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Collapsing of the Chern–Ricci flow on elliptic surfaces

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Abstract

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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Acknowledgments

The authors thank the referee for some suggestions which improved the presentation.

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA

    Valentino Tosatti, Ben Weinkove & Xiaokui Yang

Authors
  1. Valentino Tosatti
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  2. Ben Weinkove
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  3. Xiaokui Yang
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Corresponding author

Correspondence to Ben Weinkove.

Additional information

Research supported in part by NSF Grants DMS-1105373 and DMS-1236969. V. Tosatti is supported in part by a Sloan Research Fellowship.

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Tosatti, V., Weinkove, B. & Yang, X. Collapsing of the Chern–Ricci flow on elliptic surfaces. Math. Ann. 362, 1223–1271 (2015). https://doi.org/10.1007/s00208-014-1160-1

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  • DOI: https://doi.org/10.1007/s00208-014-1160-1

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