Now we pass on to Kahler geometry. We study the interaction between two structures on an underlying manifold $M$: a complex structure and a symplectic form. We require these to be algebraically compatible in the sense that the symplectic form is the imaginary part of a hermitian metric. As before there are two points of view we can take. In the first—the conventional point of view in complex differential geometry—we fix the complex structure and vary the Kahler form. If we choose a reference form $\omega_{0}$ and vary in the fixed cohomology class then (at least when $M$ is compact) any other form can be represented by a Kahler potential, in the shape

$$\omega_{\psi}=\omega_{0}+i\partial\overline{\partial}\psi.$$

For the alternative point of view we fix a symplectic form $\omega$ and consider the space ${\cal J}$ of algebraically-compatible almost-complex structures on $M$. Then the group ${\rm SDiff}$ of symplectomorphisms of $(M,\omega)$ acts on ${\cal J}$, and this is the analogue of the unitary gauge group $U(E)$ in the previous case. We consider the subset ${\cal J}_{{\rm int}}$ of integrable almost complex structures, which is preserved by ${\rm SDiff}$. This is partitioned into equivalence classes under the relation $J_{1}\sim J_{2}$ if $(M,J_{1}),(M,J_{2})$ are isomorphic as complex manifolds. Although the group ${\rm SDiff}$ does not have a true complexification one can argue that the equivalence classes in ${\cal J}_{{\rm int}}$ are formally the orbits of such a (mythical) complexified group, in the sense that they behave that way at the level of tangent spaces and Lie algebras [8].