Welcome to my homepage

About Me

I am now a post-doc at IMJ-PRG. My mentor is Sébastien Boucksom. I obtained my PhD in Chalmers Tekniska Högskola in Sweden under the supervision of Robert Berman.

My name in Chinese: 夏铭辰(Simplified)/夏銘辰(Traditional)

Email: xiamingchen2008@gmail.com

Office: Jussieu 1516-504.

I’m currently interested in Thuillier’s thesis and the Japanese language. Updated on March 24, 2024.

I am a Pastafarian.

Some Problems

This is a collection of problems arising from my own research that may be of interest to people outside my domain. If you know the solutions to any of the following problems, please let me know.

• Flat pull-back of currents on complex analytic spaces

• Compute mixed volumes of big line bundles from Okounkov bodies

By a theorem of Jow, information of all Okounkov bodies determines all numerical information of line bundles. This problem asks for explicit formulae.

Notes

The lecture notes for courses can be found on a separate page.

• LaTeX tips for working mathematicians

• Relative pluripotential theory (Chapter I to Chapter III)

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

• Radial Calabi flow

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

• Pluripotential theory on complex analytic spaces

This is integrated into the arXiv version of my paper on Mabuchi geometry. So I disabled the link.

• Note on $L^2$-methods in global pluripotential theory

My personal notes when learning the $L^2$ methods, I plan to include more details in the future. This note contains an example of a reverse Bertini theorem, which seems to be new.

• Note on relative normalisations

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

• Notes on the toroidal compactifications of Shimura varieties: I, II, III, X, XIII.

I’m organizing a seminar about Ash–Mumford–Rapoport–Tai. I will try to write more notes in the near future.

• Hausdorff convergence properties of partial Okounkov bodies.

This note has been integrated into the final version of the partial Okounkov body paper.

• Notes on Hodge theory: Carlson’s correspondence.

• Convergence of quasi-plurisubharmonic functions

A note about the d_S-topology on the space of qpsh functions. It contains a number of new results. I removed the link for the time being. It has been integrated into my lecture notes at Zhejiang university.

• Operations on transcendental non-Archimedean metrics. arXiv:2312.17150

This note is submitted to the proceeding for Bo Berndtsson’s 70th birthday. It is a trivial continuation of my joint paper with Darvas and Zhang. The only notable result is Theorem 4.21.

• Note on Duistermaat–Heckman measures of non-Archimedean metrics.

In this note, I construct the Duistermaat–Heckman measures using the theory of partial Okounkov bodies.

• Note on partial Okounkov bodies — The point of view of b-divisors.

In this note, I prove that the partial Okounkov bodies admit a natural interpretation in terms of b-divisors.

Beamers

• Pluripotential-theoretic approach to radial energy functionals Beijing university, 11/20/2020 (mm/dd/yyyy).

• Analytic Bertini theorem Oslo SCV conference, 12/18/2021.

• A mathematician’s complaint about Hermitian operators A talk for physicists during an informal seminar, 12/10/2021.

• The volume of pseudo-effective line bundles and partial Okounkov bodies YMSC Tsinghua university, 12/20/2021.

• Chern–Weil formulae of singular Hermitian vector bundles Aarhus, 05/20/2022.

• Les singularités $\mathcal{I}$-bonnes — L’intersection entre la théorie analytique et la théorie algébrique IMJ-PRG, 01/03/2023.

• A transcendental approach to non-Archimedean metrics Göteborg, 05/04/2023.

• Transcendental Okounkov bodies and the trace operator of currents Toulouse, 10/05/2023.

• Singularities in global pluripotential theory Kanazawa, 12/06/2023.

Ymir

Ymir is intended to be a Stacks Project for complex analytic spaces and non-Archimedean analytic spaces.

Research

Errare humanum est.

All my preprints can be found on arXiv. See my Google Scholar page as well.

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.

• Pluripotential-theoretic stability thresholds, IMRN, (2022). arXiv:2012.12039 Journal link

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

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 with prescribed singularities, Journal für die reine und angewandte Mathematik, (2023). arXiv:1907.07234 Journal link

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

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). arXiv:2112.03827

• Partial Okounkov bodies and Duistermaat–Heckman measures of non-Archimedean metrics. arXiv:2112.04290

• 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). arXiv:2309.07584

• The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds, (joint with Tamás Darvas). arXiv:2403.08259

A different point of view to the trace operator can be found in my lecture notes at Zhejiang university.

Non-Archimedean geometry and algebraic geometry

• On Liu morphisms in non-Archimedean geometry, Israel Journal of Mathematics, (2022). arXiv:2106.08032 Journal link

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.

• Analytic Bertini theorem, Mathematische Zeitschrift, (2022). arXiv:2110.14971 Journal link

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

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.

Links for working mathematicians

Legal links

• Online math seminars

• Tikz-cd editor

• Quiver

• Stacks Project

• Kerodon

Illegal links

• ZLibrary

Latest link.

• Libgen

If you hate Elsevier or like free knowledge, please download books from these links.

• Sci-hub

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