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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

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

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$.

Christoph Lossen