Remarks on the non-vanishing conjecture

TL;DR

This paper examines differences between rational and real non-vanishing conjectures for certain divisors and proves a log non-vanishing theorem for rationally connected varieties assuming Shokurov's ACC conjectures.

Contribution

It highlights the distinction between rational and real non-vanishing conjectures and establishes a new log non-vanishing result under specific conjectural assumptions.

Findings

Identifies differences between rational and real non-vanishing conjectures.

Proves log non-vanishing theorem for rationally connected varieties.

Assumes Shokurov's ACC conjectures for the proof.

Abstract

We discuss a difference between the rational and the real non-vanishing conjecture for pseudo-effective log canonical divisors of log canonical pairs. We also show the log non-vanishing theorem for rationally connected varieties under assuming Shokurov's ACC conjectures.