On the canonical line bundle and negative holomorphic sectional curvature

Abstract

Abstract. We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.

Kähler manifolds of semi-negative holomorphic sectional curvature

“…Additionally, due to [Kaw85a], a Brody hyperbolic projective manifold (or even one that is merely free of rational curves) has ample canonical bundle if it is of general type. Thus, up to the validity of the Abundance Conjecture, our work in [HLW10] proves the following conjecture, which the third named author learnt from S.-T. Yau in personal conversations in the early 1970s.…”

“…After the publication of our paper [HLW10], another paper on this topic appeared, namely [WWY12]. Its main result is as follows.…”

“…For the proof, we follow the basic strategy used in [HLW10]. We recall that [HLW10] was partially inspired by an earlier work of Peternell [Pet91] on CalabiYau and hyperbolic manifolds.…”

“…We recall that [HLW10] was partially inspired by an earlier work of Peternell [Pet91] on CalabiYau and hyperbolic manifolds. Note that Theorem 1.3 is an immediate corollary of Theorem 1.4(i) due the following simple lemma, applied to the case L = K M .…”

“…In our previous paper [HLW10], we investigated the implications of negative holomorphic sectional curvature for the positivity of the canonical line bundle K M on a projective Kähler manifold M . Our main results in that paper can be summed up in the following theorem.…”