Compact complex threefolds of class L associated to polynomial automorphisms of c3

Publ. Mat. 50 (2006), 401–411 COMPACT COMPLEX THREEFOLDS OF CLASS L ASSOCIATED TO POLYNOMIAL AUTOMORPHISMS OF C3 Karl Oeljeklaus and Julie Renaud Abstract We construct new families of non-K¨ahler compact complex threefolds belonging to Kato’s Class L. The construction uses certain polynomial automorphisms of C3. We also study basic properties of our manifolds. 1. Introduction and Construction We start by recalling the definition of Kato’s Class L, see [6]. Let V:= n[z0:z1:z2:z3]∈P3(C)| |z0|2+|z1|2>|z2|2+|z3|2o. A (not necessarily compact) irreducible complex space Xis said to be of Class L, if and only if Xcontains an open subset biholomorphic to U. M. Kato studied several interesting (smooth) examples and properties of Class Lspaces, see [6] and its references. The aim of the present paper is to construct new families of examples of non-K¨ahler class Lspaces and threefolds with fundamental group Z. All previously known examples of such manifolds where quotients of the complement of two disjoint linear rational curves in P3(C), which all admit projective structures (see [7]). In order to simplify the approach, we give the construction and study the properties in a special case. In the last section we give all quadratic automorphisms of C3which allow exactly the same arguments. Consider the following polynomial automorphism Hof C3 H(x, y, z) =   x2+cy2+z y2+x y  , 2000 Mathematics Subject Classification. Primary: 32J15. Key words. Polynomial automorphisms of C3, Global spherical shell, Kato’s Class L. 402 K. Oeljeklaus, J. Renaud the constant c∈Cbeing chosen arbitrarily. The inverse automorphism is given by H−1(x, y, z) =   y−z2 z x−(y−z2)2−cz2  . We extend Hand H−1to P3=[x:y:z:t]and, keeping the same notations, we get: H[x:y:z:t] = [x2+cy2+zt :y2+xt :yt :t2] H−1[x:y:z:t] = [yt3−z2t2:zt3:xt3−(yt −z2)2−cz2t2:t4]. The set of indeteminacy of His given by I+={x=y=t= 0} which is the point [0 : 0 : 1 : 0], and that of H−1equals to I−={z=t= 0} which is the projective line [x:y: 0 : 0]. It is clear that I+∩I−=∅, i.e. automorphism His regular in the sense of Sibony [8]. Definition 1.1. Let Mbe a metric space and f:M→Ma continous mapping. A compact subset K⊂Mis called attractor for fif there is an open neighbourhood V⊂Mof Ksuch that f(V)⋐Vand Tn≥0fn(V) = K. We have the following Proposition 1.2 (see [8]).Let fbe a regular polynomial automorphism of Ckof algebraic degree d≥2considered as a birational map of Pk(C). Then the set of indeterminacy I+of fis an attractor for f−1and, conversely, the set of indeterminacy I−of f−1is an attractor for f. The hyperplane at infinity minus the indeterminacy set {t= 0} \ I+ is mapped by Hto the projective line X+:= I−={z=t= 0}=n[x:y: 0 : 0]o which is an attractor for H. Let U+:= nw∈C3,lim n→∞ Hn(w)∈X+o be the bassin of attraction of Hin C3. Manifolds of Class L403 The automorphism Hbeing algebraic, it follows that its graph gr(H)⊂ C3×C3has the property that its topological closure W:= gr(H)⊂ P3×P3is a (not necessarily smooth) projective variety. We have the following commutative diagram W⊂P3×P3 pr1  pr2 '' P P P P P P P P P P P P P P3H //P3 where pri,i= 1,2 are the projections to the P3-factors. In the following we denote by pithe restrictions of prito W,i= 1,2. An easy calculation gives W= gr(H)∪D1∪D2 with D1:= n[z0:z1:z2: 0],[z2 0+cz2 1:z2 1: 0 : 0]o ∪n[0 : 0 : 1 : 0],[y0:y1: 0 : 0]o, D2:= n[0 : 0 : 1 : 0],[y0:y1:y2: 0]o. For the intersection we have l:=D1∩D2=[0 : 0 : 1 : 0],[y0:y1: 0 : 0]. Let ε > 0 and Vεthe neighbourhood of X+in P3of the form Vε:= n[z0:z1:z2:z3]∈P3| |z2|2+|z3|2< ε(|z0|2+|z1|2)o. We note also Bε:= ∂Vεthe boundary of Vε. Since X+is a attractor for Hthere is an ε > 0 such that X+⊂Vε⊂U+∪{z3= 0}\[0 : 0 : 1 : 0] and H(Vε)⋐Vε. The mapping p−1 2◦id ◦p1is biholomorphic onto its image in a open neighbourhood of p−1 1(Bε)⊂Wand therefore allows to identify holomorphically the two boundary components p−1 1(Bε) and p−1 2(Bε) of the complex space A:= p−1 2(Vε)\p−1 1(Vε). This identification gives rise to a compact complex space Xwith a global shell isomorphic Bto Bε, i.e. (X\B) = Ais connected. 404 K. Oeljeklaus, J. Renaud Let N:= pr−1 1(Vε) and M:= pr−1 2(Vε), i.e. A=M\N. The following scheme illustrates the situation and the gluing up.       @ @R p1p2 - H id @ @ BBBBB B        @ @ BBBBB B        M n a c p e N rprp × × × × V V Figure 1. Gluing up. The coordinates of the points are: a= [1 : 0 : 0 : 0] [1 : 0 : 0 : 0] n= [0 : 0 : 1 : 0] [0 : 1 : 0 : 0] c= [0 : 0 : 1 : 0] [1 : 0 : 0 : 0] p= [0 : 0 : 1 : 0] [0 : 0 : 1 : 0] e= [0 : 1 : 0 : 0] [0 : 1 : 0 : 0]. By construction, the space Xis the union of the quotient U+/hHiZ and of an irreducible divisor Dwith π1(D) = Z. Furthermore, the space Xis singular exactly in the points corresponding to l=D1∩D2. Of course it is possible to resolve the singularities by a finite sequence of modifications of P3×P3in order to obtain a smooth model of X. But since this process is not unique and does not change any of the properties discussed in the sequel, we shall work in what follows with the singular space. This space, and more generally all those that will be constructed Manifolds of Class L405 in the same way, provide examples of class L, since there are non-singular rational curves in B⊂Xadmitting an open neighbourhood isomorphic to V. Let us remark that in [7], M. Kato classified all compact manifolds of class Ladmitting a projective structure with fundamental group isomorphic to Z. We shall see in what follows that our manifolds have this fundamental group. In view of Kato’s result, it is easy to see that no smooth model of our spaces does admit any projective structure. 2. Properties We shall use the same notations as in the previous paragraph. Proposition 2.1. The fundamental group π1(X)is isomorphic to Z. Proof: Using the Mayer-Vietoris exact sequence, Van Kampen theorem and the fact that the space Ais simply connected, the proof works like in [1, pp. 11–12]. We continue by calculating the fundamental group of the attracting basin U+⊂C3of H. Proposition 2.2. π1(U+) = 1. Proof: Recall that Vε=nε(|z0|2+|z1|2)>|z2|2+|z3|2o in homogeneous coordinates [z0:z1:z2:z3] in P3. It is evident that Vε\ {z3= 0}is simply connected. Moreover, U+=∪n≥0H−n(Vε\ {z3= 0}). Let γ:S1→U+be a closed curve with K:=γ(S1). There exists n∈Nsuch as K⊂H−n(Vε\L) that is such that Hn(K)⊂Vε\{z3= 0}. Since Vε\Lis simply connected, we conclude that the curve γis trivial in π1(U+). Our next step is to give a lower bound of the complex dimension of H1(X, O). Proposition 2.3. dimCH1(X, O)≥1. Proof: We shall show that dim H1(X, O∗)0≥1. To do so, we construct a non trivial twisted holmorphic line bundle. We denote by ˜ Xthe universal cover of Xand by ˜ Hthe generator of the fundamental group π1(X) induced by the initial automorphism H. Let us consider for λ6= 1 the map: ˜ X×C−→ ˜ X×C (x, z)7−→ (˜ H(x), λz). 406 K. Oeljeklaus, J. Renaud The projection onto the first component gives e X×C/Z=: Lλ−→ e X/Z=X. The bundle Lλis a flat line bundle and therefore belongs to H1 (X,O∗ )0, since π1(X) = Z. Suppose that there exists a non-trivial section σ∈ H0(X, Lλ). The section σinduces a holomorphic function f:e X→C such that f(H(x)) = λf(x). We know that X=U+∪e D/he HiZ=e X/he HiZ, where e D, the inverse image of the divisor Din e X, which is a connected infinite chaine of rational surfaces. Here we have noted by he HiZthe group generated by the automorphism e H. The function fis necessarly equal to zero on the irreducible components of e D, hence on the whole divisor e D⊂e Xby connexity. In what follows we shall see that there is a surface with global sperical shell (GSS) naturally associated to the space X. For the sake of simplicity, we consider the case c= 0, the other cases working analogously. Then the automorphism His given by H(x:y:z:t) = (x2+tz :y2+tx :ty :t2). The point p= [1 : 1 : 0 : 0] in l=X+is a fixed point of saddle type. This is seen by an easy calculation: there are two eigenvalues equal to zero and one with absolute value strictly greater than one. At the point p, there exists locally a stable manifold noted by Ws(p)⊂Vεwhich is transversal to the attractor l, see [8]. Let Vbe a neighbourhood of 0 ∈C2 immersed onto the stable manifold. This gives rise to the commutative diagram: (V,0) −−−−→ ϕ(V,0)   yι  yι Ws(p)H −−−−→ Ws(p) where ιdenotes the immersion and ϕthe induced selfmapping of (V,0). Manifolds of Class L407 @ @ @@ BBBBBBBBBBBB                   @ @ @ @ a b c d Ws r p Figure 2. Stable manifold Ws. Coordinates: a= [1 : 0 : 0 : 0] b= [0 : 0 : 1 : 0] c= [0 : 1 : 0 : 0] p= [1 : 1 : 0 : 0] e= [0 : 0 : 0 : 1]. Let C(ϕ) = ι−1(ι(V)∩L) (where L={t= 0}is the hyperplane at infinity) be the critical set of ϕ. The restriction of ϕto V \ C(ϕ) is injective hence biholomorphic onto its image. Consequently, ϕis a strict germ of topological degree 1. According to Ch. Favre [4], ϕis a Dloussky germ and thus defines a compact complex surface Swith global spherical shell. Now the holomorphic function finduces a holomorphic function h:= ι◦fon Vwith h◦ϕ=λh, which is zero on C(ϕ). Since λ6= 1, the function his identically zero on V. This follows from the well-known fact that holomorphically non-trivial flat line bundles on surfaces with GSS have no non-trivial holomorphic sections, see e.g. [3]. We consider the increasing union Y:= [ n≥0 H−n(Ws(p)\L)⊂U+, 408 K. Oeljeklaus, J. Renaud which is a 2-dimensional connected immersed submanifold of U+and which is, as an abstract manifold, biholomorphic to the complement of the maximal divisor in the universal covering e Sof the above mentioned compact surface with GSS. Its topological closure in P3(C) contains all the projective lines of the form [x:y:t: 0] |t∈C, where x/y ∈S1 is a 2m-root of unity for some m∈N. Therefore the quotient S′:= Y/hHiZ⊂Xis not contained in any closed complex hypersurface of X. Since the function his identically zero, this is implies that the section σ∈H0(X, Lλ) vanishes identically, a contradiction to our assumption. We have proved dim H1(X, O)≥1. Now we show that there are no meromorphic functions on X. Proposition 2.4. The algebraic dimension of the space Xis equal to zero. Proof: We consider again S′:= Y/hHiZ⊂Xwhich is, as an abstract manifold biholomorphic to the complement of the maximal divisor in the GSS surface S. Let us suppose that there exists a meromorphic function fon X. Then fis constant on S′according to the properties of surfaces with global spherical shell. Since S′is not contained in any hypersurface in X, the function fis constant on X. Therefore, the algebraic dimension a(X) is equal to zero. 3. Regular quadratic automorphisms allowing the construction The example with which we have worked for the moment is just a particular case of a regular quadratic automorphism of C3having a projective line as an attractor at infinity; such automorphims belong to the fourth and fifth class in the classification of Fornæss-Wu. We shall give their list. Those of the fourth class for which we can construct a compact complex space with global shell isomorphic to Bare of the form: H4(x, y, z) =      x2+αxy +βy2+δy +γ+az y2+ν+x y the coefficients are chosen arbitrarily with a6= 0. Manifolds of Class L409 The inverse automorphism is of the form H−1 4(x, y, z) =          y−ν−z2 z 1 ax−P(y, z), P having degz(P) = 4. These automorphisms admit as set of indeterminacy the point I+= [0 : 0 : 1 : 0]; the inverse automorphisms the set I−={z=t= 0}; we check immediately that I+∩I−=∅. By the action of H4, the hyperplane at infinity {t= 0}minus I+is mapped to the P1attractor X+= [x:y: 0 : 0]. In order to get the topological closure of the graph in P3×P3one has to add the two divisors D1:= n[z0:z1:z2: 0],[z2 0+αz0z1+βz2 1:z2 1: 0 : 0]o ∪n[0 : 0 : 1 : 0],[y0:y1: 0 : 0]o D2:= n[0 : 0 : 1 : 0],[y0:y1:y2: 0]o. As for the automorphisms of the fifth class, they are of the form: H5(x, y, z) =        y2+αxy +βx2+δx +γ+az x2+θ+y x the constants are chosen arbitrarily with a6= 0. The inverse automorphism is of the form H−1 5(x, y, z) =          z y−θ−z2 1 ax−P(y, z), P having degz(P) = 4. These automorphisms admit as the indetermination set the point I+= [0 : 0 : 1 : 0]; as for the inverse automorphisms the set I−={z=t= 0}; it is again clear that I+∩I−=∅and that H5maps the hyperplane at infinity {t=0} minus I+to the P1attractor X+= [x:y: 0 : 0].