Quantisation and the Hessian of Mabuchi energy (Joel Fine)
Duration: 58 mins 15 secs
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Description: |
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Created: | 2012-04-24 10:12 | ||
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Collection: | Workshop on Kahler Geometry | ||
Publisher: | University of Cambridge | ||
Copyright: | Dr J. Ross | ||
Language: | eng (English) | ||
Distribution: | World (downloadable) | ||
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Explicit content: | No | ||
Aspect Ratio: | 16:9 | ||
Screencast: | No | ||
Bumper: | UCS Default | ||
Trailer: | UCS Default |
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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