Quantisation and the Hessian of Mabuchi energy

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• Joel Fine (Université Libre de Bruxelles)
• Thursday 12 April 2012, 14:30-15:30
• MR2.

If you have a question about this talk, please contact Dr. J Ross.

Let L be an ample line bundle over a compact complex manifold. In Kähler quantisation one approximates the space H of Kähler metrics in c_1(L) by the spaces B_k of Hermitian innerproducts on H⁰(X,L^k). Following Donaldson, we know that Mabuchi energy E on H is “quantised” by balancing energy F_k, a function on B_k.

I will explain a result in this vein, namely that the Hessian D of E, a 4th order self-adjoint elliptic operator on functions, is quantised by the Hessians P_k of the F_k, operators on the space of Hermitian endomorphisms of H^0 (X,L^k) defined purely in terms of projective embeddings. In particular, the eigenvalues and eigenspaces of P_k converge to those of D. I will explain applications of this result as well as aspects of its proof.

This talk is part of the Workshop on Kahler Geometry series.

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