Extendability of foliations

Technical Summary: Extendability of Foliations

Problem Statement
This paper addresses the extension problem for holomorphic singular foliations. Given a smooth variety $X$ embedded in a larger variety $Y$ (specifically a regular embedding of codimension 1), and a foliation $\mathcal{F}$ on $X$, the central question is whether there exists a foliation on $Y$ that restricts to $\mathcal{F}$ on $X$. The authors investigate this in both analytic and algebraic settings, considering $Y$ as a general ambient space or specifically as the total space of a deformation of $X$ (the theory of unfoldings).

Methodology
The authors employ a formal geometric approach, analyzing the extension problem at the level of infinitesimal neighborhoods $X^{(n)}$ of $X$ in $Y$. The strategy proceeds in two main stages:

1. Infinitesimal Analysis: The authors construct the obstruction theory for extending a foliation from $X^{(n-1)}$ to $X^{(n)}$. They utilize the sheaf of principal parts along the distribution, denoted $P_{\mathcal{F}, L}$, to describe the space of possible extensions.

   • They establish that the set of extensions of a distribution is acted upon transitively by $H^0(X, P_{\mathcal{F}, N_{X/Y}^{\otimes -n}})$.
   • They identify the subspace of integrable extensions via a kernel condition involving the differential of the defining 1-form, leading to the sheaf $P^{\text{int}}_{\mathcal{F}, N_{X/Y}^{\otimes -n}}$.
   • Crucially, they relate the obstructions to extending integrable foliations to the cohomology group $H^1(X, \mathcal{I}_{\text{Per}} \otimes N_{\mathcal{F}} \otimes N_{X/Y}^{\otimes -n})$, where $\mathcal{I}_{\text{Per}}$ is the ideal sheaf of persistent singularities (singularities that cannot be smoothed out along an unfolding).

2. Algebraization: Once formal extensions are established (i.e., extensions to the formal neighborhood $\hat{X}_{/Y}$), the authors apply the effective Lefschetz condition, $\text{Leff}(X, Y)$. This condition ensures that formal vector bundles and sections can be algebraized to actual algebraic (or analytic) objects on a neighborhood of $X$ in $Y$.

Key Contributions and Results

• Existence of Extensions (Theorem 1.1 & Corollaries 1.2, 1.3):
  The paper proves that if $X \subseteq Y$ is a regular embedding of codimension 1, $\mathcal{F}$ has unobstructed singularities, and the cohomology group $H^1(X, \mathcal{I}_{\text{Per}} \otimes N_{\mathcal{F}} \otimes N_{X/Y}^{\otimes -n})$ vanishes for all $n \geq 1$, then $\mathcal{F}$ admits a formal extension. If the effective Lefschetz condition $\text{Leff}(X, Y)$ holds (which is true if $X$ is an ample divisor in a smooth projective variety $Y$), this formal extension algebraizes to a global foliation on $Y$.

  • This result is applied to show that foliations on ample divisors extend under specific cohomological vanishing conditions.
  • It is also applied to unfoldings: if $H^1(X, \mathcal{I}_{\text{Per}} \otimes N_{\mathcal{F}}) = 0$, any foliation with unobstructed singularities unfolds along any deformation of $X$.

• Positivity Criteria (Theorem 1.4):
  The authors provide a criterion based on the positivity of the embedding versus the normal bundle of the foliation. If $X$ is an ample divisor in $Y$, $\dim(X) \geq 3$, and the bundle $N_{\mathcal{F}}^{\otimes 2} \otimes N_{X/Y}^{\otimes -1}$ is anti-ample, then any foliation with unobstructed singularities extends uniquely to $Y$. This recovers and generalizes previous results (e.g., from [Fig23]) regarding foliations on hypersurfaces in projective space.

• Uniqueness and the Foliated Zak-L'vovsky Problem (Section 5):
  The paper investigates the uniqueness of extensions. It introduces a foliated analogue of the Zak-L'vovsky problem (which asks if extensions of subvarieties are cones).

  • The authors define Camacho-Lins Neto regular foliations, characterized by the vanishing of extension spaces $\text{Ex}_n(\mathcal{F}, N_{X/Y})$.
  • They prove that if these extension spaces vanish, the extension of the foliation is unique.
  • A dichotomy is established: if $X$ is an ample divisor and the extension is unique, then either the extension is unique, or $X$ is a hyperplane and the extension is a cone.

• Tubular Neighborhoods and Rigidity (Theorem 1.5 & 1.6):
  The paper links the existence of tubular neighborhoods to the extendability of foliations.

  • Theorem 1.5: If $X$ has a foliation with rigid singularities (singularities where the first-order unfolding space is trivial) that extends to $Y$, and certain cohomology groups vanish, then $X$ admits a tubular neighborhood. If $Y$ is projective and $X$ is ample, this leads to the dichotomy mentioned above.
  • Theorem 1.6 (Generalization of Gómez-Mont): The authors prove that if a foliation $\mathcal{F}$ on a proper smooth variety $X$ has rigid singularities and $H^1(X, T_{\mathcal{F}}) = 0$, then every unfolding of $\mathcal{F}$ is analytically trivial. This generalizes previous results for surfaces and specific classes of foliations on $\mathbb{P}^m$.

Significance and Scope
The paper claims to provide a unified framework for understanding the extension of foliations by combining formal deformation theory with global positivity conditions. Its significance lies in:

1. Generalizing Unfolding Theory: It extends the theory of unfoldings (previously studied by Suwa, Gómez-Mont, and others) to arbitrary deformations and ambient spaces, not just specific cases.
2. Connecting Singularities to Global Geometry: It explicitly links the local nature of singularities (persistent vs. rigid) to global obstructions via the sheaf $\mathcal{I}_{\text{Per}}$.
3. Uniqueness Results: It offers new sufficient conditions for the uniqueness of foliation extensions, contributing to the moduli theory of foliations.

The authors maintain a modest tone, noting that their results rely on specific cohomological vanishing conditions and the effective Lefschetz condition. They do not claim to solve the extension problem for all foliations but rather provide affirmative answers for a broad class of "unobstructed" and "rigid" singularities under positive embedding conditions. The work is presented as a contribution to the better comprehension of the moduli theory of foliations and the geometry of restrictions/extensions.