Singularity spectrum

The Hodge filtration $F$ on $\mathcal{G}_0$ is defined by $F_k:=F_k\mathcal{G}_0:=\partial_t^k\mathcal{H}''$. The singularity spectrum $\mathrm{Sp}:\mathbf{Q}\to \mathbf{N}$, defined by

$$\mathrm{Sp}(\alpha)\,:=\,\dim_\mathbf{C}Gr_V^\alpha Gr^F_0\mathcal{G}_0$$

reflects the embedding of $\mathcal{H}''_0$ in $\mathcal{G}_0$ and satisfies the symmetry relation $\mathrm{Sp}(\alpha)=\mathrm{Sp}(n-1-\alpha)$.

Since $\mathcal{H}''_0\subset V^{>-1}$, this implies $V^{>-1}\supset\mathcal{H}''_0\supset V^{n-1}$ or, equivalently, $\mathrm{Sp}(\alpha)=0$ for $\alpha\le-1$ or $\alpha\ge n$.