Singular rationally connected threefolds with non-zero pluri-forms (Q2828004)

A complex projective variety \(X\) is rationally connected if any two general points on \(X\) can be connected by a rational curve. It is well-known that if \(X\) is a smooth rationally connected variety, then \(X\) does not carry any pluriform, that is \(H^0(X,(\Omega_X^1)^{\otimes m})=\{0\}\) for any positive integer \(m\). A long-standing conjecture of Mumford asserts that also the converse is true and this characterizes smooth complex rationally connected varieties.NEWLINENEWLINEIt is interesting to investigate similar properties for singular rationally connected varieties, that is, to investigate the existence of nonzero global sections of the reflexive hull of \((\Omega_X^1)^{\otimes m}\). In [\textit{D. Greb} et al., J. Reine Angew. Math. 697, 57--89 (2014; Zbl 1314.32014)], it was shown that if \(X\) is factorial with canonical singularities, then \(H^0(X,(\Omega_X^1)^{[\otimes m]})=\{0\}\) for any \(m>0\) (where \((\Omega_X^1)^{[\otimes m]}\) is the reflexive hull of \((\Omega_X^1)^{\otimes m}\)). In [\textit{J. Kollár}, Pure Appl. Math. Q. 4, No. 2, 203--236 (2008; Zbl 1145.14031)] and [\textit{B. Totaro}, Mathematical Sciences Research Institute Publications 59, 405--426 (2012; Zbl 1253.14040)] were provided examples of rational surfaces, with log terminal singularities, that carry pluriforms.NEWLINENEWLINEIn a previous paper [Mich. Math. J. 63, No. 4, 725--745 (2014; Zbl 1312.14121)], the author investigated the geometry of singular rational surfaces that carry pluriforms. In the paper under review the author goes further investigating singular rationally connected threefolds. Namely, the author shows that if \(X\) is \(\mathbb{Q}\)-factorial with terminal singularities then \(H^0(X,(\Omega_X^1)^{[\otimes m]})\neq\{0\}\) for some \(m>0\) if and only if there exists a Mori fibration \(p: X\rightarrow \mathbb{P}^1\) whose ramification determines the number of \(m\)-pluriforms. The author also gives some necessary conditions for the existence of global sections of \((\Omega_X^1)^{[\otimes m]}\) when \(X\) has \(\mathbb{Q}\)-factorial canonical singularities.