Asymptotic stability for K?hler–Ricci solitons

详细信息    查看全文

• 作者：Ryosuke Takahashi
• 关键词：Fano manifold ; K?hler–Ricci soliton ; Balanced metric ; 53C25
• 刊名：Mathematische Zeitschrift
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：281
• 期：3-4
• 页码：1021-1034
• 全文大小：468 KB
• 参考文献：1.Berman, R., Boucksom, S.: Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math. 181, 337-94 (2010)MATH MathSciNet CrossRef
  2.Berman, R., Boucksom, S., Guedj, V., Zeriahi, A.: A variational approach to complex Monge–Ampère equations. Publ. Math. de l’IHèS 117, 179-45 (2013)MATH MathSciNet CrossRef
  3.Boucksom, S., Eyssidieux, P.: Monge–Ampère equations in big cohomology classes. Acta Math. 205, 199-62 (2010)MATH MathSciNet CrossRef
  4.Berman, R.J., Nystr?m, D.W.: Complex optimal transport and the pluripotential theory of K?hler–Ricci solitons. arXiv:-401.-264
  5.Chen, X., Donaldson, S., Sun, S.: K?hler–Einstein metrics on Fano manifolds, III: limits as cone angle approaches \(2 \pi \) and completion of the main proof. J. Am. Math. Soc. 28, 235-78 (2015)MATH MathSciNet CrossRef
  6.Chen, X., Wang, B.: Space of Ricci flows (II). arXiv:-405.-797
  7.Donaldson, S.K.: Scalar curvature and projective embeddings, I. J. Differ. Geom. 59, 479-22 (2001)MATH MathSciNet
  8.Donaldson, S.K.: Some numerical results in complex differential geometry. Pure Appl. Math. 5, 571-18 (2009)MATH MathSciNet
  9.Fujiki, A.: On automorphism groups of compact K?hler manifolds. Invent. Math. 44, 225-58 (1978)MATH MathSciNet CrossRef
  10.Futaki, A.: Asymptotic Chow semi-stability and integral invariants. Int. J. Math. 15, 967-79 (2004)MATH MathSciNet CrossRef
  11.Mabuchi, T.: An energy-theoretic approach to the Hitching–Kobayashi correspondence for manifolds, II. Osaka J. Math. 46, 115-39 (2009)MATH MathSciNet
  12.Mabuchi, T.: Asymptotic of polybalanced metrics under relative stability constraints. Osaka J. Math. 48, 845-56 (2011)MATH MathSciNet
  13.Ono, H., Sano, Y., Yotsutani, N.: An example of asymptotically Chow unstable manifolds with constant scalar curvature. Annales de l’Institut Fourier 62, 1265-287 (2012)MATH MathSciNet CrossRef
  14.Takahashi, R.: On the modified Futaki invariant of complete intersections in projective spaces. arXiv:-410.-891
  15.Tian, G.: K-stability and K?hler–Einstein metrics. Commun. Pure Appl. Math. 68, 1085-156 (2015)CrossRef
  16.Tian, G., Zhu, X.H.: Uniqueness of K?hler–Ricci solitons. Acta Math. 184, 271-05 (2000)MATH MathSciNet CrossRef
  17.Tian, G., Zhu, X.H.: A new holomorphic invariant and uniqueness of K?hler–Ricci solitons. Comment. Math. Helv. 77, 297-25 (2002)MATH MathSciNet CrossRef
  
• 作者单位：Ryosuke Takahashi (1)
  
  1. Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

Let X be a Fano manifold. We say that a hermitian metric \(\phi \) on \(-K_X\) with positive curvature \(\omega _{\phi }\) is a K?hler–Ricci soliton if it satisfies the equation \(\mathrm{Ric}(\omega _{\phi }) - \omega _{\phi } = L_{V_{KS}} \omega _{\phi }\) for some holomorphic vector field \(V_{KS}\). The candidate for a vector field \(V_{KS}\) is uniquely determined by the holomorphic structure of X up to conjugacy, hence depends only on the holomorphic structure of X. We introduce a sequence \(\{V_k\}\) of holomorphic vector fields which approximates \(V_{KS}\) and fits to the quantized settings. Moreover, we also discuss about the existence and convergence of the quantized K?hler–Ricci solitons attached to the sequence \(\{V_k\}\). Keywords Fano manifold K?hler–Ricci soliton Balanced metric