On the connectedness of the singular set of holomorphic foliations

1. Problem Statement and Context

The paper addresses the global topological constraints on singular holomorphic foliations on complex projective manifolds. Specifically, it investigates the structure of the singular set ($Sing(\mathcal{F})$) of a foliation $\mathcal{F}$ of dimension $k > 1$ on a projective manifold $X$ of dimension $n$.

• The Core Question: Under what conditions is the union of the irreducible components of the singular set of maximal dimension ($k-1$) connected?
• Motivation: This work extends Raoul Bott's classical results (1960s) on topological obstructions to the existence of integrable foliated structures. While Bott's theorem focuses on the vanishing of characteristic classes for integrable subbundles, this paper seeks a "Bott phenomenon" for singular distributions, specifically linking the ampleness of the normal bundle's determinant to the connectivity of the singular locus.
• Specific Motivation: The paper aims to answer a question posed by D. Cerveau regarding codimension-one foliations on $\mathbb{P}^3$: Is the singular set of such foliations necessarily connected (or of a specific topological nature) under certain conditions?

2. Mathematical Framework and Definitions

• Foliation Definition: A $k$-dimensional singular holomorphic foliation on $X$ is defined by an exact sequence of sheaves:
  $$0 \to \mathcal{F} \to T_X \to N_{\mathcal{F}} \to 0$$
  where $\mathcal{F}$ is the tangent sheaf (integrable, rank $k$) and $N_{\mathcal{F}}$ is the normal sheaf (rank $n-k$, torsion-free).
• Singular Set: $Sing(\mathcal{F})$ is the locus where $N_{\mathcal{F}}$ is not locally free. It has codimension $\ge 2$.
• Assumptions:
  1. $\dim(X) = n$, $\dim(\mathcal{F}) = k > 1$.
  2. The determinant line bundle $\det(N_{\mathcal{F}})$ is ample.
  3. The singular set decomposes as $Sing(\mathcal{F}) = Z \cup Z^-$, where $Z$ consists of components of dimension exactly $k-1$, and $Z^-$ consists of lower-dimensional components.

3. Methodology

The proof relies heavily on Baum–Bott residue theory, which relates global Chern numbers to local residues at the singular set.

• Baum–Bott Formula: The authors utilize the localization of the Chern class $c_1(N_{\mathcal{F}})^{n-k+1}$ at the connected components of the singular set.
• Cohomological Approach: Instead of working purely in homology, the authors lift the residues to the cohomology group $H^{2(n-k+1)}(X)$ using Poincaré duality and Alexander duality.
• Key Logical Steps:
  1. Let $Z_1, \dots, Z_s$ be the connected components of $Z$ (dimension $k-1$).
  2. The global Chern number $c_1(N_{\mathcal{F}})^{n-k+1}[X]$ is the sum of residues localized at each $Z_i$.
  3. Using the ampleness of $\det(N_{\mathcal{F}})$, the authors construct a specific cohomological argument involving cup products with powers of the first Chern class of the line bundle.
  4. They show that the contribution of each component $Z_i$ to the total Chern number is identical and non-zero.
  5. By summing these contributions, they derive a contradiction if $s > 1$, forcing $s=1$.

4. Key Results

Theorem 1 (Main Theorem)

Let $X$ be a projective manifold of dimension $n$, and $\mathcal{F}$ a singular holomorphic foliation of dimension $k > 1$ such that $\det(N_{\mathcal{F}})$ is ample. If $Sing(\mathcal{F}) = Z \cup Z^-$ where $Z$ has dimension $k-1$, then $Z$ is a connected set.

• Sharpness: The authors prove that all assumptions are necessary:
  • If $X$ is not projective (e.g., Hopf manifolds), the result fails.
  • If $\det(N_{\mathcal{F}})$ is not ample, the result fails (e.g., foliations on products $X \times Y$).
  • If $k=1$, the singular set can be disconnected (e.g., generic curve foliations).
  • If $n \le 2(k-1)$, the connectivity is trivial due to intersection theory in $\mathbb{P}^n$ (any two subvarieties of dimension $k-1$ must intersect), making the theorem non-trivial only when $n > 2k-2$.

Corollary 1 (Topological Obstruction to Integrability)

For a codimension-one foliation on a smooth projective threefold $X$ (with $h^1(\mathcal{O}_X)=0$) where the singular set $C$ (1-dimensional) is reduced and $\det(N_{\mathcal{F}})$ is ample:
If certain cohomological vanishing conditions hold ($h^1, h^2$ of specific twists vanish), then:
$$h^2(\mathcal{F} \otimes \det(N_{\mathcal{F}})^*) = c_3(\mathcal{F}) \cap [X]$$
This equates a cohomological invariant to the number of isolated singularities. This provides a topological obstruction: if the tangent sheaf is locally free (implying no isolated singularities), specific cohomology groups must vanish, which may not happen, thus obstructing integrability.

Corollary 2 (Application to $\mathbb{P}^3$)

Let $\mathcal{F}$ be a codimension-one foliation on $\mathbb{P}^3$. If the tangent sheaf of $\mathcal{F}$ is locally free, then:

• Either $\mathcal{F}$ is a pencil of planes (degree 0), or
• $Sing(\mathcal{F})$ is a singular curve.
  This answers Cerveau's question by ruling out the existence of foliations with locally free tangent sheaves and smooth, connected singular sets (unless they are pencils of planes).

5. Significance and Contributions

1. Resolution of Cerveau's Question: The paper definitively answers a long-standing question about the structure of singular sets for foliations on $\mathbb{P}^3$, establishing that "nice" (locally free) tangent sheaves force the singular set to be singular (non-smooth) or trivial.
2. New Topological Obstruction: It introduces a new type of global obstruction to integrability. While Bott's theorem uses characteristic class vanishing, this work uses connectivity of the singular locus as an obstruction. If a distribution has a disconnected singular set of dimension $k-1$ and an ample normal determinant, it cannot be integrable.
3. Refinement of Baum-Bott Theory: The paper demonstrates a sophisticated application of Baum-Bott residues, moving from local analysis to global topological constraints via ampleness conditions.
4. Counter-examples and Sharpness: By providing explicit examples (Hopf manifolds, product manifolds, and specific foliations on $\mathbb{P}^n$), the authors delineate the precise boundaries of the theorem, clarifying the necessity of projectivity, ampleness, and dimension constraints.

6. Conclusion

The paper establishes that for singular holomorphic foliations on projective manifolds with ample normal determinants, the "main" part of the singular set (dimension $k-1$) must be connected. This result serves as a powerful tool to distinguish between integrable foliations and non-integrable distributions, offering a bridge between complex geometry, algebraic topology, and the theory of singular distributions.