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Joel Fine (Université Libre de Bruxelles)
Thursday 12 April 2012, 14:30-15:30
Quantisation and the Hessian of Mabuchi energy
- 👤 Speaker: Joel Fine (Université Libre de Bruxelles)
- 📅 Date & Time: Thursday 12 April 2012, 14:30 - 15:30
- 📍 Venue: MR2
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.
Series This talk is part of the Workshop on Kahler Geometry series.
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