Boundedness of Fano type fibrations (Q6596066)

Boundedness of Fanos is an important question in birational geometry, most notably the well-known BAB conjecture, which was proven by the author of this paper [Ann. Math. (2) 193, No. 2, 347--405 (2021; Zbl 1469.14085)]. Here he proves a relative version of that result, Theorem~1.2: Let \(d, r \in \mathbb N\) and \(\varepsilon > 0\), then in the set of all \((d, r, \varepsilon)\)-Fano type fibrations \((X, B) \to Z\), the \(X\) form a bounded family.\N\NDefinition: a \((d, r, \varepsilon)\)-Fano type fibration \((X, B) \to Z\) consists of a projective \(d\)-dimensional \(\varepsilon\)-lc pair \((X, B)\) (i.e.~\(K_X + B\) is \(\mathbb R\)-Cartier and the log discrepancies are \(\ge \varepsilon\); in particular, the pair is klt) together with a contraction (= surjective with connected fibres) \(f \colon X \to Z\) such that \(K_X + B \sim f^* L\) is a pullback from \(Z\), \(-K_X\) is big over \(Z\), and there is a very ample divisor \(A\) on \(Z\) with volume bounded by \(r\) such that \(A - L\) is ample.\N\NFurther results: if the coefficients of \(B\) are bounded below, then the set of all \((X, B)\) (and not just \(X\)) is bounded (Theorem~1.3). Theorem~1.4 gives a lower bound for the log canonical threshold of \((X, B)\), depending only on \(d, r, \varepsilon\). Theorem~1.6 proves for \((d, r, \varepsilon)\)-Fano type fibrations a conjecture of Shokurov about the discriminant (b-)divisor (in the canonical bundle formula for \(f\)). Theorem~1.7 is about klt complements.