Collapsing of the Chern-Ricci flow on elliptic surfaces
Differential Geometry
2015-07-24 v2
Abstract
We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-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 Kahler-Einstein metric from the base. Some of our estimates are new even for the Kahler-Ricci flow. A consequence of our result is that, on every minimal non-Kahler surface of Kodaira dimension one, the Chern-Ricci flow converges in the sense of Gromov-Hausdorff to an orbifold Kahler-Einstein metric on a Riemann surface.
Keywords
Cite
@article{arxiv.1302.6545,
title = {Collapsing of the Chern-Ricci flow on elliptic surfaces},
author = {Valentino Tosatti and Ben Weinkove and Xiaokui Yang},
journal= {arXiv preprint arXiv:1302.6545},
year = {2015}
}
Comments
46 pages, final version with minor changes to the presentation, to appear in Math. Ann