\documentclass{article} \usepackage[utf8]{inputenc} \usepackage{amsmath,amsthm,amssymb} \usepackage{xcolor} \title{Foliations on complex manifolds} \author{Carolina Araujo and Jo\~ao Paulo Figueredo} \date{September 2021} \newtheorem{thm}{Theorem} \begin{document} \maketitle \section{Preliminaries} A foliation on a manifold $M^n$ is a way of decomposing $M$ into a disjoint union of immersed submanifolds of the same dimension $r$, such that locally around every point, this decomposition is given by the fibers of a submersion $M \rightarrow \mathbb{R}^{n-r}$. The submanifolds decomposing $M$ are called the leaves of the foliation, and their dimension is called the rank of the foliation. Thus, although foliations are objects which are locally trivial everywhere, they can be very complicated when looked from the global viewpoint. For instance, consider an ODE $y' = f(x,y)$ where $f$ is a $C^1$ function. By the theorem of existence and uniqueness of solutions of ordinary differential equations, the solutions of $y' = f(x,y)$ define a foliation of rank $1$ on $\mathbb{R}^2$. Another instance of complication is that some manifolds might not admit foliations. Consider, for example, $M = S^2$ the $2$-dimensional sphere. If there existed a foliation $\mathcal{F}$, then, by taking vectors tangent to the leaves of $\mathcal{F}$, we would be able to define a smooth vector field $v$ on $S^2$, having no zeroes, which is a contradiction by the Poincar\'e-Hopf theorem. {\color{blue} By this same argument, we see that if $M$ is a compact surface with non-zero Euler characteristics, then $M$ has no regular foliation of rank one.} One might also be interested in topological properties of the leaves of a foliation $\mathcal{F}$. For instance, one might ask whether the leaves are embedded submanifolds of $M$. To illustrate this, let $M = \mathbb{R}^2/\mathbb{Z}^2$ be a $2$-dimensional torus. Then $M$ admits a foliation by considering, at every point $p \in \mathbb{R}^2$, the line in $\mathbb{R}^2$ of inclination angle $\theta$. Then, quotienting out by $\mathbb{Z}^2$, these lines define a foliation on $M$. By a theorem of Kronecker, if $\theta$ is rational, the leaves are compact, hence homeomorphic to $S^1$, and if $\theta$ is irrational, then the leaves are dense in $M$, hence homeomorphic to $\mathbb{R}$. {\color{blue} The two previous paragraphs show that a compact surface $M$ has a regular foliation of rank one (and so codimension one) if, and only if, $\chi(M) = 0$. In particular, this gives an existence theorem for foliations on compact surfaces, i.e. it gives conditions for a compact surface to admit a regular foliation. Another example of this type of result is the following, due to W. Thurston: \begin{thm}[\cite{thurston}] Let $M$ be a closed connected smooth manifold with $\chi(M) = 0$. Then $M$ has a $C^\infty$ foliation of codimension one. \end{thm} } Now suppose $X$ is a complex manifold. This means that $X$ admits an atlas $\{(U_i,f_i)\}$ such that $f_i\colon U_i \rightarrow \mathbb{C}^n$ are homeomorphisms onto open subsets and the transition functions $g_{ij} = f_i \circ f_j^{-1}$ are biholomorphisms whenever they are defined. The number $n$ is called (complex) dimension of $X$. Thus, in particular, a complex manifold of dimension $n$ is a differentiable manifold of real dimension $2n$. For instance, any compact orientable surface is a complex manifold of complex dimension $1$, and for this reason we call them complex curves. Topologically, compact complex curves are classified by their genus $g$, which is a non-negative integer counting ``the number of holes'' they have. It is then natural to define a holomorphic foliation on a complex manifold $X$ as a foliation $\mathcal{F}$ on $X$ whose leaves are locally fibers of holomorphic submersions $X \rightarrow \mathbb{C}^{n-r}$. We call $r$ the rank of the holomorphic foliation $\mathcal{F}$; the leaves of $\mathcal{F}$ are complex manifolds of dimension $r$. For example, let $X = \mathbb{C}^2$ be the complex plane and let $v$ be a holomorphic vector field on $X$, having no zeroes. Then the solutions of the complex ODE $\gamma'(t) = v(\gamma(t))$ define a holomorphic foliation on $X$ of rank $1$. The leaves are complex curves of $\mathbb{C}^2$ which are tangent to $v$. On the other hand, if $X = \mathbb{**}^2$, the complex projective space, then any holomorphic vector field on $\mathbb{**}^2$ has a zero. In particular, $\mathbb{**}^2$ ha no holomorphic foliation of rank $1$. However, if we take $X = \mathbb{**}^2 \setminus Z$, where $Z$ is a finite subset of $\mathbb{**}^2$, then in this case we can have foliations on $X$. This type of foliation will be a singular holomorphic foliation on $\mathbb{**}^2$. More generally, if $X$ is a complex manifold, then a singular holomorphic foliation $\mathcal{F}$ on $X$ will be a holomorphic foliation defined on $X \setminus S$, where $S$ is a closed subset of $X$ having codimension at least $2$. {\color{blue} The minimal closed subset $S'$ of $S$ for which $\mathcal{F}$ can be extended to a holomorphic foliation on $X \setminus S'$ is called the singular locus of $\mathcal{F}$.} Another way of looking at a foliation $\mathcal{F}$ on a complex manifold $X$ is considering its tangent sheaf $T_\mathcal{F}$. This is defined as the sheaf of vectors of $X$ which are tangent to the leaves of $\mathcal{F}$ (this is well defined outside $S$; we extend it by analytic continuation to the whole $X$, since $S$ has codimension at least $2$). This is a coherent sheaf on $X$, which is locally free outside $S$, and with rank $r$ (the rank of $\mathcal{F}$). If $v$ and $w$ are two local vector fields of $X$ which are local sections of $T_\mathcal{F}$, then their Lie bracket $[v,w]$ will also be a local section of $T_\mathcal{F}$ (since $[v,w]$ is tangent to the leaves of $\mathcal{F}$). Reciprocally, there is the following classical result. \begin{thm}[Frobenius] Let $X$ be a complex manifold and let $T \subset T_X$ be a coherent sheaf, which is locally free of rank $r$ outside a closed subset $S \subset X$ of codimension at least $2$. Suppose that for any local sections $v$ and $w$ of $T$, their Lie bracket $[v,w]$ is also a local section of $T$. Then there exists a singular holomorphic foliation $\mathcal{F}$ on $X$ such that $T = T_\mathcal{F}$. \end{thm} {\color{blue} Thus we see that there is a one-to-one correspondence between singular holomorphic foliations on $X$ of rank $r$ and coherent subsheaves of $T_X$ which are locally free of rank $r$ outside a closed subset of $X$ of codimension at least $2$ and which are closed by the Lie bracket. If $r= 1$, every coherent subsheaf of $T_X$ which is locally free of rank $1$ outside a closed subset of codimension at least $2$ is necessarily invertible, and moreover it is always closed by the Lie bracket (since $[v,v] = 0$ for all $v$). } \section{Local theory of singular holomorphic foliations} It is interesting to study the behavior of the leaves of a singular holomorphic foliation near its singular points. For instance, let $X$ be a complex manifold of complex dimension $2$ (a complex surface). Let $\mathcal{F}$ be a singular holomorphic foliation of rank $1$ on $X$. Let $p \in X$ be a singular point of $\mathcal{F}$. Then we may take a chart $(U,\varphi)$ of $X$, centered at $p$, such that $p$ is the only singular point of $\mathcal{F}$ in $U$, and such that the induced foliation on $\varphi(U) \subset \mathbb{C}^2$ is defined by a holomorphic vector field \[v = p(x,y) \frac{\partial}{\partial x} + q(x,y) \frac{\partial}{\partial y},\] where $x$ and $y$ are holomorphic coordinates of $\mathbb{C}^2$, and $p$ and $q$ are holomorphic functions on $\varphi(U)$ which have only the origin as a common zero. If $p_1(x,y) = ax+by$ and $q_1(x,y) = cx + dy$ are the linear parts of $p$ and $q$, respectively, then \[v_1 = p_1(x,y)\frac{\partial}{\partial x} + q_1(x,y)\frac{\partial}{\partial y}\] will be called the linear part of $v$. The leaves of the foliation induced by $v_1$ are easily described by integration of the differential equation $\gamma'(t) = v_1(\gamma(t))$. We see in particular, that the eigenvalues of the matrix \[A = \begin{pmatrix} a & b\\ c & d \end{pmatrix}\] describe the local behavior of the leaves. Thus, it would be interesting to know if the leaves of the foliation induced by $v$ are in some way related to the leaves induced by $v_1$. For instance, this would be the case if there was a change of coordinates $\psi(x,y) = (u,v)$ in $\mathbb{C}^2$, such that $\psi_*(v)$ is a linear vector field (we say that $v$ is linearizable). This happens in the following case: \begin{thm}[Poincar\'e-Dulac] In the notation above, let $\lambda_1$ and $\lambda_2$ be the two eigenvalues of $A$. If $\lambda_1/\lambda_2$ is neither zero nor a negative real number, then $v$ is linearizable. \end{thm} This theorem will imply that there exists leaves of $v$ which accumulate in the origin. In fact, in the coordinates $u$ and $v$, the curves $\{u = 0\} \setminus \{(0,0)\}$ and $\{v = 0\} \setminus \{(0,0)\}$ are both leaves of the foliation. In this case, the curves $\{u = 0 \}$ and $\{v = 0\}$ will be called separatrices of the foliation. More generally, if $\mathcal{F}$ is a foliation on a complex surface $S$ and $p \in S$ is a singular point of $\mathcal{F}$, we say that a complex curves $C \subset S$, passing through $p$, is a separatrix of $\mathcal{F}$, if locally around $p$, $C \setminus \{p\}$ is a leaf of $\mathcal{F}$. Thus, Poincaré's theorem implies that when $\lambda_1/\lambda_2 \not\in \mathbb{R}_{-} \*** \{0\}$, where $\lambda_1$ and $\lambda_2$ are the eigenvalues of the linear part of a local vector field defining $\mathcal{F}$ around $p$, then there exists separatrices of $\mathcal{F}$ through $p$. As a matter of fact, this holds for any foliation on complex surfaces: \begin{thm}[\cite{camachosad}] Let $\mathcal{F}$ be a singular holomorphic foliation on a complex surface $S$. Let $p \in S$ be a singular point of $\mathcal{F}$. Then there exists a separatrix of $\mathcal{F}$ through $p$. \end{thm} \section{Global theory of singular holomorphic foliations} In opposition to these last results on the local structure of leaves of a singular holomorphic foliation, we may study the global structure of the leaves, and thus the global properties of complex manifolds admitting singular holomorphic foliations. One of the first works in this direction was made by Jouanolou (\cite{jouanolou1979equations}). He studies foliations on the complex projective space $\mathbb{**}^n$ having codimension one (i.e. rank $n-1$). If $\mathcal{F}$ is a foliation on $\mathbb{**}^n$ of codimension one, then Jouanolou defines the degree of $\mathcal{F}$ as the number of intersection points of a generic line in $\mathbb{**}^n$ with a generic leaf of $\mathcal{F}$. Suppose $\mathcal{F}$ has degree $0$, and let $F \subset \mathbb{**}^n$ be an hyperplane containing the singular locus $S$ of $\mathcal{F}$. Then any line in $F$ intersecting $S$ has to be generically contained in a leaf of $\mathcal{F}$ (since otherwise the degree wouldn't be $0$). This shows that $F \setminus S$ is a leaf of $\mathcal{F}$. We conclude that $\mathcal{F}$ is induced by a pencil of hyperplanes containing $S$. In particular, if $\mathcal{F}$ is a foliation on $\mathbb{**}^n$ of degree $0$, then the leaves of $\mathcal{F}$ are algebraic manifolds (in this case, they are $\mathbb{**}^{n-1}\setminus \mathbb{**}^{n-2}$). In particular, $\mathcal{F}$ has infinitely many algebraic leaves. It is a remarkable result, by Jouanolou, that the converse is also true: \begin{thm}[\cite{jouanolou1979equations}] Let $\mathcal{F}$ be a codimension one foliation on $\mathbb{**}^n$. If $\mathcal{F}$ has infinitely many algebraic leaves, then $\mathcal{F}$ has a rational first integral: there exists a rational function $f \colon \mathbb{**}^n \dashrightarrow \mathbb{**}^1$ which induces the foliation $\mathcal{F}$, in other words, $T_\mathcal{F} = \ker(df)$. In particular, all the leaves of $\mathcal{F}$ are algebraic. \end{thm} In general, Jouanolou shows that the set ${\rm Fol}(d,n)$ of foliations of codimension one and degree $d$ on $\mathbb{**}^n$ has the structure of a quasi-projective variety. This is so because this set can be identified with the projectivization of the set of homogeneous polynomial one forms $\omega$ of degree $d+1$, in $n+1$ variables, satisfying $\omega \wedge d\omega = 0$ and $\iota_{R}\omega = 0$, where $\iota_R$ is the contraction with the radial vector field $R = \sum_{**0}^n x_i\frac{\partial}{\partial x_i}$, and this is naturally a quasi-projective variety. In the case $d = 0$, we saw that the singular set $S$ determines the foliation $\mathcal{F}$: this $S$ is the center of a pencil of hyperplanes in $\mathbb{**}^n$, and it is thus a $\mathbb{**}^{n-2}$. Thus we see that \[{\rm Fol}(0,n) \cong {\rm Grass}(n-2,n).\] Jouanolou studies the next case, of degree $d = 1$. He observes that these foliations correspond to Jacobi one forms, and he provides a classification of ${\rm Fol}(1,n)$: it has two irreducible components, and each of them is described explicitly\footnote{More recently, Cerveau and Lins Neto showed that ${\rm Fol}(2,n)$ has six irreducible components \cite{cerveauneto}.}. {\color{blue} Related to these results by Jouanolou, there are works of Gomez-Mont (see \cite{gm}) concerning the set of rank one foliations on a fixed compact complex manifold $X$. We observed above that this set is in a one-to-one correspondence with the set of invertible subsheaves of $T_X$. Using results of Douady, he manages to show that this set $\mathcal{D}$ has a natural structure of complex analytic space. Moreover, he shows that $\mathcal{D}$ may be decomposed as the disjoint union of the subspaces $\mathcal{D}_\alpha$ which are defined as the family of foliations $\mathcal{F}$ of rank one on $X$ having $c_1(T_\mathcal{F}) = \alpha$ (we remark however that it might happen that $\mathcal{D}_\alpha = \emptyset$ for some $\alpha$; it is thus an interesting problem to determine for which $\alpha$ we have $\mathcal{D}_\alpha \neq \emptyset$). He also characterizes $\mathcal{D}_\alpha$ for certain types of of manifolds. For instance, if $X$ is K\"ahler with vanishing first Betti number, he shows that $\mathcal{D}_\alpha$ is a projective space. In general, he shows that if $X$ is a projective manifold, then $\mathcal{D}_\alpha$ is also projective. Moreover he shows the following result: \begin{thm}[\cite{gm}] Let $X$ be a projective manifold and $\alpha \in H^2(X,\mathbb{Z})$ be such that for any holomorphic line bundle $L$ on $X$ with $c_1(L) = \alpha$, the vector space $H^0(X,{\rm Hom}(L,T_X))$ is of constant dimension $r > 0$, then $\mathcal{D}_\alpha$ has a natural structure of a $\mathbb{**}^{r-1}$-bundle over a complex torus of dimension $b_1(X)/2$. \end{thm} When $\dim(X) = 2$, and $X$ is a ruled surface, i.e. a $\mathbb{**}^1$-bundle over a smooth curve $C$, he describes, in \cite{gomez1989holomorphic}, the irreducible components of $\mathcal{D}_\alpha$ for each possible $\alpha$. Thus, we have a classification of holomorphic foliations on ruled surfaces via the description of its moduli space. Moreover, in the same paper he studies regular foliation on ruled surfaces, and shows the following result: \begin{thm}[\cite{gomez1989holomorphic}] Let $S\rightarrow C$ be a ruled surface, with $g(C) \neq 1$, and let $\mathcal{F}$ be a regular foliation of rank $1$ on $S$. Then \begin{itemize} \item Either $\mathcal{F}$ is induced by the fibration $S \rightarrow C$; \item Or $\mathcal{F}$ is transverse to the fibration $S \rightarrow C$, and constructed by suspension of a representation \[ \rho\colon \pi_1(C) \rightarrow {\rm PSL}(2,\mathbb{C}).\] \end{itemize} \end{thm} Thus we have a classification of regular foliations on ruled surfaces. For instance, this result implies that every regular surface on a Hirzebruch surface is algebraically integrable. Thus, the next step is to look for a classification of regular foliations on any complex surface, not only the ruled ones. This was done by Brunella, in the following theorem: \begin{thm} Let $X$ be a compact complex surface and let $\mathcal{F}$ be a regular holomorphic foliation of rank $1$ on $X$. Then we have one of the following possibilities (non exclusive): \begin{itemize} \item $\mathcal{F}$ is a fibration; \item $\mathcal{F}$ is transverse to a fibration and thus constructed by suspension of a representation of a fundamental group; \item $\mathcal{F}$ is a turbulent foliation on an elliptic surface; \item $\mathcal{F}$ is a linear foliation on a complex torus; \item $\mathcal{F}$ is a transversely hyperbolic foliation with dense leaves, whose universal covering is a fibration by disks over the disk. \end{itemize} \end{thm} McQuillan showed that the foliations in the last item of Brunella's theorem are obtained by the quotient of the one of the projections of $\mathbb{D} \times \mathbb{D}$ to $\mathbb{D}$, by an arithmetic group. Thus we have a very explicit classification of regular foliations on compact surfaces. The next natural step is thus to try to obtain some kind of classification for any foliation of rank $1$ on compact surfaces, not only the regular foliations. One idea for this is to look first to the classification of compact surfaces, in the famous Kodaira-Enriques classification, and try to adapt their method to foliated compact surfaces. Thus, Mendez considers the canonical bundle $K_\mathcal{F}$ of a foliation of rank $1$ on a compact surface $S$, and studies its Kodaira dimension, in order to achieve a classification of foliations in terms of this dimension. For instance, when the Kodaira dimension is minus infinity, this means that $K_\mathcal{F}$ is not pseudo-effective, which implies by an algebraicity criterion of Miyaoka (see Bogomolov-McQuillan as well), that $\mathcal{F}$ is induced but fibration by rational curves. } \bibliographystyle{alpha} \bibliography{bibliography} \end{document}