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May 31, 2026Open Access

Stability of parabolic systems of Hodge bundles over P1

Key Points

This research aims to explore the existence of semistable systems of Hodge bundles with parabolic structure over a finite subset of P1(C).
Utilizes parabolic Higgs bundles (E, θ) where E = L ⊕ V and θ(L) ⊆ V ⊗ Ω1 P1 (log S), with ranks specified.
Employs enumerative geometry to establish numerical criteria for the existence of semistable parabolic systems of Hodge bundles.
Analyses local systems derived from complex variations of Hodge structure under Simpson’s correspondence.
Establishes criteria for the existence of semistable parabolic systems of Hodge bundles with semisimple local monodromy.
Identifies C×-fixed points within the space of parabolic Higgs bundles.
Clarifies connections between enumerative geometry and stability of Hodge structures.

Abstract

We consider the problem of existence of semistable systems of Hodge bundles with parabolic structure over a finite set S ⊂ P1(C) of type (1, n), which are parabolic Higgs bundles (E, θ), where E = L ⊕ V and θ(L) ⊂ V ⊗ Ω1 P1 (log S), where rank L = 1 and rank V = n. Such systems of Hodge bundles are C×-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.

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