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A vector field E on an F -manifold (M, •, e) is an eventual identity if it is invertible and the multiplication X * Y := X • Y • E −1 defines a new F -manifold structure on M . We give a characterization of such eventual identities, this being a problem raised by Manin [12]. We develop a duality between F -manifolds with eventual identities and we show that is compatible with the local irreducible decomposition of F -manifolds and preserves the class of Riemannian F -manifolds. We find necessary and sufficient conditions on the eventual identity which insure that harmonic Higgs bundles and DChkstructures are preserved by our duality. We use eventual identities to construct compatible pair of metrics.

We discuss the construction of toric Kähler metrics on symplectic 2n-manifolds with a hamiltonian n-torus action and present a simple derivation of the Guillemin formula for a distinguished Kähler metric on any such manifold. The results also apply to orbifolds.[9] E. Lerman and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and symplectic varieties, Trans. Amer. Math. Soc. 349 (1997), 4201-4230.

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