Monge-Ampère equation, hyperkähler structure and adapted complex structure

Abstract

In the tangent bundle of $(M,g)$, it is well-known that the Monge-Ampère 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: $$\rho (x+iy)=\sum _i{y}_i^2-\frac{1}{3}\sum _{pqij}{R}_{ipjq}(0)x_p{x}_q{y}_i{y}_j+O(5).$$ At $M$, sectional curvatures of the Kähler metric $2i\partial \bar{\partial }\rho$ can be computed. This has enabled us to find a family of Kähler manifolds whose tangent bundles have admitted complete hyperkähler 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 ${\dot{T}}_1+[T_0,T_1]=0.$

21 pages