Splitting aspects of holomorphic distributions with locally free tangent sheaf

Technical Summary: Splitting Aspects of Holomorphic Distributions with Locally Free Tangent Sheaf

Problem Statement
The paper addresses the structural properties of singular holomorphic distributions $D$ of dimension $k$ on complex manifolds $M$ (specifically $\mathbb{P}^n$ for $n \ge 3$ or germs at the origin in $\mathbb{C}^n$), focusing on the case where the tangent sheaf $T_D$ is locally free. A central question in the field, previously posed in [1], is whether the tangent sheaf of a codimension-one foliation on $\mathbb{P}^3$ with a locally free tangent sheaf necessarily splits into a direct sum of line bundles (i.e., $T_F \cong \bigoplus \mathcal{O}_{\mathbb{P}^3}(e_i)$). While recent work [15] provided a negative answer for non-integrable distributions (specifically contact structures), this work investigates sufficient conditions under which such a splitting occurs, particularly when the distribution admits a tangent one-dimensional foliation $G$.

The core problem (Problem 2) is formulated as follows: Given a two-dimensional distribution $D$ with a locally free tangent sheaf and a one-dimensional foliation $G$ tangent to $D$, under what conditions does there exist another one-dimensional foliation $H$ tangent to $D$ such that $T_D = T_G \oplus T_H$?

Methodology
The author employs a combination of local algebraic geometry, sheaf theory, and combinatorial methods involving directed graphs.

1. Local Division and Linear Rank: The paper introduces the concept of "local division," where a foliation $G$ locally divides a distribution $D$ at a point $p$ if the defining form of $D$ can be expressed as the interior product of the vector field defining $G$ and another form. A key invariant used is the "linear rank" of a vector field (or foliation) at a singularity, defined as the rank of the linear part of the vector field.
2. Sheaf-Theoretic Characterization: The author establishes equivalences between the splitting of the tangent sheaf, the local freeness of the quotient sheaf $T_D/T_G$, and the property of $G$ locally dividing $D$. Lemma 4.6 and Proposition 4.8 are pivotal, linking the splitting of the sheaf to $T_G$ being a subbundle of $T_D$.
3. Global Analysis on $\mathbb{P}^n$: For the projective case, the analysis relies on the intersection properties of singular sets. The author utilizes the fact that in $\mathbb{P}^n$, irreducible components of certain dimensions must intersect, allowing global conclusions from local division properties.
4. Combinatorial Graph Theory: In the proof of Theorem C (specifically for nilpotent vector fields), the author constructs directed graphs (ss-digraphs) representing the action of the Lie derivative on spaces of homogeneous forms. By analyzing paths, cycles, sources, and sinks in these graphs, the author proves that certain coefficients in the defining forms of foliations must vanish, thereby establishing algebraic multiplicity conditions for singularities.
5. Computational Verification: The paper utilizes the software Maple to solve systems of linear equations derived from the conditions of tangency and integrability, particularly in the proofs of Proposition 6.1 and Theorem C for low-degree cases.

Key Contributions and Results

• Theorem A (Local Splitting): For a germ of a $k$-dimensional distribution $D$ with a locally free tangent sheaf and a tangent foliation $G$ with linear rank at least $k$, the paper proves that either $T_D$ splits as $T_G \oplus T_{H_1} \oplus \dots \oplus T_{H_{k-1}}$, or the distribution is regular. This splitting occurs if and only if the singular set of $G$ is contained in the singular set of $D$.
• Theorem B (Global Splitting on $\mathbb{P}^n$): For a two-dimensional distribution $D$ on $\mathbb{P}^n$ ($n \ge 3$) with a locally free tangent sheaf and a tangent foliation $G$ with linear rank at least 2, the existence of a splitting $T_D = T_G \oplus T_H$ is equivalent to the condition that every irreducible component of $\text{sing}(G)$ of dimension $n-2$ intersects $\text{sing}(D)$. In particular, if $\dim(\text{sing}(D)) \ge 2$ and $n \ge 4$, $T_D$ always splits.
• Theorem C (Codimension One on $\mathbb{P}^3$): If $F$ is a codimension-one holomorphic foliation on $\mathbb{P}^3$ with a locally free tangent sheaf, and there exists a non-zero holomorphic vector field on $\mathbb{P}^3$ tangent to $F$, then $T_F$ splits. This result is significant as it provides a positive answer to Problem 1 under the specific condition of the existence of a tangent vector field.
• Division Properties: The paper refines the De Rham-Saito Division Lemma. It provides alternative conditions for a vector field to divide a differential form based on the rank of the linear part of the vector field (Proposition 5.17) and the rank of the linear part of a 1-form (Proposition 5.19). These results are shown to be applicable in situations not covered by classical lemmas.
• Corollary 6.22: Extending the results to non-integrable distributions, the paper shows that a codimension-one distribution $D$ on $\mathbb{P}^3$ with a locally free tangent sheaf and degree $\deg(D) \ge 1$ splits if there exists a non-nilpotent holomorphic vector field tangent to it.

Significance and Claims
The paper claims to provide a comprehensive framework for understanding the splitting of tangent sheaves for holomorphic distributions. By introducing the concept of local division and characterizing the set $S(G, D)$ (points where division fails), the author establishes precise geometric criteria (intersection of singular sets) for splitting.

The work is presented as an extension of previous studies on foliations [1, 4, 8, 9, 16] and offers a partial resolution to the open problem regarding the splitting of tangent sheaves for codimension-one foliations on $\mathbb{P}^3$. The author notes that while non-integrable distributions of degree 0 (contact structures) do not split, the presence of a tangent vector field (specifically non-nilpotent for degree $\ge 1$) forces the splitting of the tangent sheaf.

The paper concludes by posing Problem 3: If a codimension-one foliation on $\mathbb{P}^3$ has a locally free tangent sheaf and no invariant hypersurface (and thus no rational first integral), does its tangent sheaf necessarily split? The author suggests that based on known counterexamples (which possess rational first integrals), the answer might be affirmative, but this remains an open question. The results are framed as providing sufficient conditions and division criteria that serve as alternatives or complements to classical results like the De Rham-Saito Division Lemma.