Extremal Kähler metrics and Donaldson’s quantisation

Yoshinori Hashimoto
I2M, Aix-Marseille Université

Date(s) : 25/04/2017   iCal
14 h 00 min - 15 h 00 min

Kähler metrics with “optimal” curvature properties have been studied intensively since the proposal of E. Calabi, who suggested that we look for Kähler metrics whose L^2-norm of the scalar curvature is minimal. These metrics are called extremal, and include as subclasses the constant scalar curvature Kähler (cscK) metrics and Kähler-Einstein metrics. Extremal Kähler metrics are actively studied, particularly in connection to the algebro-geometric stability of the underlying Kähler manifold, following the proposal of S.-T. Yau, G. Tian, S. Donaldson, and G. Székelyhidi.
A foundational result in this area is Donaldson’s quantisation, which provides a “finite dimensional” approximation of cscK metrics when the automorphism group is discrete. Several significant applications of this result will be reviewed in the talk, particularly in connection to the Chow stability of the manifold, and the numerical computation of the Calabi-Yau metrics.
On the other hand, examples were found to show that the above theory does not carry over naively to the case when the automorphism group is non-discrete. In this talk, we propose a new “quantising” equation, which generalises various key results in Donaldson’s quantisation when the automorphism group is no longer discrete, and can be applied more generally to extremal Kähler metrics.




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