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Notes on the Topology of Complex Singularities Notes on the Topology of Complex Singularities Liviu I. Nicolaescu University of Notre Dame, Indiana, USA Last modified: October 13, 2013. i Introduction The algebraic varieties have played a very important role in the development of geometry. The lines and the conics where the first to be investigated and moreover, the study of equations leads naturally to algebraic geometry. The past century has witnessed the introduction of new ideas and techniques, notably algebraic topology and complex geometry. These had a dramatic impact on the development of this subject. There are several reasons which make algebraic varieties so attractive. On one hand, it is their abundance and the wealth of techniques available to study them and, on the other hand, there are the often unexpected conclusions. These conclusions lead frequently to new research questions in other directions. The gauge theoretic revolution of the past two decades has only increased the role played by these objects. More recently, Simon Donaldson has drawn attention to Lefschetz’ old techniques of studying algebraic manifolds by extending them to the much more general context of symplectic manifolds. I have to admit that I was not familiar with Lefschetz’ ideas and this gave me the impetus to teach a course on this subject and write up semi- formal notes. The second raison d’ˆetre of these notes is my personal interest in the isolated singularities of complex surfaces. Loosely speaking, Lefschetz created a holomorphic version of Morse theory when the traditional one was not even born. He showed that a holomorphic map f from a complex manifold M to the complex projective line P1 which admits only nondegenerate critical points contains a large amount of nontrivial topological information about M. This infor- mation can be recovered by understanding the behavior of the smooth fibers of f as they approach a singular one. Naturally, one can investigate what happens when f has degenerate points as well and, unlike the real case, there are many more tools at our disposal when approaching this issue in the holomorphic context. This leads to the local study of isolated singularities. These notes cover the material I presented during the graduate course I taught at the University of Notre Dame in the spring of 2000. This course emphasized two subjects, Lef- schetz theory and isolated singularities, relying mostly on basic algebraic topology covered by a regular first year graduate course.1 Due to obvious time constraints these notes barely scratch this subject and yes, I know, I have left out many beautiful things. You should view these notes as an invitation to further study. The first seven chapters cover Lefschetz theory from scratch and with many concrete and I hope relevant examples. The main source of inspiration for this part was the beautiful but dense paper [46]. The second part is an introduction to the study of isolated singularities of holomorphic maps. We spend some time explaining the algebraic and the topological meaning of the Milnor number and we prove Milnor fibration theorem. As sources of inspiration we used the classical [6, 56]. I want to tank my friends and students for their comments and suggestions. In the end I am responsible for any shortcomings. You could help by e-mailing me your comments, corrections, or just to say hello. 1The algebraic topology known at the time Lefschetz created his theory would suffice. On the other hand, after reading parts of [48] I was left with the distinct feeling that Lefschetz’ study of algebraic varieties lead to new results in algebraic topology designed to serve his goals. ii Notre Dame, Indiana 2000 Contents Introduction ....................................... i 1 Complex manifolds 1 1.1 Basic definitions ................................. 1 1.2 Basic examples .................................. 3 2 The critical points contain nontrivial information 9 2.1 Riemann-Hurwitz theorem ............................ 9 2.2 Genus formula .................................. 12 3 Further examples of complex manifolds 15 3.1 Holomorphic line bundles ............................ 15 3.2 The blowup construction ............................. 21 4 Linear systems 25 4.1 Some fundamental constructions ........................ 25 4.2 Projections revisited ............................... 27 5 Topological applications of Lefschetz pencils 31 5.1 Topological preliminaries ............................. 31 5.2 The set-up ..................................... 33 5.3 Lefschetz Theorems ................................ 35 6 The Hard Lefschetz theorem 41 6.1 The Hard Lefschetz Theorem .......................... 41 6.2 Primitive and effective cycles .......................... 43 7 The Picard-Lefschetz formulæ 47 7.1 Proof of the Key Lemma ............................. 47 7.2 Vanishing cycles, local monodromy and the Picard-Lefschetz formula . 50 7.3 Global Picard-Lefschetz formulæ ........................ 61 8 The Hard Lefschetz theorem and monodromy 63 8.1 The Hard Lefschetz Theorem .......................... 63 8.2 Zariski’s Theorem ................................ 65 iii The topology of complex singularities 1 9 Basic facts about holomorphic functions of several variables 69 9.1 The Weierstrass preparation theorem and some of its consequences ..... 69 9.2 Fundamental facts of complex analytic geometry ............... 76 9.3 Tougeron’s finite determinacy theorem ..................... 86 10 A brief introduction to coherent sheaves 91 10.1 Ringed spaces and coherent sheaves ...................... 91 10.2 Coherent sheaves on complex spaces ...................... 98 10.3 Flatness ...................................... 102 11 Singularities of holomorphic functions of two variables 105 11.1 Examples ..................................... 105 11.2 Normalizations .................................. 108 11.3 Puiseux series and Newton polygons ...................... 112 11.4 Very basic intersection theory .......................... 127 11.5 Embedded resolutions and blow-ups ...................... 131 12 The link and the Milnor fibration of an isolated singularity 151 12.1 The link of an isolated singularity ........................ 151 12.2 The Milnor fibration ............................... 153 13 The Milnor fiber and local monodromy 169 13.1 The Milnor fiber ................................. 169 13.2 The local monodromy, the variation operator and the Seifert form of an isolated singularity173 13.3 Picard-Lefschetz formula revisited ....................... 177 14 The monodromy theorem 179 14.1 Functions with ordinary singularities ...................... 179 14.2 The collapse map ................................. 181 14.3 A’Campo’s Formulæ ............................... 190 15 Toric resolutions 195 15.1 Affine toric varieties ............................... 195 15.2 Toric varieties associated to fans ........................ 206 15.3 The toric variety determined by the Newton diagram of a polynomial . 222 15.4 The zeta-function of the Newton diagram ................... 232 15.5 Varchenko’ Theorem ............................... 234 16 Cohomology of toric varieties 239 17 Newton nondegenerate polynomials in two and three variables 241 17.1 Regular simplicial resolutions of 3-dimensional fans .............. 241 Bibliography ...................................... 241 Index .......................................... 247 2 Liviu I. Nicolaescu Chapter 1 Complex manifolds We assume basic facts of complex analysis such as the ones efficiently surveyed in [31, Sec.0.1]. We will denote the imaginary unit √ 1 by i. − 1.1 Basic definitions Roughly speaking, a n-dimensional complex manifold is obtained by holomorphically gluing open sets of Cn. More rigorously, a n-dimensional complex manifold is a locally compact, Hausdorff topological space X together with a n-dimensional holomorphic atlas. This con- sists of the following objects. An open cover (U ) of X. • α Local charts, i.e. homeomorphism h : U O where O is an open set in Cn. • α α → α α They are required to satisfy the following compatibility condition. All the change of coordinates maps (or gluing maps) • F : h (U ) h (U ), (U := U U ) βα α αβ → β αβ αβ α ∩ β defined by the commutative diagram Uαβ ³ ³ hα hβ ³ ³µ n Fβα n h (U ) C Û h (U ) C α αβ ⊂ β αβ ⊂ are biholomorphic. For a point p U , we usually write ∈ α h (p) = (z (p), , z (p)) α 1,α · · · n,α or (z (p), , z (p)) if the choice α is clear from the context. 1 · · · n 1 2 Liviu I. Nicolaescu From the definition of a complex manifold it is clear that all the local holomorphic objects on Cn have a counterpart on any complex manifold. For example a function f : X C is → said to be holomorphic if f h−1 : O Cn C, (f := f ) α ◦ α α ⊂ → α |Uα is holomorphic. The holomorphic maps X Cm are defined in the obvious fashion. → If Y is a complex m-dimensional manifold with a holomorphic atlas (V ; g ) and F : X i i → Y is a continuous map, then F is holomorphic if for every i the map −1 F F (Vi) X Û Vi Y ⊂³ ⊂ ³ −1 m ³ gi F : F (Vi) gi(Ui) C ³ gi ◦ → ⊂ gi◦F ³ ³µ Ù Cm is holomorphic. Definition 1.1.1. Suppose X is a complex manifold and x X. By local coordinates near ∈ x we will understand a biholomorphic map from a neighborhood of x onto an open subset of Cn. Remark 1.1.2. The complex space Cn with coordinates (z , , z ), z := x + iy is 1 · · · n k k k equipped with a canonical orientation given by the volume form i n dx dy dx dz = dz dz¯ dz dz¯ , 1 ∧ 1 ∧···∧ n ∧ n 2 1 ∧ 1 ∧···∧ n ∧ n and every biholomorphic map between open subsets preserves this orientation. This shows that every complex
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