Singular rationally connected threefolds with non-zero pluri-forms

Abstract

This paper is concerned with singular projective rationally connected threefolds $X$ which carry non-zero pluri-forms, \textit{i.e.} $H^0(X,({\Omega }_X^1{)}^{[\otimes m]})\ne \{0\}$ 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 $H^0({\mathbb{P}}^1,{\mathcal{O}}_{{\mathbb{P}}^1}(-2m+{\sum }_{z\in {\mathbb{P}}^1}[\frac{(m(p,z)-1)m}{m(p,z)}]))$ for all $m>0$, where $m(p,z)$ is the smallest positive coefficient in the divisor $p^{\ast }z$.

15 pages

References in corpus (4)

• Differential Forms on Log Canonical Spaces
• Algebraic surfaces and hyperbolic geometry
• On codimension 1 del Pezzo foliations on varieties with mild singularities
• Singular rationally connected surfaces with non-zero pluri-forms