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Kähler-Einstein metrics, random point processes and variational principles

Objective

In broad terms the aim of this proposal is to introduce a new probabilistic approach to the study of Kähler-Einstein (K-E) metrics on complex manifolds. A precise procedure, based on a blend of Statistical Mechanics, Pluripotential theory and Kähler Geometry. will be used to show that

• when a K-E metric exists on a complex manifold X it can be obtained from the “large
N limit” of certain canonical random point processes on X with N particles.

The canonical point processes are directly defined in terms of algebro-geometric data and the thrust of this approach is thus that it gives a new link between algebraic geometry on one and hand and complex differential (Kähler) geometry on the other. A major motivation for this project comes from the fundamental Yau-Tian-Donaldson conjecture in Kähler geometry, which aims at characterizing the obstructions to the existence of a K-E metric on a Fano manifold in terms of a suitable notion of algebro-geometric “stability”, notably K-Stability. In this project a new “probabilistic/statistical mechanical” version of stability will be introduced referred to as Gibbs stability, which also has an interesting purely algebro-geometric definition in the spirit of the Minimal Model Program in current algebraic geometry and another specific aim of this project is to prove or at least make substantial progress towards proving,

• There is a (unique) K-E metric on a Fano manifold X precisely when X is asymptotically Gibbs stable

The canonical random point processes will be defined as certain “beta-deformations” of determinantal point processes and share certain properties with the ones appearing in Random Matrix Theory and in the study of quantum chaos and zeroes of random polynomials (and random holomorphic sections) But a crucial new feature here is that the processes are independent of any back-ground data, such as a potential or a metric.

Field of science

  • /natural sciences/physical sciences/classical mechanics/statistical mechanics
  • /natural sciences/mathematics/pure mathematics/geometry
  • /natural sciences/physical sciences/theoretical physics/particles
  • /natural sciences/mathematics/pure mathematics/algebra/algebraic geometry

Call for proposal

ERC-2012-StG_20111012
See other projects for this call

Funding Scheme

ERC-SG - ERC Starting Grant

Host institution

CHALMERS TEKNISKA HOEGSKOLA AB
Address
-
41296 Goeteborg
Sweden
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 1 200 000
Principal investigator
Robert Berman (Dr.)
Administrative Contact
Helena Rafstedt (Ms.)

Beneficiaries (1)

CHALMERS TEKNISKA HOEGSKOLA AB
Sweden
EU contribution
€ 1 200 000
Address
-
41296 Goeteborg
Activity type
Higher or Secondary Education Establishments
Principal investigator
Robert Berman (Dr.)
Administrative Contact
Helena Rafstedt (Ms.)