The Hitchin morphism for K-trivial varieties

1. Problem Statement and Context

The Hitchin Morphism:
For a smooth proper scheme $X$ over an algebraically closed field $k$ of characteristic zero, the Hitchin morphism $h_X^r$ maps the moduli stack of Higgs bundles of rank $r$, denoted $\mathcal{M}_X^r$, to the Hitchin base $\mathcal{A}_X^r := \bigoplus_{i=1}^r H^0(X, \text{Sym}^i \Omega_X^1)$. The map sends a Higgs bundle $(E, \theta)$ to the characteristic polynomial of the Higgs field $\theta$.

The Surjectivity Issue:

• Curves: When $\dim(X)=1$, $\Omega_X^1$ is a line bundle, the integrability condition $\theta \wedge \theta = 0$ is automatic, and $h_X^r$ is known to be surjective.
• Higher Dimensions: When $\dim(X) > 1$, the integrability condition is non-trivial, and $h_X^r$ is generally not surjective.

The Chen-Ngô Conjecture:
Chen and Ngô introduced the spectral base $\mathcal{B}_X^r$, a closed subscheme of $\mathcal{A}_X^r$ (often of lower dimension), such that the Hitchin morphism factors through $\mathcal{B}_X^r$. They conjectured that the restricted morphism $\mathcal{M}_X^r \to \mathcal{B}_X^r$ is surjective (i.e., every point in the spectral base corresponds to at least one Higgs bundle).

The Specific Question:
The authors investigate for which varieties $X$ the restricted Hitchin morphism from the Dolbeault moduli space (the moduli space of Gieseker semistable Higgs bundles with the Hilbert polynomial of the trivial bundle, denoted $\mathcal{M}_{X, \text{Dol}}^r$) is surjective onto the spectral base $\mathcal{B}_X^r$. This is a stronger version of the Chen-Ngô conjecture, as it restricts the domain to bundles with vanishing Chern classes.

2. Methodology

The authors employ a geometric approach centered on the construction and analysis of spectral covers.

A. Spectral Covers and Normalization:

• They recall the construction of the spectral cover $X_s$ associated with a spectral datum $s \in \mathcal{B}_X^r$.
• Problem: The standard spectral cover $X_s$ may not be normal or Cohen-Macaulay, meaning the pushforward sheaf $\pi_{s*} \mathcal{O}_{X_s}$ may not be locally free (a vector bundle), only a coherent sheaf.
• Solution: The authors modify the construction by taking the normalization of the unique irreducible component of the spectral cover dominating $X$. Let $\nu: X'_s \to \tilde{X}_s$ be the normalization. They define the normalized spectral cover $X'_s$.
• Key Insight: By working with the normalized cover $X'_s$, they ensure the resulting pushforward sheaf $\pi'_{s*} \mathcal{O}_{X'_s}$ is a reflexive Higgs sheaf.

B. Branch Locus and Purity:

• They analyze the branch locus $Z$ of the spectral cover, which is determined by the vanishing locus of the discriminant $D(s)$ of the characteristic polynomial.
• They utilize the purity of the branch locus theorem: If $X$ is smooth and $X'_s$ is normal, the branch locus $Z'$ of the map $\pi'_s: X'_s \to X$ is either empty or has pure codimension 1.
• Strategy: If the vanishing locus of the discriminant $D(s)$ has codimension $\ge 2$, then the branch locus must be empty, implying the cover is étale.

C. Connection to the Dolbeault Moduli Space:

• If the spectral cover is étale, the vector bundle $\pi'_{s*} \mathcal{O}_{X'_s}$ is étale trivializable.
• A key lemma (Lemma 4.4) establishes that étale trivializable vector bundles on projective varieties are polystable and have vanishing Chern classes.
• Therefore, if the spectral cover is étale, the associated Higgs bundle lies in the Dolbeault moduli space $\mathcal{M}_{X, \text{Dol}}^r$.

D. The "r-small" Condition:
To ensure the discriminant vanishes only in codimension $\ge 2$, the authors introduce a geometric condition on the variety $X$. They define r-small varieties where the cotangent bundle $\Omega_X^1$ is slope-semistable and satisfies specific slope inequalities for its symmetric powers. This condition prevents non-zero global sections of symmetric differentials from vanishing along divisors (codimension 1).

3. Key Contributions and Results

1. Definition of r-small Varieties (Definition 1.4):
A smooth projective variety $X$ is r-small if:

• $\Omega_X^1$ is slope-semistable.
• $\mu(\text{Sym}^{k(k-1)}\Omega_X^1) < 1$ for all $1 \le k \le r$.
  This condition ensures that any non-zero global section of $\text{Sym}^{k(k-1)}\Omega_X^1$ has a vanishing locus of codimension $\ge 2$.

2. Surjectivity for r-small Varieties (Theorem 1.3 & Theorem 4.10):
The authors prove that if $X$ is r-small, the restricted Hitchin morphism
$$h_{X, \text{Dol}}^k: \mathcal{M}_{X, \text{Dol}}^k \to \mathcal{B}_X^k$$
is surjective for all $1 \le k \le r$.

• Mechanism: For any $s \in \mathcal{B}_X^k$, the discriminant $D(s)$ does not vanish on a divisor. By purity, the normalized spectral cover is étale. The resulting Higgs bundle is étale trivializable, hence polystable with vanishing Chern classes, placing it in $\mathcal{M}_{X, \text{Dol}}^k$.

3. Application to K-Trivial Varieties (Theorem 1.2 & Theorem 4.11):
Since the cotangent bundle of a smooth projective variety with numerically trivial canonical divisor ($K_X \equiv 0$) is known to be slope-semistable of slope 0, such varieties satisfy the r-small condition.

• Result: For any smooth projective variety with $K_X \equiv 0$ (e.g., Calabi-Yau varieties, Abelian varieties, Hyperkähler manifolds), the Hitchin morphism from the Dolbeault moduli space surjects onto the spectral base.

4. Refinement of the Chen-Ngô Conjecture:
The paper proves a stronger version of the Chen-Ngô conjecture for this class of varieties: not only does a Higgs bundle exist for every spectral datum, but one can specifically choose a polystable Higgs bundle with vanishing Chern classes.

5. Structural Insight on the Spectral Base (Proposition 4.12):
For varieties satisfying the hypothesis, the spectral base $\mathcal{B}_X^r$ is the union of the images of the Hitchin morphisms restricted to a finite set of étale trivializable vector bundles. This suggests a finite stratification of the spectral base based on the underlying étale covers.

4. Significance

• Resolution of a Major Conjecture: The paper provides a definitive proof of the Chen-Ngô conjecture (and its stronger Dolbeault variant) for a broad and geometrically significant class of varieties, specifically those with numerically trivial canonical bundles.
• Methodological Advancement: The introduction of normalized spectral covers and the rigorous application of the purity of the branch locus to higher-dimensional Higgs bundles offers a robust framework for studying the geometry of the Hitchin fibration beyond curves.
• Geometric Characterization: The work links the surjectivity of the Hitchin morphism directly to the stability properties of the cotangent bundle and the vanishing loci of symmetric differentials. It highlights that the "obstruction" to surjectivity in higher dimensions is often related to the existence of symmetric differentials vanishing on divisors.
• Future Directions: The paper raises questions about the irreducibility of the spectral base for K-trivial varieties and suggests that the spectral base might be a finite union of images of specific vector bundles, potentially offering a new perspective on the topology of these moduli spaces.

In summary, Patel and Weissmann successfully extend the theory of the Hitchin morphism to higher-dimensional K-trivial varieties by constructing normal spectral covers and proving that the resulting Higgs bundles naturally reside in the Dolbeault moduli space, thereby confirming the surjectivity of the morphism in this context.