A complex structure on S^6 [pdf]

The (3, 4, ∞) modular family of 2-tori, completed at its three special points, is a complex structure on S6. The base and the family. Let ∆ = ∆(3, 4, ∞) be the triangle group, acting on the upper half plane h; the quotient h/∆ is P1 minus a point, and adding the cusp gives P1 with orbifold points of orders 3 and 4 and the cusp. Let V ∼= Z4 carry automorphisms T1, T2 of orders 3 and 4 with T0 := (T1T2)−1 unipotent, (T0 − I)2 = 0: this is a representation ∆ → SL(V), and we let it act on the dual lattice Λ = V∗ by Aj = (T−1 j )t (§2). The period functions τ, μ, β on h are an equivariant period map for it (§3), and the quotient is a family of complex 2-tori over the punctured orbifold curve: the analogue of a universal family of abelian surfaces over a modular curve, with this rank-four lattice representation in place of the symplectic one. The very general fibre is not algebraic. The very general fibre F has NS(F) = Zη with η a single indefinite class, and the algebraic dimension of the completed threefold is a(X) = 1 (§9). The three special fibres. The three conjugacy classes of maximal finite and parabolic cyclic subgroups of ∆ single out the three special points, and at these three points we take the three standard completions; each is a choice, and every ℓ0 and every admissible vj below occurs. At an elliptic point of order m the local monodromy is a torus automorphism of order m, and the filling is Kodaira’s logarithmic transform: (disc × torus)/(Z/m), the generator acting by a rotation of the disc and by the automorphism followed by a translation; the reduced fibre is of bielliptic type (§5). At the cusp the monodromy is unipotent of rank two, and the filling N0 is Mumford’s toric degeneration; its central fibre W is read off the prescribed A2-triangulation of R2, the deck action being the one in which the unimodular map B0 : ¯Λ → Λtor induced by M0 − I = (T−1 0 )t − I on Λ enters. Modulo the lattice the triangulation has one vertex, three edges and two triangles, i.e. W is one toric surface with hexagonal moment polygon — the degree-six del Pezzo dP6 — glued to itself along opposite sides, with two triple points (§4). Thus there is a single family, completed in the three standard ways at the three kinds of special point of an orbifold curve with a cusp. The gluing and the fundamental group. Each filling is glued to the global family along a collar by a fibre- preserving map, and we reglue by translations by local sections (§6, Prop. 6.3); among these, π1 and H∗ depend only on three integers ℓ0 (at the cusp), ℓ1, ℓ2 (at the elliptic points) (§7). Let γ ∈ V be the linear coordinate on Λ whose kernel ker γ = ⟨ ˆu, ˆw, ˆδ⟩ is the span of all (Aj − 1)Λ; then the coinvariants Λ/ ker γ ∼= Z are detected by γ alone, and ℓj = γ(vj) for the translation vectors vj. The result is π1(X) ∼= Z/|12ℓ0 − 4ℓ1 − 3ℓ2| (§7).1 This formula admits a reading, which the body does not use. It is the formula |a1a2e0 + a2b1 + a1b2| of [Orl72] for the order of H1 of the Seifert fibred space over S2(a1, a2), evaluated at e0 = ℓ0 and bj = −ℓj — the sign coming from the convention sj ◦ gj = e−2πi/mj sj for the local sections — so here (a1, a2) = (3, 4) with Seifert invariants (3, −ℓ1), (4, −ℓ2) and obstruction ℓ0; for (ℓ0, ℓ1, ℓ2) = (0, 1, −1) the value is 1 and the space in question is S3 with the circle action (z, w) 7 → (λ3z, λ4w), whose orbit space is S2(3, 4) [Orl72]. Heuristically the twist condition says: glue so that the coinvariant circle of the fibres traces out the (3, 4) Seifert fibration of the three-sphere over the base. The homology and the Euler number. The remaining torus directions contribute nothing: ˆw and ˆδ are vanishing cycles at the cusp, collapsed in the hexagonal fibre, and the rest is killed by monodromy, the coinvariants of Λ having rank one. With the twists above, X has the integral homology of S6 and is diffeomorphic to S6 (§7, §8). The Euler number localises at the cusp: e(X) = e(W) = 2 (§7), every other fibre being a torus or a free torus quotient and so of Euler number 0 — one singular fibre carries the whole Euler characteristic of S6, as the twelve nodal fibres carry the 12 of a rational elliptic surface. Why the argument of [CDP20] does not apply. That argument requires a line bundle non-torsion on every fibre and propagates a vanishing statement through each singular fibre by way of its normalisation. Here H2(X; Z) = 0 and Pic(X) ∼= C, so every line bundle is topologically trivial; at the non-normal toric fibre the differential of the fibration itself supplies the section which the passage to the normalisation assumes away, and R2 f∗(TX ⊗ L)̸ = 0 for all L. The count of H1 in [CDP20, Lemma 4.2] (after [CDP98, Lemma 3.2]) assumes trivial monodromy, whereas the monodromy representation here is non-trivial; §10 shows that this last point is repairable for X, while the failure of the reduction at the non-normal fibre is decisive. 1§7 proves π1(X) ∼= Z/|p|, p = 12ℓ0 − 4ℓ1 − 3ℓ2, and computes H∗(X; Z) for every admissible pair (v1, v2); the value |p| = 1 is one point of that family. Setup. V := Z4, basis (γ, u, w, δ); Λ := V∗, dual ( ˆγ, ˆu, ˆw, ˆδ); T1, T2 ∈ Aut(V), orders 3, 4 (columns: images), and Π(z) (τ, μ, β below): T1 = 1 0 −6 2 0 −1 1 1 0 −1 0 1 0 0 0 1 , T2 = 1 6 0 −3 0 0 −1 1 0 1 0 0 0 0 0 1 , Π(z) := 6μ τ 1 0 β μ 0 1 T0 := (T1T2)−1 = I + N, N2 = 0, Nγ = Nu = 0, Nw = −u, Nδ = γ; on Λ: Aj := (T−1 j )t, M0 := (T−1 0 )t, Λtor := ⟨ ˆw, ˆδ⟩, Λ := Λ/Λtor, B0 : Λ ∼ −→ Λtor, ¯ˆγ 7 → − ˆδ, ¯ˆu 7 → ˆw; A1, A2 fix ε := ˆγ + 2 ˆu − 4 ˆw, ε′ := ˆγ + 3 ˆu − 3 ˆw resp. B := P1 ∋ t, tc := 1/t, p1, p2, p0 := 0, 1, ∞, B◦ := B \ {p0, p1, p2}, m1, m2 := 3, 4; π : hz → B \ {p0} (hz, h upper half-pl.) orbifold uniformisation (mj at pj, cusp p0), deck gp. ∆ = ⟨g1, g2 | g3 1 = g4 2 = 1⟩, gjzj = zj ∈ π−1(pj), g′ j(zj) = e−2πi/mj , g0 := (g1 g2)−1 parabolic; ρV : ∆ → GL(V), gj 7 → Tj, Mg := ρV (g)t; Ur ⊂ π−1{0 < |tc| < r} (r small) ⟨g0⟩-inv. cusp comp.; hol. τ : hz → h, μ, β : hz → C (j modular invariant, j(i) = 1728): (τ1) : j(τ) = 1728 t ◦ π, (τ2) : τ ◦ g1 = τ−1 τ , τ ◦ g2 = − 1 τ , (τ3) : τ(z1) = eiπ/3, τ(z2) = i, (μ1) : μ ◦ g1 = 1−μ τ , μ ◦ g2 = 1 + μ τ , (μ2) : μ bounded on Ur, (β1) : β ◦ g1 = β + 2 − 6(1−μ)2 τ , β ◦ g2 = β − 3 − 6μ2 τ , (β2) : β + τ bounded on Ur, (β3) : Im β − 6(Im μ)2 Im τ < 0 on hz. Such τ, μ, β exist by Theorem 3.4; on C2 ∋ ζ = (ζ1, ζ2), Π(z) as above with columns Π(z) ˆγ, Π(z) ˆu, Π(z) ˆw, Π(z) ˆδ: λ · (z, ζ) := (z, ζ + Π(z)λ), unique Rg(z) ∈ GL2(C): Π(gz) = Rg(z)Π(z)Mg, T := (hz × C2)/Λ, ∆ by ˜g(z, ζ) := (gz, Rg(z)ζ), J := (T |hz \∆{z1,z2})/∆ → B◦, monodromy A1, A2, M0 on H1 = Λ. On Ur, e2πiτ = tcu1(tc), u1 unit at 0; for s := τ − (2πi)−1 log u1 (any branch; §4) Π = [sB0 + C(tc) | I] in bases ( ¯ˆγ, ¯ˆu), ( ˆw, ˆδ), C hol. at 0. N′ := Z3 ∋ (y, y3), y ∈ Z2 = Λtor (e1 ↔ ˆw, e2 ↔ ˆδ), cochar. lattice of (C∗)3 ∋ (x1, x2, tc) ⊂ Y smooth toric, fan F cone over A2-triangulation of R2 × {1} (vertices Z2 × {1}, edges ±e1, ±e2, ±(e1 − e2)), xk = e2πiζk , tc character of (0, 0, 1) ∈ Hom(N′, Z), hol. on Y. Λ acts freely, prop. disc. on Yϵ := {|tc| < ϵ}, ϵ small (4.5), by Ψ ¯λ := (exp 2πi C(tc) ¯λ, 1) · Φ ¯λ, Φ ¯λ from (y, y3) 7 → (y + y3 B0 ¯λ, y3); N0 := Yϵ/Λ (N0), W := {tc = 0}/Λ, η : dP6 → W normalisation, dP6 del Pezzo of degree 6. For j = 1, 2: ∆j ∋ zj gj-inv. disc, coord. sj, sj ◦ gj = e−2πi/mj sj, smj j = tj ◦ π, tj coord. at pj, Nj := (T |∆j )/⟨(z, ζ) 7 → (gjz, Rgj (z)ζ + Π(gjz)vj/mj)⟩ (N1), (N2), v1 := ε, v2 := −ε′ (free; 5.4). Glue to J, any branch: Nj by ζ 7 → ζ + (2πi)−1(log sj)Π(z)vj (§5), (Yϵ ∩ (C∗)3)/Λ on 0 < |tc| < ϵ by (z, ζ) 7 → (e2πiζ1 , e2πiζ2 , e2πis(z)). ℓj := γ(vj) (γ ∈ V = Hom(Λ, Z)): ℓ1 = 1, ℓ2 = −1, ℓ0 := 0. X := (J ⊔ N0 ⊔ N1 ⊔ N2)/ ∼ (both gluings), f : X → B induced. Theorem. X is a compact connected complex 3-manifold and f : X → B is a surjective holomorphic map with connected fibres. (1) Over B◦ the map f is a proper submersion whose fibres are complex 2-tori. The fibre f −1(p0) is the divisor W ⊂ N0, which is reduced, irreducible and has normal crossings. The normalisation η identifies opposite sides of the hexagon of (−1)-curves of dP6, the double locus is three rational curves through two triple points, and e(W) = 2. For j = 1, 2 one has f ∗(pj) = mjSj, where Sj := ( f −1(pj))red is a smooth bielliptic surface in Nj, and OX (Sj)|Sj has order mj. (2) The threefold X is simply connected, since π1(X) ∼= Z/|12ℓ0 − 4ℓ1 − 3ℓ2| = 1, and H∗(X; Z) ∼= H∗(S6; Z); consequently X is diffeomorphic to S6 (§7ff.). (3) The algebraic dimension of X is a(X) = 1 and M(X) = f ∗C(t): every meromorphic function on X is constant along the fibres of f . (4) The canonical bundle is KX ∼= f ∗OB(−1) ⊗ OX (2S2) and is not torsion; c3(X) = 2; and Aut0(X) ∼= C∗ acts along the fibres of f with fixed locus a double curve of W (§9). Remark (on [CDP20]). The Theorem contradicts [CDP20, Cor. 2.3] as published. For L ∈ Pic(X) set A := (L∗ ⊗ KX )|W : η∗ A is trivial (H2(X; Z) = 0, Pic(dP6) discrete), and for θ trivialising it d(tc ◦ f )|W ⊗ θ vanishes on the double locus, so descends to 0̸ = σ ∈ Ω1 X |W ⊗ A dying in eΩ1 W ⊗ A ( eΩ := Ω/torsion). By Serre–Grothendieck duality on W (ωW ∼= KX |W ) and base change R2 f∗(TX ⊗ L)̸ = 0 for all L (§10); so hypothesis (1) of [CDP20, Prop. 2.4, p. 680] holds for no L. The reduction of [CDP20, p. 692] to H0( eΩ1 S ⊗ ·) = 0 on reduced fibres S is stated there provided L|S is non-torsion, a proviso met at W for general L, where the conclusion H0( eΩ1 W ⊗ A) = 0 does hold; but it does not suffice. Indeed, for general L, A̸ ∼= OW forces H0(W, N∗ W ⊗ A) = H0(W, A) = 0, so the conormal term vanishes identically, while σ gives H0(W, Ω1 X |W ⊗ A)̸ = 0: this is where the reduction fails. A compact complex threefold fibred by tori over the projective line, and the six-sphere. Abstract We construct a compact connected complex manifold X of dimension three together with a surjective holomorphic map f : X → P1 whose fibres over the complement of three points p0,