Quantisation and the Hessian of Mabuchi energy (Joel Fine)

Duration: 58 mins 15 secs

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Created:

2012-04-24 10:12

Collection:

Workshop on Kahler Geometry

Publisher:

University of Cambridge

Dr J. Ross

Language:

eng (English)

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Author:

Joel Fine


Abstract:	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,Lk). 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,Lk) 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.

Abstract:

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,Lk). 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,Lk) 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.

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