Overview

Welcome! I’m Zheng Xu. I study algebraic geometry. I am a postdoctoral fellow at BICMR, Peking University, mentored by Professor Zhiyu Tian. Before that, I got my Ph.D. from Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS) in June 2024, supervised by Professor Wenhao Ou. My interests include birational geometry and Hodge theory, especially in positive and mixed characteristics. Here is my CV.

Research


[1] Liu J., Xu Z. Non-vanishing implies numerical dimension one abundance. arXiv preprint arXiv:2505.05250, 2025.

Abstract. We show that the non-vanishing conjecture implies the abundance conjecture when $\nu\leq 1$. We also prove the abundance conjecture in dimension $\leq 5$ when $\kappa\geq 0$ and $\nu\leq 1$ unconditionally.


[2] Xu Z. Abundance for threefolds in positive characteristic when $\nu= 2$[J]. arXiv preprint arXiv:2307.03938, 2023.

Abstract. In this paper, we prove the abundance conjecture for threefolds over an algebraically closed 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.


[3] Xu Z. NOTE ON THE THREE-DIMENSIONAL LOG CANONICAL ABUNDANCE IN CHARACTERISTIC >3. Nagoya Mathematical Journal. Published online 2024:1-30. doi:10.1017/nmj.2024.3

Abstract. In this paper, we prove the non-vanishing and some special cases of the log abundance for lc threefold pairs over an algebraically closed field $k$ of characteristic $p > 3$. More precisely, we prove that if $(X,B)$ be a projective lc threefold pair over $k$ and $K_{X}+B$ is pseudo-effective, then $\kappa(K_{X}+B)\geq 0$, and if $K_{X}+B$ is nef and $\kappa(K_{X}+B)\geq 1$, then $K_{X}+B$ is semiample.

As applications, we show that the log canonical rings of projective lc threefold pairs over $k$ are finitely generated, and the log abundance holds when the nef dimension $n(K_{X}+B)\leq 2$ or when the Albanese map $a_{X}:X\to \mathrm{Alb}(X)$ is non-trivial. Moreover, we prove that the log abundance for klt threefold pairs over $k$ implies the log abundance for lc threefold pairs over $k$.