  In the present note, we introduce the $\xi-$complex singularity exponents,
which come from the asymptotic property of $\xi-$Bergman kernels on sub-level
sets of plurisubharmonic functions; give some relations (including a closedness
property) among $\xi-$complex singularity exponents, complex singularity
exponents, and jumping numbers; generalize some properties of complex
singularity exponents (such as the restriction formula and subadditivity
property) to $\xi-$complex singularity exponents.
