Holomorphic Lie Algebroid Connection

• Holomorphic Lie algebroid connections are geometric structures that generalize classical holomorphic connections to include Lie algebroid frameworks on complex manifolds.
• They elucidate the split/nonsplit dichotomy with cohomological obstructions, guiding the existence and classification of such connections via Atiyah sequences.
• Extending to principal and parabolic bundles, these connections offer practical insights into moduli spaces, logarithmic, and generalized complex structure applications.

A holomorphic Lie algebroid connection is a geometric structure that generalizes the classical notion of a holomorphic connection on vector bundles to the context of holomorphic Lie algebroids over complex manifolds, especially compact Riemann surfaces. Such connections provide a unifying framework for various geometric objects, including Higgs bundles, logarithmic connections, and connections on principal bundles, and play a central role in the study of moduli spaces, extensions, and cohomological invariants. The theory is characterized by the intricate interplay of algebroid cohomology, extension classes, and the geometry of the base complex manifold.

1. Holomorphic Lie Algebroids and Their Connections

A holomorphic Lie algebroid on a complex manifold $X$ consists of a holomorphic vector bundle $V \to X$, a $\mathbb{C}$-Lie bracket on its sheaf of holomorphic sections $\mathcal{O}(V)$, and an $\mathcal{O}_X$-linear anchor map $\phi: V \to T_X$ (where $T_X$ is the holomorphic tangent bundle), subject to the Leibniz identity: $[s, f t] = f [s, t] + \phi(s)(f) t,$ for all local sections $s, t$ of $V$ and holomorphic functions $f$ (Alfaya et al., 12 Jun 2025). The anchor induces a natural generalization: a holomorphic $V$-connection on a holomorphic vector bundle $E\to X$ is a first-order holomorphic differential operator

$D: E \longrightarrow E \otimes V^*$

satisfying

$D(f s) = f D(s) + s \otimes \phi^*(df),$

mirroring the Leibniz rule of the standard connection but with $V$ replacing $T_X$ (Alfaya et al., 12 Jun 2025, Biswas et al., 2024, He et al., 23 Jun 2025).

2. The Split/Nonsplit Dichotomy and Existence Criteria

A key structural invariant is whether the holomorphic Lie algebroid $(V, \phi)$ is split or nonsplit. It is said to be split if an $\mathcal{O}_X$-linear splitting $\gamma: T_X \to V$ exists with $\phi \circ \gamma = \mathrm{Id}_{T_X}$; otherwise, it is nonsplit (Alfaya et al., 12 Jun 2025, Biswas, 14 Nov 2025).

• Nonsplit case: Every holomorphic vector bundle $E$ on the base admits a holomorphic $V$-connection; the obstruction cohomology class always vanishes.
• Split case: A vector bundle $E = \bigoplus_i E_i$ admits a holomorphic $V$-connection if and only if each indecomposable summand $E_i$ has degree zero, a direct generalization of Atiyah’s criterion (Alfaya et al., 12 Jun 2025).

The main cohomological obstruction is the extension class

$\lambda_\phi \in H^1(X, \End(E) \otimes V^*)$

arising from the Atiyah-type short exact sequence: $0 \to \End(E)\otimes V^* \to \Diff^1_1(E, E\otimes V^*) \xrightarrow{\widehat{\sigma}} \mathcal{O}_X \to 0,$ which splits if and only if $E$ admits a $V$-connection (Alfaya et al., 12 Jun 2025, Biswas et al., 2024).

3. Cohomological and Geometric Framework

Holomorphic Lie algebroid extensions and their classification are governed by the cohomology and hypercohomology of the Chevalley–Eilenberg algebra of the Lie algebroid. Nonabelian extensions are classified by obstruction classes lying in

$$\mathbb{H}^3(X, \tau^{\geq 1}\Omega_{\mathscr{A}}^\bullet(\mathscr{Z}))$$

where $\mathscr{A}$ is the base algebroid, and $\mathscr{Z}$ is the center of the kernel $\mathscr{K}$ of the extension (Bruzzo et al., 2013). When the obstruction vanishes, the set of isomorphism classes of extensions is a torsor under

$$\mathbb{H}^2(X, \tau^{\geq 1}\Omega_{\mathscr{A}}^\bullet(\mathscr{Z})).$$

The Atiyah exact sequence generalizes for Lie algebroids, encoding the existence of connections and the corresponding characteristic classes. For a holomorphic principal $G$-bundle or a vector bundle, the relevant extension class in cohomology (sometimes known as the Atiyah class) dictates the existence of holomorphic Lie algebroid connections (He et al., 23 Jun 2025, Biswas et al., 2024, Tortella, 2011).

4. Principal Bundles, Reductions, and Logarithmic and Parabolic Extensions

The theory extends to principal bundles. A holomorphic Lie algebroid connection on a principal $G$-bundle $E_G$ is realized via the splitting of the sequence

$$0 \to \operatorname{ad}(E_G) \to \mathcal{A}(E_G) \to V \to 0,$$

where $\mathcal{A}(E_G)$ is constructed as a kernel relating $V$ and the classical Atiyah bundle (Bansal et al., 26 Jan 2026, He et al., 23 Jun 2025, Biswas, 14 Nov 2025). The corresponding cohomological obstructions and splitting criteria mirror those for vector bundles, with additional structure appearing when considering equivariant or parabolic cases.

A salient application is the existence of $V$-connections on reductions of structure group, e.g., Harder–Narasimhan reductions. Under infinitesimal rigidity (vanishing of $$H^0(X, \operatorname{ad}(E_G)/\operatorname{ad}(E_P))$$), connections exist on the reduction ($E_P$), and for logarithmic scenarios, Lie algebroid connections realize logarithmic connections with prescribed pole behavior (Bansal et al., 26 Jan 2026, Biswas, 14 Nov 2025, Alfaya et al., 2024).

On pointed Riemann surfaces and for parabolic bundles, one defines parabolic Lie algebroid connections satisfying residue compatibility with the parabolic structure, and the existence criterion generalizes accordingly (Alfaya et al., 2024).

5. Moduli Spaces and Classification Results

The moduli space of holomorphic Lie algebroid connections over curves admits an algebraic and geometric description. For a fixed Lie algebroid $\mathcal{L}$ and coprime rank and degree, the moduli space

$\mathcal{M}_{\mathcal{L}}(r, d)$

is a smooth, irreducible, quasi-projective variety, often realized as a torsor over the moduli of stable bundles determined by the space of $H^0(X, \End(E)\otimes L^*)$ (Biswas et al., 2022, Tortella, 2011). In certain cases, there exist smooth projective compactifications where the boundary is a divisor, and the Picard group structure and numerical effectiveness properties can be analyzed in detail.

Line bundles on such moduli spaces, absence of regular functions, and rational connectedness for fixed determinant loci are established (Biswas et al., 2022). The universality of the Atiyah sequence, and connections with moduli of $D$-modules and Higgs bundles, are drawn via the interpretation of enveloping algebras and spectral sequences (Mishra et al., 7 Aug 2025, Tortella, 2011).

6. Finsler, Chern–Finsler, and Differential-Geometric Structures

For holomorphic Lie algebroids equipped with additional geometric (e.g., Finsler) structures, one defines and studies Chern–Finsler connections and their associated curvature, torsion, and Laplace operators on the total space or prolongation of the algebroid (Ionescu, 2017, Ionescu, 2017, Ionescu et al., 2016). The existence and uniqueness of the Chern–Finsler connection is established for any complex Finsler metric, yielding a robust toolkit for intrinsic geometry on holomorphic Lie algebroids.

The coordinate expressions, invariance under holomorphic changes of frame, and the relationships between the nonlinear and linear connections carry over and extend the classical results of Finsler and Lagrange geometry to the algebroid setting.

7. Key Examples and Applications

Notable classes of examples include:

• Atiyah algebroid of a holomorphic bundle $F$: Split type, with connections related to degree-zero criteria and the moduli of flat or twisted objects (Alfaya et al., 12 Jun 2025).
• Logarithmic Lie algebroids with poles along divisors, providing a framework for logarithmic connections and parabolic structures (Alfaya et al., 2024, Bansal et al., 26 Jan 2026).
• Poisson and generalized complex structures: Lie algebroids such as $\Omega^1_X$ with Poisson bracket structure, yielding connections interpreted as generalized holomorphic structures (Tortella, 2011, Mishra et al., 7 Aug 2025).
• Nonabelian extensions, spectral sequences, and Hodge theory: Extensions and their cohomology control the deformation groupoids and Hodge structure on the corresponding cohomology (Bruzzo et al., 2013).

The unified perspective on Lie algebroid connections reveals deep connections between deformation theory, moduli of bundles, geometric representation theory, and the theory of integrable systems across complex geometry.