TY - JOUR
T1 - Bergman iteration and C∞-convergence towards KÄhler-Ricci flow
AU - Takahashi, Ryosuke
N1 - Funding Information:
This work was supported by Grant-in-Aid for JSPS Fellows Number 16J01211.
Publisher Copyright:
© 2018, Osaka University. All rights reserved.
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
SN - 0030-6126
VL - 55
SP - 713
EP - 729
JO - Osaka Journal of Mathematics
JF - Osaka Journal of Mathematics
IS - 4
ER -