Complex Analytic and Differential Geometry

1 Complex Analytic andDifferential GeometryJean-Pierre DemaillyUniversit e de Grenoble IInstitut Fourier, UMR 5582 du CNRS38402 Saint-Martin d H`eres, FranceVersion of Thursday June 21, 20123 Table of ContentsChapter I. Complex Differential Calculus and Differential Calculus on Currents on Differentiable Holomorphic Functions and Complex Subharmonic Plurisubharmonic Domains of Holomorphy and Stein Pseudoconvex Open Sets II. Coherent Sheaves and Analytic Presheaves and The Local Ring of Germs of Analytic Coherent Complex Analytic Sets. Local Complex Analytic Cycles and Meromorphic Normal Spaces and Holomorphic Mappings and Extension Complex Analytic Bimeromorphic maps, Modifications and III. Positive Currents and Lelong Basic Concepts of Closed Positive Definition of Monge-Amp`ere Case of Unbounded Plurisubharmonic Generalized Lelong The Jensen-Lelong Comparison Theorems for Lelong Siu s Semicontinuity Transformation of Lelong Numbers by Direct A Schwarz Lemma.

2 Application to Number IV. Sheaf Cohomology and Spectral Basic Results of Homological The Simplicial Flabby Resolution of a Cohomology Groups with Values in a Acyclic of Contents5. Cech The De Rham-Weil Isomorphism Cohomology with Cup Inverse Images and Cartesian Spectral Sequence of a Filtered Spectral Sequence of a Double Hypercohomology Direct Images and the Leray Spectral Alexander-Spanier K unneth Poincar e V. Hermitian Vector Definition of Vector Linear Curvature Operations on Vector Pull-Back of a Vector Parallel Translation and Flat Vector Hermitian Vector Bundles and Vector Bundles and Locally Free First Chern Connections of Type (1,0) and (0,1) over Complex Holomorphic Vector Chern Lelong-Poincar e Equation and First Chern Exact Sequences of Hermitian Vector Line BundlesO(k) Grassmannians and Universal Vector VI.

3 Hodge Differential Operators on Vector Formalism of PseudoDifferential Harmonic Forms and Hodge Theory on Riemannian Hermitian and K ahler Basic Results of K ahler Commutation GroupsHp,q(X,E) and Serre Cohomology of Compact K ahler Jacobian and Albanese Complex Hodge-Fr olicher Spectral Effect of a Modification on Hodge VII. Positive Vector Bundles and Vanishing Bochner-Kodaira-Nakano Basic a Priori Kodaira-Akizuki-Nakano Vanishing Girbau s Vanishing Vanishing Theorem for Partially Positive Line Positivity Concepts for Vector of Contents57. Nakano Vanishing Relations Between Nakano and Griffiths Applications to Griffiths Positive Cohomology Groups ofO(k) Ample Vector Blowing-up along a Equivalence of Positivity and Ampleness for Line Kodaira s Projectivity on Pseudoconvex Non Bounded Operators on Hilbert Complete Riemannian Theory on Complete Riemannian General Estimate ford on Hermitian Estimates on Weakly Pseudoconvex H ormander s Estimates for non Complete K ahler Extension of Holomorphic Functions from Applications to Hypersurface Skoda sL2 Estimates for Surjective Bundle Application of Skoda sL2 Estimates to Local Integrability of Almost Complex IX.

4 Finiteness Theorems forq-Convex Spaces and Stein Topological Properties in Top Andreotti-Grauert Finiteness Grauert s Direct Image IComplex Differential Calculus and PseudoconvexityThis introductive chapter is mainly a review of the basic toolsand concepts which will be employedin the rest of the book: differential forms, currents, holomorphic and plurisubharmonic functions, holo-morphic convexity and pseudoconvexity. Our study of holomorphic convexity is principally concentratedhere on the case of domains inCn. The more powerful machinery needed for the study of general com-plex varieties (sheaves, positive currents, hermitian differential Geometry ) will be introduced in ChaptersII to V. Although our exposition pretends to be almost self-contained, the reader is assumed to haveat least a vague familiarity with a few basic topics, such as differential calculus, measure theory anddistributions, holomorphic functions of one Complex variable.

5 Most of the necessary background canbe found in the books of [Rudin 1966] and [Warner 1971]; the basics of distribution theory can be foundin chapter I of [H ormander 1963]. On the other hand, the readerwho has already some knowledge ofcomplex analysis in several variables should probably bypass this chapter . 1. Differential Calculus on Manifolds Differentiable ManifoldsThe notion of manifold is a natural extension of the notion ofsubmanifold definedby a set of equations inRn. However, as already observed by Riemann during the19th century, it is important to define the notion of a manifold in a flexible way, withoutnecessarily requiring that the underlying topological space is embedded in an affine precise formal definition was first introduced by H. Weyl in [Weyl 1913].Letm Nandk N { , }. We denote byCkthe class of functions which arek-times differentiable with continuous derivatives ifk6= , and byC the class of realanalytic functions.

6 Adifferentiable manifoldMof real dimensionmand of classCkis atopological space (which we shall always assume Hausdorff and separable, possessinga countable basis of the topology), equipped with an atlas ofclassCkwith values classCkis a collection of homeomorphisms :U V , I, calleddifferentiable charts, such that (U ) Iis an open covering ofMandV an open subsetofRm, and such that for all , Ithetransition map( ) = 1 : (U U ) (U U )is aCkdiffeomorphism from an open subset ofV onto an open subset ofV (see Fig. 1).Then the components (x) = (x 1,..,x m) are called thelocal coordinatesonU definedby the chart ; they are related by the transition relationx = (x ).8 chapter I. Complex Differential Calculus and PseudoconvexityMU U U U RmV V (U U ) (U U ) Fig. I-1 Charts and transition mapsIf Mis open ands N { , }, 06s6k, we denote byCs( ,R) the set offunctionsfof classCson , such thatf 1 is of classCson (U ) for each ; if is not open,Cs( ,R) is the set of functions which have aCsextension to someneighborhood of.

7 Atangent vector at a pointa Mis by definition a differential operator acting onfunctions, of the typeC1( ,R) f7 f=X16j6m j f xj(a)in any local coordinate system (x1,..,xm) on an open set a. We then simply write =P j / xj. For everya , then-tuple ( / xj)16j6mis therefore a basis of thetangent spacetoMata, which we denote byTM,a. Thedifferentialof a functionfatais the linear form onTM,adefined bydfa( ) = f=X j f/ xj(a), TM, particulardxj( ) = jand we may consequently writedf=P( f/ xj)dxj. Fromthis, we see that (dx1,..,dxm) is the dual basis of ( / x1,.., / xm) in the cotangentspaceT M,a. The disjoint unionsTM=Sx MTM,xandT M=Sx MT M,xare called thetangentandcotangent is a vector field of classCsover , that is, a mapx7 (x) TM,xsuch that (x) =P j(x) / xjhasCscoefficients, and if is another vector field of classCswiths>1, theLie bracket[ , ] is the vector field such that( )[ , ] f= ( f) ( f).

8 In coordinates, it is easy to check that( )[ , ] =X16j,k6m j k xj j k xj xk. 1. Differential Calculus on Manifolds9 Differential FormsA differential formuof degreep, or briefly ap-form overM, is a mapuonMwithvaluesu(x) pT M,x. In a coordinate open set M, a differentialp-form can bewrittenu(x) =X|I|=puI(x)dxI,whereI= (i1,..,ip) is a multi-index with integer components,i1< .. < ipanddxI:=dxi1 .. dxip. The notation|I|stands for the number of components ofI, and isreadlengthofI. For all integersp= 0,1,..,mands N { },s6k, we denote byCs(M, pT M) the space of differentialp-forms of classCs, natural operations on differential forms can be defined. (x) =PvJ(x)dxJis aq-form, thewedge productofuandvis the form of degree (p+q) defined by( )u v(x) =X|I|=p,|J|=quI(x)vJ(x)dxI dxJ. by a tangent be viewed as an antisymmetricp-linear form onTM.

9 If =P j / xjis a tangent vector, we define thecontraction uto be the differential form of degreep 1 such that( )( u)( 1,.., p 1) =u( , 1,.., p 1)for all tangent vectors j. Then ( ,u)7 uis bilinear and we find easily xjdxI= 0ifj / I,( 1)l 1dxIr{j}ifj=il simple computation based on the above formula shows that contraction by a tangentvector is aderivation, ( ) (u v) = ( u) v+ ( 1)deguu ( v). is the differential operatord:Cs(M, pT M) Cs 1(M, p+1T M)defined in local coordinates by the formula( )du=X|I|=p,16k6m uI xkdxk , one can defineduby its action on arbitrary vector fields 0,.., formula is as followsdu( 0,.., p) =X06j6p( 1)j j u( 0,..,b j,.., p)+X06j<k6p( 1)j+ku([ j, k], 0,..,b j,..,b k,.., p).( )10 chapter I. Complex Differential Calculus and PseudoconvexityThe reader will easily check that ( ) actually implies ( ).

10 The advantage of ( )is that it does not depend on the choice of coordinates, thusduis intrinsically two basic properties of the exterior derivative (again left to the reader) are:d(u v) =du v+ ( 1)deguu dv,(Leibnitz rule)( )d2= 0.( )A formuis said to beclosedifdu= 0 andexactifucan be writtenu=dvfor someformv. Rham Cohomology that a cohomological complexK =Lp Zis a collection of modulesKpover some ring, equipped with differentials, , linearmapsdp:Kp Kp+1such thatdp+1 dp= 0. Thecocycle, coboundaryandcohomologymodulesZp(K ),Bp(K ) andHp(K ) are defined respectively by( ) Zp(K ) = Kerdp:Kp Kp+1,Zp(K ) Kp,Bp(K ) = Imdp 1:Kp 1 Kp, Bp(K ) Zp(K ) Kp,Hp(K ) =Zp(K )/Bp(K ).Now, letMbe a differentiable manifold, say of classC for simplicity. TheDe RhamcomplexofMis defined to be the complexKp=C (M, pT M) of smooth differentialforms, together with the exterior derivativedp=das differential, andKp={0},dp= 0forp <0.