Definition 6.7.4D9ACE2E4ECE5

Definition 6.7.4.D9ACE2E4ECE5 The number $\nu (u,a)$ in (6.7.2) is the Lelong number of $u$ at $a$. If $u$ is plurisubharmonic on a complex manifold $X$ that has positive dimension at $a$, we compute $\nu (u,a)$ in a holomorphic chart at $a$; by proposition 6.7.1 the result does not depend on the chart. If $X$ has dimension zero at $a$, where plurisubharmonic functions are finite, we set $\nu (u,a)=0$. For a closed positive $(1,1)$-current $T$ on $X$ we put $\nu (T,a)=\nu (u,a)$, where $u$ is a plurisubharmonic potential of $T$ near $a$, that is, $\mathrm{dd}^{\mathrm{c}}u=T$. Such potentials exist, and two of them differ near $a$ by a smooth pluriharmonic function: by theorem 5.9.1, the plurisubharmonic potentials of $T$ on a coordinate ball are exactly its canonical local potentials, to which theorem 5.4.3 applies. Since a smooth function is bounded near $a$, $\nu (T,a)$ does not depend on the choice of the potential.