The Nonvanishing Theorem

Minimal models for Kähler threefolds

“…If X is a projective threefold, Theorem 1.1 was established by the seminal work of Kawamata, Kollár, Mori, Reid and Shokurov [Mor79,Mor82,Rei83,Kaw84a,Kaw84b,Kol84,Sho85,Mor88,KM92]. Based on the deformation theory of rational curves on smooth threefolds [Kol91b,Kol96], Campana and the second-named author started to investigate the existence of Mori contractions [CP97,Pet98,Pet01].…”

Cone theorem via Deligne–Mumford stacks

“…The smoothness of X is essential in his arguments: it is used to control the deformations of maps from curves to X. When X is log terminal (and ch(k) = 0), a cohomological approach has been developed by Kawamata, Reid, Shokurov, and Kollár (see [Ka84], [R83], [Sh85] and [Ko84]). This method yields the bound 2 dim X on −K X -degree of extremal rays, instead of dim X + 1 (see Keel [Ke99] for some results in ch(k) = p and dimX = 3).…”

Finite generation of canonical ring by analytic method

“…The flat bundle situation is handled by a modification of the technique of Shokurov [Shokurov 1985] using the theorem of Riemann-Roch and the vanishing theorem for multiplier ideal sheaves. By Shokurov's technique we can get a section ofm (K X − γY ) over Y with the same divisor as σ.…”

“…This enables us to assume that, after replacing X by its blowup, J is the ideal sheaf of a divisor whose components are in normal crossing. With the blow-up, we can use the technique of the minimum center of log canonical singularity [Kawamata 1985, Shokurov 1985.…”