A note on the abundance conjecture (Q2794757)

For projective varieties in characteristic zero, the main conjecture in the Minimal Model Program (MMP) is that every Kawamata log terminal (klt) pair \((X, \Delta)\) with \(K_X+\Delta\) pseudo-effective has a minimal model \((Y, \Delta_Y)\) such that \(K_Y+ \Delta_Y\) is semiample. This minimal model is called a good model.NEWLINENEWLINEIn this paper, the authors prove three main theorems.NEWLINENEWLINE{Theorem 1.} Assume the existence of good models for klt pairs in dimensions at most \(n-1\). If the abundance conjecture holds for non-uniruled klt pairs in dimension \(n\), then the abundance conjecture holds for uniruled klt pairs in dimension \(n\).NEWLINENEWLINE{Theorem 2.} Assume the existence of good models for klt pairs in dimensions at most \(n-1\). Then the existence of good models for non-uniruled klt pairs in dimension \(n\) implies the existence of good models for uniruled klt pairs in dimension \(n\).NEWLINENEWLINEThe idea of the proof is the following. By result of Demailly, Hacon and Păun [\textit{J.-P. Demailly} et al., Acta Math. 210, No. 2, 203--259 (2013; Zbl 1278.14022)], the authors only need to deal with the situation when \((X, \Delta)\) is a uniruled klt pair, \(X\) is smooth, \(\Delta\) is a reduced simple normal crossings divisor, there is an effective \(\mathbb{Q}\)-divisor \(D\) such that \( K_X+ \Delta \sim_{\mathbb{Q}}D \) and the supports of \(\Delta\) and \(D\) are the same. Then they use ramified covers and log resolutions to construct a log smooth pair \((W, \Delta_W)\) and a generically finite morphism \(w: W\rightarrow X\) such that \(K_W\) is an effective divisor. Consequently, they reach the conclusion by the construction and the fact that the \(D\)-dimension and the numerical \(D\)-dimension are preserved if the divisor is pulled back under proper surjective morphisms.NEWLINENEWLINE({Remark:} By the classification theory developed by Japanese Algebraic Geometry School, \(\kappa(D, X)\) is called the \(D\)-dimension and \(\kappa(X)=\kappa(K_X, X)\) is called Kodaira dimension. Iitaka proved fundamental theorems for \(D\)-dimension. For reference, see [\textit{K. Ueno}, Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0299.14007)])NEWLINENEWLINEFinally, the authors apply their techniques further and obtain a more general result which implies the above two theorems.NEWLINENEWLINE{Theorem 3.} Assume the existence of good models for klt pairs in dimension at most \(n-1\). If good models exist for log smooth klt pairs \((X, \Delta)\) of dimension \(n\) such that the linear system \(|K_X|\) is not empty, then good models exist for uniruled klt pairs in dimension \(n\).