Two Non-Vanishing results concerning the Anti-Canonical Bundle

Abstract

Let $(X,\Delta )$ be a klt threefold pair with nef anti-log canonical bundle $-(K_X+\Delta )$. We show that $\kappa (X,-(K_X+\Delta ))\ge 0$. To do so, we prove a more general equivariant non-vanishing result for anti-log canonical bundles, which is valid in any dimension.

arXiv commentv3: Slight changes in accordance with suggestions made by the referee, to appear in Nagoya Math. J. Comments are very welcome!

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• On Generalised Abundance, I
• Nef anti-canonical divisors and rationally connected fibrations
• The Nonvanishing problem for varieties with nef anticanonical bundle