Some remarks on the local moduli of tangent bundles over complex surfaces

Abstract

Using the Hirzebruch's Riemann-Roch formula for endomorphism bundles over a compact complex two-fold we prove that the tangent bundle of a complex surface M of general type admits a nontrivial trace-free deformation, unless M is holomorphically covered by the euclidean ball. It follows that the tangent bundle of the Mostow-Siu surface, which is a Kähler surface with a negative definite curvature tensor, does have a nontrivial trace-free moduli. Among some other results we also point out a relationship between the Kuranishi obstruction and symmetric holomorphic two tensors on a complex surface.