Holomorphic Lie algebroid connections over rationally connected varieties

Technical Summary: Holomorphic Lie Algebroid Connections over Rationally Connected Varieties

Problem Statement
The paper investigates the existence and properties of holomorphic Lie algebroid connections on vector bundles over smooth complex projective varieties. While holomorphic connections on vector bundles over compact Riemann surfaces are well-understood (existence implies flatness, and existence is governed by degree conditions on indecomposable summands), the situation over higher-dimensional varieties is more restrictive. Specifically, the authors address the behavior of these connections when the base variety $X$ is rationally connected (where any two points can be joined by an irreducible rational curve) and the underlying structure is a general holomorphic Lie algebroid $(V, \phi)$ rather than the standard tangent bundle $(TX, \text{Id}_{TX})$.

The central question is: Under what conditions does a holomorphic vector bundle $E$ over a rationally connected variety $X$ admit a $(V, \phi)$-connection, and if such a connection exists, is it necessarily flat?

Methodology
The authors employ a combination of algebraic geometry techniques and the theory of Lie algebroids:

1. Lie Algebroid Framework: The paper utilizes the definition of a Lie algebroid $(V, \phi)$, where $V$ is a holomorphic vector bundle and $\phi: V \to TX$ is the anchor map. A $(V, \phi)$-connection is defined as a $\mathbb{C}$-linear holomorphic map $\nabla: E \to E \otimes V^*$ satisfying the Leibniz rule involving the dual of the anchor map $\phi^*$.
2. Jet Bundle Formulation: The existence of a $(V, \phi)$-connection is characterized via the splitting of a specific short exact sequence involving the first-order $V$-jet bundle $J^1_V(E)$. Specifically, $E$ admits a connection if and only if the sequence $0 \to E \otimes V^* \to J^1_V(E) \to E \to 0$ splits holomorphically.
3. Restriction to Rational Curves: A key methodological step involves restricting the problem to non-constant morphisms $f: \mathbb{P}^1 \to X$. Since $X$ is rationally connected, such curves are abundant. The authors pull back the Lie algebroid $(V, \phi)$ and the bundle $E$ to $\mathbb{P}^1$, constructing a new Lie algebroid $(V_f, \hat{\phi})$ over $\mathbb{P}^1$.
4. Ample and Strictly Nef Conditions: The proofs rely heavily on the properties of the kernel $S := \text{Ker}(\phi)$. The authors assume $S$ is strictly nef. They utilize the fact that the pullback of a strictly nef bundle to $\mathbb{P}^1$ remains strictly nef (and thus has positive degree components). This allows them to invoke vanishing theorems (via Serre duality) to show that certain extension classes vanish, forcing the relevant sequences to split.
5. Curvature Analysis: To prove flatness, the authors analyze the curvature $K(\nabla) \in H^0(X, \text{End}(E) \otimes \bigwedge^2 V^*)$. By restricting to rational curves where the pullback of $TX$ is ample, they demonstrate that the pullback of $V$ is also ample. Consequently, the bundle $\bigwedge^2 V^*$ consists of line bundles of negative degree, leading to the vanishing of global sections of the curvature on these curves. By the Zariski density of such curves, the global curvature vanishes.

Key Contributions and Results
The paper establishes two primary theorems under the assumption that $X$ is a rationally connected smooth complex projective variety and $(V, \phi)$ is a Lie algebroid with a surjective anchor map $\phi$ such that $S := \text{Ker}(\phi)$ is strictly nef:

• Theorem 3.1 (Existence Criterion): A holomorphic vector bundle $E$ over $X$ admits a $(V, \phi)$-connection if and only if $E$ is holomorphically trivial.

  • Mechanism: The proof shows that if a connection exists, its pullback to any rational curve $\mathbb{P}^1$ yields a holomorphic connection on a bundle over $\mathbb{P}^1$. Since any holomorphic connection on $\mathbb{P}^1$ implies the bundle is trivial (due to simple connectivity and flatness), $f^*E$ is trivial for all such curves. This forces $E$ to be trivial on $X$.

• Theorem 3.2 (Flatness): Any $(V, \phi)$-connection on a holomorphic vector bundle over $X$ (which must be trivial by Theorem 3.1) is flat (integrable).

  • Mechanism: The curvature of the connection, when restricted to rational curves where $f^*TX$ is ample, must vanish because the relevant cohomology group $H^0(\mathbb{P}^1, \bigwedge^2 f^*V^*)$ is zero due to the negativity of the degrees of the dual of an ample bundle. Since these curves cover a Zariski dense subset of $X$, the global curvature is zero.

• Corollary 3.3: As a specific application, if $X$ is rationally connected and $E$ admits a usual holomorphic connection (corresponding to the Lie algebroid $(TX, \text{Id}_{TX})$), then $E$ is holomorphically trivial and the connection is flat.

Significance and Claims
The paper claims to generalize known results regarding holomorphic connections on Riemann surfaces and Calabi-Yau manifolds to the broader context of rationally connected varieties and Lie algebroid connections.

• Unification: The work unifies various geometric objects (Higgs bundles, logarithmic connections, meromorphic connections) under the single framework of Lie algebroid connections, showing that for rationally connected varieties with strictly nef kernels, the existence of such structures is an extremely rigid condition.
• Rigidity: The results highlight a strong rigidity phenomenon: on rationally connected varieties with the specified Lie algebroid conditions, the only vector bundles capable of supporting a $(V, \phi)$-connection are the trivial ones, and any such connection is automatically flat.
• Scope: The authors do not claim to solve the problem for all rationally connected varieties or all Lie algebroids; the results are strictly conditioned on the surjectivity of the anchor map and the strict nefness of its kernel. The paper positions these findings as a natural extension of Atiyah's classical work on connections over curves to higher-dimensional rational settings.