zbMATH — the first resource for mathematics

Hyperkähler metrics on cotangent bundles of Hermitian symmetric spaces. (English) Zbl 0879.53051
Andersen, Jørgen Ellegaard (ed.) et al., Geometry and physics. Proceedings of the conference at Aarhus University, Aarhus, Denmark, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 184, 287-298 (1997).
Let \(\Sigma\) be a Hermitian symmetric space of compact type. It is proved that there is a unique invariant hyperkähler metric \(g\) on \(T^*\Sigma\) such that the restriction of \(g\) to the zero section is the given metric on \(\Sigma\). A formula for the Kähler form of \(g\) is also found expressing it as the sum of the pullback of the Kähler form of \(\Sigma\) and a term given by an explicit potential. In the case \(M = \mathbb{C} P^n\) this formula is contained in E. Calabi [Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 269-294 (1979; Zbl 0431.53056)]. For a Hermitian symmetric space of non-compact type, similar results are true, but \(g\) is defined in a neighbourhood of the zero section only. Also the case of a flat torus is investigated.
For a general Hermitian symmetric space \(\Sigma\), one proves that any invariant hyperkähler metric, defined on a neighbourhood of the zero section in \(T^*\Sigma\) and restricting to the given metric on \(\Sigma\), is the product of the corresponding metrics on the flat, the compact and the non-compact factors of \(\Sigma\) .
For the entire collection see [Zbl 0855.00020].

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C35 Differential geometry of symmetric spaces