Math 2.0

Several complex variables and analytic spaces Theorem

Yau–Tian–Donaldson conjecture

Builds on Kähler–Einstein metricHermitian manifoldCalabi conjectureGromov–Hausdorff convergence

Statement

Let XX be a Fano manifold, that is a compact complex manifold whose anticanonical line bundle −KX-K_X is ample. Then XX admits a Kähler–Einstein metric if and only if the pair (X,−KX)(X, -K_X) is K-polystable.

Here K-polystable means the following. A test configuration of (X,−KX)(X, -K_X) is a ℂ∗\C^*-equivariant degeneration of XX with a line bundle extending a power of −KX-K_X. For every normal test configuration, the Donaldson–Futaki invariant is nonnegative. Moreover, it vanishes only for normal test configurations that are products of XX with ℂ\C twisted by a ℂ∗\C^*-action on XX.

Remarks

This form of the conjecture, for Fano manifolds, was proved by Xiuxiong Chen, Simon Donaldson and Song Sun. The direction "Kähler–Einstein implies K-polystable" is the easier one. The converse uses the continuity method with metrics having cone singularities along a divisor, and Gromov–Hausdorff limits of these metrics, which are shown to be algebraic varieties.

Example: the blow-up of ℂP2\C P^2 at one point is Fano but has no Kähler–Einstein metric. Its automorphism group is not reductive, which rules out a Kähler–Einstein metric by Matsushima's theorem, and a test configuration coming from a ℂ∗\C^*-action has negative Donaldson–Futaki invariant. By contrast ℂPn\C P^n with the Fubini–Study metric is Kähler–Einstein and K-polystable.

The general conjecture asks for a constant scalar curvature Kähler metric in c1(L)c_1(L) on a polarized manifold (X,L)(X, L), under a suitable notion of K-stability. That general form is open, and the right stability notion for it is still discussed.

Going deeper

For a student

The negative and zero cases of the Kähler–Einstein problem follow from the Calabi conjecture methods with no obstruction. In the Fano case there are obstructions, such as the Futaki invariant and Matsushima's theorem, and K-stability collects them all. The statement links a nonlinear partial differential equation to geometric invariant theory, in the spirit of the Hitchin–Kobayashi correspondence for vector bundles. After the proof, much work has gone into deciding K-stability of explicit Fano manifolds by algebraic means.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Differential form, concept, no evidence levelDifferential formconceptGromov–Hausdorff convergence, concept, no evidence levelGromov–Hausdorff conv…conceptHermitian manifold, concept, no evidence levelHermitian manifoldconceptKähler–Einstein metric, concept, no evidence levelKähler–Einstein metricconceptRicci curvature, concept, no evidence levelRicci curvatureconceptExterior derivative, concept, no evidence levelExterior derivativeconceptClosed and exact forms, concept, no evidence levelClosed and exact formsconceptCalabi conjecture, statement, no evidence levelCalabi conjecturestatementYau–Tian–Donaldson conjecture, statement, no evidence levelYau–Tian–Donaldson co…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Differential form concept
  2. Gromov–Hausdorff convergence concept
  3. Hermitian manifold concept
  4. Kähler–Einstein metric concept
  5. Ricci curvature concept
  6. Exterior derivative concept
  7. Closed and exact forms concept
  8. Calabi conjecture statement
  9. Yau–Tian–Donaldson conjecture statement

Further reading

  1. Web https://en.wikipedia.org/wiki/K-stability
About this page
TagDHYSPSYSCB
URIm2://DHYSPSYSCB
Kindstatement, theorem
WritingW0 stub
MSC32Q20, 53C55, 14J45
StatusThis copy of the site has no build data. Run m2 status DHYSPSYSCB.
AboutKähler–Einstein metric
UsesKähler–Einstein metric, Hermitian manifold, Calabi conjecture, Gromov–Hausdorff convergence
Changed2 November 2026

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