Abundance for threefolds in positive characteristic when $\nu=2$

Zheng Xu

Abundance for threefolds in positive characteristic when $\nu=2$

Algebraic Geometry 2026-04-20 v3

Authors: Zheng Xu

Abstract

In this paper, we prove the abundance conjecture for threefolds over a perfect field $k$ of characteristic $p > 3$ in the case of numerical dimension equals to $2$ . More precisely, we prove that if $(X,B)$ be a projective lc threefold pair over $k$ such that $K_{X}+B$ is nef and $\nu(K_{X}+B)=2$ , then $K_{X}+B$ is semiample.

Keywords

abundance conjecture threefolds positive characteristic numerical dimension semiample lc pairs

Cite

@article{arxiv.2307.03938,
  title  = {Abundance for threefolds in positive characteristic when $\nu=2$},
  author = {Zheng Xu},
  journal= {arXiv preprint arXiv:2307.03938},
  year   = {2026}
}

Comments

Final version, to appear in Compositio Mathematica

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