Unstable Kodaira Fibrations

Details Unstable Kodaira Fibrations Unstable Kodaira Fibrations

2006-12-19

arxiv.org/abs/math/0612571v1

Mathematics

Differential Geometry

12 pages

Scientific paper

Being inspired by Ross' construction of unstable products of certain smooth curves, we show that the product $C\times C$ of every smooth curve $C$ of genus at least 2 is not slope semistable with respect to certain polarisations. Besides, we produce examples of Kodaira-fibred surfaces of nonzero signature, which are not slope semistable with respect to some polarisations, and so they admit K\"ahler classes that do not contain any constant scalar curvature K\"ahler metrics.

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