A criterion for parabolic vector bundles to admit a parabolic Lie algebroid connection

Abstract: Given a holomorphic Lie algebroid $(V, φ)$ on a compact connected Riemann surface $X$, we give a necessary and sufficient condition for a parabolic vector bundle on $X$, with parabolic structure over a nonzero reduced effective divisor, to admit a parabolic Lie algebroid connection for the Lie algebroid $(V, φ)$.

• The paper establishes a criterion that determines when a parabolic vector bundle admits a Lie algebroid connection based on the vanishing of the anchor map at parabolic points.
• It employs Atiyah extensions and splitting conditions of the anchor map to relate parabolic degrees of indecomposable factors to the existence of the connection.
• The work generalizes classical holomorphic and parabolic connections, offering new insights for moduli spaces, deformation theory, and applications in higher geometric structures.

Parabolic Lie Algebroid Connections on Parabolic Vector Bundles

Introduction and Background

The paper establishes a sharp criterion for the existence of parabolic Lie algebroid connections on parabolic vector bundles over a compact connected Riemann surface $X$ with a fixed parabolic divisor $S$. The notion of a parabolic Lie algebroid connection generalizes both holomorphic connections and parabolic connections by coupling the structures of parabolic vector bundles (as introduced by Mehta-Seshadri) and holomorphic Lie algebroids. The main result provides a necessary and sufficient condition relating the geometry of the anchor map of the Lie algebroid $(V,\phi)$ to the parabolic degrees of the bundle's indecomposable constituents.

A holomorphic Lie algebroid $(V, \phi)$ on $X$ is a holomorphic vector bundle $V$ equipped with both a holomorphic Lie algebra structure on its sheaf of sections and an anchor map $\phi : V \to TX$ satisfying a standard Leibniz compatibility. A parabolic structure on a vector bundle $E$ over $X$ is a flag filtration of fibers at each point $x\in S$, together with weights in $S$0, endowing $S$1 with extra structure at the parabolic points.

Given these, a parabolic Lie algebroid connection on a parabolic vector bundle $S$2 is a holomorphic differential operator $S$3 that preserves the parabolic filtration in a manner compatible with the weights and the Lie algebroid structure. This generalizes several classes of connections, including parabolic connections and (strongly) parabolic Higgs bundles for suitable choices of $S$4.

Structural Results and Main Theorem

A central observation is that the potential existence of a parabolic Lie algebroid connection imposes local constraints on the anchor map $S$5. Specifically, for such a connection to exist, the anchor map must vanish at every parabolic point: $S$6 for all $S$7. This allows the anchor to factor through $S$8, leading to the categorization of Lie algebroids as logarithmically split (i.e., admitting a splitting $S$9 such that $(V,\phi)$0) or logarithmically non-split depending on whether such a splitting exists.

The main result (Corollary 2 / Theorem \ref{cori}) gives a precise dichotomy:

A parabolic vector bundle $(V,\phi)$1 on $(V,\phi)$2 admits a parabolic Lie algebroid connection for $(V,\phi)$3 if and only if:

1. $(V,\phi)$4 is logarithmically non-split, or
2. the parabolic degree of every indecomposable factor of $(V,\phi)$5 is zero.

This theorem rests on cohomological calculations involving Atiyah extensions for parabolic bundles and a careful analysis of the obstructions to splitting the associated sequence. The vanishing of the obstruction class in the non-split case is proved for any parabolic bundle $(V,\phi)$6, while in the logarithmically split case, vanishing is equivalent to a condition on parabolic degrees—reproducing a well-known criterion in the theory of parabolic connections.

A strong claim, substantiated in the paper, is that for logarithmically non-split Lie algebroids, every parabolic vector bundle (of arbitrary degree or parabolic data) over $(V,\phi)$7 admits a parabolic Lie algebroid connection.

Flat Connections and Integrability

The paper also addresses the existence of flat (integrable) parabolic Lie algebroid connections, which are connections with vanishing curvature $(V,\phi)$8. For line bundle cases, every such connection is flat by default. For higher rank, the integrability problem reduces to the existence of the underlying (quasi-)parabolic connection, subject to similar constraints. Specifically, if the image of $(V,\phi)$9 is not $(V, \phi)$0, or the parabolic degrees vanish in the split case, then there exist flat connections of the desired type.

A corollary generalizes classical integrability results to this extended setting, showing that, regardless of indecomposability or the specific Lie algebroid, one can always construct a flat quasi-parabolic Lie algebroid connection. Existence of a flat, fully parabolic connection depends on the parabolic degrees, the image of the anchor, and whether the Lie algebroid is split.

Implications and Theoretical Significance

This work generalizes and clarifies the structure of parabolic connections in the presence of additional Lie algebroid symmetry, extending well beyond the field of previous results which were mostly limited to stable $(V, \phi)$1 and did not treat the logarithmic splitting paradigm. The criterion provides a sharp dividing line: either obstructions vanish for all $(V, \phi)$2 (non-split case) or vanish precisely when the global numerical parabolic degree condition holds (split case). This dichotomy has substantial implications for the study of moduli spaces of parabolic bundles equipped with Lie algebroid connections, impacting deformation theory, classification problems, and potential applications in higher structures (e.g., in non-abelian Hodge theory, singular connections, and parabolic $(V, \phi)$3-modules).

The methodology, notably the use of Atiyah exact sequences for parabolic bundles and a functorial approach to composition of Lie algebroid connections, ensures that these results are robust and adaptable to other generalized geometric settings.

Directions for Further Development

Several natural continuations are apparent:

• Moduli spaces: Given this criterion, one can classify moduli spaces of parabolic bundles equipped with Lie algebroid connections for various types of Lie algebroids, analyzing their geometry and intersection theory using this structural theorem.
• Deformation theory: The explicit cohomological framework provided suggests avenues for studying deformation quantization, infinitesimal deformations, and obstruction theories for such connections.
• Extensions to higher dimensions: The precise local model and the algebraic tools may be adapted for higher-dimensional varieties, where both the structure of divisors and Lie algebroids are richer.
• Non-algebraic settings and applications: Lie algebroid connections appear in generalized complex geometry, Poisson modules, and mathematical physics; the criteria here could guide the study of compatible parabolic structures in these contexts.

Conclusion

The paper provides an exhaustive, cohomologically precise characterization of when a parabolic vector bundle admits a parabolic Lie algebroid connection for a given holomorphic Lie algebroid on a compact Riemann surface, relating this to both the splitting of the anchor and the parabolic degrees of indecomposable components. This fills a gap in the understanding of connections compatible with both parabolic and Lie algebroid structures, opening the way for refined study of moduli and invariants in this setting.