Borcherds's Lie algebra and $C_2$-cofiniteness of vertex operator algebras of moonshine type

We precisely determined an $\bN$-graded structure of Zhu's poisson algebra $V/C_2(V)$ of vertex operator algebras $V$ of moonshine type. Namely, if $V$ is a vertex operator algebra of moonshine type with a central charge $24$, then $C_2(V)=\sum_{n=5}^{\infty}V_n+L(-1)V$.