Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures

Jihao Liu; Zheng Xu

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arXiv · 2510.04419

Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures

Jihao Liu·Zheng Xu

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Abstract

Assuming the abundance conjecture in dimension $d$, we establish a non-algebraicity criterion of foliations: any log canonical foliation of rank $\le d$ with $\nu\neq\kappa$ is not algebraically integrable, answering question of Ambro--Cascini--Shokurov--Spicer. Under the same hypothesis, we prove abundance for klt algebraically integrable adjoint foliated structures of dimension $\le d$ and show the existence of good minimal models or Mori fiber spaces. In particular, when $d=3$, all these results hold unconditionally. Using similar arguments, we solve a problem proposed by Lu and Wu on abundance of surface adjoint foliated structures that are not necessarily algebraically integrable.

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BibTeX RIS

Jihao Liu, Zheng Xu. 2025-10-06. Non-algebraicity of non-abundant foliations and abundance for adjoint foliated structures. https://arxiv.org/abs/2510.04419

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