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

Xingyu Cheng

TL;DR

This work addresses the existence of semistable parabolic systems of Hodge bundles on a punctured projective line, corresponding to complex variations of Hodge structure and fixed by a $\mathbb{C}^\times$ action. It develops a framework that translates stability questions into generalized Gromov–Witten (GW) counting problems on Grassmannians, using parabolic shifting to relate nongeneric configurations to GW invariants and to produce explicit numerical criteria. The main results provide GW-based inequalities that depend on degrees, parabolic weights, and the parabolic degree of the total bundle, with detailed treatments of the $(1,n)$, $(1,2)$, and $(1,1)$ cases, as well as the limiting $(1,\ldots,1)$ scenario; these extend unitary criteria to VHS contexts and connect to tropical geometry in special instances. Overall, the paper offers a concrete enumerative approach to VHS-related stability questions, enabling precise solvability criteria for the existence of semistable parabolic systems of Hodge bundles and their associated local systems with semisimple monodromy.

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 $(\mathcal E, θ)$, where $\mathcal E = \mathcal L \oplus \mathcal V$ and $θ(\mathcal L) \subset \mathcal V \otimes Ω_{\mathbb P^1}^1 (\log S)$, 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.

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

TL;DR

This work addresses the existence of semistable parabolic systems of Hodge bundles on a punctured projective line, corresponding to complex variations of Hodge structure and fixed by a action. It develops a framework that translates stability questions into generalized Gromov–Witten (GW) counting problems on Grassmannians, using parabolic shifting to relate nongeneric configurations to GW invariants and to produce explicit numerical criteria. The main results provide GW-based inequalities that depend on degrees, parabolic weights, and the parabolic degree of the total bundle, with detailed treatments of the , , and cases, as well as the limiting scenario; these extend unitary criteria to VHS contexts and connect to tropical geometry in special instances. Overall, the paper offers a concrete enumerative approach to VHS-related stability questions, enabling precise solvability criteria for the existence of semistable parabolic systems of Hodge bundles and their associated local systems with semisimple monodromy.

Abstract

We consider the problem of existence of semistable systems of Hodge bundles with parabolic structure over a finite set of type . That is, we consider parabolic Higgs bundles , where and , where and . Such systems of Hodge bundles are -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.

Paper Structure

This paper contains 36 sections, 23 theorems, 173 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Deligne-Simpson problem
  3. The unitary existence problem
  4. Preliminary notation
  5. Existence of semistable parabolic bundles
  6. The existence problem for variation of Hodge structures
  7. The (1,n) problem
  8. Generalized Gromov-Witten numbers
  9. The main (1,n) theorem
  10. Acknowledgment
  11. Basic notions
  12. Parabolic vector bundles
  13. Parabolic Higgs bundles and Simpson's correspondence
  14. Variation of Hodge Structures
  15. Strongly parabolic Higgs bundles
  16. ...and 21 more sections

Key Result

Theorem 1

A semistable parabolic bundle of parabolic degree zero and weights $\alpha_1 (p) \leq \alpha_2 (p) \leq \ldots \leq \alpha_n (p) < \alpha_1 (p) + 1$ for each $p \in S$ exists if and only if the following condition holds:

Figures (1)

  • Figure 1: Picture of monodromy

Theorems & Definitions (58)

  • Definition
  • Definition
  • Theorem : The unitary existence theorem
  • Definition 1
  • Theorem 1: Existence of semistable $(1,n)$ parabolic system of Hodge bundles with distinct weights
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 48 more