Minimal projective varieties satisfying Miyaoka's equality

Abstract

In this paper, we establish a structure theorem for minimal projective klt varieties $X$ that satisfiy Miyaoka's equality $3c_2(X)=c_1(X{)}^2$. Specifically, we prove that the canonical divisor $K_X$ is semi-ample and that the Kodaira dimension $\hat{I}º(K_X)$ is either $0$, $1$, or $2$. Furthermore, based on this abundance result, we show that a maximally quasi-étale cover of $X$ is smooth, and we describe explicitly the structure of the Iitaka fibration. Additionally, we prove a similar result for projective klt varieties with a nef anti-canonical divisor.

v4: 38pages. Subsection 4.2 and Section 6 in the previous version have been revised, and Section 5 has been removed. To appear in Proceedings of the London Mathematical Society. v3: 38pages; The title was changed; the main result was improved. v2: 33pages; minor revison. v1: 3 pages; comments are welcome