Unstable products of smooth curves

Abstract

We give examples of smooth surfaces with negative first Chern class which are slope unstable with respect to certain polarisations, and so have Kahler classes that do not admit any constant scalar curvature Kahler metrics. We also compare this to the work of Song-Weinkove on the J-flow.

Added implications for unstable Hilbert and Chow points; typos corrected

References in corpus (5)

• An obstruction to the existence of constant scalar curvature Kähler metrics
• Geometry of Kaehler metrics and holomorphic foliation by discs
• Lower bounds on the Calabi functional
• Analysis of Geometric Stability
• On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy

Cited by in corpus (16)

• On the constant scalar curvature Kähler metrics, existence results
• On the constant scalar curvature Kähler metrics, general automorphism group
• On Conformally Kaehler, Einstein Manifolds
• The J-flow on Kahler surfaces: a boundary case
• Notes on GIT and symplectic reduction for bundles and varieties
• Optimal lower bounds for Donaldson's J-functional
• The Calabi flow with small initial energy
• Slopes of smooth curves on Fano manifolds
• Twisted cscK metrics and Kähler slope stability
• Kähler non-collapsing, eigenvalues and the Calabi flow
• A criterion for the properness of the K-energy in a general Kahler class
• Unstable Kodaira Fibrations
• Coupled Kähler-Einstein metrics
• Compact Complex Surfaces and Constant Scalar Curvature Kähler Metrics
• RC-positivity and scalar-flat metrics on ruled manifolds
• A simple limit for slope instability