Negative holomorphic curvature and positive canonical bundle

Abstract

Abstract. In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

Invariant metrics on negatively pinched complete Kähler manifolds

“…The proof of Lemma 31 differs from that of Cheng-Yau [CY80] and others mainly in the complex Monge-Ampère type equation. The equation used here is inspired by the authors' work [WY16a]. This new equation is well adapted to the negative holomorphic sectional curvature and the Schwarz type lemma.…”

“…Combining (6.4) and (6.6) yields the estimates of u up to the complex second order (cf. [WY16a,WY16b]). In fact, by (6.6), e − u n = ω n ω n t 1 n ≤ S n ≤ 2 (n + 1)κ 1 .…”

“…The proof makes use of a new complex Monge-Ampère type equation, which involves the Kähler class of tω − Ric(ω) rather than that of ω. This equation is inspired by our recent work [WY16a]. Theorem 3 can be viewed as a complete noncompact generalization of [WY16a, Theorem 2] (for its generalizations on compact manifolds, see for example [TY17,DT,WY16b,YZ].…”

The equality case in Wu–Yau inequality

“…Therefore log(S) solves equation ( 1). Assume vice versa that n ≥ 2 and that log(S) solves equation ( 1), to prove the claim we look more closely at the proof of proposition 9 in [7]. For every x ∈ X let us fix holomorphic local coordinate centered at x, such that the local expression g i,j of the metric ω satisfies g i,j (0) = δ i,j and…”

Kähler manifolds of semi-negative holomorphic sectional curvature

“…After this paper had been completed and gone to press at the Journal of Differential Geometry, Wu and Yau gave a proof of Conjecture 1.2 in [WY15]. Subsequently, Tosatti and Yang in [TY15] extended that proof to the Kähler case.…”