Let \(X\) be a complex projective variety which is minimal in Mori’s sense, i.e. \(X\) is a normal \(\mathbb{Q}\)-factorial variety with only terminal singularities whose canonical divisor is nef. – The abundance conjecture, due to the author, asserts that for a given minimal model, \(X\), there exists a positive integer \(m\) such that the pluricanonical system \(| mK_ X|\) is free, i.e. free from fixed components and base points. – By previous results due to the author it follows that, in the case when \(\dim X=3\), the abundance conjecture is true if \(\nu(X)>0\) implies \(\kappa(X)>0\). Here \(\kappa(X)\) (respectively \(\nu(X))\) denotes the Kodaira (respectively numerical Kodaira) dimension of \(X\). Miyaoka proved that, in the 3-dimensional case, \(\kappa(X)\geq 0\) and that \(\nu(X)=1\) implies \(\kappa(X)>0\). – In this paper the author proves that \(\kappa(X)>0\) if \(\nu(X)=2\) and \(\dim X=3\), and therefore gives the affirmative answer to the abundance conjecture in dimension 3.
Recall that a \(\mathbb{Q}\)-Fano fiber space is uniruled, i.e. covered by a family of rational curves by Miyaoka and Mori. By combining the above results, the author shows in particular the following result:
Theorem. Let \(X\) be an algebraic variety of dimension 3, defined over the complex field. Then one of the following holds:
1. \(X\) is birationally equivalent to a \(\mathbb{Q}\)-factorial variety with only terminal singularities whose \(m\)-canonical system is free for a positive integer \(m\); or
2. \(X\) is covered by a family of rational curves.