On numerical nonvanishing for generalized log canonical pairs

TL;DR

This paper investigates the numerical nonvanishing conjecture for generalized log canonical pairs, confirming it in dimension two and providing effective results for surfaces, with partial progress in higher dimensions.

Contribution

It proves the numerical nonvanishing conjecture for generalized lc pairs in dimension two and offers new effective results for surfaces, extending the understanding in higher dimensions.

Findings

Confirmed the conjecture in dimension two.

Provided effective versions for surfaces.

Extended results to certain threefolds with rational singularities.

Abstract

The nonvanishing conjecture for projective log canonical pairs plays a key role in the minimal model program of higher dimensional algebraic geometry. The numerical nonvanishing conjecture considered in this paper is a weaker version of the usual nonvanishing conjecture, but valid in the more general setting of generalized log canonical pairs. We confirm it in dimension two. Under some necessary conditions we obtain effective versions of numerical nonvanishing for surfaces. Several applications are also discussed. In higher dimensions, we mainly consider the conjecture for generalized klt pairs $(X, B+\mathbf{M})$, and reduce it to lower dimensions when $K_X+\mathbf{M}_X$ is not pseudo-effective. Up to scaling the nef part, we prove the numerical nonvanishing for pseudo-effective generalized lc threefolds with rational singularities.