Stein neighborhood bases for complex--totally real unions

RESUMO

We study Stein neighborhood bases for unions of closed complex and totally real submanifolds, with particular attention to a linear configuration with noncompact intersection. We prove that the set $X=\{w=0\}\cup\{\Im z=\Im w=0\}\subset\C^2$ admits a Stein neighborhood basis. The proof starts from the explicit plurisubharmonic defining function $\rho(z,w)=(\Im w)^2+\sinh^2(\Im z)|w|^2$ and obtains the necessary non-uniform control at infinity by adding a locally uniformly convergent series of nonnegative plurisubharmonic terms constructed by Oka--Weil approximation. We also give an example of a closed complex submanifold and a closed totally real submanifold in $\C^2$ with compact intersection whose union has no Stein neighborhood basis, showing that the unrestricted union problem has a negative answer.