Minimal Energy Local Systems on Curves

Technical Summary: Minimal Energy Local Systems on Curves

Problem Statement
The paper investigates the structure of relative character varieties $\chi(\pi_1(\Sigma_{g,n}), G, \mathcal{C})$, where $\Sigma_{g,n}$ is an orientable surface of genus $g$ with $n$ punctures, $G$ is a reductive algebraic group (specifically real forms of $GL_r(\mathbb{C})$), and $\mathcal{C}$ is a tuple of conjugacy classes representing local monodromy around the punctures.

The central problem is the existence and characterization of specific compact components within these character varieties. Previous work by Deroin and Tholozan identified "supra-maximal" representations for $G=PSL_2(\mathbb{R})$ (or $SL_2(\mathbb{R})$) on spheres ($g=0$) that form compact components. This paper seeks to generalize these findings to higher rank ($r > 2$) and arbitrary genus, identifying a class of representations called "minimal energy local systems" that form compact components of relative character varieties for groups like $U(p,q)$ and $SU(p,q)$.

Methodology
The author employs non-abelian Hodge theory, establishing a correspondence between:

1. Semisimple local systems on $X \setminus D$ (where $X$ is a smooth proper curve and $D$ is the divisor of punctures).
2. Poly-stable parabolic Higgs bundles $(E_\bullet, \theta)$ on $X$ with parabolic divisor $D$.

The methodology proceeds through the following steps:

• Moduli Space Analysis: The study focuses on the coarse moduli space $M$ of semi-stable, strongly parabolic Higgs bundles of rank $r$ and parabolic degree 0. The "strongly parabolic" condition ensures that the local monodromy at punctures remains unitary under deformation.
• Minimal Energy Definition: A local system (or its corresponding Higgs bundle) is defined as minimal energy if the Higgs cohomology group $H^1(\text{ad}(E_\bullet) \to \overline{\text{ad}(E_\bullet)} \otimes \Omega^1_X(\log D))$ has Hodge length 1. This condition is equivalent to the tangent space of the moduli space at that point having a specific decomposition under the $\mathbb{C}^\times$-action (induced by scaling the Higgs field), where only the $p=0$ and $p=1$ components of the Hodge structure are non-zero.
• Bialynicki-Birula Stratification: The proof of existence relies on the $\mathbb{C}^\times$-action on the moduli space $M$. Using the Hitchin map (which is proper) and Bialynicki-Birula theory, the author analyzes the fixed-point locus (which corresponds to Complex Variations of Hodge Structure, or C-VHS). The minimal energy systems are identified as the components of the C-VHS locus with the maximal dimension.
• Genus-Specific Analysis:
  • For genus $g > 0$, the author uses stability arguments and Riemann-Roch to show that minimal energy local systems must correspond to unitary representations (i.e., the Higgs field $\theta$ vanishes).
  • For genus $g = 0$, the author derives explicit bounds on the number of punctures $n$ relative to the rank $r$ to constrain the complexity of the Hodge filtration.

Key Contributions and Results

1. Existence of Minimal Energy Systems (Theorem 1.2):
   If the relative character variety $\chi(\pi_1(X \setminus D), GL_r(\mathbb{C}), \mathcal{C})$ is non-empty and smooth (guaranteed by generic choices of conjugacy classes), then every connected component of this variety contains a minimal energy local system. These systems underlie polarizable complex variations of Hodge structure (C-VHS) for any choice of complex structure on the curve.

2. Characterization of Compact Components (Proposition 4.19 & 4.20):

   • Genus $g > 0$: Minimal energy local systems are necessarily unitary representations. Consequently, they form compact components of the character varieties of compact groups (or specific real forms).
   • Genus $g = 0$: Minimal energy local systems are not necessarily unitary. However, they still form compact components of the relative character varieties for real forms $U(p,q)$ and $SU(p,q)$ (where $p+q=r$). The author proves these components are compact by showing they lie within the fiber $h^{-1}(0)$ of the Hitchin map, which is compact.

3. Constraints on Hodge Filtration (Theorem 1.3):
   For the case of the projective line ($g=0$), the paper establishes a bound on the complexity of the Hodge filtration for minimal energy systems. Specifically, if the number of punctures $n$ satisfies $n > 5r^2 - 25$, any irreducible minimal energy local system of rank $r$ arises from a C-VHS with at most two steps in its Hodge filtration.

   • The paper notes that for $r \le 2$, this condition is vacuous. For $r \ge 3$, it imposes a non-trivial restriction.
   • Example 6.2 demonstrates that this bound is sharp in the sense that minimal energy systems with exactly two steps in the Hodge filtration exist for arbitrarily large $n$.

4. Applications to Gromov-Witten Theory (Section 6.3):
   The paper connects minimal energy local systems to Gromov-Witten invariants of Grassmannians. By constructing minimal energy systems with two graded pieces, the author derives non-vanishing conditions for specific Gromov-Witten invariants $\langle \sigma_{I_1}, \dots, \sigma_{I_n} \rangle_d$. This provides a geometric interpretation of the existence of these local systems in terms of rational curves intersecting Schubert varieties.

Significance and Claims
The paper claims to provide a new interpretation of the compact components found in previous works by Deroin-Tholozan, Tholozan-Toulisse, and Feng-Zhang.

• Unification: It unifies these disparate results under the single framework of "minimal energy" local systems, which are defined via a Hodge-theoretic vanishing condition (Hodge length 1) rather than by explicitly constructing Higgs bundles for specific groups.
• Generality: Unlike previous methods that required explicit construction of Higgs bundles for specific Lie groups (e.g., $SU(p,q)$, $Sp(2,\mathbb{R})$), this approach uses deformation-theoretic techniques in the moduli space of parabolic Higgs bundles. This allows the results to apply to $GL_r(\mathbb{C})$ and its real forms without needing to write down the Higgs bundles explicitly.
• Universality: The minimal energy local systems are shown to be "universal" in the sense that they underlie C-VHS for any choice of complex structure on the surface, not just a specific one.

The author explicitly states that the paper does not claim to find all compact components, but rather identifies a specific, robust class of them (the minimal energy ones) that exist under generic conditions and possess strong geometric properties (compactness, universality, and specific Hodge filtration bounds). The bound $n > 5r^2 - 25$ is acknowledged to be far from sharp, serving as a sufficient condition rather than a necessary one.