Mingchen Xia (夏铭辰)

欢迎/歡迎/Welcome/Välkommen/いらっしゃいませ

About Me

I am a vibe mathematicianvibe mathematicianA mathematician who uses AI instead of the human brain to do mathematics.↓ at the Institute of Geometry and Physics in Hefei. 主业是乳法。

Email: xiamingchen2008@gmail.com

Good to know before contacting me:
1. À quelques exceptions près, je refuse de contribuercontribuerEn tant que rapporteur et auteur aux revues françaises.
2. My given name is Mingchen instead of Xia, Xiam, Mincheng, Minchen or Mingcheng. And I am male.
3. Preferred languages: Chinese > English or French. Ich versuche, mein Deutsch zu verbessern. Du kannst mir gern auf Hochdeutsch schreiben, wenn es dich nicht stört, Antworten auf holprigem Deutsch zu bekommen.

France

Where I stand on AI and mathematics

I am interested in machine learning and LLM for the moment.

Lecture notes

Fere libenter homines id quod volunt credunt.

Archimedean pluripotential theory

  • Lectures on pluripotential theory 1, 2, 3, 4.

Lecture notes for my course at Shanghai Tech university in the summer of 2023.

Lecture notes for my course at Chinese academy of science in the summer of 2023.

Lecture notes for my course at Zhejiang university in the spring of 2024. Please let me know if you find any typos/mistakes or find any of the arguments unclear.

Non-Archimedean geometry

  • Lectures on non-Archimedean pluripotential theory 1, 2, 3, 4.

Lecture notes for my course at USTC in the spring of 2024.

Beamers

En attendant le remboursement du repa de 7 euros jusqu’à aujourd’hui.

Research

Errare humanum est.

All my preprints can be found on arXiv. See my Google Scholar page as well. All my mathematical works starting from July 2026 rely on AI.

I am glad to spend my extra tokens on helping the other mathematicians (French excluded). Feel free to contact me!

Vibe math

The articles in this section are generated by AI. I have not carefully verified every detail.

  • Subharmonicity on All Affine Complex Lines Forces Upper Semicontinuity

    Abstract. Let $\Omega\subset\mathbb{C}^n$ be a domain and let $\varphi:\Omega\to[-\infty,\infty)$ be nontrivial. We prove that if $\varphi$ is subharmonic or identically $-\infty$ on every connected component of every affine complex-line section, then $\varphi$ is automatically upper semicontinuous. Thus condition (2) in Definition 1.1.2 of [Xia] is redundant.

  • Nonuniqueness for degenerate Hermitian Monge–Ampère equations

    Abstract. We give an explicit counterexample to uniqueness for a degenerate Hermitian Monge–Ampère equation. More precisely, on $X=\mathbb{P}^3$ we construct a positive nonclosed Hermitian form $\omega$, a Hermitian reference form $\omega_X$, a smooth function $f\geq 0$, $f\not\equiv 0$, and two distinct smooth normalized $\omega$-plurisubharmonic functions $\varphi$ and $\psi$ with minimal singularities such that $(\omega+\mathrm{dd}^c\varphi)^3=f\omega_X^3=(\omega+\mathrm{dd}^c\psi)^3$. The density vanishes on a nonempty open set. This disproves uniqueness of the normalized potential in Theorem D of Boucksom–Guedj–Lu, even for a Hermitian background on a connected projective manifold.

  • The envelope property for unibranch projective varieties

    Abstract. Let $X$ be an integral unibranch projective variety over an algebraically closed field $k$, endowed with the trivial absolute value, and let $\theta\in N^1(X)_{\mathbb{R}}$ be a numerical class. We prove that for every nonempty, bounded-above family $(\varphi_i)\subseteq\operatorname{PSH}(X,\theta)$ of $\theta$-plurisubharmonic functions on the Berkovich analytification $X^{\mathrm{an}}$, the regularized supremum $\sup_i^*\varphi_i$ is again $\theta$-plurisubharmonic. This establishes the envelope conjecture of Boucksom–Jonsson.

    Available upon request.

  • Real $C^{1,1}$ Regularity of Envelopes in Big Cohomology Classes

    Abstract. Let $X$ be a compact Kähler manifold and let $\theta$ be a smooth closed real $(1,1)$-form representing a big cohomology class. We prove that for every real-valued $f\in C^{1,1}(X)$ its $\theta$-plurisubharmonic envelope is locally real $C^{1,1}$ outside the non-Kähler locus; equivalently, its first derivatives are locally Lipschitz there. The proof combines a quantitative analytic exhaustion of the complement of the non-Kähler locus, floored proper Monge–Ampère Dirichlet problems, and a pole-weighted real Hessian estimate.

    Available upon request.

Works in progress

Updated in July 2026. I proved the mixed volume formula of Okounkov bodies and partial Okounkov bodies.

Updated in July 2026. I completely solved the local version of the analytic Bertini theorem.

Updated in December 2025.

Updated in July 2026. The optimal constant in Corollary 6.17 is obtained.

I’m working on a third part of this series, where I seek to define a new Monge–Ampère measure compatible with Cao’s mixed volume.

K-stability

  • On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows, Analysis & PDE (2021). arXiv:1901.07889 Journal link

My note Radial Calabi flow might be of interest to the readers of this paper.

In arXiv version 1, Section 8, I briefly explained the second order expansion of Donaldson’s L-functionals, which might be of interest as well.

Pluripotential theory

This paper was the first proof of the integration by parts formula. However, a better approach was found later on by Lu, so this paper is no longer important. I don’t intend to submit it.

There is a slight issue in the proof of Theorem 2.11 line 10: $f^{\sharp}$ is only formally smooth, not smooth. This does not affect anything in the proof. This is corrected in this version.

As pointed out by Vasanth Pidaparthy and Prakhar Gupta, the statement of Proposition 4.12(ii) (Proposition 6.12(ii) in the arXiv version) is wrong in the generality as stated there, one needs to assume that $\varphi_0\leq \gamma\leq \varphi_1$ in addition. This mistake does not affect the other parts of the paper.

The published version contains only the special case without prescribed singularities on Kähler manifolds. The method in the general case is exactly the same.

  • The closures of test configurations and algebraic singularity types, (joint with Tamás Darvas), Advances in Mathematics (2022). arXiv:2003.04818 Journal link

  • The volume of pseudoeffective line bundles and partial equilibrium, (joint with Tamás Darvas), Geometry & Topology (2024). arXiv:2112.03827 Journal link

  • Partial Okounkov bodies and Duistermaat–Heckman measures of non-Archimedean metrics, Geometry & Topology (2025). arXiv:2112.04290 Journal link

The proof of the Hausdorff convergence property in this paper is quite messy. My book contains a more readable proof.

  • Non-pluripolar products on vector bundles and Chern–Weil formulae, Mathematische Annalen (2024). arXiv:2210.15342 Journal link

  • Transcendental Okounkov bodies, (joint with Tamás Darvas, Rémi Reboulet, David Witt Nyström and Kewei Zhang), Journal of Differential Geometry (2026). arXiv:2309.07584 Journal link

  • The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds, (joint with Tamás Darvas), Transactions of the American Mathematical Society (2026). arXiv:2403.08259 Journal link

  • A counterexample to the zero-mass conjecture, (joint with Long Li). arXiv:2607.26549

Non-Archimedean geometry and algebraic geometry

In the complex analytic setting, very similar arguments (using Fréchet algebras instead of Banach algebras) give the notion of Stein morphisms. It is of interest to see if these morphisms are useful. M. Maculan pointed out that Liu morphisms are not G-local on the target, see the corrected version. I will update arXiv in a few days.

There is minor gap in the proof: In the first step, one needs to further enlarge $\Sigma_1$ to make sure that the restriction ideal coincides with the pull-back as coherent sheaves. A corrected proof is presented in my lecture notes at Zhejiang University.

  • A transcendental approach to non-Archimedean metrics of pseudoeffective classes, (joint with Tamás Darvas and Kewei Zhang), Commentarii Mathematici Helvetici (2025). arXiv:2302.02541 Journal link

The theory of non-Archimedean psh functions we developed in this paper trivally satisfies Boucksom–Jonsson’s envelope conjecture (even on a general unibranch complex space), see my note Operations on transcendental non-Archimedean metrics.

Some notes

Just a preliminary version with potentially many mistakes. I’m slowly adding new materials.

One of my unfinished projects. It contains a number of conjectures of interest.

I collect a few well-known results about relative normalisations.

This note is a trivial continuation of my joint paper with Darvas and Zhang. The only notable result is Theorem 4.21. The editors put an earlier version in the final book by mistake. Please read the arXiv version instead.

I construct a non-pluripolar qpsh function with small unbounded locus which is not I-good.

My notes while learning Ducros’ book.

I give a proof of Cartan’s theorem about closed ideals in Stein algebras. This is a well-known result, but I cannot find the proof anywhere.

My personal notes about the theory of Stein spaces. The 1976 paper of Bingener is integrated into these notes.

I turn Bingener’s famous 1976 paper in to LaTeX format.

Updated on Nov 8, 2024.

Updated on Jul 21, 2025.

Sci-hub is getting blocked in many countries recently. If the link fails to work, please try to change the domain name.