Co-Higgs bundles of Schwarzenberger type and the determinant morphism

1. Problem Statement and Context

The Hitchin Morphism and Its Image:
In the theory of Higgs bundles on a complex algebraic curve $X$, the Hitchin morphism $H$ is a proper, surjective map to the Hitchin base $B$, with generic fibers being abelian varieties. However, when $X$ is a higher-dimensional variety (specifically $\dim X \geq 2$), the Hitchin morphism is not surjective. The conjectured image is a nonlinear closed subscheme $B^\heartsuit \subset B$. While this conjecture has been verified for vector bundles on surfaces and principal Higgs bundles for certain Lie groups, an explicit description of the image remains elusive, particularly when the twisting bundle $V$ is not a line bundle.

The Specific Challenge:
The paper focuses on co-Higgs bundles (a type of $V$-twisted Higgs bundle where $V = T_X$, the tangent bundle) on the complex projective plane $\mathbb{P}^2$. The goal is to explicitly identify the image of the determinant morphism (the nonlinear component of the Hitchin map) for trace-free co-Higgs bundles.

The Object of Study:
The author restricts the study to co-Higgs bundles modeled on Schwarzenberger bundles ($V_k^\rho$). These are rank 2 holomorphic vector bundles on $\mathbb{P}^2$ constructed via a 2:1 covering map $f_\rho: \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^2$ branched over a non-singular conic $\rho = 0$. The bundles are indexed by an integer $k \geq 0$.

The Exclusion:
The case $k=3$ is explicitly excluded because stable co-Higgs bundles on $V_3^\rho$ do not follow the structural characterization found in other cases, making a unified analysis impossible.

2. Methodology

The paper employs a combination of algebraic geometry, representation theory, and explicit cohomological calculations.

• Moduli Spaces and Stability: The author utilizes the theory of Gieseker (semi)stable sheaves to define the moduli space $M_{\mathbb{P}^2}^{ss}(V_k^\rho, T_{\mathbb{P}^2})$ of trace-free co-Higgs bundles. The analysis relies on the properties of Jordan-Hölder filtrations and the behavior of the determinant morphism on these moduli spaces.
• Structural Characterization of Fields: A core methodological step is characterizing the co-Higgs field $\phi$. For $k \neq 3$, the author establishes that any non-zero stable traceless co-Higgs bundle $\phi$ modeled on $V_k^\rho$ decomposes as a tensor product:
  $$\phi = \phi_0 \otimes C$$
  where $\phi_0 \in H^0(\text{End}_0(V_k^\rho) \otimes \mathcal{O}(1))$ and $C \in H^0(T_{\mathbb{P}^2}(-1))$.
• Determinant Computation: Using the decomposition above, the determinant is computed as:
  $$\det(\phi) = \det(\phi_0) \otimes \text{Sym}^2(C)$$
  Here, $\det(\phi_0) \in H^0(\mathcal{O}(2))$ and $\text{Sym}^2(C) \in H^0(\text{Sym}^2(T_{\mathbb{P}^2}) \otimes \mathcal{O}(-2))$.
• Algebraic Geometry of Sections: The paper performs explicit calculations on the spaces of global sections $H^0(\mathcal{O}(2))$, $H^0(T_{\mathbb{P}^2}(-1))$, and $H^0(T_{\mathbb{P}^2})$. It utilizes the Euler sequence for $T_{\mathbb{P}^2}$ and coordinate charts to prove surjectivity of specific maps and to identify the image structure.
• Quotient Spaces: The images are described as quotients of products of vector spaces by $\mathbb{C}^*$ actions, effectively parameterizing the possible determinants.

3. Key Contributions and Results

The main contribution is the explicit identification of the image of the determinant morphism for Schwarzenberger bundles $V_k^\rho$ for all $k \neq 3$. The results are presented as bijections between the image and specific algebraic varieties.

Case $k=0$ ($V_0^\rho \cong \mathcal{O} \oplus \mathcal{O}(-1)$)

• Result: The image is in bijection with the quotient space:
  $$\left( H^0(\mathcal{O}(2))^\times \times H^0(T_{\mathbb{P}^2}(-1))^\times \right) / \sim \quad \sqcup \quad \{0\}$$
  where the equivalence relation is $(q, C) \sim (\alpha^2 q, \alpha^{-1} C)$ for $\alpha \in \mathbb{C}^\times$.
• Dimension: $\dim(\text{Im}) < 9$, whereas the full target space $H^0(\text{Sym}^2(T_{\mathbb{P}^2}))$ has dimension 27. The map is not surjective.

Case $k=1$ ($V_1^\rho \cong \mathcal{O} \oplus \mathcal{O}$)

• Result: The image is more complex, involving a union of the quotient space above and the space of sections $H^0(T_{\mathbb{P}^2})$.
  $$\text{Im} \cong \left( \frac{H^0(\mathcal{O}(2))^\times \times H^0(T_{\mathbb{P}^2}(-1))^\times}{(q, C) \sim (\alpha^2 q, \alpha^{-1} C)} \sqcup \frac{H^0(T_{\mathbb{P}^2})}{\{\pm 1\}} \right) / \sim$$
  where the gluing condition identifies $q \otimes \text{Sym}^2(C)$ with $\text{Sym}^2(A)$ if they are equal.
• Dimension: $\dim(\text{Im}) < 9$. The map is not surjective.

Case $k=2$ ($V_2^\rho \cong T_{\mathbb{P}^2}$)

• Result: The image is isomorphic to the same quotient space as in the $k=0$ case:
  $$\left( H^0(\mathcal{O}(2))^\times \times H^0(T_{\mathbb{P}^2}(-1))^\times \right) / \sim \quad \sqcup \quad \{0\}$$
• Dimension: $\dim(\text{Im}) < 9$. The map is not surjective.

Case $k > 3$

• Result: The image is significantly smaller, consisting only of multiples of the branching conic $\rho$.
  $$\text{Im} \cong H^0(T_{\mathbb{P}^2}(-1)) / \{\pm 1\}$$
  Specifically, the image consists of sections of the form $\rho \otimes \text{Sym}^2(C)$.
• Dimension: $\dim(\text{Im}) \leq 3$. The map is not surjective.

4. Significance

• Explicit Image Description: This paper provides the first explicit, non-conjectural description of the image of the Hitchin morphism (specifically the determinant component) for twisted Higgs bundles where the twist is the tangent bundle (a vector bundle, not a line bundle).
• Counter-Example to Surjectivity: The results rigorously demonstrate that for co-Higgs bundles on $\mathbb{P}^2$ modeled on Schwarzenberger bundles, the determinant morphism is never surjective onto the full space of global sections $H^0(\text{Sym}^2(T_{\mathbb{P}^2}))$. The image is a proper, lower-dimensional subscheme.
• Structural Insight: The analysis reveals that the integrability condition ($\phi \wedge \phi = 0$) imposes severe constraints on the co-Higgs fields, forcing them into a tensor product structure $\phi_0 \otimes C$ for $k \neq 3$. This structural rigidity is the primary reason for the non-surjectivity.
• Moduli Theory: The work contributes to the understanding of the geometry of moduli spaces of co-Higgs bundles, showing how the underlying vector bundle structure (Schwarzenberger type) dictates the geometry of the spectral data.

In summary, Banerjee successfully overcomes the difficulty of describing the Hitchin image for higher-rank twists by fixing the underlying bundle to a well-chosen family (Schwarzenberger bundles), thereby providing a concrete, computable model for the image of the determinant morphism in a setting where general descriptions were previously unavailable.