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Positivity in Kähler–Einstein theory


Tian initiated the study of incomplete Kähler–Einstein metrics on quasi–projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1-α) for α∈ (0, 1). In this paper we study how the existence of such Kähler–Einstein metrics depends on α. We show that in the negative scalar curvature case, if such Kähler–Einstein metrics exist for all small cone-angles then they exist for every α∈((n+1)/(n+2), 1), where n is the dimension. We also give a characterisation of the pairs that admit negatively curved cone-edge Kähler–Einstein metrics with cone angle close to 2π. Again if these metrics exist for all cone-angles close to 2π, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.

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[Amb03]Ambro, F.Quasi-log varieties. Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 220239.
[And11]Andreatta, M.Minimal Model Program with scaling and adjunction theory. Internat. J. Math. 24 (2013), no. 2, 1350007, 13 pp.
[AS95]Angehrn, U. and Siu, Y.-TEffective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291308.
[AMRT10]Ash, A., Mumford, D., Rapoport, M. and Tai, Y.-S.Smooth Compactifications of Locally Symmetric Varieties. Second edition. Cambridge Mathematical Library (Cambridge University Press, Cambridge, 2010).
[AL12]Atiyah, M. F. and LeBrun, C.Curvature, cones, and characteristic numbers. Math. Proc. Camb. Phil. Soc. 155 (2013), no. 1, 1337.
[Aub78]Aubin, T.Équations du type Monge–Ampère sur les variétiés kählériennes compactes. Bull. Sci. Math 2 (1978), 6395.
[BHPV04]Barth, W., Hulek, K., Peters, C. and Van de Ven, A. Compact complex surfaces. Second edition. Ergeb. Math. Grenzgeb. (3), 4 (Springer-Verlag, Berlin, 2004).
[Ber13]Berman, R. J.A thermodinamical formalism for Monge–Ampère equations, Moser-Trudinger inequalities and and Kähler–Einstein metrics. Adv. Math. 248 (2013), 12541297.
[BCHM10]Birkar, C., Cascini, P., Hacon, C. and McKernan, J.Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), 405468.
[Bre13]Brendle, S.Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. IMRN (2013), no. 24, 57275766.
[CGP13]Campana, F., Guenancia, H. and Păun, M.Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. Ann. Sci. École Norm. Sup. (4), 46 (2013), no.6, 879916.
[CG72]Carlson, J. and Griffiths, P.A defect relation for equidimensional holomorphic mappings between algebraic varieties. Ann. of Math. 95 (1972), 557584.
[CDS13]Chen, X.-X., Donaldson, S. K. and Sun, S.Kähler–Einstein metrics on Fano manifolds, I-III. J. Amer. Math. Soc. 28 (2015), no.1, 183278.
[CY86]Cheng, S. Y. and Yau, S.-T.Inequality between Chern numbers of singular Kähler surfaces and characterisation of orbit space of discrete subgroups of SU(2, 1). Contemp. Math. 49 (1986), 3143.
[CG75]Cornalba, M. and Griffiths, P.Analytic cycles and vector bundles on non-compact algebraic varieties. Invent. Math. 28 (1975), 89120.
[Dem92]Demailly, J.-P.Regularization of closed positive currents and intersection theory. J. Alg. Geom. 1 (1992), 361409.
[Dem01]Demailly, J.-P. Complex analytic and differential geometry. (2001).
[DiC14]Di Cerbo, L. F.On Kähler–Einstein surfaces with edge singularities. J. Geom. Phys. 89 (2014), 414421.
[DD15]Di Cerbo, G. and Di Cerbo, L. F.Effective results for complex hyperbolic manifolds. J. London Math. Soc. 91 (2015), 89104.
[Don10]Donaldson, S. K.Kähler metrics with cone singularities along a divisor. Essays in Mathematics and its Applications (Springer, Heidelberg, 2012), 4979.
[ELMNP06]Ein, L., Lazarsfeld, R., Mustaţǎ, M., Nakamaye, M. and Popa, M.Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56 (2006), 17011734.
[Fuj09]Fujino, O. Introduction to the log minimal model program for log canonical pairs. arXiv:0907.1506v1 [math.AG] (2009).
[GH78]Griffiths, P. and Harris, J.Principles of Algebraic Geometry, Pure and Applied Mathematics (Wiley-Interscience, New York, 1978).
[HMX12]Hacon, C., McKernan, J. and Xu, C. ACC for log canonical thresholds. arXiv:1208.4150v1 [math.AG] (2012).
[Har70]Hartshorne, R.Ample Subvarieties of Algebraic Varieties. Lecture Notes in Math. vol. 156 (Springer-Verlag, Berlin-New York, 1970).
[Har77]Hartshorne, R.Algebraic Geometry. Graduate Texts in Math. No. 52 (Springer-Verlag, New York-Heidelberg, 1977).
[Iit82]Iitaka, S.Algebraic Geometry. An introduction to birational geometry of algebraic varieties. Graduate Texts in Math. 76 (Springer-Verlag, New York-Berlin, 1982).
[Jef96]Jeffres, T. D. Kähler–Einstein cone metrics. Ph.D. Thesis (Stony Brook University, 1996).
[JMR11]Jeffres, T. D., Mazzeo, R. and Rubinstein, Y. A. Kähler–Einstein metrics with edge singularities. arXiv:1105.5216v2 [math.DG] (2011).
[KMM94]Keel, S., Matsuki, K. and McKernan, J.Log abundance theorem for threefolds. Duke Math. J. 75 (1994), no. 1, 99119.
[KMM04]Keel, S., Matsuki, K. and McKernan, J.Corrections to: “Log abundance theorem for threefolds''. Duke Math. J. 122 (2004), no. 3, 625630.
[Kob84]Kobayashi, R.Kähler–Einstein metric on an open algebraic manifold. Osaka. J. Math. 21 (1984), 399418.
[Kol92]Kollár, al. Flips and abundance for algebraic threefolds. Astérisque, vol. 211 (1992).
[Kol97]Kollár, J. Singularities of pairs. Algebraic Geometry–Santa Cruz 1995, 221–287, Proc. Symp. Pure Math. 62, Part 1 (Amer. Math. Soc. Providence, RI, 1997).
[KM98]Kollár, J. and Mori, S.Birational geometry of algebraic varieties. Cambridge Tracts in Math. 134. (Cambridge University Press, Cambridge, 1998).
[KMM92]Kollár, J., Miyaoka, Y. and Mori, S.Rational connectedness and boundedness of Fano manifolds. J. Differential Geom. 36 (1992), no. 3, 765779.
[Laz04a]Lazarsfeld, R.Positivity in Algebraic Geometry I. Ergeb. Math. Grenzgeb. 3. Folge. A series of Modern Survys in Mathematics 48 (Springer-Verlag, Berlin, 2004).
[Laz04b]Lazarsfeld, R.Positivity in Algebraic Geometry II. Ergeb. Math. Grenzgeb. 3. Folge. A series of Modern Survys in Mathematics 49 (Springer-Verlag, Berlin, 2004).
[Laz09]Lazic, V. Adjoint rings are finitely generated. arXiv:0905.2707v3[math.AG] (2009).
[MR12]Mazzeo, R. and Rubinstein, Y. A. The Ricci continuity method for the complex Monge–Ampère equation, with applications to Kähler–Einstein edge metrics. C. R. Acad. Paris, Ser I (2012), 1–5.
[McK02]McKernan, J. Boundedness of log terminal Fano pairs of bounded index. arXiv: math/0205214v1[math.AG] (2002).
[Mum77]Mumford, D.Hirzebruch's proportionality theorem in the non-compact case. Invent. Math. 42 (1977), 239272.
[For91]Forster, O.Lecture on Riemann surfaces. Graduate Texts in Math. 81. (Springer-Verlag, New York, 1991).
[Pet06]Petersen, P.Riemannian Geometry. Second edition. Graduate Texts in Math. 171 (Springer, New York, 2006).
[Sib85]Sibony, N.Quelques problemes de prolongement de courants en analyse complexe. Duke Math. J. 52 (1985), 157197.
[Tia96]Tian, G.Kähler–Einstein metrics on algebraic manifolds. Trascendental methods in algebraic geometry (Cetraro 1994), Lecture Notes in Math. 1646 (Springer, Berlin, 1996), 143185.
[TY87]Tian, G. and Yau, S.-T.Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry. Mathematical Aspects of String Theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys. 1, (World Sci. Publishing, Singapore, 1987), 574628.
[Wu08]Wu, D.Kähler–Einstein metrics of negative Ricci curvature on general quasi-projective manifolds. Comm. Anal. Geom. 16, (2008), 395435.
[Wu09]Wu, D.Good Kähler metrics with prescribed singularities. Asian J. Math. 13 (2009), 131150.
[Yau78a]Yau, S.-T.On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. Comm. Pure Appl. Math. 31 (1978), 339411.
[Yau78b]Yau, S.-T.Métriques de Kähler–Einstein sur les variétiés ouvertes. Séminarie Palaiseau, Astérisque 58 (1978), 163167.
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Mathematical Proceedings of the Cambridge Philosophical Society
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