# Semicomplete vector fields on non-Kahler surfaces.

1. Introduction. In the beginning of the 20th century, Painleve
drew attention to the general problem of determining the algebraic
ordinary differential equations whose general solution is uniform,
starting from those of smaller orders [14]. Differential equations given
by rational vector fields on algebraic manifolds of dimension n are
natural examples of such equations (they give autonomous algebraic
differential equations of order n). Already for these, and already in
small dimensions, Painleve's program is far from being achieved.

More generally, one may consider this problem for meromorphic vector fields on general compact complex manifolds, and not only algebraic ones. For meromorphic vector fields on compact complex Kahler surfaces, the situation is well understood. These are the subject of Rebelo and the author's Theorem B in [6]:

Theorem 1.1. Let X be a semicomplete meromorphic vector field on the compact complex Kahler surface S. Then, up to a bimeromorphic transformation, X is holomorphic, X has a first integral or S has a rational or elliptic fibration preserved by X (with each component of the locus of poles of X contained in a fiber).

(In it, Rebelo's notion of semicompleteness [15] is used to formalize the notion of "vector field whose general solution is uniform"). There remained the problem of understanding those semicomplete meromorphic vector fields on complex compact surfaces which are not Kahler. The aim of this article is to extend Theorem 1.1 to the non-Kahler case. Our result is the following one:

Theorem 1.2. Let S be a compact complex non-Kahler surface, X a semicomplete meromorphic vector field on S. Up to a bimeromorphic transformation, either X is holomorphic, X has a first integral or there is an elliptic fibration preserved by X (with each component of the locus of poles of X contained in a fiber).

We will show that, under the hypothesis of the theorem, if the algebraic dimension of S is one, its algebraic reduction gives either a first integral of X or a fibration preserved by X; if it is zero, we will prove that, up to a bimeromorphic transformation, X is holomorphic (and is thus to be found within the classification of holomorphic vector fields on compact complex surfaces [4, Thm. 0.3]).

Together, Theorems 1.1 and 1.2 describe semicomplete meromorphic vector fields on all compact complex surfaces.

2. Proof of the theorem. We begin the proof of Theorem 1.2. Let S be a non-Kahler compact complex surface and X a semicomplete meromorphic vector field on S. Since semicompleteness is a birational invariant [6, Cor. 12], we may suppose that S is minimal and that X is strictly meromorphic (for otherwise it would automatically satisfy Theorem 1.2). Since S is not Kahler, its algebraic dimension, that we will denote by a(S), is strictly smaller than two [1, Ch. IV, Section 5].

If a(S) = 1, there is an algebraic reduction of S, a fibration [PI] : S [right arrow] B such that B is an algebraic curve and such that any meromorphic function on S is the pullback by n of an algebraic function on B. Its generic fiber is an elliptic curve and every curve in S is in a fiber (Theorems 4.2 and 4.3 in [9, Section 4]). The vector field X naturally induces a meromorphic vector field Y on B such that [[PI].sub.*]X = Y ([PI] is a first integral if Y [equivalent to] 0). The curve of poles of X is contained in the fibers of n and thus, since X is holomorphic in a neighborhood of a generic fiber, the fibration is naturally preserved by X. Further, since X is semicomplete, so is Y and is thus holomorphic [6, Lemma 2]. This proves Theorem 1.2 in this case. Example 3.1 will illustrate this situation.

We will henceforth suppose that a(S) = 0. Our aim is to prove that the vector field is, up to a bimeromorphic transformation, holomorphic. Let [b.sub.i](S) denote the i-th Betti number of S. A theorem of Kodaira [10, Thm. 11] states that if S is a non-Kahler compact complex surface of vanishing algebraic dimension then [b.sub.1](S) = 1 (S belongs to the class [VII.sub.0]). Two cases appear:

First case, [b.sub.2](S) = 0. Since X is strictly meromorphic, the curve of poles of X is a nonempty curve on S. A theorem of Kodaira affirms that if S is a minimal surface with no nonconstant meromorphic functions with [b.sub.1](S) = 1, [b.sub.2](S) = 0, and containing at least one curve, S is a Hopf surface [11, Thm. 34]. We will prove that a meromorphic vector field on a Hopf surface of vanishing algebraic dimension is holomorphic. The Hopf surface has a finite nonramified cover [??] that is the quotient of [C.sup.2] \{0} under the action of (z, w) [??] ([alpha]z + [lambda][w.sup.n], [beta]w), with either [lambda] = 0 or [alpha] = [[beta].sup.n] (further, since we are supposing that the algebraic dimension is zero, [[alpha].sup.j] [not equal to] [[beta].sup.k] if [lambda] = 0). Let [??] be the lift of X to [??]. The surface [??] admits two holomorphic vector fields [Y.sub.1] and [Y.sub.2], linearly independent almost everywhere: if [lambda] = 0, they are given by z[partial derivative]/[partial derivative]z and w[partial derivative]/[partial derivative]w; if [lambda] [not equal to] 0, by nz[partial derivative]/[partial derivative]z + w[partial derivative]/[partial derivative]w and [w.sup.n][partial derivative]/[partial derivative]z. There exist meromorphic functions [f.sub.1] and [f.sub.2] on [??] such that [??] = [f.sub.1][Y.sub.1] + [f.sub.2][Y.sub.2]. Since [f.sub.1] and [f.sub.2] are necessarily constants, [??] is holomorphic and thus X is holomorphic as well.

Second case, [b.sub.2](S) > 0. The semicompleteness hypothesis on X will allow us to give a more precise description of its curve of poles. The following is a key element in the proof of Theorem 1.1 and involves no global hypothesis on S:

Theorem 2.1 ([6], Thm. A). Let X be a semicomplete meromorphic vector field on the surface S, Z a compact connected component of the locus of poles of X. Up to a bimeromorphic transformation, Z is either empty, a rational curve of vanishing self-intersection, or supports a divisor D of elliptic fiber type, a divisor such that D x D = 0 and [chi](D) = 0.

Enoki constructed some minimal compact complex surfaces [S.sub.n,[alpha],t] (n > 0, 0 < [absolute value of [alpha]] < 1, t [member of] [C.sup.n]), generally called Enoki surfaces. They are non-Kahler surfaces with [b.sub.1]([S.sub.n,[alpha],t]) = 1 and [b.sub.2]([S.sub.n,[alpha],t]) = n. Each one of them has a cycle formed by n rational curves [C.sub.i], i [member of] Z/nZ, with [C.sub.i] x [C.sub.i+1] = 1, [C.sub.i] x [C.sub.i] = -2 and [C.sub.i] x [C.sub.j] = 0 if [absolute value of i - j] > 1. The surface [S.sub.n,[alpha],t] has the divisor [D.sub.n,[alpha],t] = [[summation].sub.i][C.sub.i]. It is of vanishing self-intersection and has the combinatorics of Kodaira's elliptic fiber [I.sub.n].

The particular cases [S.sub.n,[alpha],0] are unramified covers of some surfaces previously constructed by Inoue [7], and are called parabolic Inoue surfaces. All of these parabolic Inoue surfaces have holomorphic vector fields and one elliptic curve [2, Thm. 1.31]. Reciprocally, if an Enoki surface has a holomorphic vector field or a curve other than those in the support of [D.sub.n,[alpha],t], it is a parabolic Inoue surface [13, Thm. 7.1].

Enoki provided the following characterization of these surfaces [5]:

Theorem 2.2 (Enoki). Let S be a minimal compact complex surface with [b.sub.1](S) = 1 and [b.sub.2](S) = n. If S has a divisor D [not equal to] 0 with D x D = 0, then S is biholomorphic to [S.sub.n,[alpha],t] for some n, [alpha], t and D = m[D.sub.n,[alpha],t] for some m [not equal to] 0.

Theorems 2.1 and 2.2 imply that if S is a compact complex surface with [b.sub.1](S) = 1 and [b.sub.2](S) [not equal to] 0 endowed with a semicomplete meromorphic vector field X, its minimal model is an Enoki surface, and the curve of poles of X is the support of the divisor Dn, a, t. Theorem 1.2 is a consequence of the following

Proposition 2.1. Let S be an Enoki surface, X a meromorphic vector field on S. Then X is holomorphic (in particular, S is a parabolic Inoue surface).

Before proceeding to the proof of this proposition, let us recall some facts about Enoki surfaces and their foliations. Enoki surfaces belong to the class [VII.sub.0]; they contain global spherical shells and may thus be obtained by Kato's construction [8]. Let us go through this construction following [2]. Let [B.sub.0] = {v [member of] [C.sup.2]; [absolute value of v] [less than or equal to] [epsilon]} be the closed ball and consider the sequence of blowups

[mathematical expression not reproducible],

where [[PI].sub.i] is the blowup of [p.sub.i-1], [p.sub.0] = 0, [C.sub.i] = [[PI].sup.-1.sub.i]([p.sub.i-1]), [p.sub.i] [member of] [C.sub.i]. Let [PI]: [B.sub.n] [right arrow] [B.sub.0], [PI] = [[PI].sub.n] [omicron] ... [omicron] [[PI].sub.1]. Let [sigma] : [B.sub.0] [right arrow] [B.sub.n] such that, [sigma]([B.sub.0]) is in the interior of [B.sub.n]. Let [p.sub.n] = [sigma](0), and suppose that [p.sub.n] [member of] [C.sub.n]. The Kato surface S is the compact complex surface resulting from the identification of the two boundary components of [B.sub.n] \ [sigma](int([B.sub.0])) --considered as a real four-dimensional manifold-with-boundary-- by the map [sigma] [omicron] [PI].

The above data can be recovered from the germ of F = [PI] [omicron] [sigma] at its fixed point 0. There is a natural correspondence between the objects in ([C.sup.2]; 0) that are invariant by F and the objects in S.

The Kato surface of the above construction is an Enoki surface if, furthermore [2, Thm. 3.33],

(a) [p.sub.i] [not member of] [[union].sub.j<i][C.sub.j] and

(b) [p.sub.1] is not in the strict transform of [[sigma].sup.-1]([C.sub.n]) under [[PI].sub.1],

or, equivalently, if the trace of DF[|.sub.0] is nonzero [2, Thm. 3.30].

We will henceforth assume that S is an Enoki surface. In suitable coordinates (x, y), the germ of F may be written as

(1) F(x, y) = (x[y.sup.n] + P(y), ty),

for some t [member of] C, 0 < [absolute value of t] < 1, and some polynomial P of degree n - 1 [3, Thm. 1.19]. Here, y = 0 is the curve that maps via [sigma] into [C.sub.n]. The parabolic Inoue case corresponds to P [equivalent to] 0 in (1); in this case, both the holomorphic vector field x[partial derivative]/[partial derivative]x and the germ of curve x = 0 are preserved by F and induce, respectively, a holomorphic vector field Y and an elliptic curve in S. In all cases, since the meromorphic one-form dy/y on ([C.sup.2], 0) is invariant by F, it induces a global meromorphic one-form on S. On its turn, the kernel of this form induces a holomorphic foliation with singularities F on S (compare with [13, Section 3]).

According to [12, Section 2.3], an Enoki surface has no foliation other than this foliation F. Let us give a short proof of this fact. Let G be a foliation on S. By Levi's extension principle, there exists a meromorphic one-form [omega] = [alpha]dx + [beta]dy on ([C.sup.2], 0) generating it. In order for G to be preserved by F, [F.sup.*][omega] = h[omega] for some meromorphic function h. If [beta] [equivalent to] 0, [omega] = [alpha]dx and [F.sup.*]([alpha]dx) = ([alpha] [omicron] F)d[F.sub.1] = ([alpha] [omicron] F)[([partial derivative][F.sub.1]/[partial derivative]x)dx + ([partial derivative][F.sub.1]/[partial derivative]y)dy]. From formula (1), [partial derivative][F.sub.1]/[partial derivative]y does not vanish identically, and G cannot be preserved. If [beta] is not identically zero, up to multiplying by a meromorphic function, we may suppose that [omega] = [alpha]dx + dy/y. Since dy/y is invariant by F, we must have that h [equivalent to] 1 and that the one-form [alpha]dx is preserved. Repeating the previous arguments shows that this is impossible unless [alpha] [equivalent to] 0. This proves the uniqueness of the foliation.

The construction of the Enoki surface and of its foliation can be done simultaneously. Consider a nonsingular foliation [F.sub.0] on ([C.sup.2], 0). Under the blowup of 0 by [[PI].sub.1], this regular point of the foliation produces an exceptional divisor [C.sub.1] that is invariant by the induced foliation [F.sub.1]. There is only one singular point of the transformed foliation [F.sub.1] along [C.sub.1]. Let [p.sub.2] be a regular point of [F.sub.1] in [C.sub.1] and continue the construction of the Enoki surface until we have a foliation [F.sub.n] on [B.sub.n]. Let [sigma] map [F.sub.0] to [F.sub.n] (mapping {y = 0} to [C.sub.n]) and suppose that F is contracting. Through Kato's construction, this produces a general Enoki surface with a foliation (notice that condition (b) in the construction of the Enoki surface is automatically satisfied). The only singularities of the foliation are at the intersection of the divisors, where the foliation has, locally, a first integral (the holonomy is trivial).

Let us now come to the proof of Proposition 2.1. Let S be an Enoki surface, D its divisor and let X be a meromorphic vector field on S. If S is a parabolic Inoue surface, it has a holomorphic and nonzero vector field Y. Since X and Y are collinear and since S has no meromorphic functions, X and Y differ by a multiplicative constant, and X is holomorphic. Let us thus assume that S is not a parabolic Inoue surface and, in particular, that it has no curves other than those of the cycle in the support of D. Let X be a meromorphic vector field on S. By Levi's extension principle, it is associated to a meromorphic vector field in [B.sub.0]. Since the foliation it induces is the unique foliation on the Enoki surface, the vector field is of the form [X.sub.0] = g(x, y)[partial derivative]/[partial derivative]x for some meromorphic function g. In order for this vector field to induce a global one, we should have

(2) [[PI].sup.-1.sub.*][X.sub.0] = [[sigma].sub.*][X.sub.0].

In particular, the curves of zeros and poles of g other than y = 0 are preserved by F and induce curves in S different from [C.sub.1], ..., [C.sub.n]. Since we supposed that there are no further curves in S, [X.sub.0] must actually be of the form h(x, y)[y.sup.q][partial derivative]/[partial derivative]x, with h holomorphic and nonzero, q [member of] Z. In the chart of the blowup (x, y) = (sy, y), the vector field reads h(sy, y)[y.sup.q-1][partial derivative]/[partial derivative]s. (As a function of s and y, h(sy, y) is holomorphic and nonzero at the origin.) After n blowups, the order of the transformed vector field along [C.sub.n] is q - n. However, in order to satisfy (2), this number must equal q. This contradiction proves Proposition 2.1 and finishes the proof of Theorem 1.2.

3. An example and a remark.

Example 3.1. Consider the meromorphic vector field [??] = y[x.sup.-1](y[partial derivative]/ [partial derivative]y--x[partial derivative]/[partial derivative]x) on [C.sup.2]. Outside {xy = 0}, it has the solutions t [??] (-2[ct.sup.1/2], [ct.sup.-1/2]). Let [??] : [C.sup.2] \{0} [right arrow] [P.sup.1] be given by (x, y) [??] x/y. The image of [??] under [PI] is the vector field Y = 2[partial derivative]/[partial derivative][xi]. Let S be the secondary Hopf surface obtained as the quotient of [C.sup.2] \{0} under the action of the group generated by the maps (x, y) [??] (2x , 2y) and [mathematical expression not reproducible]. These maps preserve [??] and [??] and there is thus a well-defined vector field X on S and a map (elliptic fibration) [PI]: S [right arrow] [P.sup.1], [[PI].sub.*](X) = Y. The elliptic curves in S coming from {x = 0} and {y = 0} are, respectively, the curves of zeros and poles of X. Outside these curves, where X is holomorphic, the solutions are single-valued (a cancels the multi-valuedness of the solutions of [??]).

Remark 3.1. Enoki surfaces do not have strictly meromorphic vector fields, although they are not far from doing so. Consider the vector field x[y.sup.q][partial derivative]/[partial derivative]x, q [member of] Z. In the chart (x, y) = (sy, y) of the blowup of 0, the vector field reads again s[y.sup.q][partial derivative]/[partial derivative]s, so that, after n blowups, the vector field is equivalent to the original one. Equivalently, the vector field is preserved under the local ramified maps (x, y) [??] (x[y.sup.n], y). However, the derivative at the origin of these maps has trace equal to 1, and they are not contracting.

doi: 10.3792/pjaa.93.73

Acknowledgements. The author is partially funded by PAPIIT-UNAM IN108214 (Mexico). This article was finished during a sabbatical leave at the Ecole Normale Superieure (France), under the support of PASPA-DGAPA-UNAM (Mexico). The author thanks both institutions.

References

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[3] G. Dloussky and F. Kohler, Classification of singular germs of mappings and deformations of compact surfaces of class [VII.sub.0], Ann. Polon. Math. 70 (1998), 49-83.

[4] G. Dloussky, K. Oeljeklaus and M. Toma, Surfaces de la classe [VII.sub.0] admettant un champ de vecteurs. II, Comment. Math. Helv. 76 (2001), no. 4, 640-664.

[5] I. Enoki, Surfaces of class [VII.sub.0] with curves, Tohoku Math. J. (2) 33 (1981), no. 4, 453-492.

[6] A. Guillot and J. Rebelo, Semicomplete meromorphic vector fields on complex surfaces, J. Reine Angew. Math. 667 (2012), 27-65.

[7] M. Inoue, New surfaces with no meromorphic functions, in Proceedings of the International Congress of Mathematicians (Vancouver, B. C, 1974), Vol. 1, 423-426, Canad. Math. Congress, Montreal, QC, 1975.

[8] Ma. Kato, Compact complex manifolds containing "global" spherical shells. I, in Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), 45-84, Kinokuniya Book Store, Tokyo, 1978.

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[13] I. Nakamura, On surfaces of class [VII.sub.0] with curves, Invent. Math. 78 (1984), no. 3, 393-443.

[14] P. Painleve, Sur les equations differentielles du second ordre et d'ordre superieur dont l'integrale generale est uniforme, Acta Math. 25 (1902), no. 1, 1-85.

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By Adolfo GUILLOT

Instituto de Matematicas, Unidad Cuernavaca, Universidad Nacional Autonoma de Mexico, AP273. Admon. de correos 3, Cuernavaca, Morelos 62251, Mexico

(Communicated by Shigefumi MORI, M.J.A., Sept. 12, 2017)

More generally, one may consider this problem for meromorphic vector fields on general compact complex manifolds, and not only algebraic ones. For meromorphic vector fields on compact complex Kahler surfaces, the situation is well understood. These are the subject of Rebelo and the author's Theorem B in [6]:

Theorem 1.1. Let X be a semicomplete meromorphic vector field on the compact complex Kahler surface S. Then, up to a bimeromorphic transformation, X is holomorphic, X has a first integral or S has a rational or elliptic fibration preserved by X (with each component of the locus of poles of X contained in a fiber).

(In it, Rebelo's notion of semicompleteness [15] is used to formalize the notion of "vector field whose general solution is uniform"). There remained the problem of understanding those semicomplete meromorphic vector fields on complex compact surfaces which are not Kahler. The aim of this article is to extend Theorem 1.1 to the non-Kahler case. Our result is the following one:

Theorem 1.2. Let S be a compact complex non-Kahler surface, X a semicomplete meromorphic vector field on S. Up to a bimeromorphic transformation, either X is holomorphic, X has a first integral or there is an elliptic fibration preserved by X (with each component of the locus of poles of X contained in a fiber).

We will show that, under the hypothesis of the theorem, if the algebraic dimension of S is one, its algebraic reduction gives either a first integral of X or a fibration preserved by X; if it is zero, we will prove that, up to a bimeromorphic transformation, X is holomorphic (and is thus to be found within the classification of holomorphic vector fields on compact complex surfaces [4, Thm. 0.3]).

Together, Theorems 1.1 and 1.2 describe semicomplete meromorphic vector fields on all compact complex surfaces.

2. Proof of the theorem. We begin the proof of Theorem 1.2. Let S be a non-Kahler compact complex surface and X a semicomplete meromorphic vector field on S. Since semicompleteness is a birational invariant [6, Cor. 12], we may suppose that S is minimal and that X is strictly meromorphic (for otherwise it would automatically satisfy Theorem 1.2). Since S is not Kahler, its algebraic dimension, that we will denote by a(S), is strictly smaller than two [1, Ch. IV, Section 5].

If a(S) = 1, there is an algebraic reduction of S, a fibration [PI] : S [right arrow] B such that B is an algebraic curve and such that any meromorphic function on S is the pullback by n of an algebraic function on B. Its generic fiber is an elliptic curve and every curve in S is in a fiber (Theorems 4.2 and 4.3 in [9, Section 4]). The vector field X naturally induces a meromorphic vector field Y on B such that [[PI].sub.*]X = Y ([PI] is a first integral if Y [equivalent to] 0). The curve of poles of X is contained in the fibers of n and thus, since X is holomorphic in a neighborhood of a generic fiber, the fibration is naturally preserved by X. Further, since X is semicomplete, so is Y and is thus holomorphic [6, Lemma 2]. This proves Theorem 1.2 in this case. Example 3.1 will illustrate this situation.

We will henceforth suppose that a(S) = 0. Our aim is to prove that the vector field is, up to a bimeromorphic transformation, holomorphic. Let [b.sub.i](S) denote the i-th Betti number of S. A theorem of Kodaira [10, Thm. 11] states that if S is a non-Kahler compact complex surface of vanishing algebraic dimension then [b.sub.1](S) = 1 (S belongs to the class [VII.sub.0]). Two cases appear:

First case, [b.sub.2](S) = 0. Since X is strictly meromorphic, the curve of poles of X is a nonempty curve on S. A theorem of Kodaira affirms that if S is a minimal surface with no nonconstant meromorphic functions with [b.sub.1](S) = 1, [b.sub.2](S) = 0, and containing at least one curve, S is a Hopf surface [11, Thm. 34]. We will prove that a meromorphic vector field on a Hopf surface of vanishing algebraic dimension is holomorphic. The Hopf surface has a finite nonramified cover [??] that is the quotient of [C.sup.2] \{0} under the action of (z, w) [??] ([alpha]z + [lambda][w.sup.n], [beta]w), with either [lambda] = 0 or [alpha] = [[beta].sup.n] (further, since we are supposing that the algebraic dimension is zero, [[alpha].sup.j] [not equal to] [[beta].sup.k] if [lambda] = 0). Let [??] be the lift of X to [??]. The surface [??] admits two holomorphic vector fields [Y.sub.1] and [Y.sub.2], linearly independent almost everywhere: if [lambda] = 0, they are given by z[partial derivative]/[partial derivative]z and w[partial derivative]/[partial derivative]w; if [lambda] [not equal to] 0, by nz[partial derivative]/[partial derivative]z + w[partial derivative]/[partial derivative]w and [w.sup.n][partial derivative]/[partial derivative]z. There exist meromorphic functions [f.sub.1] and [f.sub.2] on [??] such that [??] = [f.sub.1][Y.sub.1] + [f.sub.2][Y.sub.2]. Since [f.sub.1] and [f.sub.2] are necessarily constants, [??] is holomorphic and thus X is holomorphic as well.

Second case, [b.sub.2](S) > 0. The semicompleteness hypothesis on X will allow us to give a more precise description of its curve of poles. The following is a key element in the proof of Theorem 1.1 and involves no global hypothesis on S:

Theorem 2.1 ([6], Thm. A). Let X be a semicomplete meromorphic vector field on the surface S, Z a compact connected component of the locus of poles of X. Up to a bimeromorphic transformation, Z is either empty, a rational curve of vanishing self-intersection, or supports a divisor D of elliptic fiber type, a divisor such that D x D = 0 and [chi](D) = 0.

Enoki constructed some minimal compact complex surfaces [S.sub.n,[alpha],t] (n > 0, 0 < [absolute value of [alpha]] < 1, t [member of] [C.sup.n]), generally called Enoki surfaces. They are non-Kahler surfaces with [b.sub.1]([S.sub.n,[alpha],t]) = 1 and [b.sub.2]([S.sub.n,[alpha],t]) = n. Each one of them has a cycle formed by n rational curves [C.sub.i], i [member of] Z/nZ, with [C.sub.i] x [C.sub.i+1] = 1, [C.sub.i] x [C.sub.i] = -2 and [C.sub.i] x [C.sub.j] = 0 if [absolute value of i - j] > 1. The surface [S.sub.n,[alpha],t] has the divisor [D.sub.n,[alpha],t] = [[summation].sub.i][C.sub.i]. It is of vanishing self-intersection and has the combinatorics of Kodaira's elliptic fiber [I.sub.n].

The particular cases [S.sub.n,[alpha],0] are unramified covers of some surfaces previously constructed by Inoue [7], and are called parabolic Inoue surfaces. All of these parabolic Inoue surfaces have holomorphic vector fields and one elliptic curve [2, Thm. 1.31]. Reciprocally, if an Enoki surface has a holomorphic vector field or a curve other than those in the support of [D.sub.n,[alpha],t], it is a parabolic Inoue surface [13, Thm. 7.1].

Enoki provided the following characterization of these surfaces [5]:

Theorem 2.2 (Enoki). Let S be a minimal compact complex surface with [b.sub.1](S) = 1 and [b.sub.2](S) = n. If S has a divisor D [not equal to] 0 with D x D = 0, then S is biholomorphic to [S.sub.n,[alpha],t] for some n, [alpha], t and D = m[D.sub.n,[alpha],t] for some m [not equal to] 0.

Theorems 2.1 and 2.2 imply that if S is a compact complex surface with [b.sub.1](S) = 1 and [b.sub.2](S) [not equal to] 0 endowed with a semicomplete meromorphic vector field X, its minimal model is an Enoki surface, and the curve of poles of X is the support of the divisor Dn, a, t. Theorem 1.2 is a consequence of the following

Proposition 2.1. Let S be an Enoki surface, X a meromorphic vector field on S. Then X is holomorphic (in particular, S is a parabolic Inoue surface).

Before proceeding to the proof of this proposition, let us recall some facts about Enoki surfaces and their foliations. Enoki surfaces belong to the class [VII.sub.0]; they contain global spherical shells and may thus be obtained by Kato's construction [8]. Let us go through this construction following [2]. Let [B.sub.0] = {v [member of] [C.sup.2]; [absolute value of v] [less than or equal to] [epsilon]} be the closed ball and consider the sequence of blowups

[mathematical expression not reproducible],

where [[PI].sub.i] is the blowup of [p.sub.i-1], [p.sub.0] = 0, [C.sub.i] = [[PI].sup.-1.sub.i]([p.sub.i-1]), [p.sub.i] [member of] [C.sub.i]. Let [PI]: [B.sub.n] [right arrow] [B.sub.0], [PI] = [[PI].sub.n] [omicron] ... [omicron] [[PI].sub.1]. Let [sigma] : [B.sub.0] [right arrow] [B.sub.n] such that, [sigma]([B.sub.0]) is in the interior of [B.sub.n]. Let [p.sub.n] = [sigma](0), and suppose that [p.sub.n] [member of] [C.sub.n]. The Kato surface S is the compact complex surface resulting from the identification of the two boundary components of [B.sub.n] \ [sigma](int([B.sub.0])) --considered as a real four-dimensional manifold-with-boundary-- by the map [sigma] [omicron] [PI].

The above data can be recovered from the germ of F = [PI] [omicron] [sigma] at its fixed point 0. There is a natural correspondence between the objects in ([C.sup.2]; 0) that are invariant by F and the objects in S.

The Kato surface of the above construction is an Enoki surface if, furthermore [2, Thm. 3.33],

(a) [p.sub.i] [not member of] [[union].sub.j<i][C.sub.j] and

(b) [p.sub.1] is not in the strict transform of [[sigma].sup.-1]([C.sub.n]) under [[PI].sub.1],

or, equivalently, if the trace of DF[|.sub.0] is nonzero [2, Thm. 3.30].

We will henceforth assume that S is an Enoki surface. In suitable coordinates (x, y), the germ of F may be written as

(1) F(x, y) = (x[y.sup.n] + P(y), ty),

for some t [member of] C, 0 < [absolute value of t] < 1, and some polynomial P of degree n - 1 [3, Thm. 1.19]. Here, y = 0 is the curve that maps via [sigma] into [C.sub.n]. The parabolic Inoue case corresponds to P [equivalent to] 0 in (1); in this case, both the holomorphic vector field x[partial derivative]/[partial derivative]x and the germ of curve x = 0 are preserved by F and induce, respectively, a holomorphic vector field Y and an elliptic curve in S. In all cases, since the meromorphic one-form dy/y on ([C.sup.2], 0) is invariant by F, it induces a global meromorphic one-form on S. On its turn, the kernel of this form induces a holomorphic foliation with singularities F on S (compare with [13, Section 3]).

According to [12, Section 2.3], an Enoki surface has no foliation other than this foliation F. Let us give a short proof of this fact. Let G be a foliation on S. By Levi's extension principle, there exists a meromorphic one-form [omega] = [alpha]dx + [beta]dy on ([C.sup.2], 0) generating it. In order for G to be preserved by F, [F.sup.*][omega] = h[omega] for some meromorphic function h. If [beta] [equivalent to] 0, [omega] = [alpha]dx and [F.sup.*]([alpha]dx) = ([alpha] [omicron] F)d[F.sub.1] = ([alpha] [omicron] F)[([partial derivative][F.sub.1]/[partial derivative]x)dx + ([partial derivative][F.sub.1]/[partial derivative]y)dy]. From formula (1), [partial derivative][F.sub.1]/[partial derivative]y does not vanish identically, and G cannot be preserved. If [beta] is not identically zero, up to multiplying by a meromorphic function, we may suppose that [omega] = [alpha]dx + dy/y. Since dy/y is invariant by F, we must have that h [equivalent to] 1 and that the one-form [alpha]dx is preserved. Repeating the previous arguments shows that this is impossible unless [alpha] [equivalent to] 0. This proves the uniqueness of the foliation.

The construction of the Enoki surface and of its foliation can be done simultaneously. Consider a nonsingular foliation [F.sub.0] on ([C.sup.2], 0). Under the blowup of 0 by [[PI].sub.1], this regular point of the foliation produces an exceptional divisor [C.sub.1] that is invariant by the induced foliation [F.sub.1]. There is only one singular point of the transformed foliation [F.sub.1] along [C.sub.1]. Let [p.sub.2] be a regular point of [F.sub.1] in [C.sub.1] and continue the construction of the Enoki surface until we have a foliation [F.sub.n] on [B.sub.n]. Let [sigma] map [F.sub.0] to [F.sub.n] (mapping {y = 0} to [C.sub.n]) and suppose that F is contracting. Through Kato's construction, this produces a general Enoki surface with a foliation (notice that condition (b) in the construction of the Enoki surface is automatically satisfied). The only singularities of the foliation are at the intersection of the divisors, where the foliation has, locally, a first integral (the holonomy is trivial).

Let us now come to the proof of Proposition 2.1. Let S be an Enoki surface, D its divisor and let X be a meromorphic vector field on S. If S is a parabolic Inoue surface, it has a holomorphic and nonzero vector field Y. Since X and Y are collinear and since S has no meromorphic functions, X and Y differ by a multiplicative constant, and X is holomorphic. Let us thus assume that S is not a parabolic Inoue surface and, in particular, that it has no curves other than those of the cycle in the support of D. Let X be a meromorphic vector field on S. By Levi's extension principle, it is associated to a meromorphic vector field in [B.sub.0]. Since the foliation it induces is the unique foliation on the Enoki surface, the vector field is of the form [X.sub.0] = g(x, y)[partial derivative]/[partial derivative]x for some meromorphic function g. In order for this vector field to induce a global one, we should have

(2) [[PI].sup.-1.sub.*][X.sub.0] = [[sigma].sub.*][X.sub.0].

In particular, the curves of zeros and poles of g other than y = 0 are preserved by F and induce curves in S different from [C.sub.1], ..., [C.sub.n]. Since we supposed that there are no further curves in S, [X.sub.0] must actually be of the form h(x, y)[y.sup.q][partial derivative]/[partial derivative]x, with h holomorphic and nonzero, q [member of] Z. In the chart of the blowup (x, y) = (sy, y), the vector field reads h(sy, y)[y.sup.q-1][partial derivative]/[partial derivative]s. (As a function of s and y, h(sy, y) is holomorphic and nonzero at the origin.) After n blowups, the order of the transformed vector field along [C.sub.n] is q - n. However, in order to satisfy (2), this number must equal q. This contradiction proves Proposition 2.1 and finishes the proof of Theorem 1.2.

3. An example and a remark.

Example 3.1. Consider the meromorphic vector field [??] = y[x.sup.-1](y[partial derivative]/ [partial derivative]y--x[partial derivative]/[partial derivative]x) on [C.sup.2]. Outside {xy = 0}, it has the solutions t [??] (-2[ct.sup.1/2], [ct.sup.-1/2]). Let [??] : [C.sup.2] \{0} [right arrow] [P.sup.1] be given by (x, y) [??] x/y. The image of [??] under [PI] is the vector field Y = 2[partial derivative]/[partial derivative][xi]. Let S be the secondary Hopf surface obtained as the quotient of [C.sup.2] \{0} under the action of the group generated by the maps (x, y) [??] (2x , 2y) and [mathematical expression not reproducible]. These maps preserve [??] and [??] and there is thus a well-defined vector field X on S and a map (elliptic fibration) [PI]: S [right arrow] [P.sup.1], [[PI].sub.*](X) = Y. The elliptic curves in S coming from {x = 0} and {y = 0} are, respectively, the curves of zeros and poles of X. Outside these curves, where X is holomorphic, the solutions are single-valued (a cancels the multi-valuedness of the solutions of [??]).

Remark 3.1. Enoki surfaces do not have strictly meromorphic vector fields, although they are not far from doing so. Consider the vector field x[y.sup.q][partial derivative]/[partial derivative]x, q [member of] Z. In the chart (x, y) = (sy, y) of the blowup of 0, the vector field reads again s[y.sup.q][partial derivative]/[partial derivative]s, so that, after n blowups, the vector field is equivalent to the original one. Equivalently, the vector field is preserved under the local ramified maps (x, y) [??] (x[y.sup.n], y). However, the derivative at the origin of these maps has trace equal to 1, and they are not contracting.

doi: 10.3792/pjaa.93.73

Acknowledgements. The author is partially funded by PAPIIT-UNAM IN108214 (Mexico). This article was finished during a sabbatical leave at the Ecole Normale Superieure (France), under the support of PASPA-DGAPA-UNAM (Mexico). The author thanks both institutions.

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By Adolfo GUILLOT

Instituto de Matematicas, Unidad Cuernavaca, Universidad Nacional Autonoma de Mexico, AP273. Admon. de correos 3, Cuernavaca, Morelos 62251, Mexico

(Communicated by Shigefumi MORI, M.J.A., Sept. 12, 2017)

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