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 $T^{\ast }\mathbb{C}\mathbb{P}(1)$ of the complex projective line $\mathbb{C}\mathbb{P}(1)\simeq {\mathbb{S}}^2$, which extends the Fubini-Study metric of $\mathbb{C}\mathbb{P}(1)$. By virtue of the Mostow decomposition theorem, $T^{\ast }\mathbb{C}\mathbb{P}(1)$ is isomorphic, as $SU(2)$-equivariant fiber bundle over $\mathbb{C}\mathbb{P}(1)$, to a complex (co-)adjoint orbit of $SL(2,\mathbb{C})$. In fact, this complex (co-)adjoint orbit is fibered over $\mathbb{C}\mathbb{P}(1)\simeq {\mathbb{S}}^2$ 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 hyperkähler 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 hyperkähler manifold $T^{\ast }\mathbb{C}\mathbb{P}(1)$ is not holomorphic.