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INTERNATIONAL SCHOOL FOR ADVANCED STUDIES …

INTERNATIONAL SCHOOLFOR ADVANCED STUDIEST riesteU. BruzzoINTRODUCTION TOALGEBRAIC TOPOLOGY ANDALGEBRAIC GEOMETRYN otes of a course delivered during the academic year 2002/2003La filosofia `e scritta in questo grandissimo libro che continuamenteci sta aperto innanzi a gli occhi (io dico l universo), ma non si pu`ointendere se prima non si impara a intender la lingua, e conoscer icaratteri, ne quali `e scritto. Egli `e scritto in lingua matematica, ei caratteri son triangoli, cerchi, ed altre figure geometriche, senzai quali mezi `e impossibile a intenderne umanamente parola; senzaquesti `e un aggirarsi vanamente per un oscuro Galilei(from Il Saggiatore )iPrefaceThese notes assemble the contents of the introductory courses I have been giving atSISSA since 1995/96.

INTERNATIONAL SCHOOL FOR ADVANCED STUDIES Trieste U. Bruzzo INTRODUCTION TO ALGEBRAIC TOPOLOGY AND ALGEBRAIC GEOMETRY Notes of a course delivered during the academic year 2002/2003

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Transcription of INTERNATIONAL SCHOOL FOR ADVANCED STUDIES …

1 INTERNATIONAL SCHOOLFOR ADVANCED STUDIEST riesteU. BruzzoINTRODUCTION TOALGEBRAIC TOPOLOGY ANDALGEBRAIC GEOMETRYN otes of a course delivered during the academic year 2002/2003La filosofia `e scritta in questo grandissimo libro che continuamenteci sta aperto innanzi a gli occhi (io dico l universo), ma non si pu`ointendere se prima non si impara a intender la lingua, e conoscer icaratteri, ne quali `e scritto. Egli `e scritto in lingua matematica, ei caratteri son triangoli, cerchi, ed altre figure geometriche, senzai quali mezi `e impossibile a intenderne umanamente parola; senzaquesti `e un aggirarsi vanamente per un oscuro Galilei(from Il Saggiatore )iPrefaceThese notes assemble the contents of the introductory courses I have been giving atSISSA since 1995/96.

2 Originally the course was intended as introduction to (complex)algebraic geometry for students with an education in theoretical physics, to help them tomaster the basic algebraic geometric tools necessary for doing research in algebraicallyintegrable systems and in the geometry of quantum field theory and string theory. Thismotivation still transpires from the chapters in the second part of these first part on the contrary is a brief but rather systematic introduction to twotopics, singular homology (Chapter 2) and sheaf theory, including their cohomology(Chapter 3).

3 Chapter 1 assembles some basics fact in homological algebra and developsthe first rudiments of de Rham cohomology, with the aim of providing an example tothe various abstract 5 is an introduction to spectral sequences, a rather intricate but very pow-erful computation tool. The examples provided here are from sheaf theory but thiscomputational techniques is also very useful in algebraic thank all my colleagues and students, in trieste and Genova and other locations,who have helped me to clarify some issues related to these notes, or have pointed outmistakes.

4 In this connection special thanks are due to Fabio Pioli. Most of Chapter 3 isan adaptation of material taken from [2]. I thank my friends and collaborators ClaudioBartocci and Daniel Hern andez Ruip erez for granting permission to use that thank Lothar G ottsche for useful suggestions and for pointing out an error and thestudents of the 2002/2003 course for their interest and constant , 4 December 2004 ContentsPart 1. Algebraic Topology1 Chapter 1. Introductory material31. Elements of homological algebra32. De Rham cohomology73. Mayer-Vietoris sequence in de Rham cohomology104.

5 Elementary homotopy theory11 Chapter 2. Singular homology theory171. Singular homology172. Relative homology253. The Mayer-Vietoris sequence284. Excision32 Chapter 3. introduction to sheaves and their cohomology371. Presheaves and sheaves372. Cohomology of sheaves433. Sheaf cohomology52 Chapter 4. More homological algebra57 Chapter 5. Spectral sequences591. Filtered complexes592. The spectral sequence of a filtered complex603. The bidegree and the five-term sequence644. The spectral sequences associated with a double complex655.

6 Some applications68 Part 2. introduction to algebraic geometry73 Chapter 6. Complex manifolds and vector bundles751. Basic definitions and examples752. Some properties of complex manifolds783. Dolbeault cohomology794. Holomorphic vector bundles795. Chern class of line bundles83iiiivCONTENTS6. Chern classes of vector bundles857. Kodaira-Serre duality878. Connections88 Chapter 7. Divisors931. Divisors on Riemann surfaces932. Divisors on higher-dimensional manifolds1003. Linear systems1014. The adjunction formula103 Chapter 8.

7 Algebraic curves I1071. The Kodaira embedding1072. Riemann-Roch theorem1103. Some general results about algebraic curves111 Chapter 9. Algebraic curves II1171. The Jacobian variety1172. Elliptic curves1223. Nodal curves126 Bibliography131 Part 1 Algebraic TopologyCHAPTER 1 Introductory materialThe aim of the first part of these notes is to introduce the student to the basics ofalgebraic topology, especially the singular homology of topological spaces. The futuredevelopments we have in mind are the applications to algebraic geometry, but alsostudents interested in modern theoretical physics may find here useful material ( ,the theory of spectral sequences).

8 As its name suggests, the basic idea in algebraic topology is to translate problemsin topology into algebraic ones, hopefully easier to deal this chapter we give some very basic notions in homological algebra and thenintroduce the fundamental group of a topological space. De Rham cohomology is in-troduced as a first example of a cohomology theory, and is homotopic invariance Elements of homological Exact sequences of a ring, and letM,M ,M beR-modules. We say that twoR-module morphismsi:M M,p:M M form anexact sequenceofR-modules, and write0 M i Mp M 0,ifiis injective,pis surjective, and kerp= , the ring of integers (recall thatZ-modules are just abeliangroups), and consider the sequence( )0 Zi Cexp C 1whereiis the inclusion of the integers into the complex numbersC, whileC =C {0}is the multiplicative group of nonzero complex numbers.

9 The morphism exp is definedas exp(z) =e2 iz. The reader may check that this sequence is morphism of exact sequences is a commutative diagram0 M M M 0 y y y0 N N N 0ofR-module morphisms whose rows are INTRODUCTORY Differential a ring, differential onMis a morphismd:M MofR-modules suchthatd2 d d= 0. The pair(M,d)is called a differential elements of the spacesM,Z(M,d) kerdandB(M,d) Imdare calledcochains,cocyclesandcoboundariesof (M,d), respectively. The conditiond2= 0 impliesthatB(M,d) Z(M,d), and theR-moduleH(M,d) =Z(M,d)/B(M,d)is called thecohomology groupof the differential module (M,d).

10 We shall often writeZ(M),B(M) andH(M), omitting the differentialdwhen there is no risk of (M,d) and (M ,d ) be morphism of differential modules is a morphismf:M M ofR-modules which commutes with the differentials,f d =d morphism of differential modules maps cocycles to cocycles and coboundaries tocoboundaries, thus inducing a morphismH(f) :H(M) H(M ). M i Mp M 0be an exact sequence of dif-ferentialR-modules. There exists a morphism :H(M ) H(M )(called connectingmorphism) and an exact triangle of cohomologyH(M)H(p)//H(M ) yytttttttttH(M )H(i) construction of is as follows: let H(M ) and letm be acocycle whose class is.


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