A counterexample to Bruzzo's curve semistability conjecture for Higgs bundles

Technical Summary: A Counterexample to Bruzzo's Curve Semistability Conjecture for Higgs Bundles

Problem Statement
The paper addresses Bruzzo's curve semistability conjecture for Higgs bundles on smooth complex projective varieties. The conjecture posits the equivalence of two conditions for a Higgs bundle $E = (E, \theta)$ of rank $r$:

1. $E$ is semistable with respect to some polarization and its discriminant vanishes ($\Delta(E) = c_2(E) - \frac{r-1}{2r}c_1(E)^2 = 0$).
2. For every smooth projective curve $C$ and every morphism $f: C \to X$, the pullback Higgs bundle $f^*E$ is semistable.

While the implication $(1) \implies (2)$ is established in the literature, the converse $(2) \implies (1)$ remains the subject of investigation. Previous results confirmed the converse for rank two bundles and under specific assumptions, but the general case for higher ranks was an open problem. This paper aims to determine whether the converse holds in rank four.

Methodology and Construction
The author constructs a specific counterexample in rank four using the geometry of symmetric products of plane curves. The construction proceeds as follows:

1. Geometric Setup: Let $\Sigma \subset \mathbb{P}^2$ be a very general smooth plane quintic curve. Let $X = \Sigma(2)$ be the second symmetric product of $\Sigma$. The author utilizes the tautological rank two bundle $F = \mathcal{O}_\Sigma(1)[2]$ on $X$.
2. Higgs Field Definition: Using the isomorphism $S^2 \cong K_\Sigma$ (where $S = \mathcal{O}_\Sigma(1)$), the author defines an $\Omega^1_X$-valued symmetric form $\beta: \text{Sym}^2(F) \to \Omega^1_X$. This induces an adjoint morphism $\psi: F \to F^* \otimes \Omega^1_X$.
3. Extension Construction: A section $s \in H^0(\Sigma, S)$ is chosen to define an extension class $e_s \in \text{Ext}^1_X(F, F^*)$. This yields a rank four vector bundle $E_s$ fitting into the exact sequence:
   $$0 \to F^* \to E_s \to F \to 0$$
   The Higgs field $\theta_s: E_s \to E_s \otimes \Omega^1_X$ is constructed as the composite of the projection $E_s \to F$, the map $\psi$, and the inclusion $F^* \hookrightarrow E_s$.
4. Chern Class Calculation: The author computes the Chern classes of $E_s$. They show that $\det(E_s) \cong \mathcal{O}_X$ and $\int_X c_2(E_s) = 10$. Consequently, the discriminant $\Delta(E_s)$ is non-zero in $H^4(X, \mathbb{Q})$.

Key Technical Analysis
To prove that $E_s$ satisfies condition (2) despite having a non-vanishing discriminant, the author analyzes the pullback $f^*E_s$ for an arbitrary morphism $f: C \to X$. The stability of the pullback depends on the existence of destabilizing Higgs subsheaves.

• Criterion for Subsheaves: A Higgs subsheaf $H \subset f^*E_s$ corresponds to subsheaves $I \subset V$ and $J \subset V^*$ (where $V = f^*F$) satisfying a compatibility condition with the contracted morphism $u_f = f^*\psi$ and a vanishing condition on a specific extension class (the restriction pushout).
• Classification of Curves: The author classifies the morphisms $f$ based on the generic rank of $u_f$:
  • Generic Rank Two: If $u_f$ has generic rank two, the author shows that any potential destabilizing subsheaf leads to a contradiction involving the simplicity of the Jacobian $J(\Sigma)$ and the non-vanishing of the section $s$.
  • Generic Rank One: The root divisor of $\beta$ is isomorphic to $\Sigma \times \Sigma$. If $u_f$ has generic rank one, the image $f(C)$ must be either a "coordinate curve" (image of $\Sigma$ under $q \mapsto p+q$) or the diagonal.
    • For coordinate curves, direct degree estimates on the kernel and image of the contracted morphism prevent destabilization.
    • For the diagonal, the restriction of the extension class corresponds to a rank two bundle $P_s$. The author selects the section $s$ such that $P_s$ is semistable of degree zero. Using the Narasimhan–Seshadri correspondence, they prove that pullbacks of $P_s$ remain semistable, thereby preventing destabilization of $f^*E_s$.

Results
The main result, Theorem 1.2, establishes the existence of a smooth complex projective surface $X$ and a rank four Higgs bundle $E_s$ such that:

1. For every smooth projective curve $C$ and every morphism $f: C \to X$, the pullback Higgs bundle $f^*E_s$ is semistable.
2. The discriminant of $E_s$ is non-zero ($\Delta(E_s) \neq 0$).

Corollary 4.7 explicitly states that this result disproves Bruzzo's curve semistability conjecture for Higgs bundles in rank four.

Significance and Claims
The paper claims to provide the first counterexample to the general implication $(2) \implies (1)$ in the context of Higgs bundles. The author notes that while the converse is known for rank two, their construction demonstrates its failure in rank four.

The work clarifies the limitations of previous arguments relying on "Higgs numerical flatness" and base change compatibility for absolute Higgs Grassmannians. The author points out that these compatibility arguments fail for pullback Higgs bundles because the contraction by the differential $df$ can produce new invariant quotients that were not present in the original bundle. Consequently, the verification of curve semistability requires a direct, case-by-case analysis of the pullback bundles, which the author performs using the specific geometry of the plane quintic and its symmetric product.

The paper does not propose future applications or experimental proposals but focuses strictly on resolving the open conjecture by exhibiting a concrete counterexample. The author acknowledges that OpenAI's ChatGPT (version 5.6 Plus) assisted in performing repeated calculations and testing a wide range of examples, a process that eventually led to the counterexample constructed in the paper.