Unstable Kodaira Fibrations
Abstract
Being inspired by Ross' construction of unstable products of certain smooth curves, we show that the product C× 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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