Stability of parabolic systems of Hodge bundles over punctured ${\mathbb{P}}^1$

We consider the problem of existence of semistable systems of Hodge bundles with parabolic structure over a finite set $S\subset {\mathbb{P}}^1$ of type $(1,n)$. That is, we consider parabolic Higgs bundles $(\mathcal{E},\theta )$, where $\mathcal{E}=\mathcal{L}\oplus \mathcal{V}$ and $\theta (\mathcal{L})\subset \mathcal{V}\otimes {\mathrm{\Omega }}_{{\mathbb{P}}^1}^1(\log S)$, where $\mathrm{rank}\mathcal{L}=1$ and $\mathrm{rank}\mathcal{V}=n$. Such systems of Hodge bundles are ${\mathbb{C}}^{\times }$-fixed points in the space of all such (parabolic) Higgs bundles and these correspond to local systems coming from complex variations of Hodge structure under Simpson’s correspondence. In the spirit of Agnihotri-Woodward and Belkale, we use enumerative geometry to give numerical criteria for the existence of such semistable parabolic systems of Hodge bundles with semisimple local monodromy.