On uniruled degenerations of algebraic varieties with trivial canonical divisor
Let \(\pi : X\to C\) be a projective morphism (surjective with connected fibers) from a normal variety with canonical singularities to a smooth curve \(C\). Let \(0\in C\) be a point and \(X_0=\sum _{i\in I} m_i F_i\) be the corresponding fiber. If the general fiber has numerically trivial canonical divisor \(K_{X_t}\equiv 0\), then there is a unique way to write \(K_{X/C}=\sum _{j\in J}r_jF_j\) where \(r_j\in \mathbb Q ^{>0}\) and \(J\neq I\). In the paper under review, the author shows that: \noindent {(0)}All components \(F_i\) are uniruled (i.e. covered by rational curves) except possibly one component \(F_j\) with \(\kappa (F_j )=0\) (i.e. \(\text{ max} \{ \dim H^0(\mathcal O (mK_{F_j}))| m>0 \}=1\)). (1) If \(| I| =1\) so that \(X_0=m_1F_1\), then if \(F_1\) is not normal or it contains a \(2\) dimensional component of the singular locus of \(X\), then \(F_1\) is uniruled; if instead \(F_1\) is normal and it does not contain any \(2\) dimensional component of the singular locus of \(X\), then \(F_1\) is uniruled iff it is not canonical. (2) If \(| I| \geq 2\) and \(J\cup \{ i\}\neq I\), then \(F_i\) is uniruled. Therefore: \(K_{X/C}\) is supported by uniruled divisors; if \(F_1\) is not uniruled, then \(I=J\cup \{1 \}\); and if \(K_{X/C}\) is \(\pi\)-nef, then all \(F_i\) are uniruled. The proof is based on the technique of extensions of pluricanonical forms due to \textit{Y.-T. Siu} [Invent. Math. 134, 661-673 (1998; Zbl 0955.32017); in: Finite or infinite dimensional complex analysis and applications. Proc. int. conf. Hanoi, Vietnam, August 8--12, 2001. Adv. Complex Anal. Appl. 2, 45--83 (2004; Zbl 1044.32016)].
scientific article; zbMATH DE number 5279019
scientific article