The universal structure of moment maps in complex geometry

Ruadhaí Dervan; Michael Hallam

doi:10.5802/jep.330

The universal structure of moment maps in complex geometry
[La structure universelle des applications moments en géométrie complexe]

Ruadhaí Dervan ¹ ; Michael Hallam ²

¹Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
²Department of Mathematics, University of Aarhus, Ny Munkegade 118, 8000 Aarhus C, Denmark

Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 399-436

Abstract (VO)
Résumé

We introduce a geometric approach to the construction of moment maps in finite and infinite-dimensional complex geometry. We apply this to two settings: Kähler manifolds and holomorphic vector bundles. We first give a new, geometric proof of Donaldson–Fujiki’s moment map interpretation of the scalar curvature. Associated to arbitrary products of Chern characters of the manifold—namely to a central charge—we further introduce a geometric PDE determining a $Z$-critical Kähler metric, and show that these general equations also satisfy moment map properties. For holomorphic vector bundles, we similarly give a geometric proof that the PDE determining a $Z$-critical connection can be viewed as a moment map. Our main assertion is that this is the canonical way of producing moment maps in complex geometry, and hence that this accomplishes one of the main steps towards producing PDE counterparts to stability conditions in large generality.

Nous introduisons une approche géométrique de la construction des applications moments en géométrie complexe de dimension finie et infinie. Nous l’appliquons dans deux contextes : variétés de Kähler et fibrés holomorphes. Nous donnons d’abord une nouvelle preuve géométrique de l’interprétation de la courbure scalaire en terme l’application moment de Donaldson-Fujiki. Associée à des produits arbitraires de caractères de Chern de la variété, à savoir à une charge centrale, nous introduisons en outre une EDP géométrique déterminant une métrique de Kähler $Z$-critique, et montrons que ces équations générales satisfont également les propriétés des applications moments. Pour les fibrés holomorphes, nous donnons de manière similaire une preuve géométrique que l’EDP déterminant une connexion $Z$-critique peut être considérée comme une application moment. Notre assertion principale est qu’il s’agit là de la manière canonique de produire des applications moments en géométrie complexe, et donc que cela constitue l’une des étapes principales vers la production d’équivalents EDP aux conditions de stabilité dans une grande généralité.

Reçu le : 2024-07-22
Accepté le : 2026-02-04
Publié le : 2026-02-23

DOI : 10.5802/jep.330

Classification : 53D20, 32Q26, 53C07
Keywords: Moment maps, canonical Kähler metrics, Hermite-Einstein metrics, Yau-Tian-Donaldson conjecture
Mots-clés : Applications moments, métrique de Kähler canoniques, métriques de Hermite-Einstein, conjecture de Yau-Tian-Donaldson

Affiliations des auteurs :

Ruadhaí Dervan ¹ ; Michael Hallam ²

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
² Department of Mathematics, University of Aarhus, Ny Munkegade 118, 8000 Aarhus C, Denmark

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2026__13__399_0,
     author = {Ruadha{\'\i} Dervan and Michael Hallam},
     title = {The universal structure of moment maps in complex geometry},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {399--436},
     year = {2026},
     publisher = {\'Ecole polytechnique},
     volume = {13},
     doi = {10.5802/jep.330},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.330/}
}

TY  - JOUR
AU  - Ruadhaí Dervan
AU  - Michael Hallam
TI  - The universal structure of moment maps in complex geometry
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2026
SP  - 399
EP  - 436
VL  - 13
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.330/
DO  - 10.5802/jep.330
LA  - en
ID  - JEP_2026__13__399_0
ER  -

%0 Journal Article
%A Ruadhaí Dervan
%A Michael Hallam
%T The universal structure of moment maps in complex geometry
%J Journal de l’École polytechnique — Mathématiques
%D 2026
%P 399-436
%V 13
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.330/
%R 10.5802/jep.330
%G en
%F JEP_2026__13__399_0

Ruadhaí Dervan; Michael Hallam. The universal structure of moment maps in complex geometry. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 399-436. doi: 10.5802/jep.330

Bibliographie
Cité par

[AB83] M. F. Atiyah & R. Bott - “The Yang-Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A 308 (1983) no. 1505, p. 523-615 | DOI | Zbl | MR

[Bal06] W. Ballmann - Lectures on Kähler manifolds, ESI Lectures in Math. and Physics, European Mathematical Society, Zürich, 2006 | Zbl | DOI | MR

[Ban06] S. Bando - “An obstruction for Chern class forms to be harmonic”, Kodai Math. J. 29 (2006) no. 3, p. 337-345 | Zbl | MR

[BGV04] N. Berline, E. Getzler & M. Vergne - Heat kernels and Dirac operators, Grundlehren Math. Wissen., Springer-Verlag, Berlin, 2004, Corrected reprint of the 1992 original | Zbl | MR

[Bri07] T. Bridgeland - “Stability conditions on triangulated categories”, Ann. of Math. (2) 166 (2007) no. 2, p. 317-345 | DOI | Zbl | MR

[Che21] G. Chen - “The J-equation and the supercritical deformed Hermitian-Yang-Mills equation”, Invent. Math. 225 (2021) no. 2, p. 529-602 | Zbl | DOI | MR

[CJY20] T. C. Collins, A. Jacob & S.-T. Yau - “ $(1, 1)$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions”, Camb. J. Math. 8 (2020) no. 2, p. 407-452 | MR | DOI | Zbl

[Cor24] A. Corradini - “Equivariant localization in the theory of $Z$ -stability for Kähler manifolds”, Internat. J. Math. 35 (2024) no. 7, article ID 2450026, 29 pages | DOI | Zbl | MR

[CY18] T. C. Collins & S.-T. Yau - “Moment maps, nonlinear PDE, and stability in mirror symmetry”, 2018 | arXiv | Zbl

[CY21] T. C. Collins & S.-T. Yau - “Moment maps, nonlinear PDE and stability in mirror symmetry, I: geodesics”, Ann. PDE 7 (2021) no. 1, article ID 11, 73 pages | DOI | Zbl | MR

[Der23] R. Dervan - “Stability conditions for polarised varieties”, Forum Math. Sigma 11 (2023), article ID e104, 57 pages | MR | DOI | Zbl

[DH23] R. Dervan & M. Hallam - “The universal structure of moment maps in complex geometry”, 2023 | arXiv | Zbl

[DInNn25] R. Dervan & A. Ibáñez Núñez - “Stability conditions in geometric invariant theory”, Q. J. Math. 76 (2025) no. 1, p. 287-311 | Zbl | DOI | MR

[DMS24] R. Dervan, J. B. McCarthy & L. M. Sektnan - “ $Z$ -critical connections and Bridgeland stability conditions”, Camb. J. Math. 12 (2024) no. 2, p. 253-355 | DOI | Zbl | MR

[Don85] S. K. Donaldson - “Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles”, Proc. London Math. Soc. (3) 50 (1985) no. 1, p. 1-26 | DOI | Zbl | MR

[Don87] S. K. Donaldson - “Infinite determinants, stable bundles and curvature”, Duke Math. J. 54 (1987) no. 1, p. 231-247 | DOI | Zbl | MR

[Don97] S. K. Donaldson - “Remarks on gauge theory, complex geometry and $4$ -manifold topology”, in Fields Medallists’ lectures, World Sci. Ser. 20th Century Math., vol. 5, World Scientific Publisher, River Edge, NJ, 1997, p. 384-403 | DOI

[Don02] S. K. Donaldson - “Scalar curvature and stability of toric varieties”, J. Differential Geom. 62 (2002) no. 2, p. 289-349 | MR | Zbl

[DP21] V. V. Datar & V. P. Pingali - “A numerical criterion for generalised Monge-Ampère equations on projective manifolds”, Geom. Funct. Anal. 31 (2021) no. 4, p. 767-814 | Zbl | MR | DOI

[DS21] R. Dervan & L. M. Sektnan - “Extremal Kähler metrics on blowups”, 2021 | arXiv | Zbl

[FM85] A. Futaki & S. Morita - “Invariant polynomials of the automorphism group of a compact complex manifold”, J. Differential Geom. 21 (1985) no. 1, p. 135-142 | MR | Zbl

[FS90] A. Fujiki & G. Schumacher - “The moduli space of extremal compact Kähler manifolds and generalized Weil-Petersson metrics”, Publ. RIMS, Kyoto Univ. 26 (1990) no. 1, p. 101-183 | Zbl | DOI

[Fuj92] A. Fujiki - “Moduli space of polarized algebraic manifolds and Kähler metrics”, Sugaku Expositions 5 (1992) no. 2, p. 173-191 | Zbl | MR

[Fut06] A. Futaki - “Harmonic total Chern forms and stability”, Kodai Math. J. 29 (2006) no. 3, p. 346-369 | Zbl | MR | DOI

[Gau17] P. Gauduchon - “Calabi’s extremal Kähler metrics: An elementary introduction”, 2017 | Zbl

[Ino24] E. Inoue - “Equivariant calculus on $μ$ -character and $μ K$ -stability of polarized schemes”, J. Symplectic Geom. 22 (2024) no. 2, p. 267-353 | Zbl | MR | DOI

[JY17] A. Jacob & S.-T. Yau - “A special Lagrangian type equation for holomorphic line bundles”, Math. Ann. 369 (2017) no. 1-2, p. 869-898 | MR | Zbl | DOI

[Koi90] N. Koiso - “On the complex structure of a manifold of sections”, Osaka J. Math. 27 (1990) no. 1, p. 175-183 | MR | Zbl

[Leg21] E. Legendre - “Localizing the Donaldson-Futaki invariant”, Internat. J. Math. 32 (2021) no. 8, article ID 2150055, 23 pages | Zbl | MR | DOI

[Lej10] M. Lejmi - “Extremal almost-Kähler metrics”, Internat. J. Math. 21 (2010) no. 12, p. 1639-1662 | Zbl | MR | DOI

[Leu98] N. C. Leung - “Bando Futaki invariants and Kähler Einstein metric”, Comm. Anal. Geom. 6 (1998) no. 4, p. 799-808 | DOI | MR | Zbl

[LYZ00] N. C. Leung, S.-T. Yau & E. Zaslow - “From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform”, Adv. Theo. Math. Phys. 4 (2000) no. 6, p. 1319-1341 | MR | Zbl | DOI

[McC22] J. B. McCarthy - “Canonical metrics on holomorphic fibre bundles”, 2022 | arXiv | Zbl

[McC23] J. B. McCarthy - “Stability conditions and canonical metrics”, 2023 | arXiv | Zbl

[MnMMS00] M. Mariño, R. Minasian, G. Moore & A. Strominger - “Nonlinear instantons from supersymmetric $p$ -branes”, J. High Energy Phys. (2000) no. 1, article ID 5, 32 pages | MR | Zbl | DOI

[Ort24] A. Ortu - “Moment maps and stability of holomorphic submersions”, 2024 | arXiv | Zbl

[OS24] A. Ortu & L. M. Sektnan - “Constant scalar curvature Kähler metrics and semistable vector bundles”, 2024 | arXiv | Zbl

[Sca20] C. Scarpa - “The Hitchin-cscK system”, 2020 | arXiv | Zbl

[Sek21] L. M. Sektnan - “Canonical metrics and stability on holomorphic vector bundles” (2021), https://sites.google.com/cirget.ca/lars-sektnan/bundles

[Son20] J. Song - “Nakai-Moishezon criterions for complex Hessian equations”, 2020 | arXiv | Zbl

[Szé12] G. Székelyhidi - “On blowing up extremal Kähler manifolds”, Duke Math. J. 161 (2012) no. 8, p. 1411-1453 | MR | Zbl | DOI

[Tia97] G. Tian - “Kähler-Einstein metrics with positive scalar curvature”, Invent. Math. 130 (1997) no. 1, p. 1-37 | MR | DOI | Zbl

[UY86] K. Uhlenbeck & S.-T. Yau - “On the existence of Hermitian-Yang-Mills connections in stable vector bundles”, Comm. Pure Appl. Math. 39 (1986) no. S, suppl., p. S257-S293, Frontiers of the mathematical sciences: 1985 (New York, 1985) | MR | Zbl | DOI

[Yau93] S.-T. Yau - “Open problems in geometry”, in Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, American Mathematical Society, Providence, RI, 1993, p. 1-28 | Zbl

Cité par Sources :