Algebraic space forms

Tammo tom Dieck

Preliminary version. November 18, 2000

1. Algebraic space forms

An n–dimensional affine hypersurface V over the complex numbers C is the zero set of a polynomial f ∈ C[z_{0}, …, z_{n}]:

V = V (f) = {z ∈ C^{n+1} | f(z) = 0}.

The variety V (f) is called regular if 0 is a regular value of the holomorphic map f : C^{n+1} → C, i. e. if for each z ∈ V (f) some partial derivative ∂f/∂z_{i} is non zero.

A classical problem is: Relate the geometry and topology of V (f) with the algebraic properties of f. Here are two view points:

1. Given f, derive the topological properties of V (f).

2. Given a topological type, find f such that V (f) has this type.

Here “topological type” can refer to different categories: homology or homotopy type, diffeomorphism type etc.

Similar questions can be asked for affine varieties or algebraic varieties in general.

A distinguished situation occurs when the underlying geometrical object is Euclidean space. We therefore consider the following properties of an n–dimensional algebraic variety V over the complex numbers:

(1.1) V is diffeomorph or homeomorphic to C^{n}.

(1.2) V is contractible.

(1.3) For a subring R of the rational numbers Q we have a homology isomorphism H_{∗}(V; R) ≅ H_{∗}(C^{n}; R).

In case (1.1) we call V an (algebraic) space form. In case (1.3) we call V an R– homology space form or R–acyclic (in case R = ZZ we do not specify R). In the case (1.3) of complex surfaces (n = 2) the term R–homology plane was introduced in tom Dieck–Petrie [1989].

(1.4) Example. Let m, n be coprime natural numbers. The affine variety

V = {(x, y) | x^{m} = y^{n}} ⊂ C^{2}

is homeomorphic to C. The map t ↦ (t^{n}, t^{m}) is a homeomorphism. But V is not regular at the origin and not a smooth submanifold of C^{2}.

The following examples were studied in tom Dieck – Petrie [1990]. The corresponding abstract surfaces were first constructed by Gurjar – Miyanishi [1987].

(1.5) Theorem. Let a ≥ b > 0 be coprime integers. Define a polynomial P = P_{a,b} by

P (x, y, z) = z^{−1} ((xz + 1)^{a} − (yz + 1)^{b}) .

Then the variety

V (a, b) = {(x, y, z) | P (x, y, z) = 1}

is a regular contractible affine hypersurface in C^{3}. If V (a, b) and V (a′, b′) are homeomorphic, then (a, b) = (a′, b′). □

Later we come back to these surfaces and study natural generalizations. In dimension two one cannot find exotic affine space forms. This is a consequence of a result of Ramanujam [1971]:

(1.6) Theorem. If the regular affine variety V is homeomorphic to C^{2}, then the variety is isomorphic to the standard affine plane C^{2}. □

Ramanujam also gave the first example of a contractible affine surface which is not homeomorphic to C^{2}. A basic tool for the study of affine varieties are compactifications. One has two different (but related) possibilities:

(1.7) An algebraic conpactification of V is a projective variety X such that V is isomorphic to the complement X \ D of a divisor D in X. We call D a boundary divisor or compactification divisor of V. The compactification divisor is called a normal crossing divisor if it is the sum (= union) of irreducible regular subvarieties D_{1}, …, D_{r} with transverse intersection.

(1.8) A topological compactification of the (regular) variety V is a compact smooth manifold B with boundary ∂B such that V is diffeomorphic to the interior of B. The fundamental group π_{1}(∂B) is called the fundamental group at infinity of V. The topology of ∂B is called the topology ov V at infinity. If D is a normal crossing divisor as in (1.7), then B can be taken to be the complement of a suitable tubular neighbourhood of D. Thus knowledge of a nice algebraic compactification implies knowledge of a topological compactification.

Here is a trivial example to illustrate these concepts.

T. tom Dieck 1. Algebraic space forms 3

(1.9) Example. Affine space C^{2} is compactified by projective space IP^{2}. The compactification divisor is a projective line L in IP^{2}. Also C^{2} is diffeomorphic to the interior of the 4–dimensional unit disk D^{4} = {z ∈ C^{2} | |z| ≤ 1}.

Because of Ramanujam’s theorem (1.6), the contractible surfaces are a natural class of varieties to be studied in this context. They can be used to produce space forms in higher dimensions as we shall demonstrate later. The basic input from differential topology is the h–cobordism theorem of Smale [1962] which leads to a characterization of the unit disk as a differentiable manifold in terms of algebraic topology (see also Milnor [1965], p. 108).

(1.10) Theorem. Let D be a k–dimensional compact contractible smooth manifold with contractible boundary ∂D. Then D is diffeomorphic to the unit disk D^{k} in IR^{k}, provided k ≥ 6. □

By the Hurewicz and Whitehead theorems of algebraic topology, the contractibility of a manifold can be detected by algebraic invariants:

(1.11) Proposition. Let D be a simply connected, acyclic manifold. Then D is contractible. □

A contractible manifold can have a boundary which is not simply connected (see Mazur [1961], Gordon [1975] for examples). A similar phenomenon cannot happen for homology:

(1.12) Proposition. Suppose the compact oriented (n + 1)–manifold B is R– acyclic. Then ∂B is an R–homology n–sphere.

Proof. The exact homology sequence (coefficient ring R) gives H_{i+1}(B, ∂B) ≅ H_{i}(∂B) for i > 0. Poincaré duality H_{i+1}(B, ∂B) ≅ H^{n−i}(B) then shows H_{i}(∂B) = 0 for n > i > 0. Similarly, we see H_{n}(∂B) ≅ R and H_{0}(∂B) ≅ R (for n > 0). □

It is an interesting topological problem to determine which homology spheres are boundaries of contractible or acyclic manifolds (compare Casson–Harer [1981]). More generally, one wants to find the “smallest” manifold which bounds a given homology sphere.

In view of (1.11) and (1.12) we can say:

(1.13) Proposition. Let D be a contractible smooth k–manifold with simply connected boundary. Then D is diffeomorphic to D^{k}, provided k ≥ 6. □

There are important cases in which a simply connected manifold will have a simply connected boundary. In order to state the result we use the handle decomposition of a smooth manifold which follows from the existence of a smooth Morse function, see Milnor [19??].

(1.14) Proposition. Suppose the compact connected m–manifold M has only handles of index i for i ≤ r ≤ \frac{m}{2}. Then the relative homotopy groups π_{j}(M, ∂M) are zero for j < m − r.

Proof. In order to prove this proposition we investigate what happens when a single handle is attached.

In order to attach an i–handle one has to choose an embedding C = S^{i−1} × D^{m−i} ⊂ ∂M and form the adjunction space M′ = M ∪_{C} (D^{i} × D^{m−i}). The space M′ has the homotopy type of M ∪D^{i}, an i–cell D^{i} attached to M along S^{i−1} ×0. Let us consider the following diagram.

(M′, ∂M′)  α_{\overline{}}(M′, ∂M′ ∪ D^{m−i})

≃

(M ∪ (D^{i} × D^{m−i}), ∂M × I ∪ (D^{i} × D^{m−i}))

≃^{}

(M, ∂M)  ^{β  \overline{}}(M ∪ D^{i}, ∂M ∪ D^{i})

The vertical maps are the obvious homotopy equivalences, whereas α and β are inclusions. We consider the induced maps on homotopy groups π_{j}. The exact homotopy sequence shows that α induces an isomorphism for j < m−i. Suppose π_{j}(M, ∂M) = 0 for j < m − r. Then homotopy excision (tom Dieck [19??], p.

178) shows that β induces an isomorphism for j < m − r + i − 2.

Now one uses this information inductively, starting with (D^{m}, S^{m−1}), and attaching successively 1–handles, 2–handles etc. □

Affine varieties have the following remarkle property (1.15); see Milnor [1963], §7 for a proof.

In order to deal with regular affine varieties V from a topological point of view it is useful to know that there exists a compact manifold B with boundary such that V is diffeomorphic to the interior of B. If one realizes V as a regular subvariety of some C^{N}, then there exists an R > 0 such that for all r > R the sphere S(r) = {z ∈ C^{N} | |z| = r} is transverse to V. This is proved in Milnor [1968], Cor 2.6. Now one can take as B the intersection of V with a large disk in C^{N}.

(1.15) Proposition. Let V be an m–dimensional regular affine variety over C. Then B has a decomposition into i–handles, i < m. □

Combining (1.13), (1.14), and (1.15) we obtain:

T. tom Dieck 2. The embedding problem 5

(1.16) Theorem. Let V be a contractible regular affine variety over C of dimension m ≥ 3. Then V is diffeomorphic to C^{m}.

Proof. From (1.14) and (1.15) we obtain that ∂B is simply connected. Now apply (1.13). □

The h–cobordism theorem has another interesting consequence.

(1.17) Proposition. Let B denote a compact contractible manifold of dimension n ≥ 4. Then B × D^{2} is diffeomorphic to D^{n+2}.

2. The embedding problem

Suppose the affine variety V ⊂ C^{n} is isomorphic to C^{k}. Can V be transformed into a linear subspace by an algebraic automorphism of C^{n}?

The answer is yes in case (n, k) = (2, 1). This was proved by Abhyankar and Moh [1975] and Suzuki [1974]; compare also Lin–Zaǐdenberg [1983] and Neumann [1989a]. The answer is also yes for high codimension (??), see Kaliman [19??]. Apparently no exotic ambedding of C^{k} into C^{n} is known.

In the case of hypersurfaces the problem can be rephrased in the following way. Given a polynomial function f : C^{n} → C. Suppose V (f) is isomorphic to C^{n−1}. Does there exist a polynomial automorphism φ : C^{n} → C^{n} such that f = p_{1} ∘ φ, with p_{1}(z_{1}, …, z_{n}) = z_{1}?

Here is a view point: Suppose φ as above exists. Then there exists a large automorphism group of C^{n} under which the polynomial f is invariant. In order to find φ try to construct invariants of f. We demonstrate this view point at length by an example which has also been considered by Dimca [1990].

Let d ≥ 1 and k ≥ 1 be natural numbers. Consider the polynomial

(2.1) p(x, y, z) = x + x^{d}y + y^{k}z.

For c ∈ C and (x, y, z) ∈ C^{3} we set

(2.2) c · (x, y, z) = (x + cy^{k}, y, z − c − y^{−k}((x + cy^{k})^{d}y − x^{d}y))

The right most term is a polynomial in (x, y, z). One verifies:

(2.3) Lemma. (1) The map

C × C^{3} → C^{3}, (c, (x, y, z)) ↦ c · (x, y, z)

is a free action of the additive group C on C^{3}.

(2) The polynomial p is invariant under this action: p(c·(x, y, z)) = p(x, y, z).□

By construction, the second component of C^{3} is also invariant under the action (2.3).

(2.4) Proposition. The map

γ : C^{3} → C^{3}, (x, y, z) ↦ (p(x, y, z), y)

is the orbit map of the action (2.3).

Proof. Each orbit through a point with y ≠ 0 contains a unique point with first coordinate zero. We can then compute z from the value p(x, y, z). We have p(x, y, z) = x, hence γ(x, 0, z) = (x, 0). In this case c · (x, 0, z) = (x, 0, z − c). Therefore {(x, 0, z) | z ∈ C} is an orbit. This shows that γ induces a bijection of the orbit space onto C^{2}. The full statement will follow from the further investigations. □

We want to exhibit γ as a trivial principal bundle. This requires to find a polynomial section. (We remark that locally trivial C–bundles over affine varieties are trivial.) If we set

(2.5) γ(x, y, z) = (x + x^{d}y + y^{k}z, y) =: (a, b),

then we have

(2.6) z = \frac{a − x − x^{d}b}{b^{k}}.

Therefore we show:

(2.7) Proposition. There exist polynomials x(a, b) ∈ ZZ[a, b] such that the polynomial a − x(a, b) − (x(a, b))^{d}b is divisible by b^{k}.

Proof. Induction over k. For k = 1 we can set a − x = b. For the induction step we show: Let q ∈ ZZ[u, a, b], r ∈ ZZ[a]. There exists u = u(a, b) ∈ ZZ[a, b] such that

(2.8) u − r(a) − bq(u, a, b)

is divisible by b^{k} if we substitute u = u(a, b). For the proof of this statement we set u = r(a) = vb. We substitute into (2.8) and obtain

b^{−1}(vb − bq(vb + r(a), a − b)) = v − q(vb + r(a), a, b).

The right side is a polynomial of the form (2.8), if v is replaced with u. By induction, we can find v(a, b) ∈ ZZ[a, b] such that this polynomial is divisible by b^{k−1}. Then u = r(a) + bv(a, b) has the desired property. □

T. tom Dieck 3. Hyperbolic modifications 7

Using (2.7), we obtain a section s of γ

s : C^{2} → C^{3}, (a, b) ↦ (x, b, b^{−k}(a − x − x^{d}b))

with x = x(a, b). This finally leads to a bijective morphism

Φ : C × C^{2} → C^{3}, (c, (a, b)) ↦ c · s(a, b).

In order to exhibit Φ as an algebraic isomorphism we begin by showing that the action (2.3) is proper, in an algebraic sense. The relation of the action

R = {(x, y, z), c · (x, y, z)} ⊂ C^{3} × C^{3}

is contained in the variety W of those ((x, y, z), (x_{1}, y_{1}, z_{1})) ∈ C^{3} × C^{3} which satisfy

(2.9) \begin{matrix}y & = & y_{1} \\ x_{1} − x & = & (z − z_{1})y^{k} − y(x_{1}^{d} − x^{d}). \end{matrix}

(2.10) Lemma. The map τ : R → C, (α, c · α) ↦ c is a morphism.

Proof. If y ≠ 0, then we can compute c

(2.11) c = y^{−k}(x_{1} − x).

If 1 + y(x_{1}^{d−1} + x_{2}^{d−2}x + … + x^{d−1}) ≠ 0, then we can again compute c

(2.12) c = \frac{z − z_{1}}{1 + y(x_{1}^{d−1} + … + x^{d−1})}.

Because of (2.9), the right hand sides of (2.11) and (2.12) define the same regular function on the intersection of their domain of definition. Therefore τ is regular on Zariski open sets which cover the variety. □

An inverse to Φ is now given by

Ψ : C^{3} → C × C^{2}, α ↦ (τ(sγ(α)), α), sγ(α)).

Finally, we remark that Φ is defined over ZZ and can therefore be used to study the diophantine properties of (2.1).

3. Hyperbolic modifications

In this section we present an inductive construction of acyclic varieties. The construction is called hyperbolic modification. It produces manifolds (varieties) with an action of the multiplicative group C^{∗}.

We start whith an n–dimensional complex manifold L. Let U ⊂ C^{n} be a connected open neighbourhood of the origin and let φ : U → L be a holomorphic chart. Set φ(0) = x_{0}. We also use the open set

Ũ = {(u, x) ∈ C × C^{n} | ux ∈ U}.

It contains C × 0 ∪ 0 × C^{n}. The holomorphic map

b : C^{∗} × (U \ 0) → Ũ, (u, x) ↦ (u, u^{−1}x)

is a holomorph embedding onto an open subset of Ũ.

We define the complex manifold V to be the pushout of the diagram

C^{∗} × (L \ x_{0}) ^{\overline{}a} C^{∗} × (U \ 0) ^{b \overline{}}Ũ

with a(u, x) = (u, φ(x)). Since a and b are holomorphic embeddings onto open subspaces and the pushout is seen to be a Hausdorff space V is a complex manifold. It is called the hyperbolic modification of L at x_{0}. If we let C^{∗} act on C^{∗} × (L \ x_{0}) and C^{∗} × (U \ 0) by scalar multiplication on the first component and on Ũ by λ · (u, x) = (λu, λ^{−1}x), then V inherits a holomorphic C^{∗}–action.

There is a canonical holomorphic projection t : V → L which forgets the first component of C^{∗} × (L \ x_{0}) and is given by (u, y) ↦ φ(uy) on Ũ. It is invariant under the C^{∗}–action. The fibres of t over L \ x_{0} are closed free orbits. The fibre t^{−1}(x_{0}) is isomorphic to C × 0 ∪ 0 × C^{n} in Ũ. It contains the origin as a single closed orbit: A fixed point of hyperbolic type. All other orbits of this fibre have the origin in its closure. The manifold L is the orbit space of the subset of closed orbits. In the algebraic category this is called the algebraic quotient V//C^{∗}, compare Kraft [1989], p. 96, and Springer [1989], p. 14. We explain this with (3.9).

(3.1) Lemma. The complex manifold V is independent of the choice of the holomorphic chart φ about x_{0}.

Proof. It is easy to see that shrinking U leads to the same manifold. Therefore it suffices to consider charts φ, ψ : U → L which differ by a holomorphic automorphism α : U → U. In this case there exists a commutative diagram

C^{∗} × (U \ 0)  ^{b \overline{}}Ũ

[figure: id × α; A]
C^{∗} × (U \ 0)  ^{b \overline{}}Ũ

with a holomorphic automorphism A. This uses the fact that (u, x) ↦ u^{−1}α(ux) has a holomorphic extension to u = 0 by the derivative of α. □

(3.2) Remark. The definition of V uses only that x_{0} ∈ L is a regular point. Apart from x_{0} the object L could have singularities. Moreover the construction itself is meaningful for differentiable manifolds. But then there is no analogue of (3.1). ♡

(3.3) Definition. Let h : C^{n+1} → C be a polynomial with the following property: 0 ∈ h^{−1}(0) =: L is a regular point of the affine hypersurface L. There exists a unique polynomial

q_{h} = q : C × C^{n+1} → C

such that:

(3.4) uq(u, x) = h(ux), u ∈ C^{∗}, x ∈ C^{n}.

(3.5) q(0, x) is a nonzero linear polynomial.

(3.6) q(λ^{−1}u, λx) = λq(u, x), λ ∈ C^{∗}.

We call q_{h} the hyperbolic modification of h. ♡

(3.7) Proposition. The affine variety q^{−1}(0) =: V together with the C^{∗}– action induced by (3.6) is the hyperbolic modification of (L, 0).

Proof. We have to show that V, as a complex space, arises by the pushout construction described above.

We select a holomorphic chart φ : U → L centered at 0 and consider the following diagram

C^{∗} × (U \ 0)  b_{\overline{}}Ũ

[figure: a; B]
C^{∗} × (L \ 0) ^{A \overline{}}V

with maps

\begin{aligned}a(u, x) & = & (u, φ(x)) \\ b(u, x) & = & (u, u^{−1}x) \\ A(u, x) & = & (u, u^{−1}x) \\ B(u, x) & = & (u, u^{−1}φ(ux)). \end{aligned}

The map B has to be interpreted as a holomorphic map — compare the proof of (3.1). By construction, the diagram is commutative.

We have to show that the diagram is a pushout. This amounts to the following verifications: The maps A and B are holomorphic embeddings onto open subsets Ã and B̃ of V. The intersection Ã ∩ B̃ is as predicted by the diagram.

The image of A is V ∩ {(u, x) | u ≠ 0}, hence open in V. An inverse of A is induced by the morphism A_{1} : (u, x) ↦ (u, ux).

Let Φ : W_{1} → W_{2} be a holomorphic isomorphism between open neighbourhoods of zero in C^{n+1} such that Φ restricts to φ : 0×U = W_{1}∩(0×C^{n}) → L∩W_{2}. Let Z_{2} = {(u, x) | ux ∈ W_{2}} ⊂ C × C^{n}.

One verifies that V ∩ Z_{2} is the image of B. An inverse of B is induced by the morphism (u, x) ↦ (u, pr(u^{−1}Φ^{−1}(ux))) with pr: C × C^{n} → C^{n} the projection.□

(3.8) Proposition. (1) Let V be a hyperbolic modification of the n–dimensional manifold L. Then the following holds: If L is acyclic, then V is acyclic.

(2) If n > 2 and π_{1}(L) = 0, then π_{1}(V) = 0.

(3) If n > 2 and L is contractible, then V is contractible.

Proof. (1) follows by applying the Mayer–Vietoris sequence to the defining pushout. (2) follows similarly from the theorem of Seifert and van Kampen. (3) follows from (1) and (2) and general results of algebraic topology [tD], III (5.11) and V (6.3)). □

The hyperbolic modification can be applied to a regular point of an arbitrary affine variety. The result is again an affine variety with C^{∗}–action.

Suppose the affine variety L = L(I) is the zero set of the ideal I ⊂ C[x_{1}, …, x_{n}]. Let 0 ∈ C^{n} be a regular point of L. Let J ⊂ C[u, x_{1}, …, x_{n}] be the ideal generated by all polynomials q_{h} such that q_{h}(u, x) = uh(u, x) for h ∈ I. Then the variety V = V (J) is the hyperbolic modification of L at zero. The proof is similar to the proof of (3.7). It uses the following fact. Suppose L ⊂ C^{n} has codimension k. Since 0 ∈ L is a regular point there exist k polynomials h_{1}, …, h_{k} ∈ I such that the Jacobi matrix of (h_{1}, …, h_{k}) has rank k at the origin. Let

π : C × C^{n} → C^{n}, (u, x) ↦ ux.

Then π^{−1}(L\0) ⊂ V and π^{−1}(L\0) is isomorphic to C^{∗} ×(L\0) as a C^{∗}–variety. Let W be an open neighbourhood of zero such that

L ∩ W = {x ∈ W | h_{1}(x) = … = h_{k}(x) = 1}.

Finally, let φ : U → L ∩ W be a holomorphic chart of L about 0. Then the complex space V is isomorphic to the pushout of

C^{∗} × (L \ 0) id × φ_{\overline{}}C^{∗} × (U \ 0) b_{\overline{}}π^{−1}(U)

with b(u, x) = (u, u^{−1}x).

There are several natural generalizations of the hyperbolic modifications which yield acyclic manifolds when applied to acyclic manifolds.

Firstly, one can introduce hyperbolic fixed points with different weights. This replaces the map b : (u, x) ↦ (u, u^{−1}x) by b_{k,l} : (u, x) ↦ (u^{k}, u^{−l}x) for a pair (k, l) of coprime natural numbers. The proof of (3.8) also applies to this case. Later we will deal with this more general construction.

Secondly, one can contemplate using other groups than C^{∗}. Consider GL(n, C) as open subset of the vector space M_{n}(C) of complex (n, n)–matrices. Let π :

M_{n}(C) × C^{n} → C, (A, x) ↦ Ax and Ũ = π^{−1}(U). Define b : (A, x) ↦ (A, A^{−1}x) and consider the pushout of

GL(n, C) × (L \ 0) id × φ_{\overline{}}GL(n, C) × (U \ 0) b_{\overline{}}Ũ.

The maps id×φ and b are holomorphic embeddings onto open subsets. Moreover (id × φ, b) is a closed embedding. Therefore the pushout is a Hausdorff complex manifold with holomorphic GL(n, C)–action.

We return to the hyperbolic modification V of an affine hypersurface L defined by h. The coordinate ring of V is C[u, x]/(q). It carries the C^{∗}–action λ · (u, x) = (λu, λ^{−1}x). Let R ⊂ C[u, x]/(q) denote the ring of invariants. The homomorphism

j : C[x] → C[u, x], x_{i} ↦ ux_{i}

induces a homomorphism

J : C[x]/(h) → R.

(3.9) Proposition. The homomorphism J is an isomorphism.

Proof. Suppose p is in the kernel of J. Then there exists a relation of the type p(ux) = a(u, x)q(u, x). We multiply by u and set u = 1. This yields p(x) = α(x)h(x). This shows J to be injective.

Let S be the ring of C^{∗}–invariant in C[u, x]. Then it is easy to see that j induces an isomorphism j : C[x] → S. Since C^{∗} is a reductive group, the surjection C[u, x] → C[u, x]/(q) induces a surjection S → R (compare Springer [1989], II). □

One expresses (3.9) by saying that L is the algebraic quotient of V under the C^{∗}–action. The map π : V → V, (u, x) ↦ ux is the quotient map.

If L is a regular hypersurface, then V is again regular. Therefore we can iterate the hyperbolic modification. Since V carries a C^{∗}–action with fixed point 0 the hyperbolic modification of V at 0 carries a C^{∗} × C^{∗}–action. The n–fold modification L(n) of L carries an action of the n–dimensional torus T (n) = C^{∗} × … × C^{∗}.

(3.10) Proposition. The algebraic quotient L(n)//T (n) is isomorphic to L.□

The general pattern of the hyperbolic modification is: Let π : X → Y be a morphism and L ⊂ Y a subvariety. Exclude some singular set S ⊂ Y of π, take π^{−1}(L \ S) and form the closure of this pre–image.

4. Acyclic affine foliations

In this section we construct polynomials q : C^{n+1} → C with the property that all fibres q^{−1}(c) are diffeomorphic to Euclidean space. The construction is based on the hyperbolic modification (section 3).

Let again h : C^{n} → C be a polynomial with h(0) = 0 such that 0 ∈ h^{−1}(0) = L is a regular point of L. Let q = q_{h} : C^{n+1} → C denote the hyperbolic modification of h.

We blow up the point 0 ∈ L and consider the complement of the proper transform. This construction will also clarify the conceptual meaning of q_{h}.

Denote by X = C ×_{C}^{∗} (C^{n} \ 0) the quotient of C × (C^{n} \ 0) under the relation (u, x) ∼ (λ^{−1}u, λx) for λ ∈ C^{∗}. The morphism p : X → C^{n}, (u, x) ↦ ux blows up the point 0 ∈ C^{n} and

p : X \ E → C^{n} \ 0, E = p^{−1}(0)

is an isomorphism. Moreover we have:

(4.1) p : X \ p^{−1}(L) → C^{n} \ L is an isomorphism.

(4.2) p^{−1}(L) = L_{1} ∪ E with L_{1} = {(u, x) | q(u, x) = 0}.

We let V = q^{−1}(1).

(4.3) Lemma. The restriction q : q^{−1}(C^{∗}) → C^{∗} is a trivial fibration with typical fibre V.

Proof. The map

φ : C^{∗} × V → q^{−1}(C^{∗}), (λ; u, x) ↦ (λ^{−1}u, λx)

satisfies q ∘ φ = pr. An inverse ψ of φ is given by

ψ(u, x) = (q(u, x); q(u, x)u, q(u, x)^{−1}x).

Therefore φ is an algebraic bundle isomorphism. □

(4.4) Lemma. Suppose h^{−1}(0) ⊂ C^{n} is a regular variety. Then q is a regular polynomial.

Proof. (4.3) shows that q is regular in all point of q^{−1}(C^{∗}). If q(u, x) = 0, then h(ux) = 0. We have

(4.5) \frac{∂q}{∂x_{j}} (u, x) = \frac{∂h}{∂x_{j}} (ux)

and since h^{−1}(0) is regular there exists j such that (4.5) is non zero. □

T. tom Dieck 4. Acyclic affine foliations 13

(4.6) Proposition. The complement X \ L_{1} is isomorphic to the regular affine hypersurface V.

Proof. We have seen in (4.3) that V is regular. Moreover V ⊂ C × (C^{n} \ 0), as seen from (3.3) and (3.4). The morphism V ⊂ C × (C^{n} \ 0) → X induces the desired isomorphism. The pre-image of X \ L_{1} in C × (C^{n} \ 0) is the open subvariety W = {(u, x) | q(u, x) ≠ 0}. The morphism W → V, (u, x) ↦ (q(u, x)u, q(u, x)^{−1}x) factors over X \ L_{1} and yields an inverse. □

Because of (4.3), we call q^{−1}(c), c ≠ 0, the general fibre of q and q^{−1}(0) the singular fibre. The singular fibre was investigated in section 3 and studied under the name of hyperbolic modification. We now deal with the topology of the general fibre V.

(4.7) Proposition. The differentiable manifold V is obtained from C^{n} \ L by attaching an open 2–handle. The homotopy type Y of V is obtained by attaching a 2–cell D^{2} to C^{n} \ L along a small normal 1–sphere about L.

Proof. Set E_{0} = E\(E∩L_{1}). Set theoretically we have X\L_{1} = (X\p^{−1}(0))∪E_{0} and X \ p^{−1}(0) ≅ C^{n} \ L =: W. Therefore we have to describe the way E_{0} is attached to W. This requires a tubular neighbourhood of E_{0}.

The normal bundle of the exceptional divisor E ≅ IP^{n−1} in X is the line bundle

π : X → IP^{n−1}, (u, x) ↦ [x].

We have L_{1} = {(u, x) | q(u, x) = 0} and [x] ∈ E_{0} is equivalent to q(0, x) ≠ 0. If we fix x, then there exists a neighbourhood U_{x} of zero, such that q(u, x) ≠ 0 for u ∈ U_{x}, i. e. (u, x) ∉ L_{1} for u ∈ U_{x}. In other words: Let π_{0} : X_{0} = π^{−1}(E_{0}) → E_{0} denote the restriction of π; then there exists an open cell subbundle U ⊂ X_{0} → E_{0} such that (u, x) ∈ U \ E_{0} implies q(u, x) ≠ 0.

The complex manifold X \L_{1} can therefore be defined by the pushout diagram

U \ E_{0}  ^{c \overline{}}U

[figure: j]
C^{n} \ L  \overline{}X \ L_{1}

where j is the embedding (u, x) ↦ ux.

Let U_{z} denote the fibre of π_{0} : U → E_{0} over z ∈ E_{0}. Then j : U_{z} → C^{n}, (u, x) ↦ ux is transverse to L in 0. Let D_{z} ⊂ U_{z} be a closed cell with boundary S_{z}. The circle S_{z} is the normal sphere about L which appears in the statement of (4.7).

The pushout diagram

S_{z}  \overline{}D_{z}

[figure: j]
C^{n} \ L  \overline{}Y

defines the attachment of the 2–cell to C^{n}\L. The space Y is homotopy equivalent to X \ L_{1}. In order to see, this consider the commutative diagram

[figure: C^{n} \ L; j; U \ E_{0}; U]

[figure: =; α; β]
[figure: C^{n} \ L; S_{z}; D_{z}.]

The inclusions α and β are homotopy equivalences. The set E_{0} is the complement of the projectivized tangent space E_{0} = IP(C^{n}) \ IP(T_{0}L), hence an affine space. Therefore the cell bundle U is contractible. The inclusion α is a morphism of fibrations which is a homotopy equivalence in the base and in the fibre and therefore a homotopy equivalence. Now apply a general result of homotopy theory (tom Dieck [1971]). □

Proposition (4.7) has some consequences for the homotopy and homology of V.

(4.8) Corollary. Let N ⊂ π_{1}(C^{n} \ L) be the normal subgroup generated by a normal sphere of L. Then π_{1}(V) ≅ π_{1}(C^{n} \ L)/N.

Proof. The theorem of Seifert and van Kampen implies that attaching a 2–cell factors out exactly the subgroup N. □

(4.9) Corollary. H_{i}(V) ≅ H_{i}(C^{n} \ L) for i ≠ 1.

Proof. Let W = C^{n} \ L. Since H_{i}(Y, W) = 0 for i ≠ 2, the exact homology sequence of (Y, W) shows that homology groups of Y and W can only differ for i = 1, 2. In this case we have the exact sequence

0 → H_{2}(W) → H_{2}(Y) → H_{2}(Y, W) ^{∂ \overline{}}H_{1}(W) → H_{1}(Y) → 0

with H_{2}(Y, W) ≅ ZZ. The map ∂ is injective. In order to see this, consider

s : H_{1}(C^{n} \ L) ≅ H_{2}(C^{n}, C^{n} \ L) ^{τ \overline{}}ZZ,

where τ gives the intersection number with L (see tom Dieck [1991], V.5). Then s ∘ ∂ is an isomorphism, since H_{2}(Y, W) is generated by a normal disk which has intersection number one with L. □

(4.10) Proposition. Suppose L has only isolated singularities and is a topological manifold. Then H_{i}(C^{n} \ L) ≅ H_{i−i}(L) for i > 0.

Proof. If L has a tubular neighbourhood in C^{n} we can apply the exact homology sequence and the Thom–Isomorphism (Spanier [1966]), H_{i}(C^{n} \ L) ≅ H_{i+1}(C^{n}, C^{n} \ L) ≅ H_{i−1}(L) and deduce the claim.

Another argument uses the fact that the sphere S^{2n−1}(r) = {z ∈ C^{n} | |z| = r} intersects L transversely for all sufficiently large r. For such r, the space C^{n} \ L is homotopy equivalent to the intersection with the disk D^{2n}(r) \ D^{2n}(r) ∩ L. We therefore study the following situation: D is an m–disk, L ⊂ D a topological submanifold with ∂D ∩ L = ∂L and transverse intersection of L and ∂D. Let S = D ∪ D′ be the double of D = D′; this is an m–sphere. Now we have the following chain of isomorphisms for i > 0:

H_{i}(D \ L) ≅ H_{i+1}(D, D \ L)

(1)

≅ H^{m−i−1}(D′ ∪ L, D′)

(2)

≅ H^{m−i−1}(∂D ∪ L, ∂D)

(3)

≅ H^{m−i−1}(L, ∂L)

(4)

≅ H_{i−1}(L).

(5)

Explanation:

(1) comes from the exact homology sequence.

(2) is Poincaré–Lefschetz duality in S, see Dold [1972], VIII.7.2.

(3) and (4) is excision.

(5) is duality in L. □

(4.11) Theorem. (1) Suppose L is an acyclic topological manifold with isolated singularities. Then V is acyclic.

(2) If, moreover, π_{1}(C^{n} \ L) is normally generated by a normal sphere (e. g. π_{1}(C^{n} \ L) ≅ ZZ), then V is contractible.

Proof. From (4.10) we see that H_{i}(C^{n}\L) is zero except for i = 1. For i = 1 the argument with intersection numbers in the proof of (4.9) shows that H_{1}(C^{n}\L) ≅ ZZ, generated by a normal sphere. The result now follows from (4.7), (4.9), and (4.10). □

Altogether we can now deduce the next result.

(4.12) Theorem. Suppose n ≥ 3. Let h : C^{n} → C be a polynomial with h(0) = 0 such that h^{−1}(0) is a regular contractible hypersurface. Then the hyperbolic modification q : C^{n+1} → C is a regular polynomial such that each fibre q^{−1}(c), c ∈ C is diffeomorphic to Euclidean space.

For the proof we need another Lemma.

(4.13) Lemma. Suppose L is regular. Then:

(1) If π_{1}(L) = 0, then π_{1}(V) = 0.

(2) If L is contractible, then V is contractible.

Proof. (1) Let U be an open tubular neighbourhood of L in C^{n}. By the theorem of Seifert and van Kampen, the diagram

π_{1}(U \ L)  \overline{}π_{1}(U)

[figure]
π_{1}(C^{n} \ L)  \overline{}π_{1}(C^{n})

is a pushout diagram. We have homotopy equivalences L ≃ U and U \L ≃ L×S^{1}. Therefore π_{1}(U) = 0, π_{1}(U \ L) ≅ π_{i}(L × S^{1}) ≅ π_{1}(S^{1}), and the pushout shows that π_{1}(C^{n} \ L) is normally generated by a normal sphere of L. Now we apply (4.8).

(2) follows from the proof of (1) and (4.11). □

Proof of (4.12). By (4.4), q is regular. By (4.13), the general fibre of q is contractible. By (3.8), the singular fibre is contractible. Now apply (1.??). □

We call regular polynomials p : C^{n} → C such that all fibres are diffeomorphic to Euclidean space slice polynomials. Linear forms are trivial slice polynomials. If remains to find nontrivial examples to which (4.12) applies. This will be the subject of the next section.

5. Brieskorn varieties

The Brieskorn polynomials are among the simplest polynomials to which the considerations of the previous section can be applied.

Let a(1), …, a(n) be positive integers. The associated Brieskorn polynomial is

(5.1) h(x_{1}, …, x_{n}) = x_{1}^{a(1)} + … + x_{n}^{a(n)}.

These polynomials have an isolated singularity at the origin. The Brieskorn manifold is the intersection

(5.2) B = B(a(1), …, a(n)) = h^{−1}(0) ∩ S^{2n−1}.

Brieskorn [1966] investigated under which condition this manifold is a topological sphere. The result is as follows (5.3).

T. tom Dieck 5. Brieskorn varieties 17

Define the graph Γ of the family (a(j)): The vertices are {1, …, n}. There is an edge connecting i and j if and only if the greatest common divisor (a(i), a(j)) > 1.

(5.3) Theorem. B(a(1), …, a(n)) is a homology sphere if and only if one of the following conditions holds:

1. Γ has at least two isolated points.

2. Γ has an isolated point and another connected component Γ′ with an odd number of vertices such that for different i, j ∈ Γ′ always (a_{i}, a_{j}) = 2. □

The Brieskorn varieties are simply connected for n ≥ 4. A simple consequence of the h–cobordism theorem asserts that they are homeomorphic to the sphere S^{2n−3}, provided they are homology spheres, Smale [1956], Milnor [1968], p.

109. For further information see also Hirzebruch–Mayer [1968], Bredon [1972], and Milnor [1968].

(5.4) Theorem. Let q : C^{n+1} → C be a hyperbolic modification of h, applied to a regular point of h^{−1}(0). Suppose the manifold B in (5.2) is homeomorphic to a sphere. Then V = q^{−1}(1) is a contractible affine variety (n ≥ 2).

Proof. By homogeneity of h, the space L = h^{−1}(0) is homeomorphic to the open cone over B. Therefore, L is homeomorphic to Euclidean space if B is a sphere. By (4.11.1), V is acyclic.

In order to show that V is contractible we derive another topological construction of V. The space C^{n} \L is homeomorphic to the product of S^{2n−1} \B with an open interval J. The inclusion B ⊂ S^{2n−1} can be considered as a (generalized) knot. In order to obtain the homotopy type of V we have to attach a 2–cell along a normal sphere (4.7). Let U ⊂ S^{2n−1} be an open tubular neighbourhood of B. Up to homotopy, attaching a 2–cell amounts to adding a fibre over x ∈ B of the tubular neighbourhood to S^{2n−1} \ U. The result is the sphere S^{2n−1} with the tubular neighbourhood W of B \ x deleted. Since B is assumed to be a sphere, W is an open cell and its complement is a disk. The resulting homotopy type is therefore contractible. □

Instead of Brieskorn polynomials one can use other weighted homogeneous polynomials with appropriate topological properties.

The simplest case of (5.4) arises for the polynomial h(x, y) = x^{a} − y^{b} for coprime integers a, b. If we apply the hyperbolic modification to the regular point (1,1) we obtain the polynomial (1.5)

P (x, y, z) = z^{−1} ((xz + 1)^{a} − (yz + 1)^{b}) .

The case (a, b) = (3, 2) leads to the contractible hypersurface in C^{3}

z^{2}x^{3} + 3zx^{2} + 3x − zy^{2} − 2y = 1.

We can use this polynomial as input for (4.12). If we apply the hyperbolic modification at the point (1,1,0) we obtain the slice polynomial

q(u, x, y, z) = uz^{2}(ux + 1)^{3} + 3z(ux + 1)^{2} + 3x − (uy + 1)^{2} − 2y.

Other simple Brieskorn polynomials to which (5.4) can be applied are

x^{p} + y^{q} + z_{2}^{2} + … + z_{n}^{2}

for coprime odd integers p, q and

z_{0}^{d} + z_{1}^{2} + … + z_{n}^{2}

for d and n odd. These polynomials are interesting because they have large symmetry groups.

We now verify that some of the contractible varieties carry an exotic algebraic structure, i. e. are not isomorphic as varieties to affine space.

6. Ramified coverings

Ramified coverings can be used to construct acyclic varieties.

We assume given an action

C^{∗} × C^{n} → C^{n}, (λ, x) ↦ λ · x

of C^{∗} on C^{n}. For the time being, the action is just assumed to be continuous, if we are just dealing with topological properties. But in an algebraic context it seems more reasonable to assume that left translations x ↦ λ · x are polynomial automorphisms.

A polynomial q : C^{n} → C (or any function) is called quasi– invariant of weight l ∈ ZZ with respect to the given action if

q(λ · x) = λ^{l}q(x), x ∈ C^{n}, λ ∈ C^{∗}

holds.

We assume given a quasi–invariant polynomial q of weight l. We denote the fibre q^{−1}(c) by F_{c}. The k–fold ramified covering of C^{n} along q^{−1}(0) is the affine variety

X = {(u, x) ∈ C × C^{n} | u^{k} + q(x) = 0}, k ≥ 1.

We shall exhibit conditions on q under which X will be diffeomorphic to Euclidean space.

We begin with some elementary remarks about ramified coverings.

(6.1) Lemma. If F_{0} is regular, then X is regular.

Proof. Compute the partial derivatives of p(u, x) = u^{k} + q(x). □

T. tom Dieck 6. Ramified coverings 19

(6.2) Lemma. The map σ : X → C^{n}, (u, x) ↦ x is a k–fold ramified covering with ramification locus F_{0}.

Proof. Given x ∈ C^{n}, there exists u such that u^{k} + q(x) = 0. Therefore σ is surjective. Consider the ZZ/k–action μ · (u, x) = (μu, x) on X. Then σ induces a homeomorphism of the orbit space X/(ZZ/k) with C^{n}. □

(6.3) Lemma. F_{0} = {(u, x) | u = 0, q(x) = 0} is a regular submanifold of X. The intersection of X with {u = 0} in C^{n+1} is transverse. □

(6.4) Proposition. The map q : C^{n} \ F_{0} → C^{∗} is locally trivial with typical fibre F_{1} and structure group ZZ/l.

Proof. Let F_{c} = q^{−1}(c). The quasi–invariance of q implies that (ω, x) ↦ ω ·x is a ZZ/l–action on F_{c}. We use this action to form C^{∗} ×_{G} F_{1}, (λ, x) ∼ (λω, ω^{−1} · x), ω ∈ G = ZZ/l. The map φ : C^{∗} ×_{G} F_{1} → C^{n} \ F_{0}, (λ, x) ↦ λ · x is well–defined and satisfies q ∘ φ = pr_{1}. Let Y = {(λ, x) | λ^{l} = q(x)} ⊂ C^{∗} × (C^{n} \ F_{0}). Then pr_{2} : Y → C^{n} \ F_{0} is a quotient map. The morphism Y → C^{∗} ×_{G} F_{1}, (λ, x) ↦ (λ, λ^{−1} · x) induces an inverse to φ. □

(6.5) Proposition. Suppose π_{1}(C^{n} \ F_{0}) ≅ ZZ and F_{1} is connected. Then π_{1}(F_{c}) = 0.

Proof. Consider the exact homotopy sequence of the fibration q : C^{n} \F_{0} → C^{∗}

0 → π_{1}(F_{0}) → π_{1}(C^{n} \ F_{0}) ^{q_{∗} \overline{}}π_{1}(C^{∗}) → 0.

The hypothesis implies that q_{∗} is an isomorphism. □

We study the topology of X via γ : X → C, (u, x) ↦ u. Let Γ_{c} = γ^{−1}(c).

(6.6) Proposition. The morphism γ : X\Γ_{0} → C^{∗} is locally trivial with typical fibre F_{1} and structure group ZZ/l.

Proof. The fibre Γ_{c} = {(u, x) | q(x) = −u^{k}, u = c} is isomorphic to F_{c^{k}}. We have the ZZ/l–action ω · (1, x) = (1, ω · x) on Γ_{1}. The map φ′ : C^{∗} × Γ_{1} → X \ Γ_{0}, (λ, (1, x)) ↦ (λ^{l}, λ^{k} · x) is well–defined. Let jk ≡ 1 mod l. Then (λ, (1, x)) and (ω^{j}λ, (1, ω^{−1}·x)) have the same image under φ′ for ω ∈ ZZ/l. We therefore use the ZZ/l–action (ω, λ) ↦ ω^{j}λ on C^{∗} in order to form C^{∗} ×_{G} Γ_{1} and obtain from φ′ a morphism φ : C^{∗}×_{G}Γ_{1} → X \Γ_{0} which satisfies γ∘φ = 0 with p : C^{∗}×_{G}Γ_{1} → C^{∗}, (λ, x) ↦ λ^{l}. We obtain an inverse to φ as follows. Write X \ Γ_{0} as quotient of

Z = {(v, u, x) | v^{l} = u, (u, x) ∈ X \ Γ_{0}}

and define

Z → C^{∗} ×_{G} Γ_{1}, (v, u, x) ↦ (v, (1, v^{−k} · x)).

If we replace v by ω^{j}v, ω ∈ G, the resulting element has the same image under this map. Thus we obtain an induced map ψ : X \ Γ_{0} → C^{∗} ×_{G} Γ_{1} which is the inverse of φ. □

(6.7) Proposition. Under the hypothesis of (6.5) we have π_{1}(X) = 0.

Proof. The exact homotopy sequence of γ gives

0 → π_{1}(Γ_{1}) → π_{1}(X \ Γ_{0}) ^{γ_{∗} \overline{}}π_{1}(C^{∗}) → 0,

because Γ_{1} ≅ F_{1} is connected. By (4.5) π_{1}(Γ_{1}) = 0, hence γ_{∗} is an isomorphism. By (6.3) and the transversality theorem (?.?) the inclusion X \ Γ_{0} → X is generated by a normal sphere about Γ_{0}. In order to see, this consider the covering σ : X \ Γ_{0} → C^{n} \ F_{0}. A normal sphere of Γ_{0} is mapped to k times a normal sphere of F_{0}. The diagram

X \ Γ_{0}  ^{σ \overline{}}C^{n} \ F_{0}

[figure: γ; q]
C^{∗}  ^{τ \overline{}}C^{∗}

with τ(z) = −z^{k} is commutative. Let U be a tubular neighbourhood of F_{0} in C^{n}. The theorem of Seifert and van Kampen, applied to U, D^{n} \ F_{0}, shows that π_{1}(C^{n} \ F_{0}) is generated by a normal sphere of F_{0}.

A normal sphere of γ_{0} is contractible in X. □

The following Theorem is the main result of this section.

(6.8) Theorem. We assume the following:

(1) q : C^{n} → C is quasi–invariant of weight l.

(2) F_{0} = q^{−1}(0) is regular and ZZ/k–acyclic.

(3) π_{1}(C^{n} \ F_{0}) ≅ ZZ,

(4) F_{c} = q^{−1}(c) for c ≠ 0 is connected. Then X = {(x, u) ∈ C^{n}×C | u^{k}+q(x) = 0} is contractible.

In the sequel we work under the hypotheses of (6.8). The proof of (6.8) uses (6.9)

– (6.11).

(6.9) Lemma. The groups H_{i}(X; ZZ), i > 0, are annihilated by k.

Proof. We have the orbit map σ : X → C^{n} of the ZZ/k–action μ · (u, x) = (μu, x), see (6.2). This action can be exended to an S^{1}–action, up to an automorphism of ZZ/k. The S^{1}–action λ(u, x) := (λ^{l}u, λ^{k} · x) restricts a ZZ/k–action, which coincides with the original action up to λ ↦ λ^{l}. This uses that k and l are coprime. As a consequence of this S^{1}–action we see that ZZ/k = K acts trivially on H_{i}(X; ZZ). We use the transfer t : H_{i}(X/K) → H_{i}(X). In our case t ∘ σ_{∗} and σ_{∗} ∘ t are multiplication by k, see ??. Since X/K ≅ C^{n} is acyclic we obtain the desired result. □

(6.10) Lemma. The inclusion X \ Γ_{0} → X induces an isomorphism

H_{i}(X \ Γ_{0}; ZZ/k) → H_{i}(X; ZZ/k)

for i ≥ 2.

Proof. We use the exact sequence, the Thom isomorphism and the acyclicity of Γ_{0} ≅ F_{0}. The diagram (coefficients in ZZ/k)

H_{i+1}(X, X \ Γ_{0}) → H_{i}(X \ Γ_{0}) → H_{i}(X) → H_{i}(X, X \ Γ_{0})

[figure: ≅; ≅]
H_{i−1}(Γ_{0})  H_{i−2}(Γ_{0})

gives the claim for i ≥ 3. For i = 2 we use in addition the isomorphism

∂ : ZZ ≅ H_{2}(X, X \ Γ_{0}; ZZ) → H_{1}(X \ Γ_{0}; ZZ) ≅ ZZ

which follows from the fact that H_{1}(X \ Γ_{0}) is generated by a normal sphere. □

In order to show that H_{i}(X \ Γ_{0}; ZZ/k) is zero, we use the Wang sequences of the bundles q and γ, or rather their restrictions to the part over S^{1}:

q : A → S^{1}, γ : B → S^{1}.

The Wang sequence of q : A → S^{1} has the form

… → H_{i}(F) ^{\overline{}α_{∗} − id} H_{i}(F) → H_{i}(A) → …

where α : F → F is the monodromy of a typical fibre F. Similarly for γ with monodromy β : Γ → Γ. The monodromy α : F → F is defined by a pullback diagram

A ^{\overline{}H} [0, 1] × F

[figure: q; pr]
S^{1} ^{\overline{}e}  [0, 1]

with H(0, x) = x, H(1, x) = α(x), F = q^{−1}(1), e(t) = exp(2πi t).

(6.11) Lemma. The monodromies α and β are given by

α : F_{1} → F_{1}, x ↦ ω · x

β : Γ_{1} → Γ_{1}, x ↦ ω^{k} · x

with ω = exp(2πi/l).

Proof. Define H : I ×F_{1} → S^{1}×_{G}F_{1} by (t, x) ↦ (exp(2πit/l), x), and similarly for γ. (Compare the proof of (6.4) and (6.6).) □

Proof of (6.8). We show that X is acyclic. By (6.7), (6.9) and (6.10) it suffices to show that H_{i}(X \ Γ_{0}; ZZ/k) ≅ 0 for i ≥ 2. We use the Wang sequences with ZZ/k–coefficients.

For q : A → S^{1} we know the structure of the Wang sequence because:

H_{i}(A) ≅ H_{i}(C^{n} \ F_{0}) ≅ H_{i+1}(C^{n}, C^{n} \ F_{0}) ≅ H_{i−1}(F_{0});

the left most isomorphism is the Thom isomorphism. By hypothesis these groups are zero. Hence α_{∗} − id is an isomorphism for i ≥ 2. Since we are dealing with finite groups, bijectivity of β_{∗} − id is equivalent to injectivity. By (6.11) β_{∗} = α_{∗}^{k}. Moreover α_{∗} has order l. Suppose (β_{∗} − id)(x) = 0. Let kj ≡ 1 mod l. Then β_{∗}(x) = α_{∗}^{k}(x) = x and α_{∗}(x) = α_{∗}^{kj}(x) = α_{∗}^{k} ∘ … ∘ α_{∗}^{k}(x) = x. Since α_{∗} − id is an isomorphism, x = 0. Therefore β_{∗} −id is an isomorphism and the Wang sequence implies H_{i}(X \ Γ_{0}) = 0 for i ≥ 2. □

(6.12) Remark. The proof of (6.8) does not really use that q is a polynomial. We leave it to the reader to formulate the corresponding result in the differentiable or holomorphic category.

7. Complements of divisors

This section is devoted to the homological properties of affine surfaces. We study the surfaces together with their compactifications.

Let X be a closed, connected, oriented, smooth 4–manifold. We consider the complement V = X \D of a compact, non–empty subset D. By Poincaré duality, we have for each subring T of the rational numbers Q an isomorphism

(7.1) H_{i}(V; T) ≅ H^{4−i}(X, D; T);

see Dold [1972], VIII.7.2. We are mainly interested in the case when D is a union of two–dimensional submanifolds (with singularities). Therefore we make the following assumptions.

(7.2) Assumptions.

(1) H_{4}(D) ≅ H^{4}(D) ≅ 0.

(2) H^{3}(D) ≅ H^{3}(D) ≅ 0.

T. tom Dieck 7. Complements of divisors 23

(3) H^{∗}(D) is finitely generated. The coefficient ring for co–homology is ZZ, unless otherwise specified.

(7.3) Proposition. Under the assumptions (8.2) the following hold

(1) H_{0}(V) ≅ H^{0}(V) ≅ ZZ

(2) H_{4}(V) ≅ H^{4}(V) ≅ 0.

(3) H_{3}(V) and H^{1}(V) are free abelian.

(4) H_{∗}(V) and H^{∗}(V) are finitely generated.

Proof. We use exact sequences of the pair (X, D), Poincaré duality and universal coefficient formulas. For the purposes of duality one has to use Čech cohomology.

The exact sequence

H^{3}(D) → H^{4}(X, D) → H^{4}(D) → H^{4}(X)

and (8.2) yield H^{4}(X, D) ≅ H^{4}(X). Since X is oriented and connected, H^{4}(X) ≅ ZZ. By duality H^{4}(X, D) ≅ H_{0}(V) ≅ ZZ. Hence V is connected. It is a general fact that for a connected, non–compact 4–manifold V the groups H_{j}(V) are zero for j ≥ 4, see Dold [1972], VIII.3. The exact sequence

H^{0}(X) → H^{0}(D) → H^{1}(X, D) → H^{1}(X)

and duality H^{1}(X, D) ≅ H_{3}(V) shows H_{3}(V) to be free abelian. The universal coefficient isomorphism yields H^{0}(V) ≅ Hom(H_{0}(V), ZZ) ≅ ZZ.

The cohomology H^{∗}(X) of a compact manifold X is finitely generated, since X is a retract of a finite simplicial complex, see Dold [1972], IV.8.10. The exact sequence of the pair (X, D) and (8.2.3.) shows H^{∗}(X, D) to be finitely generated. By duality (8.1), H_{∗}(V) is finitely generated and then, by the universal coefficient formula for V, the groups H^{∗}(V) are finitely generated. A manifold has the homotopy type of a CW–complex. Therefore singular and Čech cohomology for V agree. Since H^{1}(V) ≅ Hom(H_{1}(V), ZZ), the group H^{1}(V) is free abelian. □

We are interested in T –acyclic complements V, this means:

(7.4)  H̃_{∗}(V; T) = 0 or equivalently: H̃^{∗}(V; T) = 0).

Proposition (8.3.3) has the following consequence:

(7.5) Corollary. Under the assumptions of (8.3) the following holds:

(1) H_{3}(V; T) = 0 implies H_{3}(V) = 0.

(2) H^{1}(V; T) = 0 implies H^{1}(V) = 0, H_{1}(V) finite. □

(7.6) Proposition. Suppose (8.2) and (8.4) hold. Then we have:

(1) H_{3}(X) ≅ H^{1}(X) ≅ 0.

(2) H_{1}(X; T) ≅ H^{3}(X; T) ≅ 0.

(3) H^{1}(D; T) ≅ 0.

Proof. By duality, H^{j}(X, D; T) = 0 for j < 4. The exact sequence H^{3}(X, D; T) → H^{3}(X; T) → H^{3}(D; T) and (8.2) show H^{3}(X; T) = 0 and duality then gives H_{1}(X; T) ≅ 0.

By universal coefficients we conclude H^{1}(X; T) = 0 and, since H^{1}(X) ≅ Hom(H_{1}(X), ZZ) is free and H^{1}(X; T) ≅ H^{1}(X) ⊗_{ZZ} T, we obtain H^{1}(X) = 0 and finally H_{3}(X) = 0 by duality. The exact cohomology sequence of (X, D) and H^{j}(X, D; T) = 0 imply H^{1}(D; T) ≅ H^{1}(X; T) = 0. □

Affine varieties have special properties.

(7.7) Proposition. Let V be diffeomorphic to a regular affine surface over C. Then the following holds:

(1) V has the homotopy type of a finite two–dimensional CW–complex.

(2) H_{i}(V) = 0 for i > 2.

(3) H_{2}(V) is free abelian.

(4) D is connected.

Proof. (2), (3) and (4) follow immediately from (1). For (1) see Milnor [1963], §7. □

We now assume that D is a union of closed, connected, oriented, two– dimensional, smooth submanifolds D_{1}, …, D_{r}. We say that D_{1}, …, D_{r} have normal crossings, if the following holds:

(7.8)

(1) D_{i} and D_{j} have transverse intersection for i ≠ j.

(2) For i ≠ j ≠ h ≠ i the intersection D_{i} ∩ D_{j} ∩ D_{k} is empty.

If D_{1}, …, D_{r} have normal crossings we associate to D the dual graph

Γ(D) = (Γ_{0}(D), Γ_{1}(D)).

The set Γ_{0}(D) of vertices is {D_{1}, …, D_{r}}. Each intersection point in D_{i} ∩D_{j}, i ≠ j, is an edge connecting the vertices D_{i} and D_{j}; thus Γ_{1}(D) is ∐_{i≠j} D_{i} ∩ D_{j}. We do

not distinguish notationally between a graph Γ and its geometric realization. We apply a geometric terminology to Γ, like: cycle, subdivision, Euler characteristic.

(7.9) Lemma. Let D = D_{1} ∪…∪D_{r} have normal crossings. Then H_{1}(D; Q) = 0 (or H^{1}(D; Q) = 0) if and only if the D_{j} are spheres and Γ(D) is a tree.

Proof. Suppose H_{1}(D; Q) = 0. If D_{j} is not a sphere, then H_{1}(D_{j}; Q) is non– zero of rank twice the genus of D_{j}. Moreover, H_{1}(D_{j}; Q) is a direct summand of H_{1}(D; Q). Therefore the space D is obtained from a collection of spheres ∐ D_{j} by identifying isolated points of different spheres. By a geometric construction one shows that Γ(D) is a retract of D. Therefore H_{1}(Γ(D); Q) is a direct summand of H_{1}(D; Q). Since the rank of H_{1}(Γ(D); Q) is the number of (independent) cycles of Γ(D) we conclude that Γ(D) is a tree.

Conversely, if Γ(D) is a tree of spheres one shows by induction on r that H_{1}(D) = 0. □

(7.10) Theorem. Suppose D = D_{1} ∪ … ∪ D_{r} has normal crossings. Then the following are equivalent:

(1) V = X \ D is T –acyclic.

(2) The D_{j} are spheres and Γ(D) is a tree. The inclusion j : D → X induces an isomorphism j_{∗} : H_{2}(D; T) → H_{2}(X; T) and H_{1}(X; T) = 0.

Proof. (1) ⇒ (2). By (8.6.3) H^{1}(D; T) = 0 and hence, by (8.9), the D_{j} are spheres and Γ(D) is a tree.

The exact sequence

H^{2}(X, D; T) → H^{2}(X; T) ^{j^{∗} \overline{}}H^{2}(D; T) → H^{1}(X, D; T),

duality H^{j}(X, D; T) ≅ H_{4−j}(V; T) and (1) show, that j^{∗} is an isomorphism.

From the universal coefficient theorem Spanier [1966], p. 248, Theorem 12, we conclude that j_{∗} : H_{2}(D; T) → H_{2}(X; T) is an isomorphism.

(2) ⇒ (1). The condition H_{1}(X; T) = 0 implies H^{3}(X; T) ≅ 0 and H^{1}(X; T) ≅ 0. Moreover j_{∗} is an isomorphism on H_{2}(−; T) if and only if j^{∗} is an isomorphism on H^{2}(−; T). From (8.3) we know already that H_{0}(V) ≅ H^{0}(V) and H_{4}(V) ≅ H^{4}(V) ≅ 0. The sequence (cohomology with coefficients in T)

H^{1}(D) → H^{2}(X, D) → H^{2}(X) → H^{2}(D) → H^{3}(X, D) → H^{3}(X)

shows H^{3}(X, D; T) ≅ H_{1}(V; T) ≅ 0 and H^{2}(X, D; T) ≅ H_{2}(V; T) ≅ 0. The sequence

H^{0}(X; T) → H^{0}(D; T) → H^{1}(X, D; T) → H^{1}(X; T)

shows H^{1}(X, D; T) ≅ H_{3}(V; T) ≅ 0. □

We now collect the information about integral co–homology in the case when V = X \ D and D is a union of spheres with normal crossings.

(7.11) Proposition. Suppose V is T –acyclic. Then the following holds:

(1) H_{0}(V) ≅ H^{0}(V) ≅ ZZ.

(2) H_{4}(V) ≅ H^{4}(V) ≅ 0.

(3) H_{3}(V) ≅ H^{1}(V) ≅ 0.

(4) H_{1}(V) ≅ H^{2}(V) ≅ coker(H_{2}(D) → H_{2}(X)). This is a finite T –torsion group.

(5) H_{2}(V) ≅ H^{3}(V) is a finite T –torsion group. If V is an affine variety, this group is zero.

Proof. (1) and (2) are given by (8.3), and (3) is given by (8.5). Since V is T –acyclic, we have H_{1}(D) = 0 by (8.9) and the exact sequence and duality show

coker(H_{2}(X) → H_{2}(X)) ≅ H_{2}(X, D) ≅ H^{2}(V).

Also, by (8.5), this is a finite T –torsion group. By universal coefficients, H^{2}(V) ≅ Ext(H_{1}(V), ZZ) ≅ H_{1}(V).

Since V is T –acyclic, H_{2}(V) and H^{3}(V) are finite T –torsion groups. Exact sequence and duality shows H^{3}(V) ≅ H_{1}(X) and universal coefficients H^{3}(V) ≅ Ext(H_{2}(V), ZZ) ≅ H_{2}(V). The final statement follows from (8.7.1). □

8. Homology spheres

Let B denote a compact, connected, oriented 4–manifold with boundary S = ∂B ≠ ∅.

(8.1) Proposition. Suppose B is Q–acyclic. Then S is Q–homology sphere. More precisely, H_{2}(S) = 0 and

|H_{1}(S)| = | ker i : H_{1}(B) → H_{1}(B, S)|^{2}.

Proof. Consider the exact sequence

H_{2}(B, S; Q) → H_{1}(S; Q) → H_{1}(B; Q).

T. tom Dieck 8. Homology spheres 27

By duality H_{2}(B, S; Q) ≅ H^{2}(B; Q) = 0. Hence H_{1}(S; Q) = 0 and H_{1}(S) is therefore a finite group. This gives, by duality and universal coefficients,

H_{2}(S) ≅ H^{1}(S) ≅ Hom(H_{1}(S), ZZ) ≅ 0.

Duality and universal coefficients again yield the commutative diagram

Ext(H_{1}(B, S), ZZ) ≅ H^{2}(B, S) ≅ H_{2}(B)

[figure: i^{∗}; j]
Ext(H_{1}(B), ZZ) ≅ H^{2}(B) ≅ H_{2}(B, S).

By algebra, coker i^{∗} ≅ ker i and the exact sequence of the pair (B, S) yields the short exact sequence 0 → coker j → H_{1}(S) → ker i → 0. □

Now suppose we are in a situation of the previous section: D is a union of embedded spheres D_{1}…. , D_{r} in X with normal crossing and V = X \ D. There exists a suitable tubular neighbourhood U of D in X such that X \ U is diffeomorphic to V and B = X \ U^{∘} is a smooth manifold with boundary S and interior V = B \S. In this case H_{1}(B, S) ≅ H^{3}(V) by duality. In case H^{3}(V) = 0 we obtain from (9.1)

(8.2) |H_{1}(S)| = |H_{1}(V)|^{2}.

We recall the plumbing construction for the tubular neighbourhood U of D. We assume that D_{1}, …, D_{r} has normal crossings and that Γ(D) is a tree. One chooses closed tubular neighbourhoods U_{i} of D_{i} as smooth images t_{i} : E_{i} → U_{i} of 2–cell bundles E_{i} → D_{i}. The sets U_{i} and U_{j} intersect if and only if D_{i} ∩ D_{j} ≠ ∅. Fix (i, j) with D_{i}∩D_{j} = {x}. There exist embeddings τ_{i} : D^{2} → D_{i} and τ_{j} : D^{2} → D_{j} about x and trivializations σ_{i} : D^{2} × D^{2} → t_{i}^{−1}(U_{i} ∩ U_{j}) =: E_{ij} over τ_{i} such that

D^{2} × D^{2}  ^{σ_{i} \overline{}}E_{ij}  ^{t_{i} \overline{}}U_{i} ∩ U_{j} ^{\overline{}t_{j}} E_{ji} ^{\overline{}σ_{j}} D^{2} × D^{2}

is the switching map (u, v) ↦ (v, u). Thus U can be obtained from the bundles E_{i} by a suitable gluing process along the E_{ij}, called plumbing.

The exact sequence

H^{1}(S) → H^{2}(U, S) ^{s \overline{}}H^{2}(U) → H^{2}(S) → H^{3}(U, S)

shows that H^{2}(S) ≅ H_{1}(S) is isomorphic to the cokernel of s. Let e_{i} ∈ ZZ denote the Euler number of the oriented normal bundle E_{i}. The graph Γ becomes a weighted graph if we assign to each vertex D_{i} ∈ Γ_{0}(D) the weight e_{i}. The Euler number e_{i} is also called the self intersection number of D_{i} in U or in X. The matrix w : Γ_{0}(D) × T_{0}(D) → ZZ with w(D_{i}, D_{i}) = e_{i} and w(D_{i}, D_{j}) = |D_{i} ∩ D_{j}| is called the intersection matrix of the weighted tree Γ(D).

The inclusion D → U and S → U \D are homotopy equivalences (tom Dieck [1991], II.1.5). Let x′_{i} ∈ H^{2}(D_{i}) be dual to the fundamental class.

(8.3) Lemma. H^{2}(U) is free abelian with basis x_{1}, …, x_{r} such that under the map induced by the inclusion D_{i} → U the element x_{j} is mapped to δ_{ij}x′_{i}.

Proof. This is seen by calculating H^{2}(D) inductively from the H^{2}(D_{i}) via the Mayer–Vietoris sequence. □

We use H^{2}(U, U \ D) for the calculation of H^{2}(U, S). By excision we have

H^{2}(U, U \ D_{i}) ≅ H^{2}(U_{i}, U_{i} \ D_{i})

and, by the Thom Isomorphism, the latter group is generated by the Thom class Φ_{i}. Let y_{i} ∈ H^{2}(U, U \ D) be the image of Φ_{i} under the canonical map H^{2}(U, U \ D_{i}) → H^{2}(U, U \ D).

(8.4) Lemma. H^{2}(U, S) is a free abelian group with basis y_{1}, …, y_{r}.

Proof. The proof is by induction on r using the relative Mayer–Vietoris sequence (Dold [1972], III.8.).

(8.5) Proposition. The matrix of s : H^{2}(U, S) → H^{2}(U) with respect to the bases (8.3) and (8.4) is the intersection matrix of the weighted tree Γ(D).

Proof. The Euler number e_{i} can be defined as follows: The image of the Thom class Φ_{i} under the map

H^{2}(U_{i}, U_{i} \ D_{i}) → H^{2}(U_{i}) → H^{2}(D_{i})

is e_{i}x′_{i}. The construction of the elements x_{i}, y_{i} now shows directly, that the diagonal elements of the matrix are the e_{i}.

We now suppose i ≠ j and D_{i} ∩ D_{j} = {x}. The fibre F_{i} of U_{i} over x is then part of D_{j}. We therefore have a commutative diagram induced by inclusions

\begin{matrix}y_{i} ∈ H^{2}(U, U \ D) & \overline{} \\  \\ H^{2}(F_{i}, F_{i} \ x)^{\overline{}≅} \end{matrix} [figure: H^{2}(U); H^{2}(D_{j}) ∋ x_{j}; H^{2}(D_{j}, D_{j} \ x).]

The Thom class Φ_{i} has the characteristic property that its restriction to a fibre H^{2}(F_{i}, F_{i} \ x) is the canonical generator. Similarly, the fundamental class of H^{2}(D_{j}) is mapped to the canonical generator of H^{2}(D_{j}, D_{j} \ x). Since the orientations of F_{i} and D_{j} agree, we see that the image of y_{i} in H^{2}(D_{j}) is x_{j}.

Finally, we have to show that y_{i} maps to zero in H^{2}(D_{j}) if D_{i} ∩ D_{j} = ∅. But in this case we have a commutative diagram

H^{2}(U, U \ D_{i})  \overline{}H^{2}(U)

[figure]
[figure: 0 = H^{2}(U_{j}, U_{j}); H^{2}(U_{j}); H^{2}(D_{j})]

which yields the claim. □

We make the following general remarks. Let U be a compact, connected, oriented 4–manifold with boundary S. Assume H_{2}(S) = 0 and H_{1}(U) = 0. Then we have an exact sequence

0 → H_{2}(U) ^{s \overline{}}H_{2}(U, S) → H_{1}(S) → 0.

By duality and universal coefficients H_{2}(U, S) ≅ H^{2}(U) ≅ Hom(H_{2}(U), ZZ) is free abelian. The map s translates into a map H_{2}(U) → Hom(H_{2}(U), ZZ) and thus into a bilinear form H_{2}(U) × H_{2}(U) → ZZ on the free abelian group H_{2}(U). This bilinear form is known as the intersection form. In (9.3)–(9.5) above we have made explicit what this means in our case.

From (9.2) and (9.5) we obtain the following:

(8.6) Corollary. Let V = X \ D be Q–acyclic. The order of H_{1}(S) is the absolute value of the determinant of the intersection matrix. This order is always the square of an integer. □

If (Γ, w) is any weighted tree with weight function w : Γ_{0} → Q we have the matrix (w_{xy}) with w_{xx} = w(x, x) and w_{xy} = 1 for x ≠ y if and only if x, y are connected by an edge. Call det(w_{xy}) the determinant det(Γ) = det(Γ, w) of the weighted tree. One computes det(Γ, w) inductively via a generalized continued fraction algorithm as follows. Assume the following situation: Let e ∈ Γ_{1} be an edge with weights b and c at its boundary vertices X and Y. Let the vertex with weight b ≠ 0 be a terminal vertex of the tree Γ. We obtain a new weighted tree Γ′ by removing e and X from Γ and replacing the weight c of Y by c − b^{−1}. Then the following holds:

(8.7) Proposition. det(Γ) = b det(Γ′).

Proof. We expand the determinant

| \begin{matrix}{b 1 \atop 1 c} & 0 \\ 0 & A \end{matrix} |

with respect to the first row and get

b | \begin{matrix}c & 0 \\ 0 & A \end{matrix} | − |A| = b | \begin{matrix}c − b^{−1} & 0 \\ 0 & A \end{matrix} | .

□

It is also easy to compute what happens if b = 0 (compare Eisenbud– Neumann [1985], p. 153).

9. Combinatorial properties

Let M be a closed, connected, oriented, differentiable 4–manifold. Let (C_{1}, …, C_{n}) be a set of immersed closed, connected, oriented surfaces C_{i} in M.

(9.1) Definition. A point x ∈ C := ∪C_{i} is called tidy, if the following holds:

(1) There exists an oriented C^{∞}–chart φ : (U, 0) → (M, x), U ⊂ C^{2} open, about x such that φ^{−1}(M ∩ C) is a union of complex submanifolds B_{1}, …, B_{r} wich have pairwise transverse intersection in 0.

(2) Each φ(B_{j}) is contained in exactly one C_{i} and φ|B_{j} : B_{j} → C_{i} is orientation preserving.

Here U and the B_{j} carry the canonical orientation induced from the complex structure. We can express (10.1) by saying that the intersection pattern of C in M is locally holomorphic. We call C ⊂ M tidy if all points of C are tidy. A chart φ : (U, 0) → (M, x) exhibiting x as a tidy point is called adapted to C.

A particularly interesting example arises from a non–singular projective surface M over C and algebraic curves C_{1}, …, C_{n} in M such that C = ∪C_{i} has only ordinary multiple point singularities (number of tangents = multiplicity of the point). An arrangement of projective lines L_{1}, …, L_{n} in projective space IP^{2} over C is always tidy.

The main point for giving this definition is that the process of blowing up a point, known from algebraic geometry or complex analysis, still makes sense in this context.

Let C = (C_{1}, …, C_{n}) be a tidy arrangement of immersed oriented spheres C_{i} in the complex projective plane IP^{2}. Let M be the set of its singular points. We call the number m(x) of local branches through x ∈ M the valence of x. We denote by t_{r} the number of points in M with valence r. We also use the abbreviations

(9.2) f_{0} = ∑_{r≥2} t_{r} = |M|, f_{1} = ∑_{r≥2} rt_{r}.

We have H_{2}(IP^{2}) ≅ ZZ and the generator 1 corresponds to a cycle represented by a complex projective line with its natural orientation. Since each sphere C_{i}

T. tom Dieck 9. Combinatorial properties 31

is oriented it defines an element z(C_{i}) ∈ H_{2}(IP^{2}) and a corresponding integer d_{i}, called the degree of C_{i}. Set d = ∑_{i=1}^{n} d_{i}.

(9.3) Proposition. For the tidy arrangement C = (C_{1}, …, C_{n}) the following identity holds:

∑_{r≥2} \binom{r}{2}t_{r} = ∑_{i<j} d_{i}d_{j} + ∑_{i} \binom{d_{i} − 1}{2} = \binom{d − 1}{2} + n − 1.

If all d_{i} = 1, then the right hand side equals \binom{n}{2}.

Proof. The left hand side counts the intersection points (with multiplicity) geometrically and the right hand side homologically.

Let f_{i} : S^{2} → IP^{2} be an immersion with image C_{i}. If we deform the immersion

⟨f_{j}⟩ : S = ∐_{j} S^{2} → IP^{2}

by a regular homotopy in a neighbourhood of a multiple point into an immersion with transverse intersections in the usual sense of differential topology, then a point of valence r unfolds into \binom{r}{2} points of valence 2. The right hand side of (3.1) is invariant under regular homotopies. So let us assume that f : S → IP^{2} is an immersion which has only transverse intersections with intersection number +1. Then the number of intersection points equals

(9.4) \frac{1}{2}(s^{2} − e(ν_{f})).

Here ν_{f} is the normal bundle of f and e(ν_{f}) its Euler class (= the sum of the Euler classes of the bundles over the components); and s^{2} is the self–intersection number of the cycle s defined by f. The counting (10.4) is a general fact of differential topology, see Wall [1970], Theorem 5.2 (iii). We evaluate (10.4) in the case at hand. We obtain

(9.5)  ∑_{i<j} C_{i}C_{j} + \frac{1}{2} ∑_{i}(C_{i}^{2} − e(ν_{i})).

The intersection number C_{i}C_{j} equals d_{i}d_{j}. Let D = C_{i} and let f : D → IP^{2} have degree k. Let η_{s} be the canonical line bundle over IP^{s}. Then ν_{f} is isomorphic to

f^{∗}(T IP^{2} ⊕ ε)/f^{∗}(T IP^{2} ⊕ ε) = f^{∗}(3η_{2})/2η_{1} = (3k − 2)η_{1}.

(T = tangent bundle, ε = trivial line bundle.)

Hence e(ν_{f}) = 3k − 2. We arrive at

\frac{1}{2}(D^{2} − e(ν_{f})) = \frac{1}{2}(k^{2} − (3k − 2)) = \binom{k − 1}{2}.

□

In the case of algebraic curves (10.3) can also be proved by a counting argument and a formula of M. Noether (Brieskorn–Knörrer [1986]).

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