MATHEMATICS: CNCEPTS, AND FUNDATIN Vol I - Complex Aalytic Geometry - Tatsuo SUWA CMPLEX ANALYTIC GEMETRY Tatsuo SUWA Departmet of Iformatio Egieerig, Niigata Uiversity, Japa Keywords: holomorphic fuctio, complex maifold, aalytic variety, vector budle, Cher class, divisor, Grothediec residue, sheaf cohomology, de Rham ad Dolbeault theorems, Poicaré ad Kodaira-Serre dualities, Riema-Roch theorem Cotets Aalytic fuctios of oe complex variable 2 Aalytic fuctios of several complex variables 3 Germs of holomorphic fuctios 4 Complex maifolds ad aalytic varieties 5 Germs of varieties 6 Vector budles 7 Vector fields ad differetial forms 8 Cher classes of complex vector budles 9 Divisors Complete itersectios ad local complete itersectios Grothediec residues 2 Residues at a isolated zero 3 Examples 4 Sheaves ad cohomology 5 de Rham ad Dolbeault theorems 6 Poicaré ad Kodaira-Serre dualities 7 Riema-Roch theorem Glossary Bibliography Biographical Setch Summary The mai objective of Complex Aalytic Geometry is to study the structure of complex maifolds ad aalytic varieties (the sets of commo zeros of holomorphic fuctios) It is deeply related to various fudametal areas of mathematics, such as complex aalysis, algebraic topology, commutative algebra, algebraic geometry, differetial geometry ad sigularity theory, ad there are very rich iterplays amog them The Riema-Roch theorem, for istace, is a outcome of such iteractio The subject is also related to may other braches of scieces icludig mathematical physics ad learig theory Aalytic Fuctios of e Complex Variable Let U be a ope set i the complex plae ad f a complex valued fuctio o U The differetiability of f at a poit a i U is defied as i the case of fuctios of a Ecyclopedia of Life Support Systems (ELSS)
MATHEMATICS: CNCEPTS, AND FUNDATIN Vol I - Complex Aalytic Geometry - Tatsuo SUWA real variable Thus we say that f is differetiable at a if the limit ( ) f a+ h f a lim h h ( ) exists If f is differetiable at every poit of U, we say that f is holomorphic i U df The above limit is deoted by dz ( a ) ad is called the derivative of f at a If f is holomorphic i U, the we may thi of df dz as a fuctio o U We say that f is aalytic at a poit a i U if it ca be expressed as a power series f z = c ( z a), ( ) = which coverges at each poit z i a eighborhood of a We say that f is aalytic i U if it is aalytic at every poit of U If f is aalytic i U, it is holomorphic iu A striig fact about fuctios of a complex variable is that the coverse is also true, ie, if f is holomorphic i U, it is aalytic i U There is aother importat way of expressig this property Let z = x+ y with x ad y the real ad imagiary parts, respectively We may thi of f as a fuctio of ( x, y) We write f = u+ v with u ad v the real ad imagiary parts I geeral, r we say that a fuctio of real variables is (of class) C, if the partial derivatives exist up to order r ad are cotiuous If all the partial derivatives exist we say it is C The f is holomorphic i U if ad oly if f is C i ( x, y) ad satisfies the Cauchy- Riema equatios" i U ; u v u v =, = x y y x We fiish this sectio by recallig the Cauchy itegral formula Let f be a aalytic fuctio i a eighborhood of a ad γ the boudary of a small dis about a, orieted couterclocwise The we have 2π γ f( z) dz z a = f ( a) 2 Aalytic Fuctios of Several Complex Variables Let = { z = ( z,, z) zi } be the product of copies of For a -tuple Ecyclopedia of Life Support Systems (ELSS)
MATHEMATICS: CNCEPTS, AND FUNDATIN Vol I - Complex Aalytic Geometry - Tatsuo SUWA = (,, ) of o-egative itegers, we set z = z z, = + + ad! =!! Let U be a ope set i ad f a complex valued fuctio o U We say that f is aalytic at a poit a i U, if it ca be expressed as a power series f z = c z a = c z a z a, ( ) ( ) ( ) ( ),, which coverges absolutely at each poit z i a eighborhood of a We say that f is aalytic i U if it is aalytic at every poit of U A theorem of Hartogs says that f is aalytic i U if ad oly if f is aalytic i each variable z i i U, for i =,, I the sequel, we call aalytic fuctio also a holomorphic fuctio ad use the words aalytic" ad holomorphic" iterchageably If f is holomorphic, for arbitrary, the partial derivative f f = z z z f z c z a is a power series expasio of f, the each coefficiet c is give by exists ad is holomorphic i U If ( ) = ( ) c f = ( a)! z This series is called the Taylor series of f at a m Let U be a ope set i ad f : U a map We say that f is holomorphic if, whe we write f compoetwise as f = ( f,, fm ), each f i is holomorphic Let U ad U be two ope sets i ad f : U U a map We say that f is biholomorphic, if f is bijective ad if both f ad f are holomorphic For a holomorphic map (,, f = f fm ) from a ope set U i ito, we set f f z z ( f,, fm ) = ( z,, z ) fm fm z z Ecyclopedia of Life Support Systems (ELSS)
MATHEMATICS: CNCEPTS, AND FUNDATIN Vol I - Complex Aalytic Geometry - Tatsuo SUWA ad call it the Jacobia matrix of f with respect to z We say that a poit a i U is a regular poit of f, if the ra of the Jacobia matrix evaluated at a is maximal possible, ie, mi ( m, ) therwise we say that a is a critical (or sigular) poit of f As i the case of fuctios or mappigs of real variables, we have the iverse mappig theorem ad the implicit fuctio theorem, which are basic i aalyzig a mappig at its regular poit 3 Germs of Holomorphic Fuctios Let H be the set of fuctios holomorphic i some eighborhood of i We defie a relatio i H as follows For two elemets f ad g i H, f ~ g if they coicide o a eighborhood of The the relatio ~ is a equivalece relatio i H The equivalece class of a fuctio f is called the germ of f at, which we also deote by f for simplicity We let be the quotiet set of H by this equivalece relatio The set has the structure of a commutative rig with respect to the operatios iduced from the additio ad the multiplicatio of fuctios It has the uity which is the equivalece class of the fuctio costatly equal to If we deote by { z z },, the set of power series which coverge absolutely i some eighborhood of, this set also has the structure of a rig Sice f ~ g if ad oly if f ad g have the same power series expasio, we may idetify with { z,, z} The rig is a itegral domai, ie, if fg=, for f, g i, the f = or g = We say that a germ u i is a uit if there is a germ v such that uv =, it is equivalet to sayig that it is the germ of a fuctio u with u ( ) The followig two theorems of Weierstrass are fudametal i the aalysis of the structure of the rig First, for a germ f i, we write f = az We say that the order of f is, if a = for all with < ad with = We say that the order of f i power series i z, is a for some f,,, z, as a z is, if the order of ( ) We cosider the rig [ z ] of polyomials i { } [ z ] = f( z) = a + a z + + a z a i z with coefficiets i : Ecyclopedia of Life Support Systems (ELSS)
MATHEMATICS: CNCEPTS, AND FUNDATIN Vol I - Complex Aalytic Geometry - Tatsuo SUWA A Weierstrass polyomial i z of degree is a elemet h of [ z ] of the form h = a + a z + + a z + z, where is a positive iteger ad a, a,, a are o-uits i Note that i the above, (,,, ) ay germ f i h z = z is writte as Hece the order of h i z is I geeral, f( z) = a + a z + + a z + with a i i The order of f i z is if ad oly if a, a,, a are o-uits i ad a is a uit i a a + az + + az is a Weierstrass I this case, ( ) polyomial i z of degree The Weierstrass preparatio theorem stated below says that such a f is essetially equal to a Weierstrass polyomial of degree Weierstrass divisio theorem If h is a Weierstrass polyomial i z of degree, the for ay germ f i, there exist uiquely determied elemets q i ad r i [ z] with deg r < such that f = qh + r Weierstrass preparatio theorem Let f be a germ i whose order i z is The there is a uique Weierstrass polyomial h i z of degree such that f = uh with u a uit i Next we discuss some importat properties of the rig which follow from the above theorems We say that a germ f i is irreducible if f is ot a uit ad if the idetity f = gh for germs g ad h implies that either g or h is a uit The rig is a uique factorizatio domai, ie, every germ f that is ot or a uit ca be expressed as a product of irreducible germs ad the expressio is uique up to the order ad multiplicatios by uits For germs f ad g, there is always the greatest commo divisor gcd ( f, g ), which is uique up to multiplicatio by uits We say that f ad g are relatively prime if gcd ( f, g ) is a uit Aother importat property of the rig has a fiite umber of geerators is that it is Noetheria, ie, every ideal i Ecyclopedia of Life Support Systems (ELSS)
MATHEMATICS: CNCEPTS, AND FUNDATIN Vol I - Complex Aalytic Geometry - Tatsuo SUWA - - - T ACCESS ALL THE 34 PAGES F THIS CHAPTER, Visit: http://wwweolsset/eolss-sampleallchapteraspx Bibliography R Bott ad L Tu, Differetial Forms i Algebraic Topology, Graduate Texts i Mathematics 82, Spriger-Verlag, 982, xiv+33 pp, ISBN 3-54-963-4 [This boo explais algebraic topology i terms of differetial forms, itroduces the Čech-de Rham cohomology ad gives a accout of characteristic classes of vector budles] W Fulto, Itersectio Theory, Spriger-Verlag, 984, xiii+47 pp, ISBN -387-98549-2 [This boo explais oe of the most fudametal theories i algebraic geometry, ie, the itersectios of subvarieties It icludes may importat formulas ivolvig itersectios ad characteristic classes] P Griffiths ad J Harris, Priciples of Algebraic Geometry, Joh Wiley ad Sos, 978, xii+83 pp, ISBN -47-32792- [This boo discusses algebraic geometry from the trascedetal viewpoit, ie, complex aalytic viewpoit] R Guig ad H Rossi, Aalytic Fuctios of Several Complex Variables, Pretice-Hall, 965, xiv+37 pp [This is a text boo o aalytic fuctios of several complex variables] F Hirzebruch, Topological Methods i Algebraic Geometry, Spriger-Verlag, 966, ix+232 pp [This boo gives the proof of the author s Riema-Roch theorem It also cotais basic materials o vector budles, sheaves, characteristic classes ad so forth] K Kodaira, Complex Maifolds ad Deformatio of Complex Structures, Spriger-Verlag, 986, x+465 pp, ISBN -387-9688-7 [This boo explais the structure of complex maifolds ad deformatios of such structures] J Milor, Sigular Poits of Complex Hypersurfaces, A of Math Studies 6, Priceto Uiversity Press, 968, iii+22 pp [This is a basic moograph o sigular poits of the zero set of a holomorphic fuctio] J Milor ad J Stasheff, Characteristic Classes, A of Math Studies 76, Priceto Uiversity Press, 974, vii+33 pp, ISBN -69-822- [This is a text boo o characteristic classes of vector budles] Biographical Setch Tatsuo SUWA: The mai subject of the author s research is Complex Aalytic Geometry He started his career uder the guidace of K Kodaira Earlier wors are o the structures ad deformatios of compact complex maifolds, such as some complex surfaces, compact quotiets of the complex Euclidea spaces ad holomorphic Seifert fiber spaces The, ispired by a paper of P Baum ad R Bott, published i 972, he became iterested i sigular holomorphic foliatios (itegrable system of holomorphic vector fields or -forms with sigularities) He first costructed a ufoldig theory of codimesio oe foliatio This geeralizes the ufoldig theory for fuctios, which is developed by J Mather ad is deeply related to the catastrophe theory of R Thom e of the basic results of the author s is a versality theorem, which gives a algebraic criterio for a ufoldig to be versal ad has may applicatios He the started wor o the residues of sigular holomorphic foliatios With a umber of collaborators, J-P Brasselet, D Lehma ad J Seade to ame a few, he geeralized the idex theorem of C Camacho ad P Sad for ivariat curves of foliatios ad discovered some ew idices ad residues He Ecyclopedia of Life Support Systems (ELSS)
MATHEMATICS: CNCEPTS, AND FUNDATIN Vol I - Complex Aalytic Geometry - Tatsuo SUWA costructed a residue theory, which uify the Baum-Bott, Camacho-Sad ad the other theories, as a localizatio theory of some characteristic classes i the framewor of Cech-de Rham cohomology This was published as a boo from Herma, Paris i 998 This localizatio theory tured out to be very effective i dealig with may problems ivolvig characteristic classes, icludig the characteristic classes of sigular varieties (for which he costructed a theory of Milor classes with the above collaborators), residues of Cher classes ad explicit represetatios of these, aalytic itersectio theory i sigular varieties, ad applicatios to complex dyamical systems (for which he made cotributios with F Bracci et al) The author has bee ivited to may coutries for collaboratios ad for lectures, icludig Brazil, Caada, Chia, Frace, Germay, Hugary, Italy, Korea, Mexico, Polad, Russia, Spai, Tuisia, USA He also ivited researchers from all over the world ad orgaized a umber of iteratioal symposia I particular, he orgaized three times, with his parter J-P Brasselet, Fraco-Japaese symposia with the title Sigularities i Geometry ad Topology Ecyclopedia of Life Support Systems (ELSS)