The Minimal Model Program revisited

Abstract

Abstract. We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.

Miyaoka-Yau inequality for minimal projective manifolds of general type

“…In [7], the short time existence of the solution of (2.1) is proved. Then, in [11], [10], and [3], it is proved that the solution ω t of (2.1) exists for all time, i.e. t ∈ [0, +∞), and there exists a unique semi-positive current ω ∞ on M which satisfies that: …”

“…By [11], [10], [3], and [15], for any Kähler metric as initial metric, the solution ω t of the Kähler-Ricci flow equation exists for all time t ∈ [0, ∞), and the scalar curvature of ω t is uniformly bounded. Thus we can prove (1.1) by using the technique developed in [6], where a Hitchin-Thorpe type inequality was proved for 4-manifolds which admit a long time solution to a normalized Ricci flow equation with bounded scalar curvature.…”

Around and Beyond the Canonical Class

“…In [Laz09,CL12b], Theorem 2.3 was proved directly and without the MMP, only by using induction on the dimension and the Kawamata-Viehweg vanishing. The proof of this result is not the topic here, as it was clearly surveyed in [Cor11,CL12a]. In this paper, I take Theorem 2.3 as a black box, and build upon it.…”