This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Yau–Tian–Donaldson conjecture | |
|---|---|
| Name | Yau–Tian–Donaldson conjecture |
| Field | Differential geometry; Complex manifold theory; Algebraic geometry |
| Proposed | 1990s |
| Proponents | Shing-Tung Yau, Gang Tian, Simon Donaldson |
Yau–Tian–Donaldson conjecture The Yau–Tian–Donaldson conjecture proposes a precise equivalence between differential-geometric existence of canonical Kähler metrics on compact complex manifolds and algebro-geometric stability conditions for polarized projective varieties, connecting ideas of Shing-Tung Yau, Gang Tian, and Simon Donaldson. It links analytic constructions from Calabi conjecture work and results related to the Aubin–Yau theorem with algebraic notions influenced by the Mumford–Takemoto stability program and developments in the Geometric Invariant Theory of David Mumford. The conjecture has driven interactions among researchers from institutions such as Institute for Advanced Study, Harvard University, Princeton University, Stanford University, and international schools in Beijing, Paris, and Tokyo.
The conjecture originates in efforts to solve the Calabi conjecture and to understand canonical metrics on complex manifolds, inspired by work of Eugenio Calabi, Shing-Tung Yau, and later analytic advances by Thierry Aubin. Motivated by earlier classification programs such as the Enriques–Kodaira classification and by moduli questions considered by David Mumford and Alexander Grothendieck, the conjecture aims to characterize when a polarized variety admits a constant scalar curvature Kähler metric, a problem linked to the Calabi–Yau manifold program central to research at institutions like Princeton University and projects involving researchers including Yau, Tian, and Donaldson. The algebro-geometric side draws on notions introduced in the Mukai program and influenced by stability criteria used in work of Simon Donaldson on instantons and by the Narasimhan–Seshadri theorem connecting unitary representations with stable bundles studied by M. S. Narasimhan and C. S. Seshadri.
Roughly stated, the conjecture asserts that a polarized projective manifold admits a constant scalar curvature Kähler metric if and only if the polarization is K-stable in the sense introduced by Gang Tian and formalized by Simon Donaldson. The conjectural equivalence uses test configurations related to degenerations studied by David Mumford and stability notions that generalize ideas from the Harder–Narasimhan filtration and from work by Shigeru Mukai. The precise statement invokes invariants akin to the Futaki invariant introduced by Akito Futaki and stability notions refined by researchers including Chi Li, Simon Donaldson, and Xiaowei Wang, with technical setup paralleling constructions in the Minimal Model Program driven by Shigefumi Mori and Yujiro Kawamata.
Extensions include versions for Fano varieties predicting equivalence of Kähler–Einstein metrics with K-polystability, pioneered in contributions by Gábor Székelyhidi, Chi Li, and Vladimir Tipler. The conjecture has been adapted to twisted settings influenced by work of Jean-Pierre Demailly and Robert Berman, to conical metrics inspired by edge-cone explorations connected to Donaldson's earlier singular metric studies, and to logarithmic pairs drawing on techniques used by Kollár and János Kollár in the log minimal model program. Related formulations appear in the study of constant scalar curvature metrics on orbifolds discussed in monographs by Y. Kawamata and in analytic generalizations pursued by scholars at ETH Zurich and Imperial College London.
Major breakthroughs include proofs of the Fano case combining analytic existence theorems by Gang Tian and algebraic criteria advanced by Chi Li and Gábor Székelyhidi, as well as the complete resolution of the existence of Kähler–Einstein metrics on smooth Fano manifolds through work by Chen-Donaldson-Sun building on ideas from Perelman's techniques in Ricci flow and analytic compactness results reminiscent of methods used by Richard Hamilton. Key algebraic inputs draw on developments from the Minimal Model Program by Birkar, Cascini, and Hacon, while analytic regularity arguments parallel those in studies by Jeffres, Mazzeo, and Rubinstein. Subsequent workshops and conferences at ICM venues and seminars at IHES disseminated results by teams including Chi Li, S. Paul, Berman, and Zhang.
Methods combine complex Monge–Ampère equations studied by Yau and analytic pluripotential tools advanced by Sławomir Kołodziej, with algebraic geometry inputs from GIT as developed by David Mumford and from test configuration formalisms introduced by Donaldson. The analytic side leverages continuity methods akin to those used in the Aubin continuity path while importing compactness techniques reminiscent of Perelman’s estimates from Ricci flow theory. Algebraic approaches use valuation theory and volumes following work by Boucksom, Jonsson, and Fujita, and make contact with degeneration techniques prevalent in the Hodge theory literature associated with Peters and Steenbrink.
Constructed examples include toric varieties first analyzed by Victor Guillemin and Alan Weinstein where explicit potentials verify stability via combinatorial criteria related to the Delzant theorem, and Fano examples exhibiting K-stability tested by computations by Chi Li and Ruadhaí Dervan. Notable counterexamples to naive formulations arise from pathologies in degenerations akin to phenomena studied by Zariski and by researchers in deformation theory such as Kuranishi; these show failures of existence in the absence of refined K-polystability conditions, paralleling subtleties previously encountered in moduli problems handled by Mumford and Gieseker.
Active directions include extending the equivalence to broader classes of varieties influenced by the Minimal Model Program and understanding uniform K-stability variants pursued by Székelyhidi, Berman, and Boucksom. Researchers in groups at Princeton University, Oxford University, Cambridge University, University of California, Berkeley, and Peking University investigate bridging analytic regularity gaps, refining algebraic stability invariants via non-Archimedean geometry developed by Berkovich and Boucksom–Jonsson, and probing computational criteria inspired by toric and spherical varieties studied by Brion and Delzant. The conjecture continues to catalyze collaborations linking the legacies of Yau, Tian, and Donaldson with contemporary programs in algebraic and differential geometry.
Category:Conjectures in mathematics