Affine surfaces with locally trivial action of the additive group of complex numbers.

Tammo tom Dieck

Preliminary Version, November 18, 2000

The purpose of this note is to study the algebra and topology of regular affine surfaces over the complex numbers which carry a free algebraic action of the additive group G = (C, +) such that they become principal G–bundles.

The generating examples of such surfaces are the varieties of Danielewski [1] which he studied in connection with the cancellation problem. The simplicity of these varieties makes it desirable to have a detailed picture of them. We describe their diffeomorphism type as a complex line bundle over the projective line IP^{1}. (The complex structure is a different one.) In addition, we construct canonical equivariant compactification divisors. This is the content of section 2.

In general, the topology at infinity of the G–varieties in question is best studied by first exhibiting normal crossing compactification divisors. The boundary of a tubular neighbourhood of such divisors is the boundary of the variety as a differentiable manifold. These boundaries are given by a plumbing construction. By well known techniques, the homology and the fundamental group can be read off from the weighted dual graph of the compactification divisor.

In section 1 we describe the local model for the construction of compactification divisors. In section 3 we indicate the structure of the general case and give examples of hypersurfaces in C^{3}.

1. The local model.

We denote by G = G_{a} the additive group (C, +) of complex numbers. Let k ∈ IN_{0} and denote by C^{2}(k) the affine space C^{2} with action

(1.1) G × C^{2} → C^{2}, t · (x, y) = (x, y + tx^{k}).

The set F = 0 × C ⊂ C^{2}(k) is the fixed point set (k > 0) and the other orbits are free. The action on C^{2}(0) is free. The action (1.1) has a canonical compactification. Let Σ(k) be the Hirzebruch manifold

(1.2) Σ(k) = (C^{2} \ 0) ×_{C}^{∗} IP^{1}; it is defined as the C^{∗}–orbit space

(x_{0}, x_{1}), [y_{0}, y_{1}] ∼ (λx_{0}, λx_{1}), [λ^{k}y_{0}, y_{1}]

for λ ∈ C^{∗}, where [y_{0}, y_{1}] are homogeneous coordinates of points in the projective line IP^{1}. The compactified action is

(1.3) t · ((x_{0}, x_{1}), [y_{0}, y_{1}]) = ((x_{0}, x_{1}), [y_{0} + tx_{0}^{k}y_{1}, y_{1}]).

The action (1.1) is contained as the chart x_{1} = 1 = y_{1}. The complement of C^{2}(k) in Σ(k) consists of the divisors L = {x_{1} = 0} and D = {y_{1} = 0}. The divisor D is fixed under G and L is G–stable. For k > 0 we have E_{0} = {x_{0} = 0} as a further G–fixed divisor.

We define a standard equivariant expansion of Σ^{2}(k). An expansion of Σ(k) = X_{0} is a sequence

(1.4)  π : X_{r}  ^{π_{r} \overline{}}X_{r−1} → … → X_{1} ^{π_{1} \overline{}}X_{0}

of σ–processes π_{j} where each π_{j} blows up a point z_{j−1} ∈ E_{j−1} ⊂ X_{j−1} to the exceptional divisor E_{j} = π_{j}^{−1}(z_{j−1}). We assume that the exceptional set of π, i.e. the set

Σ_{π} = {z | π^{−1}(z) is not a point}

consists of a single point in F. Moreover, we assume that z_{j−1} is not contained in the proper transforms of the previous E_{i}, i < j − 1. The pre–image of Σ_{π} in X_{r} is a divisor E_{1} ∪ E_{2} ∪ … ∪ E_{r} where E_{j} also denotes the proper transform of E_{j} ⊂ X_{j}. The total transform of L ∪ D ∪ E_{0} in X_{r} has a linear weighted dual graph

0 −k −1 −2  −2 −1
• • • • ⋯ • •
L D E_{0} E_{1}  E_{r−1} E_{r}

The dotted part consists of (−2)–curves. This finishes the description of the standard expansion. The next lemma states its equivariance properties.

(1.5) Lemma. Suppose r ≤ k. The expansions are G–equivariant with respect to an algebraic action on the X_{j}. The divisors E_{j}, 0 ≤ j < k consist of fixed points. The divisor E_{k} carries a free action on E_{k} \ (E_{k} ∩ E_{k−1}).

Proof. The proof is by induction on r. We show that a single σ–process applied to a point of F in C^{2}(k) produces a situation which is isomorphic to C^{2}(k − 1) in a neighbourhood of the exceptional divisor (k > 0).

T. tom Dieck 2. The varieties of Danielewski 3

Let X = (C^{2} \ 0) ×_{C}^{∗} C with (x, y; u) ∼ (λx, λ^{−1}u) for λ ∈ C^{∗}. We have the G–action (1.6) t · (x, y, u) = (x, y + tx^{k}u^{k−1}; u)

on X. The map

(1.7) π : X → C^{2}(k), (x, y; u) ↦ (xu, c + yu)

is equivariant und blows up the point (0, c) ∈ F. The complement of the proper transform of F is isomorphic to C^{2}(k − 1) under the map

(1.8)  (u, y) ↦ (1, y; u).  □

Let W_{k} denote the G–variety which is the complement of the divisor L ∪ D ∪ E_{0} ∪ … ∪ E_{k−1} in X_{k}. By the previous lemma, W_{k} carries a free G–action. The restriction of π to W_{k} is a G–morphism π : W_{k} → C^{2}(k).

The next proposition gives complete insight into the model situation.

(1.9) Proposition. There exists an isomorphism φ : C × G = C^{2}(0) → W_{k} of G–varieties such that ψ = πφ : C^{2}(0) → C^{2}(k) has the form

ψ(x, u) = (x, p_{k}(x) + ux^{k})

with a polynomial p_{k} of degree less than k. The polynomials p_{k} are in bijctive correspondence with the standard expansions.

Proof. Consider the morphism

τ_{0} : C → C^{2}(k), x ↦ (x, a_{0} + a_{1}x + … + a_{k−1}x^{k−1}).

The coefficient a_{0} is determined by the fact that π_{1} blows up (0, a_{0}). The proper transform of τ_{0} can be written in terms of the chart (1.8) as

τ_{1} : C → C^{2}(k − 1), x ↦ (x, a_{1} + … + a_{k−1}x^{k−2}).

We continue in this manner. The coefficient a_{j} is determined by π_{j+1}. The proper transform τ_{k} : C → W_{k} of τ_{0} is a section of the action, i. e. meets each orbit exactly once. We obtained the desired isomorphism φ if we set φ(x, u) = u · τ_{k}(x). □

2. The varieties of Danielewski

In connection with the cancellation problem Danielewski [1] has constructed interesting simple examples of regular affine varieties with locally trivial free G– action. In this section we describe different models of the varieties, determine their differential topology, and construct compactifications divisors.

The first model is a weighted flag variety (see [4] for generalizations). The second model is an explicit description as a hypersurface in C^{3}.

For each k ∈ IN we have a regular affine variety W (k) in C^{4} given by

(2.1) W (k) = {(a, b, c, d) ∈ C^{4} | ad^{k} − bc^{k} = 1}.

Note that W (1) = SL(2, C). This fact motivates some of the subsequent constructions. The varieties carry a free right action of C^{∗}

(2.2) (a, b, c, d) · λ = (λ^{k}a, λ^{−k}b, λc, λ^{−1}d).

The following polynomial functions x, y and z on W (k) are C^{∗}–invariant:

(2.3) \begin{matrix}x = ad^{k} + bc^{k} \\ y = ab \\ z = cd. \end{matrix}

They satisfy the equation

(2.4) x^{2} − 4yz^{k} = 1.

This equation defines a regular affine variety U(k) in C^{3}. Let V (k) = W (k)/C^{∗} denote the orbit variety. The functions (2.3) induce a morphism

(2.5) φ : V (k) → U(k), (a, b, c, d) ↦ (ad^{k} + bc^{k}, ab, cd).

The following assertion is easily verified:

(2.6) Proposition. The geometric quotient V (k) equals the algebraic quotient V (k)//C^{∗}. The map (2.5) is an isomorphism of varieties. □

(2.7) Remark. If we view W (1) = SL(2, C), then the action (2.2) of λ ∈ C^{∗} on W (1) amounts to right multiplication by the diagonal matrix Dia(λ, λ^{−1}). Let D ⊂ SL(2, C) be the subgroup of such matrices. Then V (1) is the coset variety SL(2, C)/D.

Let IP^{1}(k) denote the quotient of C^{2}\0 with respect to the C^{∗}–action (a, c)·λ = (λ^{k}a, λc). This is a weighted projective space, homeomorphic to the ordinary projective line IP^{1} = IP^{1}(1). We have a projection

p : V (k) → IP^{1}(k), (a, b, c, d) ↦ (a, c).

The following diagram is a pullback

(2.8) [figure: V (k); Φ_{k}; V (1); p; p; IP^{1}(k); φ_{k}; IP^{1}.] Here Φ_{k}(a, b, c, d) = (a, b, c^{k}, d^{k}) and φ_{k}(a, c) = (a, c^{k}). Note that φ_{k} is a ramified covering and topologically a map of degre k.

The varieties V (k) have a large symmetry group. Thus is obvious for V (1) = SL(2, C)/D because SL(2, C) acts by left translation. Certain subgroups still acts on V (k) in a similar manner.

Let G(m) denote the semi–direct product C × C^{∗} with multiplication

(a, α) · b, β) = (a + α^{m}b, αβ).

If B ⊂ SL(2, C) denotes the group of upper triangular matrices we have an isomorphism

(2.9) G(2) → B, (a, α) ↦ \binom{α aα^{−1}}{0 α^{−1}}.

The group G(m) acts on C by

(a, α) · z = α^{m}z + a.

The resulting G(m)–space C will be denoted C(m). The group G(2m) acts on W (m) by

(u, α)(a, b, c, d) = (α^{m}a + uα^{−m}c^{m}, α^{m}b + uα^{−m}d^{m}, α^{−1}c, α^{−1}d).

This action commutes with the right C^{∗}–action (2.2) and induces therefore an action of G(2m) on V (m).

This G(2m)–action can be transported by the isomorphism (2.5) into an action on U(k). There exists an algebraic action of G(2m) on C^{3} which induces this action on U(k). One verifies that the following action has this property:

(u, α) · (x, y, z) = (x + 2uα^{−2m}z^{m}, α^{2m}z^{m}, α^{2m}y + ux + u^{2}α^{−2m}z^{m}, α^{−2}z)

for (u, α) ∈ G(2m), (x, y, z) ∈ C^{3}. The G(2)–action on W (1) corresponds under (2.9) to the left translation action of B on SL(2, C). The projection V (m) → IP^{1}(m) becomes G(2m)–equivariant, when G(2m)–acts by (u, α)·(a, c) = (α^{m}a+ uα^{−n}c^{m}, α^{−1}c); this is the compactified action on C(2m).

The topology of V (k) is now completely determined by the next proposition. It uses the complex line bundles H(−2k) → IP^{1}, see [2], p. 251.

(2.10) Proposition. The map V (k) → IP^{1}(k) is diffeomorphic to the line bundle H(−2k) → IP^{1}. In particular V (k) is the interior of a compact manifold M with boundary a lens space L(2k). Therefore the fundamental group of V (k) at infinity is isomorphic to ZZ/2k.

Proof. The projection V (1) → IP^{1} can be identified with the canonical quotient map

SL(2, C)/D = SL(2, C) ×_{B} B/D → SL(2, C)/B.

We identify

σ : B/D ^{≅ \overline{}}C, \binom{λ μ}{0 λ^{−1}} ↦ λμ.

The left translation by \binom{α aα^{−1}}{0 α^{−1}} ∈ B on B/D is transformed under σ into the action z ↦ α^{2}z + a. Thus B/D ≅ C(2). The differential topology of the bundle above is obtained by restriction to the compact form SL(2, C)/B ≅ SU(2)/T with maximal torus T = SU(2) ∩ B. Thus we obtain precisely the description of the bundle H(−2) over IP^{1}, namely SU(2) ×_{T} C(2) → SU(2)/T. Now we use the pullback (2.8) and the fact that a map of degree k induces from H(−2) the bundle H(−2k). □

We have already explained in [3] that the affine G(2k)–variety x^{2} − 4yz^{k} = 1 is obtained from C^{2} + C^{2} by identifying C^{∗} × C → C^{∗} × C, (x, u) ↦ (x, u + x^{−k}). We are going to describe an equivariant compactification of U(k). This uses the results of section 1.

We start with the G–action on C^{2}(k) and Σ(k) and apply two standard expansions at the same time. In the generic case, when we start by blowing up two different points in F, we obtain a compactification divisor B with dual graph

(2.11)  [figure: \begin{aligned}& & & −2 & & −2 & −1 \\ & & & • & ⋯ & • & • & E′_{k} \\ 0 & −k & −2 \\ • & • & • \\ L & D & E_{0} \\ & & & • & ⋯ & • & • & E_{k} \\ & & & −2 & & −2 & −1 \end{aligned}]  Divisor B

The dotted parts contain k − 1 curves with self–intersection −2. The complement U_{k} of B − E_{k} is isomorphic to a variety W_{k} as in (1.9). Similarly, the complement U′_{k} of B − E′_{k} is a variety W′_{k}. The variety D(k) which is defined to be the complement of B −E_{k} −E′_{k} carries a locally trivial G–action since it is the union U_{k} ∪ U′_{k}. More precisely: D(k) is obtained by gluing U_{k} and U′_{k} along an isomorphism γ : U_{k} \ E_{k} → U′_{k} \ E′_{k}. From (1.9) we can derive an explicit formula for γ. Suppose

φ : C^{2}(0) → U_{k}, φ′ : C^{2}(0) → U′_{k}

have been chosen such that

πφ(x, u) = (x, p(x) + ux^{k})

and similarly with a polynomial q(x) for φ′. Then we have isomorphisms

(C \ 0) × G ^{φ \overline{}}U_{k} \ E_{k} ^{π \overline{}}(C \ 0) × G

T. tom Dieck 3. The general case 7

and similarly for U′_{k} \ E′_{k}. The transition function is given by

(2.12) (x, u) ↦ (x, u + \frac{p(x) − q(x)}{x^{k}}).

Note that x^{−k}(p(x) − q(x)) has the form

(2.13) g(x) = c_{−k}x^{−k} + … + c_{−1}x^{−1}

with c_{−k} ≠ 0. The gluing (2.12) is exactly the one used by Danielewski. Thus we see that the affine variety U(k) is one of the varieties D(k). Therefore we have shown:

(2.14) Proposition. U(k) has the normal crossing compactification divisor B − E_{k} − E′_{k}. □

By tracing the definitions we see that the compactification of U(k) is G(2k)– equivariant.

If the two standard expansions separate at a higher infinitesimal level, then we obtain a similar picture. The only difference is that the plynomials p and q have coefficients of x^{0}, …, x^{l} equal so that x^{−k}(p(x)) has a pole of order k − l. The resulting compactification has the shape:

(2.15) [figure: \begin{aligned}& & & & & & & −2 & & −2 \\ & & & & & & & • & ⋯ & • \\ 0 & −k & −1 & −2 & & −2 & −3 \\ • & • & • & • & ⋯ & • & • \\ & & & & & & & • & ⋯ & • \\ & & & & & & & −2 & & −2 \end{aligned}]

The components of the divisor are, of course, smooth rational curves.

3. The general case

The method of the previous section is easily generalized and yields equivariant compactifications for affine varieties with locally trivial G–action. We begin with the constructive side. The method also allows for an algebraic classification of the resulting varieties.

We assume given the following data:

(3.1) Data

1. Z compact Riemann surface.

2. p : P → Z holomorphic or algebraic IP^{1}–bundle.

3. s : Z → P a section of p and L_{1}, …, L_{r} fibres of p over z_{1}, …, z_{r} such that U = Z \ {z_{1}, …, z_{r}} is affine.

4. Set W = P \(L_{1}∪…∪L_{r}∪s(Z)) and assume that p : W → U is isomorphic to the projection pr_{1} : U × C → U.

5. σ : U → C a holomorphic function with isolated zeros.

6. A G–action on P which corresponds to t·(u, z) = (u, z +σ(u)t) on U ×C ≅ W.

Suppose σ has a zero of order k in u ∈ U. We apply a finite number of k–fold standard expansions (in the sense of section 1) to points in the fibre over u. Any such process results in a locally trivial free G–action, as explained in section 1. This action has a projection onto U. If σ(u) ≠ 0, then there is a single orbit over u. If σ(u) = 0, then there are b orbits over u if we have applied b standard expansions to the fibre over u. The compactification divisor of this action has the following structure of the dual graph:

(3.2) [figure: L_{r}; L_{1}; D; B(u)]

The box B(u) contains the tree which results from the expansion of the fibre over u. The generic situation for B(u) is a star-shaped tree with center-weight −n and n strings of length k consisting of −2–curves if σ has a zero of order k in u. A typical non–generic case for k = 2 is (2.15).

By standard topology the compactification divisor determines the topology at infinity. For the fundamental group see e.g. [5]. For H_{1} one has to use the intersection matrix of the weighted tree. Each box B(u) contributes a certain torsion group; the projection onto the surface U is split surjective.

Some of the G–varieties appear as equivariant hypersurfaces in C^{3}. We communicate some explicit formulas.

Let p(x) ∈ C[x], q(x, y) ∈ C[x, y]. We consider the afffine variety V (p, q) in C^{3} given by

(3.3) P (x, y, z) ≅ p(x)z − q(x, y) = 0.

For each a(x) ∈ C[x] we have a factorization

(3.4) q(x, y + a(x)) − q(x, y) = a(x)r_{a}(x, y).

T. tom Dieck REFERENCES 9

This implies that for a(x) = p(x)h(x) ∈ p(x)C[x] the automorphism of C^{3}

(3.5) (x, y, z) ↦ (x, y + a(x), z + b(x, y))

with b(x, y) = h(x)r_{a}(x, y) leaves the polynomial P (x, y, z) and hence V (p, q) invariant. Thus V (p, q) has a rather large algebraic automorphism group. The minimal case of this construction gives the G–action

(3.6) c · (x, y, z) = (x, y + cp(x), z + r(x, y, c))

where q(x, y + cp(x)) − q(x, y) = p(x)r(x, y, c) and c ∈ C.

When is the action (3.7) free? The orbit of (x, y, z) is obviously free when p(x) ≠ 0. Suppose p(x_{0}) = 0 and q(x, y) = ∑_{r} q_{r}(x)y^{r}. Then r(x_{0}, y, c) =

c ∑ rq_{r}(x_{0})y^{r−1}. Therefore the orbit of (x_{0}, y, z) is free if and only if \frac{∂q(x_{0},y)}{∂y} ≠ 0. Since p(x_{0}) = 0 implies q(x_{0}, y) = 0, the conditions (x_{0}, y, z) ∈ V (p, q) and \frac{∂q(x_{0},y)}{∂y} = 0 imply that y is a multiple root of q(x_{0}, y). Thus if q(x_{0}, y) has no multiple roots whenever p(x_{0}) = 0, then V (p, q) carries a free G–action and, moreover, is a regulasr hypersurface in C^{3}. Suppose these conditions hold. The projection

ρ : V (p, q) → C, (x, y, z) ↦ x

is G–invariant. If p(x_{0}) ≠ 0, then the fibre is isomorphic to C and a G–orbit. If p(x_{0}) = 0, then the fibre of ρ over x_{0} consists of the orbits (x_{0}, y_{j}) × C, where y_{j} runs through the roots of q(x_{0}, y).

References

[1] Danielewski, W: On the cancellation problem and automorphism group of affine algebraic varieties. Preprint.

[2] tom Dieck, T.: Topologie. Berlin – New York, de Gruyter 1991.

[3] tom Dieck, T.: Homology planes without cancellation property. Mathematica Gottingensis 4 (1991). To appear: Archiv Math.

[4] tom Dieck, T.: Weighted flag varieties. In preparation.

[5] Lamotke, K.: Regular solids and isolated singularities. Braunschweig – Wiesbaden, Vieweg 1986.

(to be continued)