2.1 Foliated affine structures

An affine structure on a curve is an atlas for its complex structure taking values in $\mathbf {C}$ whose changes of coordinates lie within the affine group $\{z\mapsto az+b\}$.

The affine distortion of a local biholomorphism between open subsets of ${\bf C}$ is the operator

\[ \mathcal{L} (f) := \frac{f''}{f'}\,dz, \]

which plays a fundamental role in the study of affine structures. It vanishes precisely when $f$ is an affine map. A simple computation shows that, for the composition of two germs of biholomorphisms between open sets of $\mathbf {C}$,

(2.1)\begin{equation} \mathcal{L} ( f\circ g ) = \mathcal{L}(g) + g^* \mathcal{L}(f). \end{equation}

Hence, the affine distortion of a biholomorphism between open subsets of curves equipped with affine structures does not depend on the chosen affine charts. Given two affine structures on a curve $C$, the affine distortion of the identity map measured in the corresponding affine charts, namely the one-form $\mathcal {L} ( \psi \circ \phi ^{-1})$ for $\phi$ a chart of the first affine structure and $\psi$ a chart of the second, gives a globally well-defined one-form on $C$ which vanishes if and only if the affine structures agree. Reciprocally, given an affine structure and a one-form $\alpha$ on $C$, if $\alpha$ reads $a(z)\,dz$ in some affine chart of the affine structure, the maps given in this chart by the solutions $\psi$ of $\psi ''= a \psi '$ give a second globally-defined affine structure on $C$. An easy consequence of (2.1) is that this provides the moduli space of affine structures on $C$ with the structure of an affine space directed by the vector space of holomorphic one-forms on $C$ (see also [Reference GunningGun66, § 9]).

Given an affine structure on a curve, the family of vector fields which are constant in the coordinates of the affine structure is well-defined. Such a family is the one of flat sections of a holomorphic connection on the tangent bundle of $C$. Reciprocally, given a holomorphic connection on the tangent bundle of the curve, one can define the atlas of charts where the flat sections of the connection are constant vector fields. A change of coordinates of this atlas maps a constant vector field to another constant vector field, and hence it belongs to the affine group, thus retrieving an affine complex structure on the curve. We deduce that there is a canonical correspondence between affine structures on a curve and connections on its tangent bundle. In particular, the only compact curves admitting affine structures are elliptic ones (see [Reference BenzécriBen60], or Theorem 4.1 in § 4). On such a curve, there is a canonical affine structure coming from its uniformization by $\mathbf {C}$, or, equivalently, by the integration of a non-identically-zero holomorphic one-form.

We will adopt both points of view in order to extend the definition of affine structures on curves to the context of unidimensional singular holomorphic foliations on complex manifolds.

2.1.1 The foliated setting

Let us begin by recalling that a singular holomorphic foliation $\mathcal {F}$ of dimension one on a complex manifold $M$ is defined by the data of a cover by open sets $\{U_i\}_{i\in I}$ of $M$ and a family $\{Z_i\}_{i\in I}$ of holomorphic vector fields $Z_i$ on $U_i$, such that the vanishing locus of $Z_i$ in $U_i$ has codimension at least two, and that on the intersection $U_i\cap U_j$ of two open sets of the cover, the vector fields $Z_i$ and $Z_j$ are proportional, namely, $Z_i=g_{ij}Z_j$ for a function $g_{ij} :U_i \cap U_j\rightarrow {\mathbf {C}} ^*$. The set where the vector fields vanish is the singular set of the foliation, denoted by $\mathrm {Sing}(\mathcal {F})$. When defined in this way, two foliations are regarded as equivalent if the subsheafs of the sheaf of sections of the tangent bundle $TM$ of $M$ generated by the vector fields $Z_i$ are the same. This subsheaf is called the tangent sheaf of the foliation; it is locally free, and corresponds to the sheaf of sections of a holomorphic line bundle, the tangent bundle of the foliation, that we denote by $T_{\mathcal {F}}$. We then have a morphism $T_{\mathcal {F}} \rightarrow T M$, which vanishes only over the singular set of $\mathcal {F}$. This map completely characterizes $\mathcal {F}$, and can be used as an alternative definition of a foliation. The canonical bundle of the foliation is the bundle $K_{\mathcal {F}} := T^* _{\mathcal {F}}$.

A first definition of a foliated affine structure is the following.

Definition 2.1 Let $M$ be a complex manifold and $\mathcal {F}$ a singular holomorphic foliation by curves on $M$. A holomorphic foliated affine structure on $\mathcal {F}$ is an open cover $\{U_i\}$ of $M\setminus \mathrm {Sing}(\mathcal {F})$ and submersions $\phi _i:U_i\to \mathbf {C}$ transverse to $\mathcal {F}$ such that, in restriction to a leaf $L$ of $\mathcal {F}$, $(\phi _i|_L)\circ (\phi _j|_L)^{-1}$ is an affine map of $\mathbf {C}$.

No condition is explicitly imposed on the singular set of the foliation, but Hartogs's phenomena will implicitly do so. The affine geometry of the leaves as they approach the singular set is studied in § 3.

Remark 2.2 If nonempty, the set of foliated affine structures on the foliation $\mathcal {F}$ on the manifold $M$ is an affine space directed by the vector space $H^0(K_{\mathcal {F}})$. By considering a foliated version of the construction described at the beginning of § 2.1, on the restriction of $\mathcal {F}$ to $M\setminus \mathrm {Sing}(\mathcal {F})$, the difference between two foliated affine structures is a family of holomorphic one-forms along the leaves of $\mathcal {F}$ varying holomorphically in the transverse direction, this is, a section of $K_{\mathcal {F}}$ over $M\setminus \mathrm {Sing}(\mathcal {F})$, vanishing identically if and only if the affine structures coincide. By Hartogs's theorem, this section of $K_{\mathcal {F}}$ over $M\setminus \mathrm {Sing}(\mathcal {F})$ extends holomorphically to a section of $K_{\mathcal {F}}$ over $M$.

There are foliations without any foliated affine structure, e.g. those having a compact leaf of genus different from one. Notwithstanding, and in contrast with the scarcity of curves having affine structures, there are many foliations that support them.

Example 2.3 A holomorphic vector field with isolated singularities on a manifold of dimension at least two, e.g. a holomorphic vector field on a compact Kähler manifold [Reference KobayashiKob72], induces a foliated affine structure whose changes of coordinates are not only affine but are actually translations (we call these foliated translation structures).

Example 2.4 The orbits of a homogeneous polynomial vector field on $\mathbf {C}^{n+1}$ are preserved by homotheties, and the vector field defines a foliation on $\mathbf {P}^n$; moreover, the vector field also endows this foliation with a foliated affine structure: it induces a translation structure along its phase curves, and the homotheties of $\mathbf {C}^{n+1}$ act affinely in the translation charts. As we shall see in Proposition 4.3, for a foliation $\mathcal {F}$ on $\mathbf {P}^n$ of degree strictly greater than one, every foliated affine structure on $\mathcal {F}$ may be obtained in this way: if $X$ is a homogeneous vector field on $\mathbf {C}^{n+1}$ of degree $d>1$ generating $\mathcal {F}$, the foliated affine structures on $\mathcal {F}$ are those induced by the vector fields of the form $X+PR$, where $R=\sum _iz_i\partial /\partial z_i$ is the radial vector field and $P$ is a homogeneous polynomial of degree $d-1$.

Example 2.5 (Inoue surfaces)

The Inoue surfaces $S_M$, $S_N^{(+)}$, and $S_N^{(-)}$ are compact complex non-Kähler surfaces which are quotients of $\mathbf {H}\times \mathbf {C}$ by groups of affine transformations of $\mathbf {C}^2$ (see [Reference InoueIno74]). For the surfaces $S_M$, the associated action preserves the two foliations of $\mathbf {H}\times \mathbf {C}$ and is affine on the leaves of both of them (with respect to the tautological affine structure on $\mathbf {C}$, to the one inherited from the inclusion $\mathbf {H}\subset \mathbf {C}$ on $\mathbf {H}$). The two foliations induced in $S_M$ thus admit foliated affine structures. For the surfaces $S_N^{(+)}$ and $S_N^{(-)}$, the action on $\mathbf {H}\times \mathbf {C}$ preserves the foliation given by the fibers of the first factor, and acts affinely upon its leaves. The induced foliations are also endowed with foliated affine structures.

Example 2.6 (Hopf surfaces)

Hopf surfaces are compact complex surfaces whose universal covering is biholomorphic to $\mathbf {C}^2\setminus \{0\}$ (see [Reference Barth, Hulek, Peters and Van de VenBHPV04, Ch. V, Section 18]). The primary ones are quotients of $\mathbf {C}^2\setminus \{0\}$ by contractions of the form

(2.2)\begin{equation} (x,y)\mapsto (\alpha x+\lambda y^n,\beta y), \end{equation}

with $n\geq 1$ and $\lambda =0$ if $\alpha \neq \beta ^n$. In the elliptic case (when $\alpha =\beta ^n$ and $\lambda =0$), when $x$ and $y$ are given weights $n$ and $1$, respectively, every quasihomogeneous polynomial vector field on $\mathbf {C}^2$ is preserved by the contraction up to a constant factor. The associated foliations on the Hopf surface have thus a foliated affine structure (which may be a translation one in particular cases). In the general (nonelliptic) case, the linear diagonal vector fields $Ax\partial /\partial x+By\partial /\partial y$ ($AB\neq ~0$) if $\lambda =0$, or the ‘Poincaré–Dulac’ ones $(nx+\mu y^n)\partial /\partial x+y\partial /\partial y$ ($\mu \in \mathbf {C}$) if $\lambda \neq 0$ are preserved by the contraction, and induce nowhere-vanishing vector fields on the Hopf surface. Further, the coordinate vector field $\partial /\partial x$ is preserved up to a constant factor, and the foliation it induces has a natural foliated affine structure. By Brunella's classification [Reference BrunellaBru97, Section 5], there are no further foliations on primary Hopf surfaces: every foliation on a primary Hopf surface supports a foliated affine structure.

2.1.2 Foliated connections

Now let us turn to a more intrinsic equivalent definition of a foliated affine structure in terms of the notion of foliated connection, which enables the construction of more examples.

Given a foliation $\mathcal {F}$ and a sheaf $\mathcal {S}$ of $\mathcal {O}_S$-modules, a foliated connection on $\mathcal {S}$ relative to $\mathcal {F}$ is a differential operator $\nabla :\mathcal {S} \rightarrow \mathcal {O} (K_{\mathcal {F}}) \otimes \mathcal {S}$ which, away from the singular points of $\mathcal {F}$, satisfies the Leibniz rule

\[ \nabla ( f s ) = d_{\mathcal{F}} f \otimes s + f \nabla s , \]

for every $f\in \mathcal {O}$ and every $s\in \mathcal {S}$, where $d_{\mathcal {F}}$ denotes the differential along the leaves of $\mathcal {F}$. (In general, we will consider $\mathcal {F}$ as fixed and omit it from the discussion.) A foliated connection on a holomorphic vector bundle is a foliated connection on its sheaf of sections. Foliated connections appear in the work of Baum and Bott under the name of partial connections (see [Reference Baum and BottBB70, Section 3]).

In particular, after extending to the singularities of $\mathcal {F}$ via Hartogs's theorem, a foliated connection on $T_{\mathcal {F}}$ is a map $\nabla : T_{\mathcal {F}} \rightarrow T_{\mathcal {F}} \otimes K_{\mathcal {F}} = \mathcal {O} _M$ which to a vector field $Z$ tangent to $\mathcal {F}$ assigns a holomorphic function $\nabla (Z)$, its Christoffel symbol, satisfying the Leibniz rule

(2.3)\begin{equation} \nabla(fZ)=Zf+f\nabla(Z). \end{equation}

Let us see that such a connection is equivalent to a foliated affine structure.

Given a foliated connection $\nabla$ on $T_{\mathcal {F}}$ and $p\notin \mathrm {Sing}(\mathcal {F})$, if $Z$ is a vector field tangent to $\mathcal {F}$ that does not vanish at $p$ and such that $\nabla (Z)\equiv 0$ (if $Z$ is parallel), if $\phi$ is a function such that $d\phi (Z)\equiv 1$, $\phi$ is part of an atlas of a foliated affine structure that depends only on $\nabla$ (it is not difficult to see that such a $Z$ and such a $\phi$ always exist).

For the other direction, let $\mathcal {F}$ be a foliation endowed with a foliated affine structure $\sigma _0$. Let $Z$ be a vector field defined on the open set $U\subset M$, tangent to $\mathcal {F}$ (with a singular set of codimension two), and denote by $\sigma _Z$ the foliated affine structure induced by $Z$ on $U$. As in Remark 2.2, the difference $\sigma _Z-\sigma _0$ is a section $\alpha$ of $K_{\mathcal {F}}$ over $U$. Consider the holomorphic function $\alpha (Z)$ on $U$, and define a foliated connection $\nabla$ on $T_{\mathcal {F}}$ by setting $\nabla (Z)=\alpha (Z)$. Let us verify that Leibniz's rule (2.3) takes place. We will do so locally on a curve, in a coordinate $z$ where $Z=\partial /\partial z$. A chart for the affine structure induced by $f(z)\partial /\partial z$ is $\int ^z d\xi /f(\xi )$ and, thus, $\sigma _{Z}-\sigma _{fZ}= -f'/f \,dz$. Hence, the contraction of $\sigma _{fZ}-\sigma _0$ with $fZ$ yields $Zf+f\alpha (Z)$, in agreement with formula (2.3).

Observe that the definition of foliated affine structures via foliated connections has the advantage of not needing to distinguish between regular and singular points of the foliation.

Lemma 2.7 (Extension Lemma [Reference Guillot and RebeloGR12, Proposition 8])

Let $M$ be a manifold, $\mathcal {F}$ a foliation on $M$, $p\in M$. Let $X$ be a meromorphic vector field defined on a neighborhood of $p$ whose divisor of zeros and poles $D$ is invariant by $\mathcal {F}$ and which is tangent to $\mathcal {F}$ away from it. Then, on a neighborhood of $p$, the foliated affine structure induced by $X$ away from $D$ extends to $D$ in a unique way.

Proof. Let $Z$ be a nonvanishing holomorphic vector field defining $\mathcal {F}$ on a neighborhood of $p$. Let $X=f_1^{n_1}\cdots f_k^{n_k}Z$, for $n_i\in \mathbf {Z}$ and reduced holomorphic functions $f_i$ such that $Zf_i$ divides $f_i$, say $Zf_i=h_if_i$ for some holomorphic function $h_i$. In the complement of the divisor of zeros and poles of $X$, $X$ induces a foliated connection $\nabla$ such that $\nabla (X)\equiv 0$, for which

\[ \nabla(Z)=\nabla\Big(\Big(\prod f_i^{-n_i}\Big)X\Big)=\Big(\prod f_i^{n_i}\Big)Z\Big(\prod f_i^{-n_i}\Big)=-\sum_{i=1}^k \frac{n_i}{f_i} Zf_i =-\sum_{i=1}^k n_i h_i, \]

and $\nabla$ extends holomorphically to a full neighborhood of $p$.

Let us see how one can concretely apply this lemma to produce foliated affine structures.

Example 2.8 (Elliptic fibrations)

Let $\mathcal {F}$ be an elliptic fibration. There is a natural foliated affine structure in the complement of the singular fibers: a smooth elliptic curve carries a canonical affine structure (given by the integration of any nowhere-vanishing holomorphic form) which varies holomorphically with the curve. (For the universal elliptic curve, the existence of a foliated affine structure is just a corollary of the fact that the Hodge bundle of abelian differentials on the fibers exists [Reference ZvonkineZvo12]; the fact that the Chern class of this bundle does not vanish shows that one cannot reduce this affine structure to a translation one.) This structure can be extended to the singular fibers as follows. First, build a non-identically-zero meromorphic section of $T_{\mathcal {F}}$ whose divisor of zeros and poles is supported on a union of fibers. (To do so, one can apply Corollary 12.3 in [Reference Barth, Hulek, Peters and Van de VenBHPV04, Ch. V] to get a meromorphic volume form $\omega$ on the total space whose divisor of zeros and poles is supported on a finite union of fibers, the desired section of $T_{\mathcal {F}}$ being the symplectic gradient of a meromorphic function defined on the base with respect to $\omega$.) On the fibers on which this vector field is regular, it induces the canonical affine structure. Lemma 2.7 then shows that this foliated affine structure extends to the whole surface.

2.1.4 More on foliated connections

For the problem of establishing the existence of foliated connections on the tangent bundle of a foliation $\mathcal {F}$, investigating the existence of foliated connections on other line bundles might prove rewarding, since the set of isomorphism classes of line bundles admitting foliated connections forms a group and is closed under the operations of taking powers and extracting roots: foliated connections on other line bundles might propagate up to $T_{\mathcal {F}}$. An interesting problem is thus that of determining, for a singular holomorphic foliation on a complex manifold, which are the holomorphic line bundles having foliated connections.

A fundamental example of a foliated connection is the Bott connection [Reference BottBot72] on $K_{M/ \mathcal {F}}$: it is defined through the exterior derivative operator

\[ d : K_{M/ \mathcal{F}} \rightarrow \mathcal{O} (K_M) \simeq \mathcal{O} (K_{\mathcal{F}} ) \otimes K_{M/ \mathcal{F}}, \]

where the last isomorphism is given by adjunction.

On a closed Kähler manifold, every holomorphic line bundle with trivial first Chern class carries a flat unitary connection, which, by restriction, induces a foliated connection. Hence, in this setting, the problem consists in determining which are the Chern classes of line bundles which carry foliated connections. This set is a subgroup of the Néron–Severi group which contains all the torsion points. As we have shown, it contains the first Chern class of $K_{M/\mathcal {F}}$, but, in general, it seems difficult to say more. There are, however, situations where this point of view allows the existence of foliated affine structures to be established. Let us give some examples.

Lemma 2.10 On a compact Kähler manifold with vanishing first Chern class, any singular holomorphic foliation carries a foliated affine structure.

Proof. Since the manifold has vanishing first Chern class, its canonical bundle has a unitary flat connection. By the adjunction formula, the tensor product of this connection with the Bott connection produces a flat connection on the cotangent bundle of the foliation and, hence, by duality, a foliated affine structure.

Corollary 2.11 Any foliation on a Calabi–Yau manifold has a foliated affine structure.

Lemma 2.12 If the Picard number of a compact Kähler manifold $M$ is one, then:

1. − if the first Chern class of $K_{M/ \mathcal {F}}$ is not a torsion element in the Néron–Severi group, there is a foliated affine structure;

2. − otherwise, $\mathcal {F}$ has a transverse invariant pluriharmonic form.

Proof. If the first Chern class of $K_{M/\mathcal {F}}$ is not a torsion element in the Néron–Severi group, then any line bundle over $S$ has a foliated connection; this is due to the fact that having a foliated connection is stable under taking power or roots. In particular, the tangent bundle carries a foliated affine structure. If not, $K_{M/\mathcal {F}}$ carries a unitary flat connection over $S$. Given a flat section $\omega$, naturally considered as a holomorphic form of degree $n-1$, the product $\omega \wedge \overline {\omega }$ is a well-defined pluriharmonic form on $S$ which vanishes on the foliation $\mathcal {F}$. Such a form is closed because $S$ is Kähler and, hence, defines a family of transverse pluriharmonic forms.

Example 2.13 (Hypersurfaces of ${\mathbf {P}}^3$)

Well-known examples of surfaces having Picard number one are generic hypersurfaces of ${\mathbf {P}}^3$ of degree at least four, by a theorem of Noether, see [Reference DeligneDel73]. These are simply connected by the hyperplane section theorem of Lefschetz and, in particular, it is impossible in this case for the normal bundle to a foliation to have a torsion first Chern class. Indeed, if it were the case, the normal bundle would be holomorphically trivial, and so would be its dual, and consequently we would have a holomorphic form on the surface vanishing on the foliation. However, such a form does not exist because the surface has a vanishing first Betti number. In other words, we have proved that on a generic surface in $\mathbf {P}^3$, every singular holomorphic foliation carries a foliated affine structure. Note that this property holds on the explicit examples produced in [Reference ShiodaShi81], namely the surfaces defined in homogeneous coordinates by $w^m+ xy^{m-1}+y z^{m-1}+zx^{m-1}=0$ for $m\geq 5$ a prime number.

2.2 Foliated projective structures

A projective structure on a curve is an atlas for its complex structure taking values in $\mathbf {P}^1$ whose changes of coordinates lie within the group of projective transformations $\{z\mapsto (az+b)/ (cz+d)\}$. In this case, the Schwarzian derivative

(2.4) \begin{equation} \{f(x),x\}=\frac{f'''}{f'}-\frac{3}{2}\biggl(\frac{f''}{f'}\biggr)^2 =\biggl(\frac{f''}{f'}\biggr)'-\frac{1}{2}\biggl(\frac{f''}{f'}\biggr)^2, \end{equation}

plays a role analogous to the one played by the affine distortion in the context of affine structures.

Given two projective structures on a curve $C$ with charts $\{(U_i,\phi _i)\}$ and $\{(V_j,\psi _j)\}$, the quadratic form on $U_i\cap V_j$ given by

(2.5)\begin{equation} \{f(z),z\}\,dz^2, \end{equation}

for $f=\psi _j\circ \phi _i^{-1}$, gives a globally well-defined quadratic form on $C$, which vanishes if and only if the projective structures coincide. This is due to the fact that the operator (2.5) satisfies

\[ \{f\circ g,z\}\,dz^2=\{g,z\}\,dz^2+g^*(\{f,w\}\,dw^2). \]

Reciprocally, given a projective structure with charts $\{(U_i,\phi _i)\}$ and a quadratic form $\beta$ on $C$, if $\beta$ reads $\beta _i(z)\,dz^2$ in $U_i$, the charts locally given by the solutions of the Schwarzian differential equation $\{f,z\}=\beta _i$ give a globally well-defined projective structure on $C$. In this way, on a curve, the projective structures form an affine space directed by the vector space of holomorphic quadratic differentials.

Projective structures are much more flexible than affine ones: they exist on any curve, and, for instance, for curves of genus $g\geq 2$, their moduli is an affine space of dimension $3g-3$. Projective structures associated to particular geometries (spherical for genus zero, Euclidean in the case of genus one, and hyperbolic for genus at least two) are given by the Uniformization Theorem [Reference de Saint-GervaisdSG10]. Nevertheless, the existence of unrestricted projective structures can be very easily established independently from it, as Poincaré was well aware of; see [Reference GunningGun66, § 9] for a modern presentation.

Foliated projective structures may also be defined in terms of foliated projective connections: a foliated projective connection (on $T_{\mathcal {F}}$) is a map $\Xi :T_{\mathcal {F}}\to \mathcal {O}(M)$ that to a vector field $Z$ tangent to $\mathcal {F}$ associates a holomorphic function $\Xi (Z)$, its Christoffel symbol, satisfying the modified Leibniz rule:

(2.6)\begin{equation} \Xi(fZ)=f^2\Xi(Z)+fZ^2(f)-\frac{1}{2}(Zf)^2. \end{equation}

(When restricted to curves, this definition is equivalent to those found in [Reference TyurinTyu78, Def. 1.3.1] and [Reference GunningGun67, Section 4].)

For instance, if $\nabla :T_{\mathcal {F}}\to \mathcal {O}(M)$ is a foliated connection on $T_{\mathcal {F}}$, the associated foliated projective connection $\Xi$ is

(2.7)\begin{equation} \Xi(Z)=-\frac{1}{2}(\nabla(Z))^2+Z(\nabla(Z)). \end{equation}

Let us see that a foliated projective structure is equivalent to a foliated projective connection. Let $\mathcal {F}$ be a foliation endowed with a foliated projective structure $\rho _0$. Let $Z$ be a vector field tangent to $\mathcal {F}$ with singular set of codimension at least two, and consider the foliated projective structure $\rho _Z$ that it defines. From Remark 2.15, the difference $\rho _Z-\rho _0$ is a section $\alpha$ of $K_{\mathcal {F}}^2$. Define $\Xi (Z)$ as $\alpha (Z^{\otimes 2})$. Let us prove that it satisfies condition (2.6). As before, it is sufficient to do so locally in a curve. Consider a curve endowed with a projective structure $\rho _0$, $Z$ a holomorphic vector field and $z$ a local coordinate in which $Z=\partial /\partial z$. Let $\alpha (z)\,dz^2$ be the quadratic form $\rho _Z-\rho _0$. The projective structure defined by $fZ$ has $\int ^zd\xi /f(\xi )$ as a chart and, thus,

\[ \rho_Z-\rho_{fZ} =\biggl(\frac{1}{2}\biggl(\frac{f'}{f}\biggr)^2-\frac{f''}{f}\biggr)\,dz^2. \]

Hence, the contraction of $\rho _{fZ}-\rho _0$ with $(fZ)^{\otimes 2}$ gives $f^2\alpha (Z^{\otimes 2})+Z^2f-\frac {1}{2}(Zf)^2$, establishing (2.6). Reciprocally, if $\Xi$ is a foliated projective connection, $p\notin \mathrm {Sing}(\mathcal {F})$ and $Z$ is a holomorphic vector field tangent to $\mathcal {F}$ that does not vanish at $p$, and such that $\Xi (Z)\equiv 0$, if $\phi$ is a function defined in a neighborhood of $p$ such that $d\phi (Z)\equiv 1$, $\phi$ defines a foliated projective structure in the sense of Definition 2.14 that depends only on $\Xi$.

Example 2.17 (Turbulent foliations)

Let $S$ be a compact surface, $\pi :S\to C$ an elliptic fibration, $\mathcal {F}$ a turbulent foliation on $S$ adapted to $\pi$, i.e. almost every fiber of $\pi$ is everywhere transverse to $\mathcal {F}$. We refer the reader to [Reference Barth, Hulek, Peters and Van de VenBHPV04, Ch. 5, § 7] and [Reference BrunellaBru04, Ch. 4, Section 3] for facts around elliptic fibrations and turbulent foliations that we will use leisurely. We will restrict to the cases where $\pi$ is relatively minimal, and show that $\mathcal {F}$ admits a foliated projective structure.

Let us begin with the case where $\mathcal {F}$ is regular and where the fibers of $\pi$ are simple. This implies that, as a foliation, the fibration is also regular, and, in particular, that it does not have singular fibers. Let $C_0\subset C$ be the subset above which $\pi$ and $\mathcal {F}$ are transverse. By the arguments in Example 2.16, the projective structures on $C_0$ and the foliated ones on $\pi ^{-1}(C_0)$ are in correspondence. We will establish a condition on the projective structure on the base that guarantees that the foliated one extends to the invariant fibers. Let $F$ be a fiber such that, for $p=\pi (F)$, $p\notin C_0$. The fibration around $F$ is given by the projection $\pi :\mathbf {D}\times \mathbf {C}/\Lambda \to \mathbf {D}$ for some elliptic curve $\mathbf {C}/\Lambda$. For some local coordinates $z$ and $w$ in $\mathbf {D}$ and $\mathbf {C}/\Lambda$ centered at $p$, $\mathcal {F}$ is given by the nonvanishing holomorphic vector field $Z=z^n\partial /\partial z+B(z) \partial /\partial w$, with $B$ a holomorphic nonvanishing function, and where $n$ is the multiplicity with which $F$ appears in the tangency divisor between $\mathcal {F}$ and the fibration. Let $\Xi _0$ be a projective connection on $\mathbf {D}\setminus \{0\}$, and let $\Xi$ be the corresponding foliated projective connection on $\pi ^{-1}(\mathbf {D}\setminus \{0\})$. In the spirit of Lemma 2.7, by formula (2.6), since $\pi _*Z=z^n\partial /\partial z$,

(2.8)\begin{equation} \Xi( Z)=\Xi_0\biggl(z^n\frac{\partial}{\partial z}\biggr)=z^{2n}\Xi_0\biggl(\frac{\partial}{\partial z}\biggr)+\frac{1}{2}n(n-2)z^{2(n-1)}. \end{equation}

If this expression is holomorphic (in particular, if the projective structure on $C_0$ extends as a regular one to $p$), the foliated projective structure extends to $F$; this extension is unique.

If $\Gamma$ is a finite group acting on $S$ with isolated fixed points, preserving the fibration, the foliation and the foliated projective structure, it induces a foliated projective structure on the nonsingular part of $S/\Gamma$. Let us prove that the foliated projective structure extends to the minimal resolution of the singular points. Let $q\in S$ be a point with nontrivial stabilizer. In a neighborhood of $q$, this stabilizer is a cyclic group $\langle g\rangle$ of order $m$ of biholomorphisms fixing $q$ such that the derivative of $g$ at $q$ has as eigenvalues two primitive $m$th roots of unity. Suppose that, in a neighborhood of $q$, the nonvanishing vector field $Z$ gives both the foliation and its projective structure, and choose local coordinates where $Z=\partial /\partial x$. The action of $g$ on the leaf space of $Z$ may be linearized while preserving the expression of $Z$, and we may suppose that it is given by $y\mapsto \omega y$ for a primitive $m$th root of unity $\omega$. Since the foliation induced by $Z$ is preserved, the action of $g$ on $Z$ must consist in multiplying it by a function $f$ such that $f^m \equiv 1$ and, thus, $f \equiv \omega ^n$ for some $n< m$, $(m,n)=1$. In particular, the vector field $y^{m-n}Z$, which by the Leibniz formula (2.3) induces the same affine structure than $Z$ away from $y=0$, is preserved by $g$. It induces a vector field on the quotient as well as on its minimal resolution. Since in this resolution the divisor contracting to the singular point is invariant by the foliation coming from $Z$, the affine structure extends to this divisor by Lemma 2.7.

All turbulent foliations adapted to relatively minimal elliptic fibrations are constructed through such a process of taking quotients and resolving singularities. Thus, by suitably choosing a possibly singular projective structure on the base of the associated minimal elliptic fibration, we obtain a foliated projective structure: every turbulent foliation adapted to a relatively minimal elliptic fibration admits a foliated projective structure.

Not all foliations support foliated projective structures. As we mentioned in the introduction, by the work of Zhao [Reference ZhaoZha19], no Kodaira fibration admits one (we will give another proof of this fact through Corollary 6.1; one more has recently appeared in [Reference Eriksson, Wentworth and Freixas i MontpletEWF21]). Despite the generality of this result, we thought it worthwhile to include a concrete, hands-on, self-contained instance of it.

The existence of a foliated projective structure is equivalent to the vanishing of a class $\beta _{\mathcal {F}}$ in $H^1 (M, K_{\mathcal {F}}^2)$ whose definition mimics the definition of the class $\alpha _{\mathcal {F}}$ introduced in the context of foliated affine structures. Namely, take a covering of $M$ by open sets $U_i$ on which we have foliated projective connections $\Xi _i$, and consider the cocycle $\beta =(\beta _{ij}) _{ij}$, where $\beta _{ij}=\Xi _i-\Xi _j$ is a section of $K_{\mathcal {F}}^2$ over $U_i\cap U_j$. Its cohomology class $\beta _{\mathcal {F}}\in H^1 (M,K_{\mathcal {F}}^2)$ is well-defined. To construct a globally defined foliated projective connection, one needs to modify each $\Xi _i$ in $U_i$ by adding some section $\beta _i$ of $K_{\mathcal {F}}^2$, in such a way that the resulting connections on the $U_i$ coincide in the intersection of their domains. This is equivalent to solving the equation $\beta _i - \beta _j = \beta _{ij}$, so there exists a foliated projective structure if and only if $\beta _{\mathcal {F}}=0$.

If $M$ is a curve and $\mathcal {F}$ is the foliation whose only leaf is $M$, then by Serre duality $h^1 (M, K_M^2) =h^0 (M, TM)$, and we recover the fact that every compact curve of higher genus has a projective structure. Notice, however, that this argument does not allow to conclude that rational and elliptic curves have such structures.

Despite Example 2.18, it is quite common for a singular holomorphic foliation to carry a foliated projective structure. The following criteria is a consequence of Kodaira's vanishing theorem.

Let us illustrate the use of this lemma.

Proof. Let $C$ be a curve of genus $g$, $S=C\times \mathbf {P}^1$, and let $\mathcal {F}$ be a foliation on $S$. Curves of the form $\{*\}\times \mathbf {P}^1$ will be called vertical; those of the form $C\times \{*\}$, horizontal. Let $V\in H^2(S,\mathbf {Z})$ be the Poincaré dual of a vertical curve, $H\in H^2(S,\mathbf {Z})$ that of a horizontal one. These generate $H^2(S,\mathbf {Z})$. If $\mathcal {F}$ is either the vertical or the horizontal foliation, the proposition follows from the existence of projective structures on curves, so we suppose that we are in neither case. Let us denote by $n_h$ (respectively, $n_v$) the number of tangencies of $\mathcal {F}$ with a generic horizontal (respectively, vertical) curve. We call $n_h$ the horizontal degree and $n_v$ the vertical one. They are both nonnegative. We claim that

(2.9)\begin{equation} n_h\geq 2g-2. \end{equation}

The foliation $\mathcal {F}$ is defined by a morphism $i: T_\mathcal {F} \rightarrow T S$ that vanishes on the singular set of $\mathcal {F}$ (a finite number of points). Since $TS= \mathrm {pr}_1^* (TC) \oplus \mathrm {pr}_2^* (T{\mathbf {P}}^1)$, the morphism $i$ is given by sections of $K_{\mathcal {F}} \otimes \mathrm {pr}_1^* (T_C )$ and of $K_{\mathcal {F}}\otimes \mathrm {pr}_2^* (T{\mathbf {P}}^1)$ that vanish simultaneously on a finite set. Since the foliation is not the vertical one, the first section does not vanish identically. Since $K_{\mathcal {F}}=K_S\otimes N_{\mathcal {F}}$, $c_1( K_{\mathcal {F}} \otimes \mathrm {pr}_1^* (T C ))= n_v H + (n_h-2g+2) V$, and such a section can only exist if both $n_v\geq 0$ and $n_h-2g+2\geq 0$, proving the claim.

Since $K_S=\mathrm {pr}_1^*(K_C)\otimes \mathrm {pr}_2^*(K_{\mathbf {P}^1})$, $c_1(K_S ) = (2g-2) V - 2 H$. Let $c_1(N_{\mathcal {F}} )= a H+b V$ for some $a,b\in \mathbf {Z}$. On a horizontal curve that is not invariant by $\mathcal {F}$, a meromorphic section of $N_{\mathcal {F}}^*$ induces a meromorphic one-form having $n_h-(aH+bV) \cdot H$ zeros, so $n_h-b= 2g-2$. The same reasoning shows that $n_v-a = -2$. To sum up,

\[ c_1 (N_{\mathcal{F}}) = (n_v+2) H+ (n_h -2g+2) V. \]

From $K_{\mathcal {F}}=K_S\otimes N_{\mathcal {F}}$, $c_1 (K_{\mathcal {F}}) = n_v H + n_h V$, so

(2.10)\begin{equation} c_1 (K_{\mathcal{F}}^2 \otimes K_S^*) = (2n_v+2) H + (2n_h - (2g-2)) V. \end{equation}

If $g\neq ~1$ or $n_h>0$, we infer from (2.9) and (2.10) that $K_{\mathcal {F}} ^2 \otimes K_S^*$ intersects positively $H$ and $V$ and, hence, every algebraic curve in $S$. This implies, by Nakai's criterion [Reference Barth, Hulek, Peters and Van de VenBHPV04, Ch. IV, Corollary 6.4], that it is ample, and, by the previous lemma, $\mathcal {F}$ admits a foliated projective structure.

Finally, if $g= 1$ and $n_h=0$, the foliation is turbulent, adapted to a nonsingular elliptic fibration. We have already shown in Example 2.17 that these foliations admit foliated projective structures as well.

3.1 The affine case

3.1.2 The foliated case

Let $\mathcal {F}$ be a foliation defined on a neighborhood of $0$ in $\mathbf {C}^n$ and endowed with a foliated affine structure induced by the foliated connection $\nabla$. Let $Z$ be a vector field tangent to $\mathcal {F}$ and $\gamma =\nabla (Z)$ its Christoffel symbol, as defined in § 2.1, and recall that $\gamma$ is a holomorphic function at $0$. It follows from (2.3) that if $\lambda _1,\ldots, \lambda _n$ are the eigenvalues of $Z$ at $0$, the ratio $[\lambda _1:\cdots : \lambda _n:\gamma (0)]$ is an invariant of the foliated affine structure.

In dimension one, this invariant may be expressed in terms of the previously defined affine ramification index. Consider a singular affine structure on a neighborhood of $0$ in $\mathbf {C}$ given by the connection $\nabla$. Let $\gamma =\nabla (\lambda z\partial /\partial z)$. The difference between the affine structure induced by the coordinate $z$ and the first one is $(\gamma (z)/\lambda -1)\,dz/z$ and, thus, for the ramification index $\nu$ of the original affine structure,

(3.1)\begin{equation} \nu=\frac{\lambda}{\gamma(0)}. \end{equation}

In the foliated case in $(\mathbf {C}^n,0)$, if the eigenvalues at $0$ of the vector field are $\lambda _1$, …, $\lambda _n$ and its Christoffel symbol $\gamma$ does not vanish at $0$ we say that $\nu _i=\lambda _i/\gamma (0)$ is a principal ramification index. From (3.1),

(3.2)\begin{equation} [\lambda_1:\cdots:\lambda_n:\gamma(0)]=[\nu_1:\cdots:\nu_n:1]. \end{equation}

In the generic nondegenerate case there will be $n$ curves $C_1,\ldots,C_n$ through $0$, invariant by $\mathcal {F}$, pairwise transverse, and tangent to the eigenspaces of the linear part of the vector field, and $\nu _i$ will be the ramification index of the affine structure on $C_i$ at $0$.

Generically, the ratio (3.2) determines the foliated affine structure.

Theorem 3.2 Let $\mathcal {F}$ be a foliation on a neighborhood of $0$ in $\mathbf {C}^n$, with a singularity at $0$, tangent to a nondegenerate vector field $Z$ satisfying Brjuno's condition ($\omega$). For a generic foliated connection $\nabla$ on $T_{\mathcal {F}}$, there exist coordinates around $0$ where $\mathcal {F}$ is tangent to a linear vector field $Z'$ whose Christoffel symbol $\nabla (Z')$ is constant.

We refer the reader to [Reference ArnoldArn80, Ch. 5] for details on Brjuno's condition ($\omega$), and only mention that it is satisfied by generic (in a measure-theoretic sense) linear parts. The genericity of the affine structure will be made precise further on. Note that this theorem introduces a notion of equivalence that is different from that of having the same foliated atlas (Remark 2.2), in which no change of coordinates is involved. The proof of our theorem is an application of the following general result.

Theorem 3.3 (Brjuno and Pöschel)

Let $Z=\sum _i \lambda _iz_i\partial /\partial z_i$ be a linear vector field on $(\mathbf {C}^n,0)$. Let $F$ be a holomorphic function defined in the neighborhood of $(0,0)$ in $\mathbf {C}\times \mathbf {C}^n$ such that $F(0,0)=0$, and consider the differential equation $Zf=F(f,z)$ subject to the condition ${f(0)=0}$. Let $\mu =\partial F/\partial f|_{(0,0)}$ and suppose that $\mu \neq \langle K,\lambda \rangle$ for every $K\in (\mathbf {Z}_{\geq 0})^n$ with $|K|\geq 2$. Let

\[ \omega'(m)=\min_{2\leq |K|\leq m} |\langle K,\lambda \rangle-\mu|. \]

Then, if

(3.3)\begin{equation} -\sum_{\nu\geq 0} 2^{-\nu}\log\omega'(2^{\nu+1})<\infty, \end{equation}

the equation has a holomorphic solution (which is, moreover, unique).

In this theorem, the function $f$ will be a solution of the differential equation if and only if the vector field $Z\oplus F(\zeta,z)\partial /\partial \zeta$, defined in a neighborhood of the origin of $\mathbf {C}^{n}\times \mathbf {C}$, has $\zeta =f(z)$ as an invariant manifold. The condition $\mu \neq \langle K,\lambda \rangle$ guarantees the existence of a formal solution, and (3.3) guarantees its convergence. For $n=1$, the hypothesis on $(\omega ')$ is superfluous, and the result reduces to Briot and Bouquet's theorem [Reference InceInc44, §12.6].

Theorem 3.3 does not exactly appear in the literature in the above formulation. Brjuno's announcement [Reference BrjunoBrj74] gives a similar statement, and we can find in [Reference PöschelPös86] an analogous result by Pöschel in the context of invariant manifolds for germs of diffeomorphisms; the proof of the latter may be adapted in a straightforward way to give a complete proof of the above theorem. (For the case where the $\lambda _i$ belong to the Poincaré domain, see also [Reference KaplanKap79, Reference Carrillo and SanzCS14]; see [Reference CharrièreCha88, § IX] for an analogous result under Siegel-type Diophantine conditions.)

Proof of Theorem 3.2 Since $Z$ satisfies Brjuno's condition ($\omega$), it is linearizable, so we may suppose that it is already linear. Suppose that $f$ is a function such that

(3.4)\begin{equation} Zf=\gamma(0)-f\gamma,\quad f(0)=1. \end{equation}

The existence of such a function follows, generically, from Theorem 3.3, which we may apply to (3.4). In terms of the statement of Theorem 3.3, $\mu =-\gamma (0)$; generically, $\mu \neq \sum _i m_i\lambda _i$, and condition (3.3) is satisfied. The Christoffel symbol of the vector field $Z'=fZ$, is, by construction, the constant $\gamma (0)$. It remains constant in the coordinates where $Z'$ is linear.

Note that the condition $-\gamma (0)\neq \sum _i m_i\lambda _i$ may be expressed solely in terms of the principal affine ramification indices.

3.2 The projective case

3.2.1 Projective structures with singularities on curves

Let $U\subset \mathbf {C}$ be a neighborhood of $0$, $U^*=U\setminus \{0\}$, and consider a projective structure on $U^*$. Let $\beta$ be the quadratic form in $U^*$ measuring the difference from an auxiliary projective structure on $U$ to this one. We say that $0$ is a singularity for the projective structure if $\beta$ does not extend holomorphically to $0$. A singularity of a projective structure is said to be Fuchsian if $\beta$ has at most a double pole at $0$. The quadratic residue $\mathrm {Q}(\beta,0)$ of the quadratic form $\beta$ at $0$, $\mathrm {Q}((r/z^2+\cdots )\,dz^2,0)=r$, does not depend on the choice of the auxiliary projective structure. In this case, we define the (normalized) projective angle at $0$ of the projective structure with singularities as $\measuredangle (0)=\sqrt {1-2\mathrm {Q}(\beta,0)}$. It is only well-defined up to sign. The normalized projective angle of the projective structure with developing map $z\mapsto z^\theta$ is $\pm \theta$. We define the projective ramification index at $0$ as the reciprocal of the normalized projective angle. Again, it is only well-defined up to sign.

We also have a local classification of projective structures with Fuchsian singularities in dimension one.

Proposition 3.4 Consider a projective structure on a neighborhood of $0$ in $\mathbf {C}$ having a Fuchsian singularity at $0$ with normalized projective angle $\theta \in \mathbf {C}$. Then, there exists a singular affine Fuchsian structure in its class. In particular, there exists a coordinate $z$ around $0$ where the developing map is given as in Proposition 3.1.

Proof. The difference from the projective structure induced by a local coordinate $z$ to the singular one has the form $S(z)\,dz^2$, $S(z)=\frac {1}{2}(1-\theta ^2)z^{-2}+\cdots$. From (2.4), the affine structure with invariant $g(z)\,dz$ is in the projective class of the original projective structure if $g$ is a solution to the Riccati equation $g'=S+\frac {1}{2}g^2$ (if there is some $f$ for which $f''/f'=g$ and $\{f,z\}=S$). For $u=zg$, this equation reads

(3.5)\begin{equation} zu'=z^2S(z)+u+\tfrac{1}{2}u^2. \end{equation}

Let $\theta$ be a root of $\theta ^2$ that is not a strictly positive integer. By the theorem of Briot and Bouquet [Reference InceInc44, §12.6], (3.5) has a holomorphic solution $u(z)$ with $u(0)=\theta -1$. The affine structure induced by $g\,dz=u\,dz/z$ is, thus, Fuchsian and induces the original projective structure.

3.2.2 The foliated case

Let $\mathcal {F}$ be a foliation tangent to a nondegenerate vector field defined in a neighborhood of $0$ in $\mathbf {C}^n$ and endowed with a foliated projective structure induced by the projective connection $\Xi$. Let $Z$ be a vector field tangent to $\mathcal {F}$ and $\rho =\Xi (Z)$ its Christoffel symbol, as defined in § 2.2. From (2.6), if $\lambda _1,\ldots,\lambda _n$ are the eigenvalues of $Z$ at $0$, in the weighted projective space $\mathbf {P}(1,\ldots,1,2)$, the ratio $[\lambda _1:\cdots :\lambda _n:\rho (0)]$ is an invariant of the foliated projective structure. Let us relate this invariant, in dimension one, to the previously defined projective ramification index. Consider a singular projective structure on $(\mathbf {C},0)$ and let $\rho =\Xi (\lambda z\partial /\partial z)$. The difference of the projective structure with coordinate $z$ and the singular one is $\frac {1}{2}(1+2\rho /\lambda ^2)\,dz^2/z^2$ and, thus, for the projective ramification index $\nu$, $\nu ^2=-\frac {1}{2}\lambda ^2/\rho (0)$. In particular,

(3.6)\begin{equation} [\lambda_1:\cdots:\lambda_n:-2\rho(0)]=[\nu_1:\cdots:\nu_n:1] \quad \text{in } \mathbf{P}(1,\ldots,1,2). \end{equation}

The numbers $\nu _i$ are said to be the principal projective ramification indices of the foliated projective structure at $0$.

Remark 3.5 The individual principal projective ramification indices are only well-defined up to sign. More generally, only their even functions are well-defined.

Remark 3.6 If a foliated affine structure is considered as a projective one, its affine and projective ramification indices coincide (within the limitations given by the previous remark).

The analogue of Theorem 3.2 for foliated projective structures is the following one.

Theorem 3.7 Let $\mathcal {F}$ be a foliation on a neighborhood of $0$ in $\mathbf {C}^n$, with a singularity at $0$, generated by a nondegenerate vector field $Z$ satisfying Brjuno's condition ($\omega$). For a generic foliated projective structure on $\mathcal {F}$:

1. − there exists a foliated affine structure in its class; and

2. − there exist coordinates where $\mathcal {F}$ is tangent to a linear vector field having a constant Christoffel symbol.

Proof. Let $\mathcal {F}$ be a foliation endowed with a foliated projective structure with foliated projective connection $\Xi$. Let $Z$ be a vector field tangent to $\mathcal {F}$, and suppose that it is linear. Let $\rho =\Xi (Z)$ be its Christoffel symbol. From formula (2.7), if $\gamma$ is a function such that $Z\gamma =\frac {1}{2}\gamma ^2+\rho$, there exists, like in Proposition 3.4, a foliated affine structure inducing the given projective one, with connection $\nabla$, such that $\nabla (Z)=\gamma$. We may resort to Theorem 3.3 to establish the existence of a solution to this equation with one of the initial conditions $\gamma (0)$ such that $\gamma ^2(0)+2\rho (0)=0$. For the hypothesis of the theorem, $\mu =\gamma (0)$, and according to it, we have solutions to the equation whenever $\gamma (0)\neq \sum _i m_i\lambda _i$ and condition (3.3) is satisfied. Theorem 3.2 and formula (2.7) establish the second part of our claim.

4.1 The geodesic vector field

Consider a foliated affine structure along the foliation $\mathcal {F}$. For every $v\in T_{\mathcal {F}}$ such that $\pi (v)$ is not a singular point of $\mathcal {F}$, for some $U\subset \mathbf {C}$, with $0\in U$, there is a geodesic of the associated connection, $c:(U,0)\to (M,\pi (v))$, tangent to $\mathcal {F}$, such that $c'(0)=v$ (the germ of $c$ at $0$ is unique). The derivative gives a lift $\widetilde {c}:U\to T_{\mathcal {F}}$ with $\pi (\widetilde {c}(t))=c(t)$ and $\widetilde {c}(t)=c'(t)$. The vector field on $T_{\mathcal {F}}\setminus \pi ^{-1}(\mathrm {Sing}(\mathcal {F}))$ that has this curve as its integral one through $v$ extends, by Hartogs's theorem, to all of $T_{\mathcal {F}}$. This is the geodesic vector field of the foliated affine structure. Since, for $\alpha \in \mathbf {C}^*$, $c(\alpha t)$ is a geodesic whenever $c(t)$ is, the geodesic vector field enjoys the following equivariance property: if $X$ denotes the geodesic vector field and $h_\alpha :T_{\mathcal {F}}\to T_{\mathcal {F}}$ the homothety by $\alpha$ on the fibers,

(4.1)\begin{equation} X(h_\alpha v)=\alpha Dh_\alpha (X(v)). \end{equation}

This characterization allows some foliated affine structures to be classified.

Local expressions for geodesic vector fields may be given as follows. Let $\{U_i\}_{i\in I}$ be a cover of $M$ by open subsets such that, in $U_j$, $\mathcal {F}$ is given by the vector field $Z_j$. If $U_i\cap U_j\neq \emptyset$, let $g_{ij}:U_i\cap U_j\to \mathbf {C}^*$ be the function such that $Z_i=g_{ij}Z_j$. The line bundle $T_{\mathcal {F}}$ is obtained by gluing the sets in $\{U_i\times \mathbf {C}\}_{i\in I}$ by means of the identification

(4.2)\begin{equation} (u,\zeta_j)=(u,g_{ij}\zeta_i) \end{equation}

if $U_i\cap U_j\neq \emptyset$. Now let the foliated affine structure come into play. Let $\gamma _j:U_j\to \mathbf {C}$ be the Christoffel symbol $\nabla (Z_j)$. Consider, in $U_j\times \mathbf {C}$, the vector field

(4.3)\begin{equation} X_j=\zeta_j Z_j- \gamma_j\zeta_j^2\frac{\partial}{\partial \zeta_j}. \end{equation}

In $(U_i\times \mathbf {C})\cap (U_j\times \mathbf {C})$, under (4.2), this vector field reads $g_{ij}\zeta _i Z_j- (g_{ij}\gamma _j+Z_jg_{ij})\zeta _i^2\partial /\partial \zeta _i$, which, by Leibniz's rule (2.3), equals $X_i$. This shows that (4.3) defines a global holomorphic vector field $X$ on the total space of $T_{\mathcal {F}}$. We now establish that this is the geodesic vector field of $\nabla$.

The vector field $H$ on $T_{\mathcal {F}}$ given by $\zeta _j\partial /\partial \zeta _j$ in $U_j\times \mathbf {C}$ is globally well-defined. We have the relation $[H,X]=X$. In its integral form, it implies that if $(z(t),\zeta _j(t))$ is a solution to $X_j$ and $\alpha \in \mathbf {C}$, $(z(\alpha t),\alpha \zeta _j(\alpha t))$ is also a solution, which is exactly condition (4.1). Since all the solutions above a given point may be constructed in this way, the vector field $X$ gives a class of parametrizations of the leaves of $\mathcal {F}$ that is invariant under precompositions by affine maps. The inverses of these parametrizations form the atlas of charts of a foliated affine structure.

Let us prove that the foliated affine structure associated to $X$ is exactly the one we started with, that the parametrized solutions of $X$ project onto the geodesics of our original foliated affine structure. If the vector field $Z_j$ is such that $\nabla (Z_j)$ vanishes identically, then, on the one hand, the geodesics of $\nabla$ are the integral curves of $Z_j$ (with their natural parametrization) and its constant multiples; on the other hand, $X_j$ reduces to $\zeta _jZ_j$ ($\zeta _j$ is a first integral), and the integral curves of the latter project also onto the integral curves of the constant multiples of $Z_j$. We conclude that $X$ is the geodesic vector field of $\nabla$.

The vector field $X$ induces a foliation by curves $\mathcal {G}$ on $T_{\mathcal {F}}$ that leaves the zero section invariant. This is exactly the setting of Lehmann's theorem.

4.2 Lehmann's theorem

For the proof of Theorem 4.1, we will use an index theorem due to Lehmann which generalizes the Camacho–Sad index theorem to higher dimensions [Reference LehmannLeh91]. Let us recall it in the generality that will suit our needs. We follow the normalizations and sign conventions found in [Reference SuwaSuw98].

Let $V$ be a manifold of dimension $n+1$, $M\subset V$ a codimension-one smooth compact submanifold with normal bundle $N_M$, and $\mathcal {G}$ a foliation by curves on $V$ leaving $M$ invariant. Suppose that the singularities of the foliation induced by $\mathcal {G}$ on $M$ are isolated. For such a singularity $p$, in coordinates $(z_1,\ldots,z_n,w)$ centered at $p$, where $M$ is given by $w=0$ and $\mathcal {G}$ is induced by $X=\sum _{i=1}^n a_i(z,w)\partial /\partial z_i+w b(z,w) \partial /\partial w$, define

\[ \mathrm{Res}_\mathcal{G}(c_1^n,M, p)=(-1)^n\biggl(\frac{i}{2\pi}\biggr)^n\int_T \frac{b^n(z,0)}{\prod_{i=1}^n a_i(z,0)}\,dz_1\wedge\cdots\wedge dz_n, \]

with $T=\{w=0\}\cap (\bigcap _{i=1}^n\{\|a_i(z,0)\|=\epsilon \})$ for some sufficiently small $\epsilon$. (This number is well-defined.) Lehmann's theorem affirms that

\[ \sum_{p\in\mathrm{Sing}(\mathcal{G}|_M)}\mathrm{Res}_\mathcal{G}(c_1^n,M,p)=c_1^n(N_M), \]

where $c_1(N_M)\in H^2(M,\mathbf {Z})$ denotes the first Chern class of $N_M$.

If the restriction of $X$ to $w=0$ is nondegenerate at $p$ and the eigenvalues of the linear part of this restriction are $\lambda _1,\ldots,\lambda _n$, we have that $\mathrm {Res}_\mathcal {G}(c_1^n,M, p)=b^n(0)(\lambda _1\cdots \lambda _n)^{-1}$.

Proof of Theorem 4.1 Let $\mathcal {G}$ be the foliation on $T_{\mathcal {F}}$ induced by the geodesic vector field $X$. It leaves the zero section $M$ invariant. If $\mathcal {F}$ is generated by $Z=\sum _i a_i(z)\partial /\partial z_i$ in a neighborhood of $p$, $\mathcal {G}$ is, in a neighborhood of $p$ in $T_{\mathcal {F}}$, tangent to the vector field $\sum _{i=1}^n a_i(z)\partial /\partial z_i-\zeta \gamma (z)\partial /\partial \zeta$. If $\gamma (0)\neq 0$ and $\nu _1$, …, $\nu _n$ are the principal affine ramification indices of the foliated affine structure at $p$, then, by (3.1),

\[ \mathrm{Res}_\mathcal{G}(c_1^n,M, p)=\frac{(-\gamma(0))^n}{\lambda_1\cdots\lambda_n}=\frac{(-1)^n}{\nu_1 \cdots \nu_n }. \]

On the other hand, by construction, $N_M$ is exactly $T_{\mathcal {F}}$. A straightforward application of Lehmann's theorem yields Theorem 4.1.

5.1 Some foliated jet bundles

From a vector bundle over a manifold $V\to M$, one may construct the vector bundle $J^kV\to M$ of $k$-jets of sections of $V$ (see, for instance, [Reference Kumpera and SpencerKS72]). In the presence of a singular foliation by curves $\mathcal {F}$ on $M$, we may consider the bundle $J_{\mathcal {F}} ^k V$ of $k$-jets of sections of $V$ ‘along the leaves of $\mathcal {F}$’, which exists, at least, away from the singularities of $\mathcal {F}$. The bundle need not extend to the singular points of the foliation, but if it does, this extension is unique. (See Theorem 1 and Proposition 7 in [Reference SerreSer66].)

For $V=T_{\mathcal {F}}$, the bundle $J_{\mathcal {F}} ^k T_{\mathcal {F}}$ can be defined as a vector bundle (of rank $k+1$) on the whole of $M$. Let us show this concretely by exhibiting explicit local trivializations. Let $Z$ be a vector field tangent to $\mathcal {F}$ defined on an open set $U\subset M$, which vanishes only on the singular set of $\mathcal {F}$, and let $A=\sum ({1}/{l!}) a_lz^l \in \mathbf {C}[z]$ be a polynomial of degree at most $k$. Build a holomorphic section $\sigma (Z,A)$ of $J^k_{\mathcal {F}} T_{\mathcal {F}}$ over $U$ in the following way: at a regular point $p$ of $\mathcal {F}$ in $U$, for the vector field $fZ$ such that $Z^l( f)|_p = a_l$ for $l=0,\ldots, k$, $\sigma (Z,A)(p)$ is the $k$-jet along $\mathcal {F}$ at $p$ of the vector field $f Z$. The section $\sigma (Z,1)$ is the one obtained by taking the $k$-jet of the vector field $Z$ itself. By construction, the sections $\sigma (Z,1), \sigma (Z,z),\ldots, \sigma (Z,z^k)$ are linearly independent at every regular point of $\mathcal {F}$ in $U$, so they induce a trivialization of $J^k_{\mathcal {F}} T_{\mathcal {F}}$ on $U\setminus \mathrm {Sing} (\mathcal {F})$. The existence of these local trivializations suffices to show that the bundle $J_{\mathcal {F}} ^k T_{\mathcal {F}}$ can be extended over the singular set in a unique way, because the cocycle that expresses one trivialization in terms of the other consists of holomorphic functions $U \setminus \mathrm {Sing} (\mathcal {F})\rightarrow \mathrm {GL}(k+1,\mathbf {C})$ that, by Hartogs's theorem, extend to the singular set as functions taking values in the set of invertible matrices (the determinant cannot vanish only on a set of codimension two or more). We thus construct $J^k_{\mathcal {F}}T_{\mathcal {F}}\to M$, the vector bundle of $k$-jets of $T_{\mathcal {F}}$ along $\mathcal {F}$. We have the linear projections $j_{k-i}:J^k_{\mathcal {F}}T_{\mathcal {F}}\to J^{k-i}_{\mathcal {F}}T_{\mathcal {F}}$, and, naturally, $J^0_{\mathcal {F}}T_{\mathcal {F}}=J^0T_{\mathcal {F}}=T_{\mathcal {F}}$.

We are interested in the case $k=1$, for which it is useful to have an explicit expression for the cocycle defined above. Let $\{U_i\}_{i\in I}$ be a cover of $M$ by open subsets such that $\mathcal {F}$ is generated by the holomorphic vector field $Z_i$ in $U_i$. Let $g_{ij}:U_i\cap U_j\to \mathbf {C}^*$ be such that $Z_i=g_{ij}Z_j$; it is the cocycle associated to $T_{\mathcal {F}}$. Let $p\in U\setminus \mathrm {Sing}(\mathcal {F})$, and let $z$ be a coordinate along the leaf $L$ of $\mathcal {F}$ through $p$, centered at $p$, where the restriction of $Z_j$ to $L$ reads $\partial /\partial z$. In restriction to $L$, $Z_i={g_{ij}}(z)\partial /\partial z$ and, thus, along $L$,

\[ Z_i=\sum_{l=0}^\infty \frac{1}{l!}\biggl(\frac{d^l}{d\zeta^l}g_{ij}(\zeta)\bigg|_{\zeta=0}\biggr)z^l\frac{\partial}{\partial z} =\sum_{l=0}^\infty \frac{1}{l!} Z_j^l(g_{ij})|_{p} z^l\frac{\partial}{\partial z}. \]

On the other hand, $\sigma (Z, z)$ is independent of $Z$: if a vector $X$ field tangent to $\mathcal {F}$ vanishes at a nonsingular point $p$ in the leaf $L$ of $\mathcal {F}$, its first jet along $\mathcal {F}$ at $p$ is the eigenvalue of the linear part at $p$ of its restriction to $L$. From these facts, on $U_i\cap U_j$,

\[ \biggl(\begin{array}{@{}c@{}} \sigma(Z_i,1) \\ \sigma(Z_i,z) \end{array}\biggr)=\biggl(\begin{array}{@{}cc@{}} g_{ij} & Z_j(g_{ij}) \\ 0 & 1 \end{array}\biggr)\biggl(\begin{array}{@{}c@{}} \sigma(Z_j,1) \\ \sigma(Z_j,z) \end{array}\biggr). \]

Summing up, $J^1_{\mathcal {F}}T_{\mathcal {F}}$ can be obtained by gluing the sets in $\{U_i\times \mathbf {C}^2\}_{i\in I}$ through the identifications

(5.3) \begin{equation} \biggl(\begin{array}{@{}c@{}} \zeta_j \\ \xi_j \end{array}\biggr)=\biggl(\begin{array}{@{}cc@{}} g_{ij} & 0 \\ Z_j(g_{ij}) & 1 \end{array}\biggr)\biggl(\begin{array}{@{}c@{}} \zeta_i \\ \xi_i \end{array}\biggr), \end{equation}

with $(\zeta _i,\xi _i)$ linear coordinates on $\mathbf {C}^2$. In these, the section of $J^1_{\mathcal {F}}T_{\mathcal {F}}$ over $U_i$ generated by $fZ_i$ is given by

(5.4)\begin{equation} fZ_i\mapsto (f,Z_i(f)). \end{equation}

The projections $(\zeta _i,\xi _i)\mapsto \zeta _i$ glue into the natural linear projection $j_0:J^1_{\mathcal {F}}T_{\mathcal {F}}\to T_{\mathcal {F}}$. The line subbundle $\ker (j_0)$ is trivial.

5.2 The geodesic vector field and its projectivization

If $\mathcal {F}$ is endowed with a foliated projective structure and $p\in M\setminus \mathrm {Sing}(\mathcal {F})$, a geodesic through $p$ is a parametrized curve $f:(U,0)\to (M,p)$, $U\subset \mathbf {C}$, $0\in U$, which is tangent to $\mathcal {F}$ and which induces the given projective structure on the leaf of $\mathcal {F}$ through $p$ (which is the inverse of a projective chart). For the tautological projective structure on $\mathbf {P}^1$, in the affine chart $[z:1]$, the geodesics through $0$ are those of the form $t\mapsto at/(1-bt)$ with $a\in \mathbf {C}^*$, $b\in \mathbf {C}$. Their corresponding velocity vector fields are $a^{-1}(a+bz)^2\partial /\partial z$. Each one of them is characterized by its $1$-jet at $0$. Furthermore, every $1$-jet of vector field with a nonvanishing $0$-jet is realized by the velocity vector field of a geodesic. In this way, above the regular part of $\mathcal {F}$, the geodesics of a foliated projective structure lift into $J^1_{\mathcal {F}} T_{\mathcal {F}} \setminus \ker (j_0)$ through their velocity vector fields, and there is a lift of a unique geodesic through every point in $J^1_{\mathcal {F}} T_{\mathcal {F}}\setminus \ker (j_0)$. There is, thus, a natural vector field on $J^1_{\mathcal {F}} T_{\mathcal {F}} \setminus \ker (j_0)$ associated to a foliated projective structure. This is its geodesic vector field.

If $f:U\to M$ is a geodesic defined in a neighborhood of $t=0$ and $\big (\begin {smallmatrix}a & b \\ c & d \end {smallmatrix}\big )\in \mathrm {SL}(2,\mathbf {C})$ is sufficiently close to the identity, $\displaystyle f(({at+b})/({ct+d}))$ is also a geodesic: we have a local action of $\mathrm {SL}(2,\mathbf {C})$ on $J^1_{\mathcal {F}} T_{\mathcal {F}} \setminus \ker (j_0)$ induced by the foliated projective structure. Let us restrict this action to the one-parameter subgroups of the standard basis of $\mathfrak {sl}(2,\mathbf {C})$. The vector fields on $J^1_{\mathcal {F}} T_{\mathcal {F}}$ associated to these will satisfy the same Lie-algebraic relations. Let $f:U\to M$ be a geodesic. That the action via reparametrization of the one-parameter subgroup $\big (\begin {smallmatrix} 1 & s \\ 0 & 1 \end {smallmatrix}\big )$ preserves geodesics is equivalent to the fact that if $f(t)$ is a parameterized geodesic, so is $f(t+s)$ for every (sufficiently small) fixed $s$. This reparametrization comes thus from the flow of the geodesic vector field. The one induced by the one-parameter subgroup $\big (\begin {smallmatrix}e^{s/2} & 0 \\ 0 & e^{-s/2} \end {smallmatrix}\big )$ yields the geodesic $t\mapsto f(e^st)$. It multiplies the velocity vector field of $f$ by the factor $e^s$. In coordinates, it multiplies each one of the two components of (5.4) by $e^s$, and is thus induced by the vector field $H$ on $J^1_{\mathcal {F}} T_{\mathcal {F}}$ that, in $U_j\times \mathbf {C}^2$, reads $\zeta _j\partial /\partial \zeta _j\oplus \xi _j\partial /\partial \xi _j$ (it retains its expression in the other charts and is globally well-defined). Now let us consider the more interesting case of the one-parameter subgroup $\big (\begin {smallmatrix} 1 & 0 \\ -s & 1 \end {smallmatrix}\big )$. Let us suppose that we are on a curve where we have a local coordinate $z$ and that $f(0)=0$. The reparametrization gives the geodesic $\displaystyle f( {t}/({1-st}))$, whose velocity vector field is $(1+sf^{-1}(z))f'(f^{-1}(z))\partial /\partial z$. With respect to the vector field $Z=v(z)\partial /\partial z$, the $1$-jet of its velocity vector field, as in (5.4), is

\[ \biggl(\frac{f(0)}{v(0)},-\frac{v'(0)}{v(0)}+\frac{f''(0)}{f'(0)}+2s\biggr). \]

The action of this one-parameter subgroup induces the vector field $Y$ on $J^1_{\mathcal {F}} T_{\mathcal {F}}$ that reads $2\partial /\partial \xi _j$ on $U_j\times \mathbf {C}^2$. As expected, $[H,Y]=-Y$.

If $X$ is a vector field on $J^1_{\mathcal {F}} T_{\mathcal {F}}$ giving the geodesic vector field of a foliated projective structure, its integral curves project to curves of $\mathcal {F}$ and, together with the vector fields $H$ and $Y$, satisfies the $\mathfrak {sl}(2,\mathbf {C})$ relations

(5.5)\begin{equation} [Y,X]=2H,\quad [H,X]=X,\quad [H,Y]=-Y. \end{equation}

The integral form of these relations gives the reparametrization of the geodesics on $J^1_{\mathcal {F}} T_{\mathcal {F}}$: if $(z(t),\zeta _j(t),\xi _j(t))$ is a solution of $X_j$ and $\big (\begin {smallmatrix} a & b \\ c & d \end {smallmatrix}\big )\in \mathrm {SL}(2,\mathbf {C})$ is sufficiently close to the identity,

\[ \biggl(z\biggl(\frac{at+b}{ct+d}\biggr), \frac{1}{(ct+d)^2}\zeta_j\biggl(\frac{at+b}{ct+d}\biggr), \frac{1}{(ct+d)^2}\xi_j\biggl(\frac{at+b}{ct+d}\biggr)-\frac{2c}{ct+d}\biggr) \]

is also a solution of $X$.

Let us explicitly construct the vector field $X$ associated to a foliated projective connection $\Xi :T_{\mathcal {F}}\to \mathcal {O}_M$. Let $\rho _i:U_i\to \mathbf {C}$ be the Christoffel symbol $\Xi (Z_i)$ of $Z_i$. Consider, on $U_j\times \mathbf {C}^2$, the vector field

\[ X_j=\zeta_j Z_j\oplus \zeta_j\xi_j\frac{\partial}{\partial \zeta_j}\oplus\biggl(\frac{1}{2}\xi_j^2-\rho_j\zeta_j^2\biggr)\frac{\partial}{\partial \xi_j}. \]

In $(U_i\times \mathbf {C}^2)\cap (U_j\times \mathbf {C}^2)$, under (5.3), $X_j$ reads

\[ g_{ij}\zeta_i Z_j\oplus \zeta_i\xi_i\frac{\partial}{\partial \zeta_i}\oplus\biggl(\frac{1}{2}\xi_i^2-\left[g_{ij}^2 \rho_j+g_{ij}Z_j^2 g_{ij} -\frac{1}{2}(Z_jg_{ij})^2 \right]\zeta_i^2\biggr)\frac{\partial}{\partial \xi_i}, \]

which, by the Leibniz rule (2.6), is exactly $X_i$. Thus, these vector fields glue into a globally-defined holomorphic vector field $X$ on $J^1_{\mathcal {F}} T_{\mathcal {F}}$. It satisfies the relations (5.5); since the projections of its integral curves onto $M$ differ by precompositions with fractional linear transformations, it induces a foliated projective structure.

Let us identify the foliated projective structure induced by the vector field $X$ just defined. Let us do so in dimension one, in a coordinate $z$ where $Z$ is $\partial /\partial z$, this is, for the vector field $X=\zeta \partial /\partial z+ \zeta \xi \partial /\partial \zeta +(\frac {1}{2}\xi ^2-\rho \zeta ^2)\partial /\partial \xi$ and $\widehat {\pi }(z,\zeta,\xi )=z$. Let $(z(t),\zeta (t),\xi (t))$ be a solution to $X$. Comparing the projective structures induced by $z$ and $t$ on the base, we have

\[ \{t,z(t)\}=-\frac{1}{(z'(t))^2}\{z(t),t\}=\rho(z). \]

It follows from this formula that for the projective structure induced by $X$, the Christoffel symbol of $\partial /\partial z$ is $\rho$, and coincides, as we sought to establish, with the one induced by the projective connection $\Xi$.

Note that $X$ is defined as a holomorphic vector field on all of $J^1_{\mathcal {F}} T_{\mathcal {F}}$ and that it is transverse to $\pi$ over $M\setminus \mathrm {Sing}(\mathcal {F})$ away from $\ker (j_0)$.