# 欢迎/歡迎/Welcome/Välkommen/いらっしゃいませ ## About Me I am a ↓ at the Institute of Geometry and Physics in Hefei. 主业是乳法。 - In 2022, I obtained my PhD degree at Chalmers Tekniska Högskola in under the supervision of Robert Berman. - Afterwards, I was a Wallenberg postdoc from 2022 to 2025. In 2023 and 2024 I applied for the position twice without getting a single interview. Email:

Good to know before contacting me:
1. À quelques exceptions près, je refuse de contribuer 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.

[Where I stand on AI and mathematics](ai-and-mathematics.html) I am interested in vibe coding and Lean4 for the moment.

## Vibe coding Two small tools I built with AI assistance:

arXivpecker

An AI-assisted mathematics paper reader with interactive structure, editable TeX, proof expansion, citation lookup, notes, version comparison, and theorem-level dependency maps.

FindReferee

An assistant for investigating who may have written a referee report, with uncertainty-aware probabilities and an evidence report.

## Lecture notes > Fere libenter homines id quod volunt credunt. ### Archimedean pluripotential theory - Lectures on pluripotential theory [1](Lectures/ShanghaiTech1.pdf), [2](Lectures/ShanghaiTech2.pdf), [3](Lectures/ShanghaiTech3.pdf), [4](Lectures/ShanghaiTech4.pdf). > Lecture notes for my course at Shanghai Tech university in the summer of 2023. - [Characterizations of I-good singularities](Lectures/CAS1.pdf). > Lecture notes for my course at Chinese academy of science in the summer of 2023. - [Singularities in global pluripotential theory](Lectures/SGPT_final.pdf). This is (almost) the final version. Nonsense comments in Frenglish are not welcome. > 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](Lectures/USTC_NA_1.pdf), [2](Lectures/USTC_NA_2.pdf), [3](Lectures/USTC_NA_3.pdf), [4](Lectures/USTC_NA_4.pdf). > Lecture notes for my course at USTC in the spring of 2024. ### Complex geometry - [Lectures on complex geometry](Lectures/CG.pdf) and [Sample papers](Lectures/CG_papers.pdf) for the final presentation. > Lecture notes for my course at USTC in the fall of 2026.

## Beamers - [Pluripotential-theoretic approach to radial energy functionals](Beamers/PTA.pdf) Beijing university, 11/20/2020 (mm/dd/yyyy). - [Analytic Bertini theorem](Beamers/ABT.pdf) Oslo SCV conference, 12/18/2021. - [A mathematician’s complaint about Hermitian operators](Beamers/MC.pdf) A talk for physicists during an informal seminar, 12/10/2021. - [The volume of pseudo-effective line bundles and partial Okounkov bodies](Beamers/VPPO.pdf) YMSC Tsinghua university, 12/20/2021. - [Chern--Weil formulae of singular Hermitian vector bundles](Beamers/CWF.pdf) Aarhus, 05/20/2022. - [Les singularités $\mathcal{I}$-bonnes --- L'intersection entre la théorie analytique et la théorie algébrique](Beamers/IBS.pdf) IMJ-PRG, 01/03/2023. > En attendant le remboursement du repa de 7 euros jusqu'à aujourd'hui. - [A transcendental approach to non-Archimedean metrics](Beamers/TAnA.pdf) Göteborg, 05/04/2023. - [Transcendental Okounkov bodies and the trace operator of currents](Beamers/TOB.pdf) Toulouse, 10/05/2023. - [Singularities in global pluripotential theory](Beamers/SGPT.pdf) Kanazawa, 12/06/2023. - [Partial Okounkov bodies and toric geometry](Beamers/POB.pdf) Cambridge, 05/17/2024. - [A brief history of potential theory — From Poisson to Lelong](Beamers/HPT.pdf) Shanghai, 09/04/2024 - [Transcendental b-divisors](Beamers/TBD.pdf) Göteborg, 05/28/2025. - [Zero-mass conjecture --- AI-assisted counterexample](Beamers/ZMC.pdf) Harbin, 08/14/2026.

## Research >Errare humanum est. All my preprints can be found on arXiv. See [my Google Scholar page](https://scholar.google.se/citations?user=1GbYhEMAAAAJ) 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. - [A Gram-Cone Obstruction to Positive Mirror Coordinates](Papers/Pop20_vibe.pdf) > **Abstract.** We examine the positive mirror map associated in [Pop20] with the four-dimensional essential deformation space of the Iwasawa manifold. After arranging the deformation parameters into a matrix $T\in M_2(\mathbb{C})$, the map factors through the Gram matrix $T^*T$. It is therefore constant on every left $U(2)$-orbit, has zero differential at the central fibre, and is not locally injective. This gives an explicit counterexample to the positive-image coordinate correspondence asserted in [Pop20, Theorem 7.3(v)]. We identify the exact local replacement: the Gram map identifies the quotient by left $U(2)$ with the cone of positive-semidefinite Hermitian $2\times 2$ matrices, on which the induced map is locally the restriction of a real-analytic diffeomorphism. We also record the independent dimension and parity obstruction to the metric-side coordinate construction used in the same assertion. - [A Determinant-Zero Obstruction to Pointwise Stability](Papers/Pop26_vibe.pdf) > **Abstract.** We show that the strict pointwise stability condition of [Pop26, Definition 3.4] is empty on compact Riemann surfaces for holomorphic vector bundles of rank at least two. Indeed, every such bundle contains a line subbundle $L$, while the coherent subsheaf $L(-p)\subset E$ has a determinant section with a simple zero at $p$. Its stability function therefore vanishes at $p$ for every smooth Hermitian metric on $E$. An explicit irreducible $\mathrm{SU}(2)$ representation of a genus-two surface group yields counterexamples to [Pop26, Theorem 1.4(2)] and to the decomposition assertion of [Pop26, Theorem 6.1]. The example remains uniformly semistable. On curves the obstruction disappears after restricting the test class to saturated subsheaves, whereas in higher dimension even a saturated inclusion may have a vanishing determinant. - [Generic Nonuniqueness for Subcritical Positive Kähler–Einstein Dirichlet Problems](Papers/Non_uniq_Dir_vibe.pdf) > **Abstract.** Let $X = \mathbb{C}^3 / \{\pm 1\}$ with its isolated klt point $p$, and let $m$ be the natural adapted measure on $X$. We exhibit a nonempty open set of smoothly bounded Stein neighborhoods $\Omega$ of $p$ on which the normalized positive Kähler--Einstein Dirichlet problem admits two distinct bounded solutions for one and the same exponent $\gamma$, which is strictly smaller than the Moser--Trudinger critical exponent of $(\Omega,p)$. The domains are obtained by joining three balls by thin necks, the two solutions are produced by a Leray--Schauder degree argument, and a uniform Moser--Trudinger inequality keeps the exponent subcritical on the whole family. Consequently uniqueness cannot hold for generic Stein neighborhoods of $p$ in the $C^\infty$ topology on domains, even after imposing countably many prescribed transversality conditions on the boundary. - [A Counterexample to the Global Kähler–Current Conjecture for Weak Kähler–Ricci Flows](Papers/GKC_vibe.pdf) > **Abstract.** We disprove the global Kähler-current part of the higher-dimensional geometric-smoothing conjecture formulated by Di Nezza, Guedj, and Lu. We construct a compact Kähler surface and singular initial datum for which the maximal weak fixed-background Kähler–Ricci flow fails to dominate any positive multiple of the background form at every time in a genuine Arnold-multiplicity jumping interval. The surface is the blow-up of the projective plane at one point, and the initial current is purely divisorial with unequal weights. An exact formula for the flow reduces the obstruction to a positive current with zero Lelong numbers in a nef and big class of degree zero on the exceptional curve. - [Distinct Polar and Non-Bounded Loci on a Projective Fourfold](Papers/Polar_vibe.pdf) > **Abstract.** We prove that a big real $(1,1)$-class on a smooth projective fourfold can have a non-bounded locus strictly larger than its polar locus, giving a negative answer to Question 42 in the survey of Dinew--Guedj--Zeriahi. - [Uniform Green Functions and Sobolev Geometry in Families of Singular Kähler Varieties](Papers/Green_vibe.pdf) > **Abstract.** We prove uniform estimates for Green's functions, diameters, volumes of balls, and Sobolev inequalities for Kähler metrics with bounded potentials on the fibres of a proper family of normal irreducible compact Kähler spaces, assuming only a uniform $L^p$ bound, $p>1$, on the normalized volume densities. The estimates hold on every fibre, singular fibres included, and no uniform bound on the Kähler potentials is assumed. Green's functions are defined with poles in the regular locus of the metric, while the diameter and non-collapsing estimates hold at every point of the metric completion; the Sobolev inequalities hold in the full subcritical range allowed by the gradient estimate for the Green's function. Two examples—a reducible conic degeneration and a non-normal nodal cubic—show that irreducibility cannot be dropped, and that a Green's function with pole at a singular point is not canonically defined. - [A Proposed Construction of a Complex Structure on the Six-Sphere](Papers/S6_vibe.pdf) > **Abstract.** This is a streamlined version of the original manuscript: we reconstruct only the chain of arguments in the source manuscript [1] that is claimed to produce an integrable complex structure on the standard smooth six-sphere $S^6$. The retained material consists of the period family over the $(3,4,\infty)$ orbifold, its three analytic fillings, the fundamental-group calculation, the integral nearby-cycle and Leray calculations, and recognition of the resulting manifold. All general parameter families, duplicated matrix appendices, and invariants unused by this conclusion are removed. The delicate cusp-properness and integral-saturation arguments remain explicit. - [Errata to Guedj--Zeriahi's book *Degenerate Complex Monge–Ampère Equations*](Papers/Errata_GZ_vibe.pdf) - [Errata to Demailly's book *Complex Analytic and Differential Geometry*](Papers/Errata_Demailly_vibe.pdf) ([annotated version with errata](Papers/Demailly_CADG_annotated.pdf)) > The annotated version should be downloaded and opened locally in a PDF reader that supports annotations. - [Subharmonicity on All Affine Complex Lines Forces Upper Semicontinuity](Papers/PSHD_vibe.pdf) > **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. > 0 sorry Lean4 formalization is available here . - [Nonuniqueness for degenerate Hermitian Monge--Ampère equations](Papers/NUMA_vibe.pdf) > **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. - **Transcendental Nakamaye theorem** > **Abstract.** Let $X$ be a compact Kähler manifold and let $\alpha$ be a pseudoeffective real $(1,1)$-class. We prove that every positive-dimensional irreducible component $V$ of the non-Kähler locus of $\alpha$ has vanishing numerical restricted volume: $\langle\alpha^{\dim V}\rangle_{X\mid V}=0$. Consequently, the non-Kähler locus of $\alpha$ agrees with the null locus defined by numerical restricted volumes. > Available upon request. ### Works in progress - [Mixed volumes of Okounkov bodies](Papers/MVOB.pdf) > Updated in July 2026. I proved the mixed volume formula of Okounkov bodies and partial Okounkov bodies. - [Transcendental b-divisors](Papers/TB.pdf) > Updated in December 2025. - [Transcendental b-divisors II](Papers/TB2.pdf) > 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](https://arxiv.org/abs/1901.07889) [Journal link](https://msp.org/apde/2021/14-6/p12.xhtml) > My note [Radial Calabi flow](Notes/RCF.pdf) might be of interest to the readers of this paper. - Pluripotential-theoretic stability thresholds, ***IMRN*** (2022). [arXiv:2012.12039](https://arxiv.org/abs/2012.12039) [Journal link](https://academic.oup.com/imrn/advance-article-abstract/doi/10.1093/imrn/rnac186/6644735) > 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 - Integration by parts formula for non-pluripolar product. [arXiv:1907.06359](https://arxiv.org/abs/1907.06359) > 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. - Mabuchi geometry of big cohomology classes, ***Journal für die reine und angewandte Mathematik*** (2023). [arXiv:1907.07234](https://arxiv.org/abs/1907.07234) [Journal link](https://www.degruyter.com/document/doi/10.1515/crelle-2023-0019/html) > 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](Papers/MG.pdf). > 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](https://arxiv.org/abs/2003.04818) [Journal link](https://www.sciencedirect.com/science/article/pii/S0001870822000147) - The volume of pseudoeffective line bundles and partial equilibrium, (joint with Tamás Darvas), ***Geometry & Topology*** (2024). [arXiv:2112.03827](https://arxiv.org/abs/2112.03827) [Journal link](https://msp.org/gt/2024/28-4/p09.xhtml) - Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics, ***Geometry & Topology*** (2025). [arXiv:2112.04290](https://arxiv.org/abs/2112.04290) [Journal link](https://doi.org/10.2140/gt.2025.29.1283) > The proof of the Hausdorff convergence property in this paper is quite messy. My [book](Lectures/SGPT2.pdf) contains a more readable proof. - Non-pluripolar products on vector bundles and Chern--Weil formulae, ***Mathematische Annalen*** (2024). [arXiv:2210.15342](https://arxiv.org/abs/2210.15342) [Journal link](https://link.springer.com/article/10.1007/s00208-024-02838-4) - Transcendental Okounkov bodies, (joint with Tamás Darvas, Rémi Reboulet, David Witt Nyström and Kewei Zhang), ***Journal of Differential Geometry*** (2026). [arXiv:2309.07584](https://arxiv.org/abs/2309.07584) [Journal link](https://doi.org/10.4310/jdg/1766433802) - 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](https://arxiv.org/abs/2403.08259) [Journal link](https://doi.org/10.1090/tran/9785) - A counterexample to the zero-mass conjecture, (joint with Long Li). [arXiv:2607.26549](https://arxiv.org/abs/2607.26549) - A counterexample to the bounded mass property, (joint with Kewei Zhang). [arXiv:2608.21053](https://arxiv.org/abs/2608.21053) ### Non-Archimedean geometry and algebraic geometry - On Liu morphisms in non-Archimedean geometry, ***Israel Journal of Mathematics*** (2022). [arXiv:2106.08032](https://arxiv.org/abs/2106.08032) [Journal link](https://link.springer.com/article/10.1007/s11856-022-2456-6) > 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](Papers/Liu.pdf). I will update arXiv in a few days. - Analytic Bertini theorem, ***Mathematische Zeitschrift*** (2022). [arXiv:2110.14971](https://arxiv.org/abs/2110.14971) [Journal link](https://link.springer.com/article/10.1007/s00209-022-03103-7) > 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. - Analytic Bertini theorem II --- The local case. [arXiv:2607.25230](https://arxiv.org/abs/2607.25230) - A transcendental approach to non-Archimedean metrics of pseudoeffective classes, (joint with Tamás Darvas and Kewei Zhang), ***Commentarii Mathematici Helvetici*** (2025). [arXiv:2302.02541](https://arxiv.org/abs/2302.02541) [Journal link](https://doi.org/10.4171/CMH/586) > 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](Notes/OTNA.pdf).

## Some notes - [LaTeX tips for working mathematicians](Notes/LaTeX_tips.pdf) - [Relative pluripotential theory](Notes/RPT.pdf) (Chapter I to Chapter III) > Just a preliminary version with potentially many mistakes. I'm slowly adding new materials. - [Radial Calabi flow](Notes/RCF.pdf) > One of my unfinished projects. It contains a number of conjectures of interest. - [Note on relative normalisations](Notes/RN.pdf) > I collect a few well-known results about relative normalisations. - Notes on the toroidal compactifications of Shimura varieties: [I](Notes/SV1.pdf), [II](Notes/SV2.pdf), [III](Notes/SV3.pdf), [X](Notes/SV10.pdf), [XIII](Notes/SV13.pdf). - Notes on Hodge theory: [Carlson's correspondence](Notes/Hodge1.pdf). - [Operations on transcendental non-Archimedean metrics](https://bookstore.ams.org/view?ProductCode=CONM/810). [arXiv:2312.17150](https://arxiv.org/abs/2312.17150). > 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. - [Non-pluripolar currents are not necessarily I-good](Notes/IGE.pdf) > I construct a non-pluripolar qpsh function with small unbounded locus which is not I-good. - Notes on Ducros' book "La structure des courbes analytiques" [Chapter 3](Notes/Duc3.pdf), [Chapter 4](Notes/Duc4.pdf) > My notes while learning Ducros' book. - [A proof of Satz von H. Cartan](Notes/SC.pdf) > 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. - [Stein spaces](Notes/SS.pdf) > My personal notes about the theory of Stein spaces. The 1976 paper of Bingener is integrated into these notes. - [Schemata über Steinschen Algebren](LaTeX/Bingener.pdf) > I turn Bingener's famous 1976 paper in to LaTeX format. - [Analytic singularities are not birationally invariant](Notes/AS_Bir.pdf)

## My favourite links ### Legal links - [Online math seminars](https://researchseminars.org) - [Tikz-cd editor](https://tikzcd.yichuanshen.de) - [Quiver](https://q.uiver.app) - [Stacks Project](https://stacks.math.columbia.edu) - [Kerodon](https://kerodon.net) - [Almost Sure](https://almostsuremath.com) - [MacTutor](https://mathshistory.st-andrews.ac.uk) ### Illegal links - [ZLibrary](https://z-library.sk) > Updated on Nov 8, 2024. - [Libgen](https://libgen.vg) > Updated on Jul 21, 2025. - [Sci-hub](http://sci-hub.vkif.top) > Sci-hub is getting blocked in many countries recently. If the link fails to work, please try to change the domain name.