TL;DR
This paper demonstrates that products of smooth curves of genus at least 2 and certain Kodaira-fibred surfaces are not slope semistable under specific polarisations, indicating the existence of Kähler classes without constant scalar curvature metrics.
Contribution
It introduces new examples of unstable Kodaira fibrations and analyzes their slope stability properties under various polarisations.
Findings
Product of smooth curves of genus ≥ 2 is not slope semistable.
Constructs examples of Kodaira-fibred surfaces with nonzero signature that are slope unstable.
Shows these surfaces lack Kähler metrics with constant scalar curvature.
Abstract
Being inspired by Ross' construction of unstable products of certain smooth curves, we show that the product of every smooth curve 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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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Nonlinear Waves and Solitons