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Positivity in Kähler–Einstein theory

Published online by Cambridge University Press:  25 June 2015

GABRIELE DI CERBO  and
LUCA F. DI CERBO
GABRIELE DI CERBO
Affiliation:
Department of Mathematics, Princeton University, NJ 08544-1000, U.S.A. e-mail: gdi@math.princeton.edu
LUCA F. DI CERBO
Affiliation:
Department of Mathematics, Duke University, Durham NC 27708-0320, U.S.A. e-mail: luca@math.duke.edu
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 2 , September 2015 , pp. 321 - 338
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[Amb03] Ambro, F. Quasi-log varieties. Tr. Mat. Inst. Steklova 240 (2003), Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 220–239.Google Scholar
[And11] Andreatta, M. Minimal Model Program with scaling and adjunction theory. Internat. J. Math. 24 (2013), no. 2, 1350007, 13 pp.Google Scholar
[AS95] Angehrn, U. and Siu, Y.-T Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291–308.Google Scholar
[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).Google Scholar
[AL12] Atiyah, M. F. and LeBrun, C. Curvature, cones, and characteristic numbers. Math. Proc. Camb. Phil. Soc. 155 (2013), no. 1, 13–37.Google Scholar
[Aub78] Aubin, T. Équations du type Monge–Ampère sur les variétiés kählériennes compactes. Bull. Sci. Math 2 (1978), 63–95.Google Scholar
[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).Google Scholar
[Ber13] Berman, R. J. A thermodinamical formalism for Monge–Ampère equations, Moser-Trudinger inequalities and and Kähler–Einstein metrics. Adv. Math. 248 (2013), 1254–1297.Google Scholar
[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), 405–468.Google Scholar
[Bre13] Brendle, S. Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. IMRN (2013), no. 24, 5727–5766.Google Scholar
[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, 879–916.Google Scholar
[CG72] Carlson, J. and Griffiths, P. A defect relation for equidimensional holomorphic mappings between algebraic varieties. Ann. of Math. 95 (1972), 557–584.Google Scholar
[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, 183–278.Google Scholar
[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), 31–43.Google Scholar
[CG75] Cornalba, M. and Griffiths, P. Analytic cycles and vector bundles on non-compact algebraic varieties. Invent. Math. 28 (1975), 89–120.Google Scholar
[Dem92] Demailly, J.-P. Regularization of closed positive currents and intersection theory. J. Alg. Geom. 1 (1992), 361–409.Google Scholar
[Dem01] Demailly, J.-P. Complex analytic and differential geometry. www-fourier.ujf-grenoble.fr/~demailly/lectures.html (2001).Google Scholar
[DiC14] Di Cerbo, L. F. On Kähler–Einstein surfaces with edge singularities. J. Geom. Phys. 89 (2014), 414–421.Google Scholar
[DD15] Di Cerbo, G. and Di Cerbo, L. F. Effective results for complex hyperbolic manifolds. J. London Math. Soc. 91 (2015), 89–104.Google Scholar
[Don10] Donaldson, S. K. Kähler metrics with cone singularities along a divisor. Essays in Mathematics and its Applications (Springer, Heidelberg, 2012), 49–79.Google Scholar
[ELMNP06] Ein, L., Lazarsfeld, R., Mustaţǎ, M., Nakamaye, M. and Popa, M. Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56 (2006), 1701–1734.Google Scholar
[Fuj09] Fujino, O. Introduction to the log minimal model program for log canonical pairs. arXiv:0907.1506v1 [math.AG] (2009).Google Scholar
[GH78] Griffiths, P. and Harris, J. Principles of Algebraic Geometry, Pure and Applied Mathematics (Wiley-Interscience, New York, 1978).Google Scholar
[HMX12] Hacon, C., McKernan, J. and Xu, C. ACC for log canonical thresholds. arXiv:1208.4150v1 [math.AG] (2012).Google Scholar
[Har70] Hartshorne, R. Ample Subvarieties of Algebraic Varieties. Lecture Notes in Math. vol. 156 (Springer-Verlag, Berlin-New York, 1970).Google Scholar
[Har77] Hartshorne, R. Algebraic Geometry. Graduate Texts in Math. No. 52 (Springer-Verlag, New York-Heidelberg, 1977).Google Scholar
[Iit82] Iitaka, S. Algebraic Geometry. An introduction to birational geometry of algebraic varieties. Graduate Texts in Math. 76 (Springer-Verlag, New York-Berlin, 1982).Google Scholar
[Jef96] Jeffres, T. D. Kähler–Einstein cone metrics. Ph.D. Thesis (Stony Brook University, 1996).Google Scholar
[JMR11] Jeffres, T. D., Mazzeo, R. and Rubinstein, Y. A. Kähler–Einstein metrics with edge singularities. arXiv:1105.5216v2 [math.DG] (2011).Google Scholar
[KMM94] Keel, S., Matsuki, K. and McKernan, J. Log abundance theorem for threefolds. Duke Math. J. 75 (1994), no. 1, 99–119.Google Scholar
[KMM04] Keel, S., Matsuki, K. and McKernan, J. Corrections to: “Log abundance theorem for threefolds''. Duke Math. J. 122 (2004), no. 3, 625–630.Google Scholar
[Kob84] Kobayashi, R. Kähler–Einstein metric on an open algebraic manifold. Osaka. J. Math. 21 (1984), 399–418.Google Scholar
[Kol92] Kollár, J. et al. Flips and abundance for algebraic threefolds. Astérisque, vol. 211 (1992).Google Scholar
[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).Google Scholar
[KM98] Kollár, J. and Mori, S. Birational geometry of algebraic varieties. Cambridge Tracts in Math. 134. (Cambridge University Press, Cambridge, 1998).Google Scholar
[KMM92] Kollár, J., Miyaoka, Y. and Mori, S. Rational connectedness and boundedness of Fano manifolds. J. Differential Geom. 36 (1992), no. 3, 765–779.Google Scholar
[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).Google Scholar
[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).Google Scholar
[Laz09] Lazic, V. Adjoint rings are finitely generated. arXiv:0905.2707v3[math.AG] (2009).Google Scholar
[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.Google Scholar
[McK02] McKernan, J. Boundedness of log terminal Fano pairs of bounded index. arXiv: math/0205214v1[math.AG] (2002).Google Scholar
[Mum77] Mumford, D. Hirzebruch's proportionality theorem in the non-compact case. Invent. Math. 42 (1977), 239–272.Google Scholar
[For91] Forster, O. Lecture on Riemann surfaces. Graduate Texts in Math. 81. (Springer-Verlag, New York, 1991).Google Scholar
[Pet06] Petersen, P. Riemannian Geometry. Second edition. Graduate Texts in Math. 171 (Springer, New York, 2006).Google Scholar
[Sib85] Sibony, N. Quelques problemes de prolongement de courants en analyse complexe. Duke Math. J. 52 (1985), 157–197.Google Scholar
[Tia96] Tian, G. Kähler–Einstein metrics on algebraic manifolds. Trascendental methods in algebraic geometry (Cetraro 1994), Lecture Notes in Math. 1646 (Springer, Berlin, 1996), 143–185.Google Scholar
[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), 574–628.Google Scholar
[Wu08] Wu, D. Kähler–Einstein metrics of negative Ricci curvature on general quasi-projective manifolds. Comm. Anal. Geom. 16, (2008), 395–435.Google Scholar
[Wu09] Wu, D. Good Kähler metrics with prescribed singularities. Asian J. Math. 13 (2009), 131–150.Google Scholar
[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), 339–411.Google Scholar
[Yau78b] Yau, S.-T. Métriques de Kähler–Einstein sur les variétiés ouvertes. Séminarie Palaiseau, Astérisque 58 (1978), 163–167.Google Scholar