Complex Surfaces and Coverings
Complex Surfaces and Coverings
Abstract and Keywords
This chapter deals with complex surfaces and their finite coverings branched along divisors, that is, subvarieties of codimension 1. In particular, it considers coverings branched over transversally intersecting divisors. Applying this to linear arrangements in the complex projective plane, the chapter first blows up the projective plane at non-transverse intersection points, that is, at those points of the arrangement where more than two lines intersect. These points are called singular points of the arrangement. This gives rise to a complex surface and transversely intersecting divisors that contain the proper transforms of the original lines. The chapter also introduces the divisor class group, their intersection numbers, and the canonical divisor class. Finally, it describes the Chern numbers of a complex surface in order to define the proportionality deviation of a complex surface and to study its behavior with respect to finite covers.
Keywords: complex surface, finite covering, linear arrangement, projective plane, intersection point, transversely intersecting divisor, divisor class group, canonical divisor class, Chern numbers, proportionality deviation
We now turn our attention from Riemann surfaces to surfaces of two complex dimensions. Once again, we will study coverings branched along subvarieties of codimension 1. Since Riemann surfaces have complex dimension 1, divisors on them are merely finite sums of points with integer coefficients. In the case of two complex dimensions, however, divisors are finite sums of one-dimensional subvarieties with integer coefficients. We are now faced with a complication: two or more of these subvarieties may have a point in common. We begin our study of branched coverings in §3.1, where we suppose the branch locus consists of subvarieties that intersect transversally, that is, no more than two subvarieties meet at a common point. In order to treat coverings branched over subvarieties with more than double intersection points, we use the technique of blowing up a point to reduce the problem to the case of transverse intersections. This blowing-up process is discussed in some detail at the end of the chapter in §3.5. In §3.2, we introduce the Chern numbers of a complex surface. This enables us to define the proportionality deviation of a complex surface, denoted by Prop, in §3.3, and to study its behavior with respect to finite covers. The vanishing of Prop is intimately related to the existence of finite covers of line arrangements that are ball quotients. This is the main topic of this book and is the subject of Chapters 5 and 6. We show how solutions of Prop=0 arise from the complete quadrilateral arrangement. Finally, we discuss the signature of a complex surface in §3.4.
3.1 Coverings Branched Over Subvarieties With Transverse Intersections
Let X denote a complex surface, that is, a complex manifold of complex dimension 2. Unless otherwise stated, we assume from now on that our complex surfaces are compact, connected, and algebraic. Let D_{i}i , i ∈ I for some finite index set I , be a set of smooth complex one-dimensional irreducible subvarieties. These have the underlying structure of a compact Riemann surface. In a neighborhood of any point p ∈ D_{i}i, we can choose local complex coordinates (u, v) on X such that D_{i}i is given locally by the equation u = 0, and u is called a normal coordinate to D_{i}i at p. Assume that any two distinct (p.48) subvarieties in the system intersect transversally. That is, for any i ≠ j with i, j ∈ I and any p ∈ D_{i}i ∩ D_{j}j , we have a normal crossing at p, that is, we can find local coordinates (u, v) at p such that D_{i}i is given locally by u = 0 and D_{j}j is given locally by v = 0. The cardinality of D_{i}i ∩ D _{j}j , i = j, is therefore finite and is by definition the intersection number D_{i}i · D _{j}j . We shall say more about the definition of intersection numbers in §3.2. In addition, we assume that no more than two of the D_{i}i intersect at one point; in other words, the union ∪_{i}i D_{i}i has only ordinary double points.
We shall deal mainly with good covers as given by the following (compare with [75], p. 150, [137], and [138]):
Definition 3.1 Let Y be a complex surface that is a finite covering
of X. Therefore, the map π is holomorphic and surjective, and above any point of X there are only a finite number of points of Y. We suppose that π is branched along a system {D_{i}i }_{i}i∈_{I}I of one-dimensional subvarieties of X intersecting transversally. The covering is defined to be a good covering if, in addition, there are integers N ≥ 1 and b_{i}i ≥ 2, i ∈ I , such that
(i) for i ∈ I , we have b_{i}i | N and there are N/b_{i}i points of Y over each point of D_{i}i \ ∪_{j}j=_{i}i D_{j}j ∩ D_{i}i : centered at each such point q of Y, there are local coordinates (s, t) such that , υ = t are local coordinates centered at π(q), with u a normal coordinate to D_{i}i at π(q); themap π is given locally by the quotient of an open neighborhood of q by the action of for m ∈ Z/b_{i}iZ;
(ii) for i, j ∈ I , i = j andD_{i}i∩ D_{j}j = φ, we have b_{i}ib _{j}j | N and there are N/b_{i}i b _{j}j points over each point of D_{i}i ∩ D_{j}j ; centered at each such point q of Y, there are local coordinates (s, t) such that are local coordinates centered at π(q), with u a normal coordinate to D_{i}i at π(q) and v a normal coordinate to D_{j}j at π(q); themap π is given locally by the quotient of an open neighborhood of q by the action of for m ∈ Z/b_{i}iZ, n ∈ Z/b _{j}jZ;
(iii) over the points not appearing in (i) and (ii) there are N points of Y, and N is called the degree of the covering; at any such point of Y, the map π is locally biholomorphic.
Consider the following example of a covering of the complex projective plane 𝕡_{2} by itself that has degree n^{2}2. Let (w_{0} : w_{1} : w_{2}) and (z_{0} : z_{1} : z_{2}) be projective coordinates on 𝕡_{2} and consider the map, i = 0, 1, 2. The _{i}i
subvarieties w_{i}i = 0, i = 0, 1, 2, on X intersect transversally at one point. (p.49) In terms of affine coordinates on the open subset w_{0} ≠ 0 of 𝕡_{2}, we have a covering branched along u = w_{1}/w_{0} = 0, v = w_{2}/w_{0} = 0 on which the functions become single-valued.
We can compute the Euler number of a good covering Y using the properties of the Euler number given in Chapter 1, §1.1. Namely,
In the first part of the sum, we compute the contribution to the Euler number of the complement of the ramification locus and, in the second part, we take into account the contribution from the ramification locus. Let Then, the above formula for the Euler number takes the simple form
Compare this to the analogous one-dimensional formula for a covering Y_{1} of a Riemann surface X_{1} of degree N ramifying with degree b_{i}i over points P_{i}i , with N/b_{i}i points of Y_{1} over P_{i}i (see Chapter 2, §2.4):
In higher dimensions we can check that, under suitable assumptions, the expression for the Euler number resembles a Taylor formula. This might be expected on consideration of the one-dimensional and two-dimensional cases.
3.2 Divisor Class Group And Canonical Class
Recall the notion of divisor from Chapter 1, §1.4. The group of divisors on X is the abelian group generated freely by the irreducible analytic subvarieties of X of complex dimension 1, called the prime divisors. These subvarieties are therefore curves on X, which are not necessarily smooth. Let be a divisor, where the m_{i}i are integers and the C_{i}i are prime divisors. There is an open covering of X such that, for every open set U in the covering, the intersection C_{i}i ∩ U is UUiithe locus of the equation , where generates a prime ideal in the ring of regular functions on U. Therefore, on U, the divisor D is given by Uthe ilocus of zeros and poles of counted with multiplicity, and this is independent of the chosen open covering. Therefore, (p.50) for some open covering of X, and for every open set U in the covering, the divisor is given on U by the zeros and poles, counted with multiplicity, of a meromorphic function fU on U, which is not identically zero on U.Moreover, on the intersection U ∩ V of two sets U and V belonging to the covering, the function fU / fV is holomorphic and nonzero. Addition and subtraction of divisors corresponds to multiplication and division of their local functions. A nonzero global meromorphic function f on X determines a divisor, ( f ), by letting fU = f for every U. We say that two divisors D_{1} and D_{2} are linearly equivalent if there is a global meromorphic function f on X such that D_{1} − D_{2} = ( f ). The corresponding equivalence classes form the group Div(X) of divisor classes on X. If π : Y → X is a surjective map between complex surfaces, then the pullback on functions induces a homomorphism π^{∗}∗ : Div(X) → Div(Y).
Let ω be a nonzeromeromorphic differential 2-form on X. In terms of local complex coordinates (z_{1}, z_{2}), it has the form
where a(z_{1}, z_{2}) is a meromorphic function. For an arbitrary complex surface, such a form need not exist. However, all algebraic surfaces have such forms. We use these forms to define the canonical class and to derive properties of the canonical class under ramification. Nonetheless, the theory applies for all complex surfaces. Under a change of complex coordinates, dz_{1} ∧ dz_{2} is multiplied by the determinant of the Jacobian of the coordinate change. Therefore, we may use the meromorphic function a(z_{1}, z_{2}) to locally describe a divisor. The quotient of any two meromorphic differential 2-forms is a meromorphic function, and so they define linearly equivalent divisors. Therefore, in this way we obtain a well-defined divisor class, which we call the canonical class K = KX of X (see also Chapter 1, §1.4).
Let D be a representative of a divisor class on X. Then D is also an oriented two-dimensional real cycle on the real four-dimensional manifold X, and determines an element [D] of the homology group H_{2}(X,ℤ). As linearly equivalent divisors give rise to homologous elements of H_{2}(X,ℤ) (see [55]), [D] depends only on the divisor class of D. The homology class [K] is called the canonical homology class. On a four-dimensional oriented real compact manifold X, we can always choose a smooth representative for every class in H_{2}(X,ℤ). The intersection number of two homology classes is defined by moving their smooth representatives topologically so that they are in transverse positions, and then using the definition of intersection number for transversally intersecting cycles. Namely, let A and B be two smooth oriented real two-dimensional submanifolds of X representing elements α and β of H_{2}(X,ℤ). We move A and B topologically until they intersect transversally. At each intersection point, their orientations together generate either a positive or (p.51) a negative orientation in the real four-dimensional tangent space at the point. The intersection number of α and β is then
where ∑_{+} denotes the sum over the intersection points where A and B give
a positive orientation, and ∑_{−} the sum over the intersection points where
A and B give a negative orientation. Consider the example X = 𝕡_{2} with projective coordinates (z_{0} : z_{1} : z_{2}) and divisors given by the lines z_{1} = 0 and z_{2} = 0. Let real coordinates (x_{1}, y_{1}) and (x_{2}, y_{2}) be given by z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2}, and consider the orientation determined by this complex structure, that is, by the order x_{1}, y_{1}, x_{2}, y_{2}. The intersection number of the divisors z_{1} = 0 and z_{2} = 0 is then 1. If we orient 𝕡_{2} in the opposite way, their intersection number becomes −1. By Poincaré duality [49], p. 53, we have a natural isomorphism between H_{2}(X,ℤ) and the cohomology group H^{2}2(X,ℤ). The image of [D] under this isomorphism is the first Chern class c_{1}(D) (see also Chapter 1, §1.4). The intersection product in homology becomes the cup product in cohomology, composed with integration over the manifold, that is, with pairing with the fundamental homology class.
Let A be a smooth subvariety of complex codimension 1 with homology class α ∈ H_{2}(X,ℤ). The normal bundle NA^{/X}/X to A in X is given by the quotient bundle T_{X}X|_{A}A/T_{A}A, where for M a manifold, T_{M}M denotes its tangent bundle. Consider any continuous section of a bundle with isolated zeros and poles. The number of zeros and poles of this section, counted with multiplicity (which is negative for the poles), is called the degree of the bundle. The self-intersection α · α is the degree of N_{A}A/_{X}X. We can assume that the section of N_{A}A/_{X}X used to compute the degree is differentiable and gives a smooth differentiable submanifold homologous to A and intersecting A transversally.
In complex geometry, two transversally intersecting smooth subvarieties D_{1} and D_{2} of codimension 1 always define a positive orientation at any intersection point. Their intersection number [D_{1}] · [D_{2}] is therefore simply the number of intersection points. Notice that on a complex surface X, this means an irreducible curve with negative self-intersection number is the only irreducible curve in its homology class.
There is an algebraically defined intersection pairing on Div(X) that agrees with the topological intersection form under the induced map from Div(X) to H_{2}(X,Z). In fact, there is a unique pairing on the divisor group Div(X) of X,
a) if D_{1} and D_{2} are smooth subvarieties of codimension 1 that meet transversally, then D_{1} · D_{2} = card(D_{1} ∩ D_{2}),
b) D_{1} · D_{2} = D_{2} · D_{1}
c) (D_{1} + D_{2}) · D = D_{1} · D + D_{2} · D
d) if D_{1} and D_{2} are linearly equivalent, then D_{1} · D = D_{2} · D for every D.
The first step in the construction of this pairing is the definition of the local intersection of two curves with no component in common. Suppose that D_{1} and D_{2} are two divisors with no component of dimension 1 in common and that they are defined by holomorphic local functions. Let p be a point in D_{1} ∩ D_{2}. Then there are functions f and g in the local ring O_{X}X,_{p}p of holomorphic functions at p giving, respectively, the local equations of D_{1} and D_{2}. We define the local intersection number I_{p}p(D_{1}, D_{2}) to be the complex dimension of the space O_{X},_{p}p/( f, g) where ( f, g ) is the ideal generated by f and g. If p ∉ D_{1} ∩ D_{2}, we let I_{p}(D_{1}, D_{2}) = 0. Then,
The divisors D_{1}, D_{2} meet transversally at p if and only if I_{p}p(D_{1}, D_{2}) = 1, as then ( f, g ) is the maximal ideal of O_{X}X,_{p}p. In this way, we recover property (a) of the pairing. We have to show that there is a unique symmetric bilinear pairing from Div(X) to ℤ, which factors through linear equivalence and agrees with the above definition for curves meeting transversally. For a proof of this, see, for example, [44],[51], and [130].
By way of example, consider a smooth curve C and the diagonal
in C × C. The Euler number of C equals the self-intersection number ofΔ . Namely, the normal bundle of ∆ in C × C is isomorphic to the tangent bundle of C. The number of zeros of a vector field on C, counted with multiplicity, equals the Euler number of C by the Poincaré–Hopf theorem. Hence, ∆ · ∆ = 2 − 2g , which is negative when g ≥ 2, so that is then isolated in its homology class. Therefore, the automorphism group of C is discrete. In fact, it is finite.
Returning to the canonical class K, the first Chern class c_{1}(X) of X is defined as −c_{1}(K). Therefore, we have
The Euler number e(X) is also denoted by c_{2}(X). We therefore have the two Chern numbers, and c_{2}(X).
(p.53) In complex dimension 1, we define, in an analogous way, the canonical divisor K using the meromorphic 1-forms. It is a linear combination of points and has degree 2 g − 2, where g is the genus of the underlying Riemann surface. Here, the Euler number 2 − 2g is the only Chern number.
Let us see how the canonical class behaves under covering maps. Let X and Y be smooth compact algebraic surfaces. Let π : Y ↦ X be a good covering as defined in Definition 3.1 of §3.1. We retain the notation of that definition. Let D be the divisor that is the locus of vanishing of the determinant of the Jacobian of π on Y. Then, looking at the meromorphic 2-forms of X and Y, we have
In order to express this formula in terms of the branching locus D_{i}i on X, we use rational coefficients, and obtain a two-dimensional Hurwitz formula,
where, as before, To understand this formula, recall that the lifting π^{∗}∗ of divisors corresponds to the lifting of
the local meromorphic functions. Over the points of D_{i}i \ ∪_{i≠j}j D_{i}i ∩ D_{j}j, we have a local map , υ = s, with The lift of D_{i}i is given by
Consequently,
In (3.7), we used the cohomological relations
for α, β ∈ H^{2}2(X,ℤ).
Consider, for example, the complex projective plane 𝕡_{2}. It has Euler number e(𝕡_{2}) = 3 since it is the disjoint union of a copy of ℂ^{2}2 and a copy of 𝕡_{1}, viewed as the projective line at ∞. Its divisor class group is generated by a projective line l , corresponding, by Poincaré duality, to an element h ∈ H ^{2}2(𝕡_{2}), where h^{2}2 pairs with the fundamental class to give 1. Choosing homogeneous coordinates [u_{0} : u_{1} : u_{2}] on 𝕡_{2}, we let z_{1} = u_{1}/u_{0}, z_{2} = u_{2}/u_{0} be affine coordinates on u_{0} ≠ 0 and we let w_{1} = u_{0}/u_{1},w_{2} = u_{2}/u_{1} be affine coordinates on u_{1} ≠ 0. Then, the 2-form given on u_{0} ≠ 0 by dz_{1} ∧ dz_{2} has a pole of order 3 at w_{1} = 0, since
(p.54) Hence, the canonical class has representative K = −3l≃ −3h, and K has self-intersection K^{2}2 = 9. Recall that and that we have c_{2}(𝕡_{2}) = e(𝕡_{2}) = 3; therefore,
3.3 Proportionality
The equality , satisfied by 𝕡_{2}, will occupy us for a great deal of the remainder of this book, especially in Chapters 5 and 6. This comes from its relation to ball quotients. The complex 2-ball B = B_{2} is the bounded homogeneous domain contained in 𝕡_{2} that is given in projective coordinates by
The group of automorphisms of B, which act on the projective coordinates by fractional linear transformations, is denoted by PU(2, 1). In fact, PU(2, 1) is the group of all biholomorphic automorphisms acting on B (see [78], §4A.1, p. 16). It is naturally contained in the automorphism group PGL(3,ℂ) of 𝕡_{2}. In 1956, F. Hirzebruch [54] proved a theorem, of which the following is a special case.
Theorem 3.1 Assume that
(i) is a discrete subgroup of PU(2, 1),
(ii) operates freely, that is, an element that is not the identity has no fixed point,
(iii) B /is compact.
For such a ball quotient, the Chern numbers are proportional to those of 𝕡_{2}. Therefore, we have the following equation:
The quotient complex surface B /is algebraic, according to a result of Kodaira.
Yoichi Miyaoka and Shin-Tung Yau independently proved that for surfaces of general type (see Chapter 4, §§4.2, 4.3). Ball quotients satisfying the conditions of Theorem 3.1 are of general type. By a famous result of Yau, they are characterized by equality in the Miyaoka–Yau inequality. This motivates the following definition.
(p.55) Definition 3.2 The proportionality deviation of a complex surface Y is defined by the following formula:
Our goal is to calculate Prop for ramified covers and to find examples where it vanishes. This is the main objective of Chapter 5.
Recall the generalization of the Hurwitz formula given by (3.2). Rewriting it in terms of the second Chern number, we have
Expressing the self-intersection formula (3.7) for the canonical class in terms of the first Chern number, we have
In order to simplify this formula, we use the following.
Adjunction formula: If D is a smooth submanifold of X of dimension 1, then
For a proof of the adjunction formula, see [51], p. 361. By way of example, let D be a smooth curve of degree n in 𝕡_{2}. That is, D is given by a homogeneous polynomial of degree n in homogeneous coordinates of 𝕡_{2}, not all of whose derivatives vanish simultaneously. Then, as a divisor, D is equivalent to nl, where l is a projective line in 𝕡_{2}. Therefore, D · D = n^{2}2 and · D = −3n. By the adjunction formula, we have e(D) = 3n − n^{2}2, and we also know that e(D) = 2 − 2g, where g is the genus of the curve. Therefore, 3n − n^{2}2 = 2 − 2g , which implies
Theorem 3.2 With the notations of §3.1, for good coverings
(p.56) of degree N, the proportionality deviation is given by
Proof. The formula (3.12) is obtained by combining (3.9), (3.10), and (3.11).
If the D_{i}i are k lines in general position in 𝕡_{2}, then e(D_{i}i ) = 2, D_{i}i · D_{i}i = 1, and D_{i}i · D_{j}j = 1, i ≠ j. Therefore, from (3.12), we obtain
Of course, for i = 1, … , k. For k = 3, the far right-hand side of the preceding expression equals
and, as for all i, j, we have Notice that any vector of the form (x, x, x) is a solution of S = 0. We now proceed inductively, writing
Now, as k ≥ 4,
Hence, Prop(Y) > 0 for all k ≥ 4. For k = 3, the example following Definition 3.1 shows Prop(Y) > 0 is not valid.
The example of the complete quadrilateral
An arrangement that will be important in our study of Appell’s hypergeometric function in Chapter 6, and that gives rise to examples where Prop does vanish, is the complete quadrilateral. It is the arrangement of six lines having four triple intersection points, no three of which are collinear (see Figure 1). Any four points with this property are equivalent up to a projective transformation. (p.57)
The six lines are the ways of connecting these four points by lines. This arrangement has three double and four triple intersection points. Any three of its lines not having a common triple point give an affine coordinate system on an open subset of 𝕡_{2}, and, in suitable projective coordinates (z_{0} : z_{1} : z_{2}), the arrangement is given by
The discussion in this chapter assumed that the branching locus consists of divisors with normal crossings. This is certainly not the case for the complete quadrilateral. In order to apply the considerations of the present chapter, we construct a new algebraic surface by blowing up the triple points of the complete quadrilateral. For a more complete discussion of this blowing-up process, see §3.5. Consider a triple intersection point and assume it is at the origin. Roughly speaking, when we blow up the origin, we replace each neighboring point not at the origin by the line joining it with the origin, and replace the origin itself by the family of lines through the origin. Therefore, in blowing up 𝕡_{2} at the four triple points of the complete quadrilateral, which we index by {0i}, i = 1, … , 4, we replace them by projective lines. We therefore obtain a new surface X with a total of ten divisors Dαβ , indexed by α, β ∈ {0, 1, … , 4}. Six of these divisors are transforms of the original lines of the arrangement and four of them are the divisors coming from the blown-up points. For example, the divisor D_{12} is the proper transform of the line passing through the points 03 and 04, whereas D_{0i} for i = 1, … ,4 is the divisor obtained by blowing up the point 0i. The ten divisors have only transverse intersection points. There are fifteen such points. The intersection numbers (p.58) for two divisors, Dαβ and Dγ δ, {αβ} = {γ δ}, are given by
and for {αβ} = {γ δ} by
For each of the ten divisors D, we have e(D) = 2. Notice that for each of the original lines in the arrangement on 𝕡_{2}, the self-intersection is +1. This self-intersection goes down by 1 on X for each blown-up point lying on the line. Therefore, we end up with self-intersection −1: again, we refer to our discussion of blow-ups in §3.5.When we blow up 𝕡_{2} to obtain X, each blown-up point augments c_{2} by +1, as a point becomes a copy of S^{2}2, whereas the extra curve with self-intersection −1 decreases c^{2}2_{1} by −1. Hence, we have
Using (3.12) of Theorem 3.2, we deduce that a good covering Y of X of degree N, as given by Definition 3.1 of §3.1, satisfies
We now exhibit a formal solution to Prop = 0. Namely, let μ_{0},μ_{1}, … , μ_{4} be real numbers with μ_{0} + … + μ_{4} = 2 and let xαβ = μα + μβ . Therefore,
and
If we take, for example, α = 0, β = 2, the expression in the brackets {·} in the formula above gives
Table 3.1
n_{01} |
n_{02} |
n_{03} |
n_{04} |
n_{12} |
n_{13} |
n_{14} |
n_{23} |
n_{24} |
n_{34} |
---|---|---|---|---|---|---|---|---|---|
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
8 |
8 |
8 |
8 |
4 |
4 |
4 |
4 |
4 |
4 |
9 |
9 |
9 |
3 |
9 |
9 |
3 |
9 |
3 |
3 |
6 |
6 |
6 |
4 |
6 |
6 |
4 |
6 |
4 |
4 |
12 |
12 |
6 |
6 |
6 |
4 |
4 |
4 |
4 |
3 |
12 |
12 |
12 |
4 |
6 |
6 |
3 |
6 |
3 |
3 |
15 |
15 |
15 |
5 |
5 |
5 |
3 |
5 |
3 |
3 |
24 |
24 |
24 |
8 |
4 |
4 |
3 |
4 |
3 |
3 |
In this case, we see that
A geometrically interesting situation occurs in the case x_{αβ}αβ = 1 − 1/b_{αβ}αβ for a positive integer b_{αβ}αβ , already understood in Definition 3.1 of §3.1. Hence, we have the restriction, which we call INT,
Up to permutation, the eight solutions of INT with 0 < μ_{α}α < 1 and μ_{α}α = 2 are displayed in Table 3.1.
We refer to Chapters 5 and 6 for a discussion of the existence of the covering Y in these cases. The example μ_{α}α = 2/5, and hence b_{αβ}αβ = 5, gives rise to a Kummer covering Y of X (see Appendix B).
3.4 Signature
In the preceding discussion, the numerical invariants and c_{2}(X) of an algebraic surface X played a fundamental role. The Euler number e(X) = c_{2}(X) is a topological invariant. It turns out that is an invariant of the underlying oriented four-dimensional manifold. To see this, we first recall the definition of the signature of a compact oriented four-dimensional manifold X, which we assume to be differentiable. Consider the homology group H_{2}(X,ℤ). We have a bilinear symmetric pairing
(p.60) given by the intersection number
The number α · β vanishes if α or β is a torsion element of the group H_{2}(X,ℤ). The pairing is therefore defined on L = H_{2}(X,ℤ)/H_{2}(X,ℤ)_{tors}tors, which is a lattice of rank b_{2}(X), where b_{2}(X) is the second Betti number of X. Thus, we have an integral quadratic form
which is unimodular by Poincaré duality. Over the reals, the form can be diagonalized and on the diagonal it has a number of positive entries, say and a number of negative entries, say .We have
By definition, the signature of X is
The signature depends only on the real quadratic form and can be defined in the same way for a differentiable oriented 4k-dimensional manifold X. According to the signature theorem ([55], §8), sign(X) can be expressed in terms of Pontryagin characteristic classes depending only on the real tangent bundle of X. The Pontryagin classes can be expressed in terms of Chern classes when X is a complex manifold. For real dimension 4, this gives the formula
Previously it had been conjectured that sign(X) ≤ 1 for an algebraic surface. However, the ball quotients of §3.3, Theorem 3.1, give examples of surfaces with arbitrarily large positive signatures. Indeed, for a ball quotient,
Let Y → X be a good covering. The formulas (3.9) for together with the adjunction formula (3.11), give
(p.61) Recall that The fact that this formula for the signature involves only the numbers , but not D_{i} · D_{j} for i ≠ j, is related to special properties of the signature expressed by the Atiyah-Bott-Singer fixed point theorem [56].
In the case of the complete quadrilateral, the surface X is 𝕡_{2} with four points blown up. There are ten divisors on X given by D_{αβ}αβ with
_{αβ}αβ
We renumber them as D_{i} , (i = 1, … , 10).We have
In the case where Prop(Y) = 0 (see the above list),
If, for example, we have b_{i}i = 5 for all i, then
3.5 Blowing Up Points
Let X be a complex surface. Blowing up a point p ∈ X is a local process, so, using local coordinates, we need only consider the case of blowing up the origin 0 = (0, 0) ∈ ℂ^{2}2. The result is independent of the local coordinates used. Let (z_{1}, z_{2}) be the standard coordinates of ℂ^{2}2, and let (u_{1} : u_{2}) be projective coordinates in 𝕡_{1}. Let U be a neighborhood of (0, 0) ∈ ℂ^{2}2. The blow-up at (0, 0) ∈ ℂ^{2}2 is given by the subset V of U × P_{1}, with coordinates (z_{1}, z_{2}) × [u_{1} : u_{2}] satisfying z_{1}u_{2} = z_{2}u_{1}. For each choice of z_{1}, z_{2}, the corresponding points in the blow-up consist of the lines passing through (z_{1}, z_{2}) and the origin. If (z_{1}, z_{2}) is not the origin, there is a unique such line. If (z_{1}, z_{2}) is the origin, the line is not unique; rather, we have all the lines through the origin, corresponding to a copy of 𝕡_{1}. In this way, by blowing up at p ∈ X, we construct a new surface Y and a map
We call π−^{1}1(p) = E the exceptional divisor. It is a copy of 𝕡_{1}, and
is biholomorphic. Locally, Y may be described in terms of affine coordinates. Let U be a coordinate neighborhood of p with local coordinates z_{1},z_{2} centered (p.62) at p. Then, W = π−^{1}1(U) is covered by two coordinate charts W_{1} and W_{2}. On W_{1} we have coordinates (u, v), and π is given by
On W_{2} we have coordinates (u, v), and π is given by
We see that u is an affine coordinate for (u_{1} : u_{2}) on u≠_{2} = 0 and that υ′ is an affine coordinate for (u_{1} : u_{2}) on u_{1} = 0. We identify W_{1} \ {u = 0} with W_{2} \ {υ′ = 0} by the equations , υ = u′υ′. The exceptional divisor is given in W_{1} by υ = 0 and in W_{2} by u′ = 0. If L is a smooth irreducible curve through p, then we can suppose that it is given by z_{1} = 0 in a neighborhood of p, where z_{1}, z_{2} are local coordinates. After blow-up, it is transformed into another curve , called its proper transform. The curve is given by the closure in Y of π−^{1}1(L \ {p}). Working in W_{1}, the exceptional divisor E has the equation υ = 0, and has the local equation u = 0. Therefore, we have
and, more generally, the divisor class group of Y is given by
the direct sum indicating orthogonality with respect to the intersection pairing. For divisors D_{1} and D_{2} on X, we have π∗D_{1} · π∗D_{2} = D_{1} · D_{2}. These facts hold because, topologically, homology classes on X can be given by cycles not passing through p. Hence,
and so we see that E has self-intersection −1:
On an algebraic surface, a smooth curve of genus 0, that is, a rational curve, which also has self-intersection −1, is called an exceptional curve.
Let us consider the blowing-up process topologically. We have
(p.63) and, as the Betti numbers are the ranks of the homology groups, this implies that
All other homology groups, and hence Betti numbers, are unchanged.
Now consider two real oriented compact differential 4-manifolds M and N, and let p ∈ M and q ∈ N. Let U(p) be an open neighborhood of p and U(q) be an open neighborhood of q. Suppose that both and are compact balls with respect to a coordinate system. Then, the boundaries of both M \ U(p) and N \ U(q) are diffeomorphic to a 3-sphere S^{3}3. By definition, the manifold M#N is that obtained by identifying these boundaries by an orientation reversing diffeomorphism. The blow-up of a complex surface X at a point p gives a complex surface Y topologically isomorphic to Here, the orientation on 𝕡_{2}(C) is reversed and, in particular, a line in 𝕡_{2}(C) with self-intersection +1 becomes a line with self-intersection −1. Only one of these lines is realized complex analytically: the blown-up point. This topological interpretation explains why (3.24) and (3.25) hold.
For manifolds of real dimension 2, the blow-up of the origin over a compact disk around the origin in ℝ^{2}2 yields a Möbius strip, the origin becoming a copy of 𝕡_{1}(ℝ), the “backbone” of the Möbius strip, and the proper transforms of the lines through the origin intersecting the backbone at the point corresponding to their slope. The Möbius strip is fibered by intervals over 𝕡_{1}(ℝ)≃ S^{1}1, just as the blown-up four-dimensional disk is fibered by 2-disks over 𝕡_{1}(ℂ) ≃S^{2}2.
Returning to the complex case, we have, for the canonical class,
If we write a meromorphic 2-form ω in local coordinates as
then, using the affine coordinates introduced earlier,
which proves (3.26). It also follows that
We can check this using the adjunction formula, which gives
(p.64) We can also calculate the self-intersection of in Y, namely,
since E and π∗L are orthogonal with respect to the intersection product. Hence, the self-intersection of L decreases by 1 when we pass to the proper transform. Using (3.22), (3.24), and (3.26), we also have
whereas, for the Euler numbers,
since blowing up removes a point and inserts a copy of S^{2}2. This also follows from the behavior of the Betti numbers in (3.25). Therefore,