A note on $\xi -$Bergman kernels

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.