This is primarily a linear algebra question, but for motivation I want to state this question in its natural, global context. Whenever we have a non-relativistic quantum field theory (renormalized, of course), we can take the classical phase space with polarization as a high-dimensional Kahler manifold $K$ (A priori, one would get $K=\mathbb C^N$, but physical considerations can yield more interesting parameter spaces, e.g. modding-out by gauge transformations). 


I am interested in seeing the local picture in all of this: take the classical field Hamiltonian $H$, which is a real-valued $C^\infty$-function on $K$. The canonical quantization of the Hessian of this function at a point $x\in K$ (roughly) represents the mean-field approximation for the statistical dynamics of a field configuration localized near $x\in K$. How do I provide the following?:


1. A proof that $d^2H$ is pointwise diagonalizable by a symplectic transformation.
2. An algorithm/formula for said symplectic diagonalization.
3. A formula for the resulting change in complex structure on the tangent space.


Any help would be appreciated, including references (Kahler geometry is new to me).



Hessians on Kahler Manifolds
Williamson's theorem explains the normal forms of matrices under symplectic linear transformation. <http://www.ime.usp.br/~piccione/Downloads/LecturesIME.pdf> p. 23


