Rational Connectedness of Log Q-fano Varieties

Let X be a log Q-Fano variety, i.e, there exists an effective Q-divisor D such that (X,D) is Kawamata log terminal (klt) and −(KX + D) is nef and big. By a result of Miyaoka-Mori [15], X is uniruled. The conjecture ([10], [16]) predicts that X is rationally connected. In this paper, apply the theory of weak (semi) positivity of (log) relative dualizing sheaves f∗(KX/Y + ∆) (which has been developed by Fujita, Kawamata, Kollár, Viehweg and others), we show that a log Q-Fano variety is indeed rationally connected. Remark: IfX is a smooth Fano variety, Campana [1] and Kollár-MiyaokaMori [13] had proved that X is rationally connected. However their methods depend on the (relative) deformation theory which seem quite difficult to apply to the singular case.