Singular rationally connected threefolds with non-zero pluri-forms

TL;DR

This paper classifies singular rationally connected threefolds with non-zero pluri-forms, showing they admit a fibration over the projective line and describing the structure of their pluri-forms explicitly.

Contribution

It establishes the existence of a fibration to 1 for such threefolds with terminal singularities and provides an explicit isomorphism for their pluri-forms.

Findings

Existence of a fibration from the threefold to 1.

Explicit description of the space of pluri-forms in terms of divisors on 1.

Characterization of the structure of non-zero pluri-forms on these threefolds.

Abstract

This paper is concerned with singular projective rationally connected threefolds $X$ which carry non-zero pluri-forms, \textit{i.e.} $H^0(X,(\Omega_X^1)^{[\otimes m]}) \neq \{0\}$ for some $m > 0$, where $(\Omega_X^1)^{[\otimes m]}$ is the reflexive hull of $(\Omega_X^1)^{\otimes m}$. If $X$ has $\mathbb{Q}$-factorial terminal singularities, then we show that there is a fibration $p$ from $X$ to $\mathbb{P}^1$. Moreover, there is a natural isomorphism from $H^0(X, (\Omega_X^1)^{[\otimes m]})$ to $H^0(\mathbb{P}^1, \mathscr{O}_{\mathbb{P}^1}(-2m+\sum_{z\in \mathbb{P}^1} [\frac{(m(p,z)-1)m}{m(p,z)}]))$ for all $m>0$, where $m(p,z)$ is the smallest positive coefficient in the divisor $p^*z$.