Calabi–Yau (p+1)(p+1)-folds from pp-folds

We establish the general formalism for constructing metrics of Calabi–Yau (p+1)(p+1)-folds in terms of that of a pp-fold by adding a complex-line bundle. We present a few explicit low-lying examples. We further consider holomorphic linearization and obtain the six-dimensional analogue of the Gibbons–Hawking instanton. Whilst the Kähler potential for the Gibbons–Hawking instanton is given by the harmonic function of a three-dimensional flat space, for the generalized solution it is related to the harmonic functions of certain three-dimensional non-flat spaces that are direct products of RR and two-dimensional Kähler spaces.