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The aim of this paper is twofold. First we give an explicit construction of the in¯nitesimal deformations of the category Coh(X) of coherent sheaves on a smooth projective variety X. Secondly,we show that any Fourier-Mukai transform ©: Db(X) ! Db(Y ) extends to an equivalence between the derived categories of the deformed Abelian categories.

We give an elementary proof of the existence of an asymptotic expansion in powers of$k$of the Bergman kernel associated to$L$^{$k$}, where$L$is a positive line bundle over a compact complex manifold. We also give an algorithm for computing the coefficients in the expansion.

In the context of recently proposed holographic dualities between higher spin theories in AdS3 and 1+1-dimensional CFTs with W-symmetry algebras, we revisit the definition of higher spin black hole thermodynamics and the dictionary between bulk fields and dual CFT operators. We build a canonical formalism based on three ingredients: a gauge-invariant definition of conserved charges and chemical potentials in the presence of higher spin black holes, a canonical definition of entropy in the bulk, and a bulk-to-boundary dictionary aligned with the asymptotic symmetry algebra. We show that our canonical formalism shares the same formal structure as the so-called holomorphic formalism, but differs in the definition of charges and chemical potentials and in the bulk-to-boundary dictionary. Most importantly, we show that it admits a consistent CFT interpretation. We discuss the spin-2 and spin-3 cases in detail and generalize our construction to theories based on the hs[\lambda] algebra, and on the sl(N,R) algebra for any choice of sl(2,R) embedding.