Hyperkaehler Marriage of the two sphere with the hyperbolic space

The Eguchi-Hanson metric is a natural metric on the total space of the cotangent bundle $T^*\mathbb{CP}(1)$ of the complex projective line $\mathbb{CP}(1) \simeq \mathbb{S}^2$, which extends the Fubini-Study metric of $\mathbb{CP}(1)$. By virtue of the Mostow decomposition theorem, $T^*\mathbb{CP}(1)$ is isomorphic, as $SU(2)$-equivariant fiber bundle over $\mathbb{CP}(1)$, to a complex (co-)adjoint orbit of $SL(2, \mathbb{C})$. In fact, this complex (co-)adjoint orbit is fibered over $\mathbb{CP}(1)\simeq \mathbb{S}^2$ with each fiber isomorphic to the hyperbolic disc $\mathbb{H}^2$. In this paper, we are interested in the complex structure inherited on the hyperbolic disc $\mathbb{H}^2$ by the hyperk\"ahler extension of the $2$-sphere. Contrary to what is generally believed, we show that it differs from the natural complex structure of $\mathbb{H}^2\subset \mathbb{C}$ inherited from its embedding in $\mathbb{C}$. In other words, the embedding of $\mathbb{H}^2$ with its Hermitian-symmetric structure into the hyperk\"ahler manifold $T^*\mathbb{CP}(1)$ is not holomorphic.