Let \(V\) be a compact complex manifold of dimension \(n\) which admits a Kähler metric \(\omega_0\). Any other Kähler metric cohomologous to \(\omega_0\) can be expressed via a Kähler potential. Define \(\mathcal H\) to be the space of Kähler potentials, \({\mathcal H}=\{\phi\in C^{\infty}(V):\omega_{\phi}=\omega_0+i\overline{\partial}\partial\phi>0\}\). Each Kähler potential \(\phi\in\mathcal H\) gives a measure \(d\mu_{\phi}=\frac{1}{n!}\omega^n_{\phi}\) on \(V\). Define a Riemannian metric on the infinite dimensional manifold \(\mathcal H\) using the \(L^2\)-norms furnished by theses measures (a tangent vector \(\delta\phi\) to \(\mathcal H\) at a point \(\phi\in\mathcal H\) is just a function on \(V\)) and set \(\|\delta\phi\|^2_{\phi}=\int_V (\delta\phi)^2 d\mu_{\phi}\). This is the main result of the paper: The Riemannian manifold \(\mathcal H\) is an infinite dimensional symmetric space; it admits a Levi-Civita connection whose curvature is covariant constant. At a point \(\phi\in\mathcal H\) the curvature is given by \(R_{\phi}(\delta_1\phi,\delta_2\phi)\delta_3\phi=-\frac 14\{\{\delta_1\phi,\delta_2\phi\}_{\phi},\delta_3\phi\}_{\phi}\), where \(\{.,.\}_{\phi}\) is the Poisson bracket on \(C^{\infty}(V)\) of the symplectic form \(\omega_{\phi}\). The author presents two proofs of this theorem. Moreover, it is investigated the geometry of the space \(\mathcal H\), in particular the geodesic equation. A number of conjectures and questions is stated and it is outlined the relevance of these ideas to well-established problems in Kähler geometry.
The theory is an infinite dimensional analogy of the ordinary symmetric spaces of finite dimension.
For the entire collection see [Zbl 0930.00050].

Reviewer: Zbigniew Olszak (Wrocław)