalar curvature," Invent. Math., vol. 130, iss. 1, pp. 1-37, 1997.

Show bibtex

@article {Tian97, MRKEY = {1471884}, AUTHOR = {Tian, Gang}, TITLE = {Kähler-{E}instein metrics with positive scalar curvature}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {130}, YEAR = {1997}, NUMBER = {1}, PAGES = {1--37}, ISSN = {0020-9910}, CODEN = {INVMBH}, MRCLASS = {53C25 (32L07 53C55)}, MRNUMBER = {1471884}, MRREVIEWER = {Thalia D. Jeffres}, ZBLNUMBER = {0892.53027}, DOI = {10.1007/s002220050176}, }
[Tian2] $Go to document$ G. Tian, "Existence of Einstein metrics on Fano manifolds," in Metric and Differential Geometry. The Jeff Cheeger Anniversary Volume, Basel: Birkhäuser/Springer Basel AG, 2012, vol. 297, pp. 119-159.

Show bibtex

@incollection{Tian2, author={Tian, Gang}, TITLE = {Existence of {E}instein metrics on {F}ano manifolds}, BOOKTITLE={Metric and Differential Geometry. The Jeff Cheeger Anniversary Volume}, PAGES={119--159}, SERIES={Progr. in Math.}, VOLUME={297}, PUBLISHER={Birkhäuser/Springer Basel AG}, ADDRESS={Basel}, YEAR={2012}, ZBLNUMBER = {1250.53044}, DOI = {10.1007/978-3-0348-0257-4_5}, }
[Tian12] G. Tian, K-stability and Kähler-Einstein metrics.

Show bibtex

@misc{Tian12, author = {Tian, Gang}, TITLE = {{K}-stability and {K}ähler-{E}instein metrics}, SORTYEAR={2013}, ARXIV={1211.4669}, }
[Tianp] G. Tian, Private communication.

Show bibtex

@misc{Tianp, author = {Tian, Gang}, TITLE = {Private communication}, SORTYEAR={2014}, }
[Wang] $Go to document$ X. Wang, "Height and GIT weight," Math. Res. Lett., vol. 19, iss. 4, pp. 909-926, 2012.

Show bibtex

@article {Wang, MRKEY = {3008424}, AUTHOR = {Wang, Xiaowei}, TITLE = {Height and {GIT} weight}, JOURNAL = {Math. Res. Lett.}, FJOURNAL = {Mathematical Research Letters}, VOLUME = {19}, YEAR = {2012}, NUMBER = {4}, PAGES = {909--926}, ISSN = {1073-2780}, MRCLASS = {14L24 (32Q26)}, MRNUMBER = {3008424}, ZBLNUMBER = {06165862}, DOI = {10.4310/MRL.2012.v19.n4.a14}, }
[Zhang06] $Go to document$ Q. Zhang, "Rational connectedness of log ${\bf Q}$-Fano varieties," J. Reine Angew. Math., vol. 590, pp. 131-142, 2006.

Show bibtex

@article {Zhang06, MRKEY = {2208131}, AUTHOR = {Zhang, Qi}, TITLE = {Rational connectedness of log {${\bf Q}$}-{F}ano varieties}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik}, VOLUME = {590}, YEAR = {2006}, PAGES = {131--142}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {14E30 (14J45)}, MRNUMBER = {2208131}, MRREVIEWER = {Stefan Kebekus}, ZBLNUMBER = {1093.14059}, DOI = {10.1515/CRELLE.2006.006}, }

Special test configuration and K-stability of Fano varieties

Abstract

Authors