Spectral and Logarithmic Atiyah Classes for Higgs Bundles

Technical Summary: Spectral and Logarithmic Atiyah Classes for Higgs Bundles

Problem Statement
The paper addresses the relationship between the Atiyah class of a Higgs bundle $(E, \theta)$ and the Atiyah class of the underlying spectral line bundle $L$ defined on the spectral curve $Y$. In the context of the Beauville–Narasimhan–Ramanan (BNR) correspondence, a regular semisimple Higgs bundle on a smooth complex projective curve $X$ is identified with a line bundle $L$ on a spectral curve $Y$ via a finite morphism $\pi: Y \to X$, such that $E \cong \pi_* L$.

A fundamental difficulty arises when comparing the Atiyah classes of these objects: they reside in different cohomology groups with distinct coefficient sheaves. Specifically, the Atiyah class of the Higgs bundle lies in $H^1(X, \Omega^1_X \otimes \text{End}(E))$, while the Atiyah class of the spectral line bundle lies in $H^1(Y, \Omega^1_Y)$. The authors seek to establish a precise bridge between these classes, determining how the obstruction to a holomorphic connection on $E$ is induced by the spectral data of $L$, particularly across the discriminant locus where the spectral cover ramifies.

Methodology
The authors employ a combination of jet bundle theory, finite morphism properties, and logarithmic differential geometry.

1. Jet Comparison via Finite Pushforward: The core technical machinery involves analyzing the first jet sequences of $E$ on $X$ and $L$ on $Y$. By constructing a canonical morphism $\Psi_{\pi, L}: J^1_X(E) \to \pi_* J^1_Y(L)$, the authors relate the jet sequences of the underlying bundle and the spectral line bundle. They utilize the exactness of the pushforward functor $\pi_*$ for finite morphisms between smooth curves and the standard exact sequence of Kähler differentials to identify the discrepancy between these jet sequences.
2. Spectral Action and Centralizers: The study restricts to the étale locus $X^\circ = X \setminus \Delta$ (where $\Delta$ is the discriminant divisor). Here, the spectral cover is unramified, allowing the identification of the centralizer of the Higgs field with the pushforward of the structure sheaf of the spectral curve.
3. Logarithmic Refinement: To extend results across the discriminant, the authors assume $\Delta$ is reduced. They construct a "regularized centralizer sheaf" $Z_\theta$ by saturating the image of the spectral algebra $\pi_* \mathcal{O}_Y$ inside $\text{End}(E)$. They then introduce logarithmic differentials $\Omega^1_X(\log \Delta)$ to handle the singularities at the branch points, constructing a canonical morphism that maps differentials on $Y$ to logarithmic differentials on $X$ tensored with the spectral algebra.

Key Contributions and Results

• The Jet Comparison Theorem (Proposition 2.1): The authors construct a canonical morphism $\Psi_{\pi, L}$ fitting into a short exact sequence:
  $$0 \to J^1_X(E) \xrightarrow{\Psi_{\pi, L}} \pi_* J^1_Y(L) \to \pi_*(\Omega^1_{Y/X} \otimes L) \to 0$$
  This sequence demonstrates that the discrepancy between the first jets of $E$ and the pushed-forward jets of $L$ is measured exactly by the ramification sheaf $\pi_*(\Omega^1_{Y/X} \otimes L)$.

• The Spectral Atiyah Class (Corollary 2.2): The authors define a "spectral Atiyah class" $\beta_{\text{spec}}(L, \pi) \in \text{Ext}^1_X(E, \pi_*(\Omega^1_Y \otimes L))$ represented by the pushed-forward jet sequence of $L$. They prove that the ordinary Atiyah class $\text{At}(E)$ maps to this spectral class via the natural coefficient morphism $\iota: \Omega^1_X \otimes E \to \pi_*(\Omega^1_Y \otimes L)$.

• Abelianization on the Étale Locus (Theorem 3.3): On the étale locus $X^\circ$, the authors show that the Atiyah class of $E$ is completely induced by the scalar Atiyah class of the spectral line bundle $L^\circ$. Specifically, the restriction $j^*\text{At}(E)$ is the image of the scalar spectral class $\beta^\circ_{\text{spec}}(L, \pi)$ under the spectral action map $id \otimes m^\circ$. This confirms that away from the discriminant, the non-abelian Atiyah class is governed entirely by the abelian spectral data.

• Logarithmic Refinement (Theorem 4.6): Assuming the discriminant divisor $\Delta$ is reduced, the authors construct a "logarithmic spectral Atiyah class":
  $$\text{At}^{\log}_\theta(E) \in H^1(X, \Omega^1_X(\log \Delta) \otimes Z_\theta)$$
  where $Z_\theta$ is the regularized centralizer sheaf. They prove a global compatibility relation:
  $$(k \otimes \text{id})^* \text{At}(E) = (id \otimes q)^* \text{At}^{\log}_\theta(E)$$
  where $k$ and $q$ are natural inclusions. This result establishes that the ordinary Atiyah class, when viewed with logarithmic poles along the discriminant, is represented by a class taking values in the regularized centralizer.

Significance
The paper provides a rigorous cohomological framework for understanding Higgs bundles through their spectral data. It clarifies that while the Atiyah class of a Higgs bundle is generally non-abelian, it is "abelianized" by the spectral correspondence on the étale locus. The primary significance lies in the extension of this abelianized description across the branch divisor. By introducing the logarithmic spectral Atiyah class with values in the regularized centralizer, the authors demonstrate that the failure of the spectral description at the level of ordinary jets is controlled and can be reformulated using logarithmic geometry. This offers a precise mechanism for tracking the obstruction to holomorphic connections in Higgs bundles via the geometry of the spectral curve, even in the presence of ramification.