[Paper Review] Foliations on complex projective surfaces

Marco Brunella|ArXiv.org|Dec 5, 2002

Geometry and complex manifolds11 references22 citations

TL;DR

This paper provides a systematic classification of foliations on complex projective surfaces using an orbifold-theoretic approach, focusing on Kodaira dimension and canonical bundle properties. It extends McQuillan's framework with complements from Brunella and others, showing that foliations on Kähler surfaces—especially K3 and torus surfaces—are either Riccati, turbulent, or Kummer foliations arising from Kronecker foliations under group quotients.

ABSTRACT

These are lecture notes of a course given in Pisa, SNS, in february 2002. They provide a classification of holomorphic foliations of nongeneral type on compact Kaehler surfaces.

Motivation & Objective

• To classify holomorphic foliations on complex projective surfaces by their Kodaira dimension, extending McQuillan's foundational work.
• To reformulate classical results on foliations using an orbifold perspective, particularly in the presence of cyclic quotient singularities.
• To extend the classification to compact Kähler nonprojective surfaces, including tori and K3 surfaces, by analyzing canonical bundles and Kähler metrics.
• To show that on K3 surfaces, foliations with pseudoeffective canonical bundles must be numerically trivial and arise as quotients of Kronecker foliations.
• To establish that all such foliations on Kähler surfaces are either turbulent, Kummer, or holomorphic (Kronecker) foliations, depending on the surface's algebraic dimension.

Proposed method

• Uses an orbifold framework to define foliations on surfaces with cyclic quotient singularities, treating the canonical bundle $K_{\cal F}$ as a $\mathbb{Q}$-bundle.
• Applies the identity $K_X = K_{\cal F} \otimes N_{\cal F}^*$ to relate the surface's canonical bundle to the foliation's canonical and conormal bundles.
• Employs Zariski decomposition for $\mathbb{Q}$-bundles on K3 surfaces to write $K_{\cal F} = P + N$, with $P$ pseudoeffective and $N$ effective with contractible support.
• Applies the semistability of the tangent bundle on K3 surfaces via Yau's theorem, leading to $c_1(K_{\cal F}) \cdot [\omega] \geq 0$ for all Kähler forms $\omega$.
• Uses the fact that on K3 surfaces, only the trivial bundle satisfies $L \cdot L \geq 0$ and $h^0(L) \leq 1$, implying $P$ is numerically trivial.
• Constructs a finite étale cover $Y \to X'$ where $K_{{\cal F}^\prime}$ becomes holomorphically trivial, leading to $Y$ being a torus and $\cal F$ a Kummer foliation.

Experimental results

Research questions

• RQ1 How can the classification of foliations on complex projective surfaces be systematically organized via Kodaira dimension?
• RQ2 What role does the orbifold structure play in extending foliation theory to surfaces with cyclic quotient singularities?
• RQ3 What are the possible types of foliations on compact Kähler nonprojective surfaces, particularly K3 and torus surfaces?
• RQ4 How does the pseudoeffectivity of $K_{\cal F}$ constrain the geometry of foliations on K3 surfaces?
• RQ5 Can every foliation on a Kähler surface with trivial algebraic dimension be realized as a quotient of a Kronecker foliation?

Key findings

• On Kähler nonprojective surfaces with algebraic dimension 1, foliations transverse to the elliptic fibration are turbulent, and such foliations are classified as such.
• On tori, the only possible foliations are Kronecker foliations generated by constant vector fields, due to the absence of curves.
• On K3 surfaces, the canonical bundle $K_{\cal F}$ must be numerically trivial after a finite base change, implying $K_{\cal F}^{\otimes n} \cong \mathcal{O}_X(E)$ for some effective divisor $E$.
• The only $\mathbb{Q}$-bundle $P$ on a K3 surface with $P \cdot P \geq 0$ and $h^0(P) \leq 1$ is the trivial bundle, forcing $P$ to be numerically trivial.
• After contracting the support of the negative part in Zariski decomposition, the foliation lifts to a finite étale cover $Y$ of a surface $X'$, where $Y$ is a torus and the pullback foliation is Kronecker.
• Thus, all foliations on K3 surfaces arise as Kummer foliations—quotients of Kronecker foliations by finite cyclic groups—providing a complete classification.