I am (virtually) unemployed for the time being.
Previously, I was a post-doc at IMJ-PRG. My mentor was Sébastien Boucksom. I obtained my PhD degree at Chalmers Tekniska Högskola in Sweden under the supervision of Robert Berman.
My name in Chinese: 夏铭辰(Simplified)/夏銘辰(Traditional)
Email: [email protected] (The IMJ email address is still in use, but will expire soon.)
Office: None.
I'm currently interested in the reverse Andreotti--Grauert theorems and Swedish. Updated on Aug 21, 2024.
I am a Pastafarian.
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.
By a theorem of Jow, information of all Okounkov bodies determines all numerical information of line bundles. This problem asks for explicit formulae.
The lecture notes for courses can be found on a separate page.
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Relative pluripotential theory (Chapter I to Chapter III)
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.
- Pluripotential theory on complex analytic spaces
This is integrated into the arXiv version of my paper on Mabuchi geometry. So I disabled the link.
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.
I collect a few well-known results about relative normalisations.
I'm organizing a seminar about Ash--Mumford--Rapoport--Tai. I will try to write more notes in the near future.
This note has been integrated into the final version of the partial Okounkov body paper.
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Notes on Hodge theory: Carlson's correspondence.
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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.
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. The arXiv version contains a mistake in the proof of Lemma 2.9, which is fixed in the current version.
In this note, I construct the Duistermaat--Heckman measures using the theory of partial Okounkov bodies.
In this note, I prove that the partial Okounkov bodies admit a natural interpretation in terms of b-divisors.
In this note, I give an example illustrating the idea that the trace operator could systematically improve known inequalities in the literature.
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Pluripotential-theoretic approach to radial energy functionals Beijing university, 11/20/2020 (mm/dd/yyyy).
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Analytic Bertini theorem Oslo SCV conference, 12/18/2021.
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A mathematician’s complaint about Hermitian operators A talk for physicists during an informal seminar, 12/10/2021.
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The volume of pseudo-effective line bundles and partial Okounkov bodies YMSC Tsinghua university, 12/20/2021.
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Chern--Weil formulae of singular Hermitian vector bundles Aarhus, 05/20/2022.
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Les singularités $\mathcal{I}$-bonnes --- L'intersection entre la théorie analytique et la théorie algébrique IMJ-PRG, 01/03/2023.
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A transcendental approach to non-Archimedean metrics Göteborg, 05/04/2023.
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Transcendental Okounkov bodies and the trace operator of currents Toulouse, 10/05/2023.
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Singularities in global pluripotential theory Kanazawa, 12/06/2023.
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Partial Okounkov bodies and toric geometry Cambridge, 05/17/2024.
Ymir is intended to be a Stacks Project for complex analytic spaces and non-Archimedean analytic spaces.
Errare humanum est.
All my preprints can be found on arXiv. See my Google Scholar page as well.
- 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.
- 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.
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.
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The closures of test configurations and algebraic singularity types, (joint with Tamás Darvas), Advances in Mathematics, (2022). arXiv:2003.04818 Journal link
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The volume of pseudoeffective line bundles and partial equilibrium, (joint with Tamás Darvas), Geometry & Topology, (2024). arXiv:2112.03827 Journal link
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Partial Okounkov bodies and Duistermaat--Heckman measures of non-Archimedean metrics. arXiv:2112.04290
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Non-pluripolar products on vector bundles and Chern--Weil formulae, Mathematische Annalen, (2024). arXiv:2210.15342 Journal link
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Transcendental Okounkov bodies, (joint with Tamás Darvas, Rémi Reboulet, David Witt Nyström and Kewei Zhang). arXiv:2309.07584
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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. Given the strong analogy with the usual trace opeartor, it is natural to ask if one could solve the Dirichlet problem for our trace operators: Is it possible to extend (in the sense of trace operators, up to I-equivalence) any Kähler current within a given big cohomology class?
- 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
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). 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.
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