Negatively curved metrics on Kodaira surfaces

Abstract

It has been known that Kodaira surfaces of general type have negative holomorphic tangent bundles (cf. [! 1]). This definition of negativity is algebraic-geometric in nature, i.e. negative in the sense of Grauert [5]. Griffiths [7] discussed the relations between various notions of negativity in algebraic geometry and those in differential geometry, and raised the question of whether a Grauert-negative holomorphic vector bundle also admits a hermitian metric of negative holomorphic bisectional curvature. If it does, we would say that this metric is negatively curved, and the vector bundle is negative in Griffiths' sense. The aim of this work is to show that the tangent bundles of certain type of surfaces considered in [11] by Schneider, including Kodaira surfaces of general type, are negative in Griffiths' sense. We also remark here that Cheung [2] constructed on Kodaira surfaces of general type a "K/ihler metric" of negative holomorphic sectional curvature.