## Tokyo Journal of Mathematics

### On the $C^\alpha$-convergence of the Solution of the Chern-Ricci Flow on Elliptic Surfaces

Masaya KAWAMURA

#### Abstract

We will study the Chern-Ricci flow on non-Kähler properly elliptic surfaces. These surfaces are compact complex surfaces whose first Betti number is odd, Kodaira dimension is equal to 1 and admit an elliptic fibration to a smooth compact curve. We will show that a solution of the Chern-Ricci flow converges in $C^\alpha$-topology on these elliptic surfaces by choosing a special initial metric.

#### Article information

Source
Tokyo J. Math., Volume 39, Number 1 (2016), 215-224.

Dates
First available in Project Euclid: 30 March 2016

https://projecteuclid.org/euclid.tjm/1459367266

Digital Object Identifier
doi:10.3836/tjm/1459367266

Mathematical Reviews number (MathSciNet)
MR3543140

Zentralblatt MATH identifier
1357.53079

#### Citation

KAWAMURA, Masaya. On the $C^\alpha$-convergence of the Solution of the Chern-Ricci Flow on Elliptic Surfaces. Tokyo J. Math. 39 (2016), no. 1, 215--224. doi:10.3836/tjm/1459367266. https://projecteuclid.org/euclid.tjm/1459367266

#### References

• Barth, W. P., K. Hulek, C. A. M. Peters and A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, Berlin, 2004.
• Belgun, F. A., On the metric structure of non-Kähler complex surfaces, Math. Ann. 317 (2000), no. 1, 1–40.
• Brîinzănescu, V., Néron-Severi group for nonalgebraic elliptic surfaces. II. Non-Kählerian case, Manuscripta Math. 84 (1994), no. 3–4, 415–420.
• Fang, S., V. Tosatti, B. Weinkove and T. Zheng, Inoue surfaces and the Chern-Ricci flow, preprint, arXiv: 1501.07578v1.
• Gill, M., Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds, Communications in Analysis and Geometry 19 (2011), no. 2, 277–304.
• Kodaira, K., On the structure of compact complex analytic surfaces, II, Amer. J. Math. 88 (1966), no. 3, 682–721.
• Sherman, M. and B. Weinkove, Local Calabi and curvature estimates for the Chern-Ricci flow, New York J. Math. 19 (2013), 565–582.
• Song, J. and B. Weinkove, Lecture notes on the Kähler-Ricci flow, preprint, arXiv: 1212.3653.
• Tosatti, V. and B. Weinkove, On the evolution of a Hermitian metric by its Chern-Ricci form, J. Differential Geom. 99 (2015), no. 1, 125–163.
• Tosatti, V. and B. Weinkove, The Chern-Ricci flow on complex surfaces, Compos. Math. 149 (2013), no. 12, 2101–2138.
• Tosatti, V., B. Weinkove and X. Yang, Collapsing of the Chern-Ricci flow on elliptic surfaces, preprint, arXiv: 1302.6545v1, to appear in Math. Ann.
• Vaisman, I., Non-Kähler metrics on geometric complex surfaces, Rend. Sem. Mat. Univ. Politec. Torino 45 (1987), no. 3, 117–123.
• Wall, C. T. C., Geometric structure on compact complex analytic surfaces, Topology 25 (1986), no. 2, 119–153.