Collapsing of the Chern-Ricci flow on elliptic surfaces

Valentino Tosatti Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA Ben Weinkove Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA Xiaokui Yang Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA

Complex Variables and Complex Analysis Differential Geometry mathscidoc:2207.08001

Mathematische Annalen, 362, 1223–1271, 2014.12
We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kähler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kähler–Einstein metric from the base. Some of our estimates are new even for the Kähler–Ricci flow. A consequence of our result is that, on every minimal non-Kähler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kähler–Einstein metric on a Riemann surface.
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@inproceedings{valentino2014collapsing,
  title={Collapsing of the Chern-Ricci flow on elliptic surfaces},
  author={Valentino Tosatti, Ben Weinkove, and Xiaokui Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220703155042143960531},
  booktitle={Mathematische Annalen},
  volume={362},
  pages={1223–1271},
  year={2014},
}
Valentino Tosatti, Ben Weinkove, and Xiaokui Yang. Collapsing of the Chern-Ricci flow on elliptic surfaces. 2014. Vol. 362. In Mathematische Annalen. pp.1223–1271. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220703155042143960531.
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