TY - JOUR

T1 - Bergman iteration and C∞-convergence towards KÄhler-Ricci flow

AU - Takahashi, Ryosuke

N1 - Funding Information:
2010 Mathematics Subject Classification. 53C25. This work was supported by Grant-in-Aid for JSPS Fellows Number 16J01211.
Funding Information:
Acknowledgements. The author would like to express his gratitude to his advisor Professor Shigetoshi Bando and Professor Ryoichi Kobayashi for useful discussions on this article. The author also would like to thank Professor Shin Kikuta, Satoshi Nakamura and Yusuke Miura for several helpful comments. This research is supported by Grant-in-Aid for JSPS Fellows Number 16J01211.

PY - 2018/10

Y1 - 2018/10

N2 - On a polarized manifold (X, L), the Bergman iteration φ(m) k is defined as a sequence of Bergman metrics on L with two integer parameters k, m. We study the relation between the Kähler-Ricci flow φt at any time t ≥ 0 and the limiting behavior of metrics φ(m) k when m = m(k) and the ratio m/k approaches to t as k → ∞. Mainly, three settings are investigated: the case when L is a general polarization on a Calabi-Yau manifold X and the case when L = ±KX is the (anti-) canonical bundle. Recently, Berman showed that the convergence φ(m) k → φt holds in the C0-topology, in particular, the convergence of curvatures holds in terms of currents. In this paper, we extend Berman’s result and show that this convergence actually holds in the smooth topology.

AB - On a polarized manifold (X, L), the Bergman iteration φ(m) k is defined as a sequence of Bergman metrics on L with two integer parameters k, m. We study the relation between the Kähler-Ricci flow φt at any time t ≥ 0 and the limiting behavior of metrics φ(m) k when m = m(k) and the ratio m/k approaches to t as k → ∞. Mainly, three settings are investigated: the case when L is a general polarization on a Calabi-Yau manifold X and the case when L = ±KX is the (anti-) canonical bundle. Recently, Berman showed that the convergence φ(m) k → φt holds in the C0-topology, in particular, the convergence of curvatures holds in terms of currents. In this paper, we extend Berman’s result and show that this convergence actually holds in the smooth topology.

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M3 - Article

AN - SCOPUS:85055146146

VL - 55

SP - 713

EP - 729

JO - Osaka Journal of Mathematics

JF - Osaka Journal of Mathematics

SN - 0030-6126

IS - 4

ER -