Remarks on the abundance conjecture.

*(English)*Zbl 1365.14019Let \(\pi :X\to U\) be a projective morphism of varieties over \(\mathbb C\) and \((X,\Delta )\) a log canonical pair. The abundance conjecture says that if \(K_X+\Delta \) is \(\pi \)-nef (so that \((K_X+\Delta )\cdot C\geq 0\) for any curve \(C\) contained in fibers of \(\pi\)), then \(K_X+\Delta\) is \(\pi\)-semiample i.e. there exists a morphism \(f:X\to Y\) over \(U\) such that \(K_X+\Delta \sim _{\mathbb R }f^* D\) for some \(\mathbb R\)-divisor \(D\) which is ample over \(U\). This is one of the most important conjectures in higher dimensional birational geometry. It is known to hold if \(\dim X=3\).

The main result of the paper under review is to show that the abundance conjecture holds for \(n\)-dimensional varieties \(X\) such that \(K_X+\Delta \) is \(\pi\)-big assuming that it holds in full generality for \((n-1)\)-dimensional varieties. In particular the abundance conjecture holds for \(4\)-folds \(X\) such that \(K_X+\Delta \) is \(\pi\)-big. Recall that \(K_X+\Delta \) is \(\pi\)-big if it is \(\mathbb R\)-linearly equivalent to the sum of a \(\pi\)-ample \(\mathbb R\)-divisor and an effective \(\mathbb R\)-divisor.

The main result of the paper under review is to show that the abundance conjecture holds for \(n\)-dimensional varieties \(X\) such that \(K_X+\Delta \) is \(\pi\)-big assuming that it holds in full generality for \((n-1)\)-dimensional varieties. In particular the abundance conjecture holds for \(4\)-folds \(X\) such that \(K_X+\Delta \) is \(\pi\)-big. Recall that \(K_X+\Delta \) is \(\pi\)-big if it is \(\mathbb R\)-linearly equivalent to the sum of a \(\pi\)-ample \(\mathbb R\)-divisor and an effective \(\mathbb R\)-divisor.

Reviewer: Christopher Hacon (Salt Lake City)