## Osaka Journal of Mathematics

### Bergman iteration and $C^{\infty}$-convergence towards Kähler-Ricci flow

Ryosuke Takahashi

#### Abstract

On a polarized manifold $(X,L)$, the Bergman iteration $\phi_k^{(m)}$ 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 $\phi_t$ at any time $t \geq 0$ and the limiting behavior of metrics $\phi_k^{(m)}$ when $m=m(k)$ and the ratio $m/k$ approaches to $t$ as $k \to \infty$. Mainly, three settings are investigated: the case when $L$ is a general polarization on a Calabi-Yau manifold $X$ and the case when $L=\pm K_X$ is the (anti-) canonical bundle. Recently, Berman showed that the convergence $\phi_k^{(m)} \to \phi_t$ holds in the $C^0$-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.

#### Article information

Source
Osaka J. Math., Volume 55, Number 4 (2018), 713-729.

Dates
First available in Project Euclid: 10 October 2018

https://projecteuclid.org/euclid.ojm/1539158667

Mathematical Reviews number (MathSciNet)
MR3862781

Zentralblatt MATH identifier
06985308

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

#### Citation

Takahashi, Ryosuke. Bergman iteration and $C^{\infty}$-convergence towards Kähler-Ricci flow. Osaka J. Math. 55 (2018), no. 4, 713--729. https://projecteuclid.org/euclid.ojm/1539158667

#### References

• H. Amann: Linear and Quasilinear Parabolic Problems, Abstract Linear Theory, Monographs in Mathematics 89, Birkhäuser Boston Inc., Boston, MA, 1995.
• R. Berman and S. Boucksom: Growth of balls of holomorphic sections and energy at equilibrium, Invent. Math. 181 (2010), 337–394.
• R. Berman, S. Boucksom, V. Guedj and A. Zeriahi: A variational approach to complex Monge-Ampère equations, Publ. Math. l'IHÈS 117 (2013), 179–245.
• R. J. Berman, B. Berndtsson and J. Sjöstrand: A direct approach to Bergman kernel asymptotics for positive line bundles, Ark. Mat. 46 (2008), 197–217.
• R. J. Berman: Relative Kähler-Ricci flows and their quantization, Anal. PDE 6 (2013), 131–180.
• T. Bouche: Convergence de la métrique de Fubini-Study d'un fibré linéaire positif, Ann. Inst. Fourier (Grenoble) 40 (1990), 117–130.
• H.D. Cao: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds, Invent. Math. 81 (1985), 359-372.
• D. Catlin: The Bergman kernel and a theorem of Tian; in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math., Birkhäuser Boston, Boston, MA, 1999, 1–23.
• X. Dai, K. Liu and X. Ma: On the asymptotic expansion of Bergman kernel, J. Differ Geom. 72 (2006), 1–41.
• S.K. Donaldson: Scalar curvature and projective embeddings, I, J. Differ. Geom. 59 (2001), 479–522.
• S.K. Donaldson: Some numerical results in complex differential geometry, Pure and Appl. Math. Q. 5 (2009), 571–618.
• J. Fine: Calabi flow and projective embeddings, J. Differ. Geom. 84 (2010), 489–523.
• Y. Hashimoto: Quantization of extremal Kähler metrics, preprint (2015), arXiv:1508.02643.
• S. Saito and R. Takahashi: Stability of anti-canonically balanced metrics, preprint (2016), arXiv:1607.05534.
• G. Székelyhidi: An Introduction to Extremal Kähler Metrics, Graduate Studies in Mathematics 152, American Mathematical Society, 2014.
• G. Tian: On a set of polarized Kähler metrics on algebraic manifolds, J. Differ. Geom. 32 (1990), 99–130.
• G. Tian and X. Zhu: Convergence of Kähler-Ricci flow, J. Amer. Math. Soc. 20 (2007), 675–699.
• S. Zelditch: Szegö kernels and a theorem of Tian, Int. Math. Res. Not. IMRN (1998), 317–331.