The trace operator of quasi-plurisubharmonic functions on compact K\"ahler manifolds

We introduce the trace operator for quasi-plurisubharmonic functions on compact K\"ahler manifolds, allowing to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish. Using this construction we obtain novel Ohsawa--Takegoshi extension theorems and give applications to restricted volumes of big line bundles.