On the height of foliated surfaces with vanishing Kodaira dimension

Publ. Mat. 49 (2005), 363–373 ON THE HEIGHT OF FOLIATED SURFACES WITH VANISHING KODAIRA DIMENSION Jorge Vit´ orio Pereira Abstract We prove that the height of a foliated surface of Kodaira dimension zero belongs to {1,2,3,4,5,6,8,10,12}. We also construct an explicit projective model for Brunella’s very special foliation. 1. Introduction Afoliated surface consists of a pair (S, F), where Sis a complex surface and Fis a saturated singular holomorphic foliation. By a singular holomorphic foliation we mean an element of PH0(S, TS ⊗ L), for some line bundle L. The line bundle Lis the cotangent bundle of Fand will be denoted by T∗F. We say that a singular holomorphic foliation is saturated if any represent of Fin H0(S, T S ⊗T∗F) has a finite singular set. On this paper all the foliated surfaces will be projective. The sections of T∗Fcan be interpreted as the 1-forms along the leaves of F. Keeping in mind the analogous case of projective manifolds, it is natural to ask whether the integers h0(S, T S ⊗ T ∗F⊗k) for k∈Z>0, are birational invariants of (S, F). The answer turns out to be no as in the case of projective surfaces with arbitrary singularities. One needs to restrict to a class with mild singularities. A nice surprise, pointed out in [6], is that the reduced foliated surfaces singularities (in the sense of Seidenberg) form such class. Moreover Seidenberg proved that every foliated surface is birationally equivalent to a reduced foliated surface. A birational classification of reduced foliated surfaces according to theirs Kodaira dimension have been carried out recently, cf. [2], [5], [6]. Recall that the Kodaira dimension of a reduced foliated surface (S, F), kod(S, F) for short, is defined as kod(S, F) = lim sup k→∞ log h0(S, T∗F⊗k) log k. 2000 Mathematics Subject Classification. 37F75. Key words. Foliations, Kodaira dimension. 364 J. V. Pereira For a foliated surface (S, F) of non-negative Kodaira dimension the height of (S, F), h(S, F) for short, is defined in [7] as the smallest integer ksuch that T∗F⊗khas kod(S, F) + 1 algebraically independent sections. In particular, if kod(S, F) = 0 then h(S, F) = min{k∈Z>0|h0(S, T∗F⊗k)6= 0}. The purpose of this note is to prove the following Theorem 1. If (S, F)is a reduced foliated surface of Kodaira dimension zero then h(S, F)∈ {1,2,3,4,5,6,8,10,12}. Moreover, if Sis not rational or Fadmits a rational first integral then h(S, F)∈ {1,2,3,4,6}. It is interesting to note that for projective surfaces Swith Kodaira dimension zero it is well known that (cf. [1]) min{k∈Z>0|h0(S, KS⊗k)6= 0} ∈ {1,2,3,4,6}. For foliated surfaces the exceptional cases where h(S, F)∈ {5,8,10,12} correspond to foliated rational surfaces obtained as quotients of linear foliations on Abelian surfaces with complex multiplication. 2. Height versus transformation groups 2.1. Minimal and relatively minimal foliated surfaces. A foliated surface (S, F) is a relatively minimal foliated surface if, and only, it satisfies the following universal property: any bimeromorphic morphism (S, F)→(S0,F0) onto a reduced foliated surface is in fact a biholomorphism. It turns out that every foliated surface admits a relatively minimal model, i.e., is birationally equivalent to a relatively minimal foliated surface. To verify this it is sufficient to apply Seidenberg’s Theorem to obtain a reduced foliation and then do successive contractions of the so called F-exceptional curves, i.e., the smooth rational curves of selfintersection −1 whose contraction still yields a reduced foliation, cf. [2, Proposition 1, p. 73]. In general we don’t have the uniqueness of the relatively minimal model and when we do have the uniqueness we say that the foliated surface admits a minimal model, or equivalently, it is birationally equivalent to a minimal foliated surface. The minimal foliated surfaces can also be characterized by an universal property, namely: any bimeromorphic map (S0,F0)99K (S, F) onto a reduced foliated surface is in fact a morphism. On the Height of Foliated Surfaces 365 It has to be noted that the definition of minimal model above is not the only one in the literature. In [5] a more functorial definition, buildup on the concepts of Q-gorenstein and canonical singularities, is made and from the point of view of Mori-Theory it is much more natural. We have adopted the definitions above (the same that are presented in [2]) since they are build-up on the concept of reduced singularities which is widely known in the theory of foliations of surfaces. 2.2. Foliated surfaces with vanishing Kodaira dimension. Concerning the classification of foliated surfaces of Kodaira dimension zero the key fact is the following Theorem 2.1. Let (S, F)be a relatively minimal foliated surface. If the Kodaira dimension of (S, F)is zero then there exists a ramified covering π:S0→Sand a birational morphism ρ:S0→S00 such that the foliation ρ∗π∗Fis generated by a global holomorphic vector field v. Moreover vis in at least one of the following classes: (a) vis tangent to a smooth elliptic fibration; (b) vgenerates a Kroenecker foliation on an Abelian Surface; (c) vis tangent to the suspension of a representation π1(E)→Aut(P1), where Eis an elliptic curve; (d) vis birationally equivalent to a linear vector field on P2. The above result first appeared, in a slightly different form, as Theorem 5 in McQuillan’s paper [5]. The statement above, as it is, can be found in [2, pp. 88, 110 and 119]. On the proof of Theorem 2.1 presented in [2, pp. 110–112, 119–127] the ramified covering πis constructed using the ramified covering trick: if σis a non-zero section of T∗F⊗h(F)then S0is the minimal resolution of e Sthe preimage of the graph of σunder the map E(T∗F)⊗h(F) −−−−→ E(T∗F⊗h(F)), where E(T∗F) is the total space of T∗F. It follows that πis ramified along the zero set of σ. Note also that by construction [k(S0) : π∗k(S)] = h(F) and that ξis a cyclic covering, i.e., k(S0) is a cyclic extension of π∗k(S). In particular we have that the ramified covering π is Galois. More explicitly, observe that C∗acts naturally on E(T∗F) and the action of the subgroup generated by ξ, where ξis a primitive root of unity of order h(F), leaves e Sinvariant. Thus, after resolving the singularities of e S,ξinduces an automorphism gξof order h(F) of the foliated surface (S0, π∗F). 366 J. V. Pereira Note that in general π∗Fis not tangent to a holomorphic flow. This will be the case for a relatively minimal reduction ρ∗π∗Fof F, where ρ:S0→S00 is a birational morphism obtained by successive contractions of π∗F-exceptional curves. In general the birational map ρ◦gξ◦ρ−1is not an automorphism. Although we can guarantee in cases (a), (b) and (c) of Theorem 2.1 that it will be an automorphism since ρ∗π∗Fis a minimal foliation. We close this section remarking that every reduced foliated surface of Kodaira dimension zero and tangent to a holomorphic flow has trivial cotangent bundle. In fact if (S, F) is foliated surface tangent to a global holomorphic vector field vthen TF=OS((v)0). If the divisor (v)0is non trivial then h0(S, T∗F⊗k) = 0 for every k∈Z>0and consequently kod(S, F) = −∞. In particular the fixed points of the holomorphic flow (equivalently the zeros of v) are isolated. 2.3. The height of quotients of holomorphic actions. Let (S, F) be a reduced foliated surface tangent to a holomorphic vector field v. If φ:S→Sis an automorphism of (S, F) then there exists a rational function g∈k(S) such that φ∗(v) = g·v. When kod(S, F) = 0 then from the triviality of TFit follows that gis in fact a constant. Thus, in the case of Kodaira dimension zero, the automorphism group of (S, F) admits a natural character λ: Aut(S, F)→C∗defined by the relation: if φ:S→Sis an automorphism of (S, F) then φ∗(v) = λ(φ)·v. Proposition 2.1. Let (S0,F0)be a reduced foliated surface of Kodaira dimension zero tangent to a holomorphic flow and Gbe a finite subgroup of Aut(S0,F0). If (S, F)is the minimal resolution of (S0,F0)/G then (i) kod(S, F) = −∞ and (S, F)is a rational fibration, or (ii) kod(S, F) = 0 and h(S, F) = [G: ker λ]. Proof: Let (S00,F00) = (S0,F0)/G and ρ:S0→S00 be the natural quotient map. Note that in general S00 will be singular, but with mild singularities: all its singularities are cyclic quotient singularities. Thus the sheaf T∗F00 might fail to be a line bundle: it may be not locally free around the singularities of S0/G, cf. [5], [2]. Anyway, some power of T∗F00 is a line-bundle and, for most purposes, we can deal with T∗F00 as if it were a line-bundle. In particular, if we denote by Fix(G) the set {p∈S0|there exists g6= Id ∈Gsuch that g(p) = p} then we can apply the formulas of [2, p. 29] to assure that (1) T∗F0=ρ∗T∗F00 ⊗ OS0(R) On the Height of Foliated Surfaces 367 where Ris an effective Q-divisor with support equal to the union of irreducible components of Fix(G) which are not F0-invariant. Suppose first that Ris a nontrivial divisor. Since T∗F0is the trivial bundle then it follows from (1) that ρ∗T∗F00 =OS0(−R). Therefore ρ∗T∗F00 and (consequently) T∗F00 are not pseudo-effective. From Miyaoka’s Theorem, see [2, p. 89], we deduce that (S, F) is a (maybe singular) rational fibration, i.e., we are in case (i). Suppose now that Ris a trivial divisor. Let v∈H0(S0, TF0) be a nontrivial vector field and k= [G: ker λ]. From the definition of λ:G→C∗it follows that v⊗k∈H0(S0, T F0⊗k) is invariant under the action of G. Thus ρ∗v⊗kis meaningful and it follows from (1) that it defines a trivialization of TF00⊗k: seing ρ∗v⊗kas a section of TF00⊗k it is a nowhere vanishing section. In particular h0(S00, T∗F00⊗k) = 1. If π: (S, F)→(S00,F00) denotes the minimal resolution of (S00,F00) then using the fact that (S0, T ∗F0) is reduced it can be easily verified that T∗F=π∗T∗F00 ⊗ OS00 (E) for some effective Q-divisor Esupported on the exceptional locus of π. Thus, T∗F⊗kis trivial, h0(S, T∗F⊗k) = h0(S00, T ∗F00⊗k) = 1 and, consequently, h(S, F)≤[G: ker λ]. From the definition of λit follows that h(S, F)≥[G: ker λ], establishing the proposition. 3. Quotients of holomorphic actions Having at hand Theorem 2.1 and Proposition 2.1 we will deduce Theorem 1 from a case-by-case analysis. Let (S, F) be a foliated surface tangent to vector field vand G⊂ Aut(S, F) be a finite subgroup. Let us study the different possibilities: 3.1. Case (a): Elliptic fibrations. In this case there exists a holomorphic map π:S→Bfrom Sto an algebraic curve Bwith connected fibers. If B∗is the set of regular values of πand S∗=π−1(B∗) then the restriction of πto S∗is a locally trivial fibration. If u:e B→B∗is the universal covering of B∗then e S, the fibered product of uand π, is a trivial fibration over e B, i.e., e S=e B×Efor some elliptic curve E. 368 J. V. Pereira The action of Gon Slifts to an action e S=e B×E. Moreover if g∈G then ϕg, the automorphism induced by gin e B×E, is of the form ϕg(x, y) = (αg(x), βg·y+γg(x)) for coordinates (x, y) where x∈e Band y∈C/Γ = E. It follows that the morphism λ:G→C∗is given by g7→ βg. This is sufficient to establish that [G: ker λ]∈ {1,2,3,4,6}. 3.2. Case (b): Kroenecker foliations. Here S=Ais an abelian surface. First recall that the automorphism group of Afits into the exact sequence 0−→ A−→ Aut(A)−→ Hol(A)−→ 0 where Hol(A) is the holonomy part of Aut(A) which can be identified with a subgroup of GL2(C) and Aacts on itself by translations. If G⊂Aut(S, F)⊂Aut(A) is a finite subgroup then the character λ:G→C∗factors through the natural projection G→G/(G∩A) since the translations act trivially on vector fields. In general the holonomy part of Aut(A) is a finite group of order 1, 5, 10 or 2m·3n, where m≤5 and n≤2, see [8]. Although since finite subgroups of C∗are cyclic we have just to bound the order of the cyclic subgroups of GL2(C) which preserves a lattice Γ on C2. Forgetting the complex structure of Awe are lead to bound the order of elements of GL4(Z). Let g∈GL4(Z) be an element of finite order. Thus all the eigenvalues are roots of the unity with minimal polynomial of degree at most 4 and if kdenotes the order of gthen φ(k)∈ {1,2,3,4}, where φis Euler’s function. We have the following possibilities: φ(k) = 1 and k= 1; φ(k) = 2 and k∈ {2,3,4,6}; or φ(k) = 4 and k∈ {5,8,10,12}. We conclude that in case (b) [G: ker λ]∈ {1,2,3,4,5,6,8,10,12}. 3.3. Case (c): Suspension over an elliptic curve. Suppose now that Sis a P1-bundle over an elliptic curve E. The vector field vinduces a Riccati foliations without invariant fibers. Here we are in a dual situation to the case (a). If u:C×P1→Sis the universal covering of Sthen we can choose coordinates (x, y)∈C×P1in such a way that vlifts to the vector field ∂x. We can therefore conclude that the character λ:G→C∗ factors through the morphism G⊂Aut(S)→Aut(E). In particular, as On the Height of Foliated Surfaces 369 in case (a), it follows that [G: ker λ]∈ {1,2,3,4,6}. 3.4. Case (d): Rational surfaces. Suppose now that Sis a rational surface and vis a holomorphic vector field. on this case we do not have in general that (S, F) admits a minimal model. Although we can suppose without loss of generality that S=P2and that vis a vector field on P2 we have to consider Gas a finite subgroup of Bir(P2) instead of Aut(P2). Up to the end of this section (x, y) will stand for the coordinates of aC2⊂P2. We will distinguish two cases: (d.1) v=x∂x+λy∂yfor some λ∈C\Q; (d.2) v=x∂x+∂y. 3.4.1. Case (d.1): Quotients of C∗×C∗.If the vector field vis of the form (d.1) then the only algebraic curves invariant by vare the lines x= 0, y= 0 and the line at infinity. Thus if ϕ:P299K P2is a birational map preserving the foliation induced vthen ϕbelongs to Aut(C∗×C∗), the group of algebraic automorphisms of C2\ {x·y= 0}∼ =C∗×C∗. We now find ourselves on a situation completely similar to the case (b); the group Aut(C∗×C∗) fits on the splitting exact sequence 0−→ C∗×C∗−→ Aut(C∗×C∗)−→ GL2(Z)−→ 0, where the homomorphism Aut(C∗×C∗)→GL2(Z) is given by the action on fundamental group of C∗×C∗. We remark, for the sake of clearness, that this homomorphism admits a right inverse given by GL2(Z)−→ Aut(C∗×C∗) a b c d7−→ (x, y)7−→ (xa·yb, xc·yd). We can check that λ:G→C∗factors through the morphism G⊂ Aut(C∗×C∗)→GL(2,Z) and, in complete analogy with case (b), we have reduced our problem to bound the order of cyclic elements of GL2(Z). Therefore [G: ker λ]∈ {1,2,3,4,6} in case (d.1). 370 J. V. Pereira 3.4.2. Case (d.2): Quotients of C∗×C.It remains to treat the case v=x∂x+∂y. Note that the only algebraic curves invariant by v are {x= 0}and the line at infinity. Note also that vis tangent to rational 1-form ω=dx x+dy. As in case (d.1) if ϕleaves F, the foliation induced by v, invariant then φmust be biregular when restricted to C2\ {x= 0}∼ =C∗×C. A simple argument shows that every ϕ∈Aut(C∗×C) is of the form ϕ(x, y) = (α1·x, α2xβ·y+f(x)), where α1, α2∈C∗,∈ {−1,1},β∈Zand f∈Cx, 1 x. Moreover, since Fdoes not admit a rational first integral, if ϕpreserves F then ϕ∗ω=κ·ωfor some κ∈C∗. Comparing the dy component of ωand ϕ∗ωwe see that β= 0 and comparing the dx component we deduce that fmust be constant. Thus, as a simple computation shows, if ϕis a birational map of P2which preserves Fthen it must be of the form ϕ(x, y) = (α·x,  ·y+β) where α∈C∗,β∈Cand ∈ {−1,1}. In particular [G: ker λ]∈ {1,2} in case (d.2). 3.5. Proof of the Main Theorem. Let (S, F) be a foliated surface of Kodaira dimension zero. From Theorem 2.1, Proposition 2.1 and the analysis of quotient of foliated surfaces generated by vector fields just made it follows at once that h(S, F)∈ {1,2,3,4,5,6,8,10,12}. If Fadmits a rational first integral then Fis an elliptic fibration and the arguments in §3.1 allow us to conclude that h(S, F)∈ {1,2,3,4,5,6}. Suppose now that h(S, F)∈ {5,8,10,12}. Again from the analysis above it follows that (S, F) is birationally equivalent to the quotient of a Kroenecker foliation. We can apply [8, Theorem 2.1] to conclude that Sis rational. On the Height of Foliated Surfaces 371 4. Some examples 4.1. Examples with h(S, F) = 2. If Fis a Kroenecker foliation of an abelian surface Athen quotient of Fby the canonical involution of A(multiplication by −1) is a foliation of Kodaira dimension zero on aK3 surface. Another example already appeared in case (e.2) of §3. 4.2. Examples with h(S, F)∈ {3,4,6}.Examples of rational foliated surfaces with h(S, F)∈ {3,4,6}have already appeared in the literature. For h(S, F) = 3 we have the very special foliation of Brunella [2, pp. 57–59]. For h(S, F)∈ {3,4,6}we have the pencils of foliations studied by Lins Neto in [4], see also the appendix of [3]. They are obtained through quotients of E×Eby a diagonal automorphism of order 3, 4 or 6. In §5 we will determine an explicit model for Brunella’s very special foliation on the projective plane. To construct examples with h(S, F)∈ {3,4,6}with Snot rational we fix k∈ {3,4,6}and Ean elliptic curve with an automorphism gof order k. Let Cbe an algebraic curve with a cyclic automorphism hof order ksuch that C/hhiis not rational. It is a trivial matter to verify that quotient of the natural elliptic fibration on C×Eby the cyclic group generated by h×gis a foliated surface with the wanted property. 4.3. Examples with h(S, F)∈ {5,8,10,12}.Let ξnbe a primitive root of the unity of order n,n∈ {5,8,10,12}. Settle Γn= 1ξnξ2 nξ3 n 1ξ2 nξ2k nξ3k n! with k= 2,3,3,5 corresponding to each value nrespectively. Then An=C2/Γkis an abelian surface (in fact it is an abelian surface of CM-type, cf. [8]) and ϕ:C2−→ C2 (x, y)7−→ (ξ·x, ξk·y) induces an automorphism of Anof order nwhich we still denote by ϕ. The foliations Fxand Fyinduced, respectively, by the vector fields ∂x and ∂yare invariant under the action of ϕ. Moreover ϕ∗∂x=ξn∂x and ϕ∗∂y=ξk n∂y. Since the foliations Fxand Fydo not admit rational first integrals then from Proposition 2.1 we have that h(An,Fx) hϕi=h(An,Fy) hϕi=n.