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

TL;DR

This paper investigates the existence of semistable parabolic Hodge bundles over punctured projective lines, providing numerical criteria using enumerative geometry linked to complex variations of Hodge structures.

Contribution

It introduces new numerical criteria for the existence of semistable parabolic Hodge bundles with semisimple monodromy, extending previous geometric methods.

Findings

Numerical criteria for existence of semistable parabolic Hodge bundles.

Connection between Hodge bundles and complex variations of Hodge structure.

Application of enumerative geometry to stability conditions.

Abstract

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 $\theta (\mathcal L) \subset \mathcal V \otimes \Omega_{\mathbb P^1}^1 (\log S)$, where $\operatorname{rank} \mathcal L = 1$ and $\operatorname{rank} \mathcal V = n$. 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.