Singularities on the base of a Fano type fibration

الأصل · EN

Let f X→ Z be a Mori fibre space. McKernan conjectured that the singularities of Z are bounded in terms of the singularities of X. Shokurov generalised this to pairs: let (X,B) be a klt pair and f X→ Z a contraction such that Kₓ+B 0/Z and that the general fibres of f are Fano type varieties; adjunction for fibre spaces produces a discriminant divisor BZ and a moduli divisor MZ on Z. it is then conjectured that the singularities of (Z,BZ+MZ) are bounded in terms of the singularities of (X,B). We prove Shokurov conjecture when (F, BF) belongs to a bounded family where F is a general fibre of f and KF+BF=(Kₓ+B)|F.