Topological Invariants and Differential Geometry

In this chapter, we compile some prerequisites from topology and differential geometry needed in later chapters. For the most part we do not provide proofs since there are many good references for this material, for example. In §1.1, for a topological space X, we define singular homology and cohomology, as well as the Euler number e(X). The Euler number is the topological invariant that we will encounter the most often in the subsequent chapters. For a complex surface X, it coincides with the second Chern number c2 (X), as we shall see in Chapter 3 (we assume there that X is smooth compact connected algebraic). In that chapter, we also introduce for such a surface the first Chern number c2/1(X), which can be defined as the self-intersection number of the canonical divisor. Some generalities on the first Chern class c1 (X) as well as necessary background on the canonical divisor are given in §1.4, although intersection theory for surfaces is only introduced in Chapter 3. The Miyaoka-Yau inequality for minimal smooth compact connected algebraic surfaces of general type, which is of deep importance for the material of this book, is derived in Chapter 4, and is the inequality c21(X)[less than or equal to]3c2(X) relating the Chern numbers and c2/1(X) and c2(X). In Chapter 6, we derive a version of this inequality for surfaces with an orbifold structure that are not necessarily compact (we touch on the non-compact situation also at the end of Chapter 4). When this inequality is an equality, X is a quotient of the complex two-dimensional ball B2 by a discrete subgroup of the automorphisms of B,2, acting without fixed points in Chapter 4 and with fixed points in Chapter 6. For the summaries of the proofs of the Miyaoka-Yau inequalities in these chapters, we use techniques due to Aubin, S.-T. Yau, and R. Kobayashi coming from differential geometry and partial differential equations. Some of the differential geometry can be found in §1.4 and the rest is derived as needed in Chapters 4 and 6.

In this book, we discuss the Miyaoka-Yau inequality only for surfaces, as our interest is in weighted line arrangements in the complex projective plane. A suitably generalized Miyaoka-Yau inequality due to Aubin and Yau holds, for example, for compact Kähler manifolds of dimension n whose first Chern class vanishes or is negative, meaning it is represented by a real closed negative definite (1, 1)-form (see §1.4 for definitions). For a statement, see p. 323.

1.1 TOPOLOGICAL INVARIANTS

Let X be a topological space. We briefly recall the definition, using singular chains, of the singular homology groups Hi(X, Z) with integer coefficients (see, for example). Viewing Rn as the subset of Rn+1 consisting of the vectors with (n + 1)th coordinate equal to 0, we can consider the union R8 = [union]n[greater than or equal to]1Rn. Let en, n [greater than or equal to] 1 be the vector whose nth coordinate is 1 and whose other coordinates are 0, and let e0 be the vector with all its coordinates 0. For i = 0, the standard simplex ?i of dimension i is given by the set

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A singular i-dimensional simplex in X is a continuous map from ?i onto X. The singular i-chains Ci(X) in X are the finite linear combinations, with integer coefficients, of the singular i-dimensional simplices. They form an abelian group.

For i = 1 and k = 0, ... i, we define the kth face map [partial derivative]ki: ?i-1 [arrow right]?i by

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with

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and

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The boundary operator

[partial derivative] : Ci(X) [right arrow] Ci-1(X)

is defined as the alternating sum

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It satisfies [partial derivative][partial derivative] = 0, and so we have a differential complex with homology group

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This definition depends only on X. No triangulation or similar structure is needed. If X is triangulated, we can use the chain groups c(X) consisting of finite linear combinations of the simplices of the triangulation; otherwise the definition is as before. It is known that the homology groups obtained in this way are the groups Hi(X, Z) defined above. If the abelian group Hi (X, Z) is finitely generated, then the rank of Hi (X, Z) is called the ith Betti number, bi(X). The subgroup of elements of finite order is the torsion subgroup Ti(X). If the space X admits a finite triangulation, then the Hi (X, Z) are all finitely generated and trivial for i sufficiently large. If Hi (X, Z) is finitely generated, then Hi (X, Z) [cross product] R is a real vector space of dimension bi(X), which is also denoted by Hi (X, R). The homology groups are homotopy invariants.

The dual construction gives singular cohomology. A singular i-cochain on X is a linear functional on the Z-module Ci(X) of singular i-chains. The group of singular i-cochains is therefore Ci(X) = Hom(Ci(X), Z). The coboundary operator is defined by (d?)(s) = ? ( [partial derivative]s), ? [member of] C*(X), and satisfies dd = 0. The graded group of singular cochains C*(X) = [direct sum] iCi( X) is therefore a differential complex whose homology is called the singular cohomology of X with integer coefficients. The ith cohomology group is denoted by Hi(X, Z) (for more details, see).

The Euler-Poincaré characteristic, which we also call the Euler number, is an important invariant of a topological space X, and is denoted by e(X). Assume H(Xi Z) is finitely generated and trivial for i sufficiently large. Then

e(X) = [summation over (i)](- 1)ibi(X) = [summation over (i)] (-1)idim Hi(X,R). (1.1)

The number e(X) is a topological invariant and a homotopy invariant. If the space has a finite triangulation, then

e(X) = [summation over (i)](-1)i rank (ci(X)).

In...