  In the tangent bundle of $(M,g)$, it is well-known that the Monge-Amp\`ere
equation $(\partial\bar\partial \sqrt\rho)^n=0$ has the asymptotic expansion $
\rho(x+iy)=\sum_{ij} g_{ij} (x) y_{i} y_{j} + O(y^4)$ near $M$. Those 4th order
terms are made explicit in this article: $$\rho(x+iy)=\sum_{i}y_{i}^2-\frac
13\sum_{pqij} R_{i p j q}(0)x_p x_q y_{i}y_{j}+O(5).$$ At $M$, sectional
curvatures of the K\"ahler metric $2i\partial\bar\partial\rho$ can be computed.
This has enabled us to find a family of K\"ahler manifolds whose tangent
bundles have admitted complete hyperk\"ahler structures whereas the adapted
complex structure can only be partially defined on the tangent bundles.
  In these cases, the study of the adapted complex structure is equivalent to
the study of some gauge transformations on the baby Nahm's equation $\dot
T_1+[T_0,T_1]=0.$
