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Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)$

Paula Tretkoff

Print publication date: 2016

Print ISBN-13: 9780691144771

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691144771.001.0001

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Complex Surfaces and Coverings

Complex Surfaces and Coverings

Chapter:
(p.47) Chapter Three Complex Surfaces and Coverings
Source:
Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)
Author(s):

Paula Tretkoff

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691144771.003.0004

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 Dii , iI 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 pDii, we can choose local complex coordinates (u, v) on X such that Dii is given locally by the equation u = 0, and u is called a normal coordinate to Dii at p. Assume that any two distinct (p.48) subvarieties in the system intersect transversally. That is, for any ij with i, jI and any pDiiDjj , we have a normal crossing at p, that is, we can find local coordinates (u, v) at p such that Dii is given locally by u = 0 and Djj is given locally by v = 0. The cardinality of DiiD jj , i = j, is therefore finite and is by definition the intersection number Dii · D jj . We shall say more about the definition of intersection numbers in §3.2. In addition, we assume that no more than two of the Dii intersect at one point; in other words, the union ∪ii Dii 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

π:YX

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 {Dii }iiII 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 bii ≥ 2, iI , such that

  1. (i) for iI , we have bii | N and there are N/bii points of Y over each point of Dii \ ∪jj=ii DjjDii : centered at each such point q of Y, there are local coordinates (s, t) such that Complex Surfaces and Coverings , υ‎ = t are local coordinates centered at π‎(q), with u a normal coordinate to Dii at π‎(q); themap π‎ is given locally by the quotient of an open neighborhood of q by the action of Complex Surfaces and Coverings for m ∈ Z/biiZ;

  2. (ii) for i, jI , i = j andDiiDjj = φ‎, we have biib jj | N and there are N/bii b jj points over each point of DiiDjj ; centered at each such point q of Y, there are local coordinates (s, t) such that Complex Surfaces and Coveringsare local coordinates centered at π‎(q), with u a normal coordinate to Dii at π‎(q) and v a normal coordinate to Djj at π‎(q); themap π‎ is given locally by the quotient of an open neighborhood of q by the action of Complex Surfaces and Coverings for m ∈ Z/biiZ, n ∈ Z/b jjZ;

  3. (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 𝕡2 by itself that has degree n22. Let (w0 : w1 : w2) and (z0 : z1 : z2) be projective coordinates on 𝕡2 and consider the map,Complex Surfaces and Coverings i = 0, 1, 2. The ii

subvarieties wii = 0, i = 0, 1, 2, on X intersect transversally at one point. (p.49) In terms of affine coordinates on the open subset w0 ≠ 0 of 𝕡2, we have a covering branched along u = w1/w0 = 0, v = w2/w0 = 0 on which the functions Complex Surfaces and Coveringsbecome 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,

(3.1)
e(Y)=N( e(X) i e( D i )+ 1 2 i = j D i D j )+N i 1 b i ( e( D i ) j = i D i D j )+ N 2 i = j 1 b i b j D i D j .

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 Complex Surfaces and CoveringsThen, the above formula for the Euler number takes the simple form

(3.2)
e(Y) N =e(X) i x i e( D i )+ 1 2 i = j x i x j D i D j .

Compare this to the analogous one-dimensional formula for a covering Y1 of a Riemann surface X1 of degree N ramifying with degree bii over points Pii , with N/bii points of Y1 over Pii (see Chapter 2, §2.4):

(3.3)
e( Y 1 ) N =e( X 1 ) i x i e( P i ).

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 Complex Surfaces and Coverings be a divisor, where the mii are integers and the Cii are prime divisors. There is an open covering of X such that, for every open set U in the covering, the intersection CiiU is UUiithe locus of the equation Complex Surfaces and Coverings, where Complex Surfaces and Coveringsgenerates 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 Complex Surfaces and Coveringscounted 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 UV 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 D1 and D2 are linearly equivalent if there is a global meromorphic function f on X such that D1D2 = ( f ). The corresponding equivalence classes form the group Div(X) of divisor classes on X. If π‎ : YX 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 (z1, z2), it has the form

ω( z 1 , z 2 )=a( z 1 , z 2 )d z 1 d z 2 ,

where a(z1, z2) 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, dz1dz2 is multiplied by the determinant of the Jacobian of the coordinate change. Therefore, we may use the meromorphic function a(z1, z2) 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 H2(X,ℤ). As linearly equivalent divisors give rise to homologous elements of H2(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 H2(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 H2(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

αβ= + 1 1,

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 = 𝕡2 with projective coordinates (z0 : z1 : z2) and divisors given by the lines z1 = 0 and z2 = 0. Let real coordinates (x1, y1) and (x2, y2) be given by z1 = x1 + iy1 and z2 = x2 + iy2, and consider the orientation determined by this complex structure, that is, by the order x1, y1, x2, y2. The intersection number of the divisors z1 = 0 and z2 = 0 is then 1. If we orient 𝕡2 in the opposite way, their intersection number becomes −1. By Poincaré duality [49], p. 53, we have a natural isomorphism between H2(X,ℤ) and the cohomology group H22(X,ℤ). The image of [D] under this isomorphism is the first Chern class c1(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 α‎H2(X,ℤ). The normal bundle NA/X/X to A in X is given by the quotient bundle TXX|AA/TAA, where for M a manifold, TMM 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 NAA/XX. We can assume that the section of NAA/XX 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 D1 and D2 of codimension 1 always define a positive orientation at any intersection point. Their intersection number [D1] · [D2] 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 H2(X,Z). In fact, there is a unique pairing on the divisor group Div(X) of X,

Div(X)×Div(X) ( D 1 , D 2 ) D 1 D 2 ,

(p.52) such that

  1. a) if D1 and D2 are smooth subvarieties of codimension 1 that meet transversally, then D1 · D2 = card(D1D2),

  2. b) D1 · D2 = D2 · D1

  3. c) (D1 + D2) · D = D1 · D + D2 · D

  4. d) if D1 and D2 are linearly equivalent, then D1 · D = D2 · 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 D1 and D2 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 D1D2. Then there are functions f and g in the local ring OXX,pp of holomorphic functions at p giving, respectively, the local equations of D1 and D2. We define the local intersection number Ipp(D1, D2) to be the complex dimension of the space OX,pp/( f, g) where ( f, g ) is the ideal generated by f and g. If pD1D2, we let Ip(D1, D2) = 0. Then,

D 1 D 2 = p I p ( D 1 , D 2 )0.

The divisors D1, D2 meet transversally at p if and only if Ipp(D1, D2) = 1, as then ( f, g ) is the maximal ideal of OXX,pp. 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

Δ={(x,x)|xC}

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 c1(X) of X is defined as −c1(K). Therefore, we have

c 1 2 (X)=[K][K]=KK.

The Euler number e(X) is also denoted by c2(X). We therefore have the two Chern numbers, Complex Surfaces and Coverings and c2(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 : YX 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

(3.5)
K Y = π K X +D.

In order to express this formula in terms of the branching locus Dii on X, we use rational coefficients, and obtain a two-dimensional Hurwitz formula,

(3.6)
K Y = π K X + i b i 1 b i π D i = π ( K X + i x i D i ),

where, as before, Complex Surfaces and CoveringsTo understand this formula, recall that the lifting π‎∗ of divisors corresponds to the lifting of

the local meromorphic functions. Over the points of Dii \ ∪i≠jj DiiDjj, we have a local map Complex Surfaces and Coverings, υ‎ = s, with Complex Surfaces and Coverings The lift of Dii is given by Complex Surfaces and Coverings

Consequently,

(3.7)
K Y K Y =N ( K X + i x i D i ) 2 .

In (3.7), we used the cohomological relations

( π α π β)[Y]= π (αβ)[Y]=(αβ)[ π Y]=(αβ)N[X]

for α‎, β‎H22(X,ℤ).

Consider, for example, the complex projective plane 𝕡2. It has Euler number e(𝕡2) = 3 since it is the disjoint union of a copy of ℂ22 and a copy of 𝕡1, viewed as the projective line at ∞. Its divisor class group is generated by a projective line l , corresponding, by Poincaré duality, to an element hH 22(𝕡2), where h22 pairs with the fundamental class to give 1. Choosing homogeneous coordinates [u0 : u1 : u2] on 𝕡2, we let z1 = u1/u0, z2 = u2/u0 be affine coordinates on u0 ≠ 0 and we let w1 = u0/u1,w2 = u2/u1 be affine coordinates on u1 ≠ 0. Then, the 2-form given on u0 ≠ 0 by dz1dz2 has a pole of order 3 at w1 = 0, since

d z 1 d z 2 = w 1 3 d w 1 d w 2 .

(p.54) Hence, the canonical class has representative K = −3l≃ −3h, and K has self-intersection K22 = 9. Recall that Complex Surfaces and Coverings and that we have c2(𝕡2) = e(𝕡2) = 3; therefore,

c 1 2 ( 2 )=3 c 2 ( 2 )=9.

3.3 Proportionality

The equality Complex Surfaces and Coverings, satisfied by 𝕡2, 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 = B2 is the bounded homogeneous domain contained in 𝕡2 that is given in projective coordinates by

| z 1 | 2 +| z 2 | 2 <| z 0 | 2 .

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 𝕡2. In 1956, F. Hirzebruch [54] proved a theorem, of which the following is a special case.

Theorem 3.1 Assume that

  1. (i) is a discrete subgroup of PU(2, 1),

  2. (ii) operates freely, that is, an element that is not the identity has no fixed point,

  3. (iii) B /is compact.

For such a ball quotient, the Chern numbers are proportional to those of 𝕡2. Therefore, we have the following equation:

c 1 2 (B/Γ)=3 c 2 (B/Γ).

The quotient complex surface B /is algebraic, according to a result of Kodaira.

Yoichi Miyaoka and Shin-Tung Yau independently proved that Complex Surfaces and Coveringsfor 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:

(3.8)
Prop(Y)=:3 c 2 (Y) c 1 2 (Y).

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

(3.9)
c 2 (Y)/N= c 2 (X) i x i e( D i )+ 1 2 i = j x i x j D i D j .

Expressing the self-intersection formula (3.7) for the canonical class in terms of the first Chern number, we have

(3.10)
c 1 2 (Y)/N=( K X + i x i D i ) 2 = c 1 2 (X)+2 i x i K X D i + i x i 2 D i D i + i = j x i x j D i D j .

In order to simplify this formula, we use the following.

Adjunction formula: If D is a smooth submanifold of X of dimension 1, then

(3.11)
e(D)= K X DDD.

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 𝕡2. That is, D is given by a homogeneous polynomial of degree n in homogeneous coordinates of 𝕡2, not all of whose derivatives vanish simultaneously. Then, as a divisor, D is equivalent to nl, where l is a projective line in 𝕡2. Therefore, D · D = n22 and Complex Surfaces and Coverings · D = −3n. By the adjunction formula, we have e(D) = 3nn22, and we also know that e(D) = 2 − 2g, where g is the genus of the curve. Therefore, 3nn22 = 2 − 2g , which implies

g= (n1)(n2) 2 .

Theorem 3.2 With the notations of §3.1, for good coverings

π:YX

(p.56) of degree N, the proportionality deviation is given by

(3.12)
1 N Prop(Y)= 1 N ( 3 c 2 (Y) c 1 2 (Y) )=3 c 2 (X) c 1 2 (X)+ i x i ( e( D i )+2 D i D i )+ 1 2 i = j x i x j D i D j i x i 2 D i 2 .

Proof. The formula (3.12) is obtained by combining (3.9), (3.10), and (3.11).

If the Dii are k lines in general position in 𝕡2, then e(Dii ) = 2, Dii · Dii = 1, and Dii · Djj = 1, ij. Therefore, from (3.12), we obtain

1 N Prop(Y)= 1 N (3 c 2 (Y) c 1 2 (Y))= 1 2 i = j x i x j i x i 2 .

Of course, Complex Surfaces and Coveringsfor i = 1, … , k. For k = 3, the far right-hand side of the preceding expression equals

S= 1 2 {( x 1 x 2 ) 2 + ( x 2 x 3 ) 2 + ( x 1 x 3 ) 2 },

and, as Complex Surfaces and Coveringsfor all i, j, we have Complex Surfaces and CoveringsNotice that any vector of the form (x, x, x) is a solution of S = 0. We now proceed inductively, writing

1 N Prop(Y)= 1 2 i,j=1,,k1;i = j x i x j i = k x i 2 + x k ( x 1 ++ x k1 x k ).

Now, as k ≥ 4,

x k ( x 1 ++ x k1 x k ) x k ( k1 2 x k ) 1 2 .

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)

Complex Surfaces and Coverings

Figure 3.1 The complete quadrilateral

The six lines are the Complex Surfaces and Coverings 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 𝕡2, and, in suitable projective coordinates (z0 : z1 : z2), the arrangement is given by

z 0 z 1 z 2 ( z 2 z 1 )( z 2 z 0 )( z 0 z 1 )=0.

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 𝕡2 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 D12 is the proper transform of the line passing through the points 03 and 04, whereas D0i 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

D αβ D γδ =1if{αβ}{γδ}=ϕand D αβ D γδ =0if{αβ}{γδ} = ϕ,

and for {αβ‎} = {γ‎ δ‎} by

D αβ D αβ =1.

For each of the ten divisors D, we have e(D) = 2. Notice that for each of the original lines in the arrangement on 𝕡2, 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 𝕡2 to obtain X, each blown-up point augments c2 by +1, as a point becomes a copy of S22, whereas the extra curve with self-intersection −1 decreases c221 by −1. Hence, we have

3 c 2 (X) c 1 2 (X)=16.

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

(3.13)
1 N Prop(Y)=16+ x αβ (4)+ x αβ 2 + 1 2 x αβ ( {αβ} = {γδ} x γδ D αβ D γδ ).

We now exhibit a formal solution to Prop = 0. Namely, let μ‎0,μ‎1, … , μ‎4 be real numbers with μ‎0 + + μ‎4 = 2 and let xαβ‎ = μα‎ + μβ‎ . Therefore,

αβ x αβ =8,

and

(3.14)
1 N Prop(Y)=16+ 1 2 x αβ { ( {αβ} = {γδ} x γδ D αβ D γδ )+2 x αβ }.

If we take, for example, α‎ = 0, β‎ = 2, the expression in the brackets {·} in the formula above gives

x 34 + x 13 + x 14 +2 x 02 =4.
(p.59)

Table 3.1

n01

n02

n03

n04

n12

n13

n14

n23

n24

n34

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

1 N Prop(Y)=16+ 1 2 αβ 4 x αβ =16+16=0.

A geometrically interesting situation occurs in the case xαβ‎αβ‎ = 1 − 1/bαβ‎αβ‎ for a positive integer bαβ‎αβ‎ , already understood in Definition 3.1 of §3.1. Hence, we have the restriction, which we call INT,

(3.15)
n αβ = ( 1( μ α + μ β ) ) 1 1 INT.

Up to permutation, the eight solutions of INT with 0 < μ‎α‎α‎ < 1 and μ‎α‎α‎ = 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 μ‎α‎α‎ = 2/5, and hence bαβ‎αβ‎ = 5, gives rise to a Kummer covering Y of X (see Appendix B).

3.4 Signature

In the preceding discussion, the numerical invariants Complex Surfaces and Coverings and c2(X) of an algebraic surface X played a fundamental role. The Euler number e(X) = c2(X) is a topological invariant. It turns out that Complex Surfaces and Coverings 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 H2(X,ℤ). We have a bilinear symmetric pairing

H 2 (X,)× H 2 (X,)

(p.60) given by the intersection number

(α,β)αβ.

The number α‎ · β‎ vanishes if α‎ or β‎ is a torsion element of the group H2(X,ℤ). The pairing is therefore defined on L = H2(X,ℤ)/H2(X,ℤ)torstors, which is a lattice of rank b2(X), where b2(X) is the second Betti number of X. Thus, we have an integral quadratic form

L×L,

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 Complex Surfaces and Coverings and a number of negative entries, say Complex Surfaces and Coverings .We have

b 2 + + b 2 = b 2 (X).

By definition, the signature of X is

sign(X):= b 2 + b 2 .

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

(3.16)
sign(X)= 1 3 ( c 1 2 (X)2 c 2 (X)).

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,

(3.17)
sign(X)= c 2 (X) 3 = e(X) 3 .

Let YX be a good covering. The formulas (3.9) for Complex Surfaces and Coverings together with the adjunction formula (3.11), give

(3.18)
3sign(Y)/N=3sign(X)2 i x i D i 2 + i x i 2 D i 2 =3sign(X) i (1 1 b i 2 ) D i 2 .

(p.61) Recall that Complex Surfaces and CoveringsThe fact that this formula for the signature involves only the numbers Complex Surfaces and Coverings , but not Di · Dj for ij, 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 𝕡2 with four points blown up. There are ten divisors on X given by Dαβ‎αβ‎ with Complex Surfaces and Coverings

αβ‎αβ‎

We renumber them as Di , (i = 1, , 10).We have

(3.19)
3sign(Y)/N=1 i=1 10 1 b i 2 .

In the case where Prop(Y) = 0 (see the above list),

(3.20)
e(Y)/N=1 i=1 10 1 b i 2 .

If, for example, we have bii = 5 for all i, then

c 2 (Y)/N=e(Y)/N= 3 5 .

3.5 Blowing Up Points

Let X be a complex surface. Blowing up a point pX is a local process, so, using local coordinates, we need only consider the case of blowing up the origin 0 = (0, 0) ∈ ℂ22. The result is independent of the local coordinates used. Let (z1, z2) be the standard coordinates of ℂ22, and let (u1 : u2) be projective coordinates in 𝕡1. Let U be a neighborhood of (0, 0) ∈ ℂ22. The blow-up at (0, 0) ∈ ℂ22 is given by the subset V of U × P1, with coordinates (z1, z2) × [u1 : u2] satisfying z1u2 = z2u1. For each choice of z1, z2, the corresponding points in the blow-up consist of the lines passing through (z1, z2) and the origin. If (z1, z2) is not the origin, there is a unique such line. If (z1, z2) is the origin, the line is not unique; rather, we have all the lines through the origin, corresponding to a copy of 𝕡1. In this way, by blowing up at pX, we construct a new surface Y and a map

π:YX.

We call π‎11(p) = E the exceptional divisor. It is a copy of 𝕡1, and

π:Y\EX\{p}

is biholomorphic. Locally, Y may be described in terms of affine coordinates. Let U be a coordinate neighborhood of p with local coordinates z1,z2 centered (p.62) at p. Then, W = π‎11(U) is covered by two coordinate charts W1 and W2. On W1 we have coordinates (u, v), and π‎ is given by

z 1 =uv, z 2 =v.

On W2 we have coordinates (u, v), and π‎ is given by

z 1 = u , z 2 = u v .

We see that u is an affine coordinate for (u1 : u2) on u2 = 0 and that υ‎′ is an affine coordinate for (u1 : u2) on u1 = 0. We identify W1 \ {u = 0} with W2 \ {υ‎′ = 0} by the equations Complex Surfaces and Coverings , υ‎ = u′υ‎′. The exceptional divisor is given in W1 by υ‎ = 0 and in W2 by u′ = 0. If L is a smooth irreducible curve through p, then we can suppose that it is given by z1 = 0 in a neighborhood of p, where z1, z2 are local coordinates. After blow-up, it is transformed into another curve Complex Surfaces and Coverings, called its proper transform. The curve Complex Surfaces and Coverings is given by the closure in Y of π‎11(L \ {p}). Working in W1, the exceptional divisor E has the equation υ‎ = 0, and Complex Surfaces and Coverings has the local equation u = 0. Therefore, we have

π L= L ˜ +E, L ˜ E=1,

and, more generally, the divisor class group of Y is given by

Div(Y)= π (Div(X))E,

the direct sum indicating orthogonality with respect to the intersection pairing. For divisors D1 and D2 on X, we have π‎D1 · π‎D2 = D1 · D2. These facts hold because, topologically, homology classes on X can be given by cycles not passing through p. Hence,

(3.23)
( L ˜ +E)E=0,

and so we see that E has self-intersection −1:

(3.24)
EE=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

H 2 (Y,)= H 2 (X,)[E],

(p.63) and, as the Betti numbers are the ranks of the homology groups, this implies that

(3.25)
b 2 (Y)= b 2 (X)+1.

All other homology groups, and hence Betti numbers, are unchanged.

Now consider two real oriented compact differential 4-manifolds M and N, and let pM and qN. Let U(p) be an open neighborhood of p and U(q) be an open neighborhood of q. Suppose that both Complex Surfaces and Coverings and Complex Surfaces and Coverings 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 S33. 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 Complex Surfaces and Coverings Here, the orientation on 𝕡2(C) is reversed and, in particular, a line in 𝕡2(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 ℝ22 yields a Möbius strip, the origin becoming a copy of 𝕡1(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 𝕡1(ℝ)≃ S11, just as the blown-up four-dimensional disk is fibered by 2-disks over 𝕡1(ℂ) ≃S22.

Returning to the complex case, we have, for the canonical class,

(3.26)
K Y = π K X +E.

If we write a meromorphic 2-form ω‎ in local coordinates as

ω=a( z 1 , z 2 )d z 1 d z 2 ,

then, using the affine coordinates introduced earlier,

π ω=a(uv,v)vdudv,

which proves (3.26). It also follows that

(3.27)
K Y E=1.

We can check this using the adjunction formula, which gives

(3.28)
K Y EEE=e(E)=e( 1 )=2.

(p.64) We can also calculate the self-intersection of Complex Surfaces and Coverings in Y, namely,

(3.29)
L ˜ L ˜ =( π LE ) 2 =LL+EE= L 2 1,

since E and π‎L 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

(3.30)
K Y 2 = c 1 2 (Y)= K X 2 1= c 1 2 (X)1,

whereas, for the Euler numbers,

(3.31)
c 2 (Y)= c 2 (X)+1,

since blowing up removes a point and inserts a copy of S22. This also follows from the behavior of the Betti numbers in (3.25). Therefore,

(3.32)
3 c 2 (Y) c 1 2 (Y)=3 c 2 (X) c 1 2 (X)+4.