Abstract
The continuity method is used to deform the cone angle of a weak conical Kähler–Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson’s Openness Theorem on deforming cone angle Donaldson (Essays in Mathematics and Its Applications, 2012) by combining it with the regularity result of Guenancia–Păun (arXiv:1307.6375 2013) and Chen–Wang (arXiv:1405.1201 2014). This continuity method uses relatively less regularity of the metric (only weak conical Kähler–Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a weak conical Kähler–Einstein metric along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).
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References
Aubin, T.: Réduction du cas positif de l’équation de Monge-Ampère sur les variété Kähleriennes compactes à la démonstration d’une inégalité. J. Funct. Anal. 57, 143–153 (1984)
Berman, R.: A thermodynamical formalism for Monge-Ampère equations Moser-Trudinger inequalities and Kähler-Einstein metrics. Adv. Math. 248(25), 1254–1297 (2013). doi:10.1016/j.aim.2013.08.024
Berndtsson, B.: A Brunn–Minkowski type inequality For Fano Manifolds and some uniqueness Theorems in Kähler geometry. Invent. Math. (2014). doi:10.1007/00222-014-0532-1
Berman, R.J., Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Kähler-Einstein metrics and the Kähler-Ricci flow on Log Fano varieties. arXiv:1111.7158
Chern, S.-S.: On holomorphic mappings of Hermitian manifolds of the same dimension. Proceedings of Symposium in Pure Mathematics 11, American Mathematical Society, pp. 157–170 (1968)
Chen, X.-X.: On the lower bound of the Mabuchi energy and its application. Internat. Math. Res. Notices, no. 12, 607–623 (2000)
Chen, X.-X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities. J. Amer. Math. Soc. 28, 183–197 (2015)
Campana, F., Guenancia, H., Păun, M.: Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. Annales scientifiques de l’ENS 46, fascicule 6, 879–916 (2013)
Chen, X.-X., Wang, Y.-Q.: On the regularity problem of complex Monge-Ampère equations with conical singularities (2014). arXiv:1405.1201
Donaldson, S.K.: Kähler metrics with cone singularities along a divisor. In: Essays in Mathematics and Its Applications, pp. 49–79. Springer, Heidelberg (2012). doi:10.1007/978-3-642-28821-0_4
Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math. 35, 333–363 (1982)
Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler–Einstein metrics. J. Amer. Math. Soc. 22, 607–639 (2009)
Guenancia, H., Păun, M.: Conic singularities metrics with Prescribed Ricci curvature: The case of General cone angles along normal crossing divisors. (2013) arXiv:1307.6375
Jeffres, T.D., Mazzeo, R., Rubinstein, Y.: Kähler-Einstein metrics with Edge singularities (with an appendix by C. Li and Y.A. Rubinstein), preprint, 2011, arxiv:1105.5216. In revision for Annals of Math
Kołodziej, S.: The Complex Monge-Ampère equation. Acta. Math. 180, 69–117 (1998)
Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 46, 487–523 (1982)
Li, C., Sun, S.: Conical Kähler Einstein metrics revisited. Comm. Math. Phys. 331(3), 927–973 (2014)
Lu, Y.-C.: Holomorphic Mapping of Complex Manifolds. J. Diff. Geom. 2, 299–312 (1968)
Mabuchi, T.: K-energy maps integrating Futaki invariant. Tôhoku Math. J. 38, 575–593 (1986)
Song, J., Wang, X.-W.: The Greatest Ricci Lower Bound, Conical Einstein Metrics and the Chern Number Inequality. arXiv: 1207.4839
Székelyhidi, G.: Greatest lower bounds on the Ricci curvature of Fano manifolds. Compositio Math. 147, 319–331 (2011)
Tian, G.: Kähler–Einstein metrics with Positive Scalar curvature. Invent. Math. 137, 1–37 (1997)
Yao, C.-J.: Existence of weak conical Kähler-Einstein metrics along smooth hypersurfaces. Math. Ann. (2014). doi:10.1007/s00208-014-1140-5
Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31, 339–441 (1978)
Acknowledgments
The author is very grateful to his advisor, Professor Xiuxiong Chen, for introducing this problem about openness and giving insight on the possibility of bypassing the implicit function theorem for singular metrics. Great thanks also go to Dr. Song Sun, Yuanqi Wang and Long Li for helpful discussions and useful suggestions. He would also like to thank the referee for careful reading and useful suggestions for the organization of this paper.
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Yao, C. The Continuity Method to Deform Cone Angle. J Geom Anal 26, 1155–1172 (2016). https://doi.org/10.1007/s12220-015-9586-6
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DOI: https://doi.org/10.1007/s12220-015-9586-6
Keywords
- Conical Kähler–Einstein metrics
- Weak conical Kähler–Einstein metrics
- Continuity path
- Log-Mabuchi-functional
- Modified log-Mabuchi-functional