Full page photo 143-198 SQU Journal For Science, 12 (2007) © 2007 Sultan Qaboos University Kähler Manifolds Positive Ricci Curvature Boudjemaa Anchouche Department of Mathematics and Statistics, College of Science, Sultan Qaboos University,P.O.Box 36, PC 123, Al-Khodh, Muscat, Sultanate of Oman, Email: anchouch@squ,edu.om. ات ریتشي الموجبة تشعبات كایلر لمنحنی بوجمعة عنشوش الھدف من ورقة العمل التالیة ھي إعطاء نبذة عن تشعبات كایلر لمنحنیات ریتشي الموجبة. :خالصة ABSTRACT: The aim of this paper is to give an overview of some results obtained in the field of Kähler manifolds of positive Ricci curvature. KEYWORDS: Kähler manifolds, positive Ricci curvature, kodaira dimension. 1. Introduction ifferential geometry is a cornerstone of modern mathematics. Besides the natural beauty of the subject, it has found many applications in various branches of mathematics such as number theory, analysis, topology, algebraic geometry and also in some areas of physics, ranging from general relativity to string theory. Recently one of the most famous conjectures in topology, the Poincaré conjecture1, was solved by Perelman2, a Russian mathematician, using differential geometry, a work for which he was awarded the Fields medal at the International Congress of Mathematics in Spain, August 2006. An excellent account of Perelman’s work is given by Morgan and Tian in 2007. The aim of this paper is to give an overview of an active research area of differential geometry, namely Kähler manifolds of positive Ricci curvature. Since most of the proofs are either omitted or sketched and since we have in mind a reader who is not necessarily an expert in the field, we made an effort to introduce most of the basic notions and definitions, and we included references whenever needed, so that he/she can fill the gaps and go through the details by himself/herself. The paper is organized as follows. In Part 1 we collect some basic Riemannian geometry results mostly needed 1 The Poincaré conjecture states that every simply connected closed three dimensional manifold is homeomrphic to S3 , where S3 is the three dimensional sphere. 2 Building on R. Hamilton’s work on the Ricci flow. D BOUDJEMAA ANCHOUCHE in the following sections (readers with a background in Riemannian geometry can skip it). For the sake of completeness we included some results3 in Riemannian geometry even though they are not needed in the following sections. The main reasons for doing that are; first we believe that they are fundamental results and therefore should be known, and secondly it makes a smooth transition to Kähler manifolds introduced in 2, since it gives a characterization of Kähler manifolds in terms of holonomy groups. Part 2 deals with Kähler manifolds. Part 3 deals with compact Kähler manifolds of positive Ricci curvature. Part 4 deals with Einstein-Kähler metrics on Fano manifolds. Part 5 deals with noncompact complete Kähler manifolds of positive Ricci curvature. 1.1 Riemannian manifolds This part is a collection of some definitions and results from Riemannian geometry which will be needed in the following parts. The reason for including a section on holonomy groups, even though we don’t need it in the following parts, is our belief that it is a fundamental concept in Riemannian geometry and therefore worth mentioning. Some of the material covered in this part can be found in most differential geometry textbooks, such as Kobayashi and Nomizu, 1969, Aubin 2001 and Gallot and Hulin, 1990. For an account of the development of Riemannian geometry during the second half of the twentieth century, and its main contributors we suggest a book, recently published, by Marcel Berger(2000). We expect the reader to have some knowledge of general topology (Munkres, 1975) is an excellent reference). As we said in the introduction, readers with a background in Riemannian geometry can skip this part. 1.2 Differentiable manifolds Definition 1. A locally Euclidean space M of dimension n is a Hausdorff topological space that is locally Euclidean, i.e., each point of M has a neighborhood which is homeomorphic to an open subset of n . Definition 2. A C ∞ (resp. real analytic) structure Ψ on an n − dimensional locally Euclidean space M is a collection Ψ ( ) IUα α αϕ ∈= , of pairs ( )Uα αϕ, such that 1. ( ) IUα α∈ are open sets of M satisfying I U Mα α∈∪ = . 2. U Vα α αϕ : → , are homeomorphisms, where Vα is an open subset of nR . 3. 1oα βϕ ϕ − : ( ) ( )U UU Uβ ββ α α αϕ ϕ→∩ ∩ is a ( )resp real analyticC ∞ − . diffeomorphism. 4. The collection ( ) IUα α αϕ ∈, is maximal with respect to 3., i.e., given any pair ( )U ϕ, , where U is an open subset of M and U Vϕ : → is a homeomorphism, where V is some open subset of nR , such that ( ) ( ) 1o U U U Uα ααα ϕ ϕϕ ϕ − : → ,∩ ∩ 5. is a C ∞ − (resp. real analytic) diffeomorphism for all Iα ∈ , then ( )U ϕ, ∈ Ψ . The collection ( )i i i IU ϕ ∈, satisfying 1. , 2. and 3. above is called a C ∞ -atlas. Definition 3. A C ∞ (resp. real analytic) manifold of dimension n is a pair ( )M , Ψ , where M is a locally Euclidean space of dimension n and Ψ is a C ∞ − (resp. real analytic) differentiable structure on M . The manifold ( )M , Ψ will be denoted simply by M . The pair ( )i iU ϕ, ( or ( iU , 1 i i nx x, ..., ) where 3 Holonomy groups. KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE 1 i i i nx xϕ       = , ..., ) is called a local chart and 1 i i nx x       , ..., is called a local coordinate system (defined in i U ) associated to i ϕ . A manifold M is said to be compact (resp. noncompact) if the underlying topological space is compact (resp. noncompact). Example 4. 1. Every vector space has the structure of a C ∞ , noncompact manifold. In particular n has the structure of a C ∞ , noncompact manifold. 2. The unit sphere ( ) 1 1 2 1 1 1 1 n n n in i S x R xx + + + =   = ∈ | = ,, ...,    ∑ is a compact real analytic manifold. 3. The projective space ( )n P defined by ( ) { }( )1 0n n +:= ,  ‚P ∼ù where ( ) ( ) ( ) ( )0 0 0 0such thatn nn nx y x yy yx xλ λ ∗⇐⇒ ∃ ∈ = ,, ..., , ...,, ..., , ...,∼ is a compact actually real analytic manifold. 4. An open subset of a C ∞ − manifold is a C ∞ − manifold. 5. The set of p× q matrices with real entries ( )p,qM  is a finite dimensional real vector space, hence a C ∞ − manifold. 6. If 1 M and 2 M are C ∞ − manifolds, then 1 2M M× is a C ∞ − manifold. Notation 5. 1. Let U be an open subset of a manifold M and let K be either  or  . In what follows we will denote by ( )C U∞ , K (resp. ( )C U∞ , K ) the set of infinitely differentiable (resp. r − times differentiable) functions on U with values in .| 2. Sometimes we use the word "smooth" to mean of class C ∞ . Definition 6. A Lie group G is a C ∞ − manifold endowed with a group structure which is compatible with its differentiable structure, i.e. G is a group and the map ( ) ( ) 1 G G G g h g h gh ν ν − : × → , , = ,a is of class C ∞ . Let G and H be two Lie groups. A map, or a morphism G Hϕ : → , is a differentiable map, which is a group homomorphism. BOUDJEMAA ANCHOUCHE Example 7. The following are Lie groups. 1. Every finite dimensional real vector space with its underlying abelian group structure is a Lie group. 2. The torus ( )nn n/ /≅  R is a Lie group4. 3. The linear group ( )GL n ,  is the set of all n n× invertible matrices with real entries. 4. Special linear group: ( ) ( ){ }nSL n , A M det A 1 .= ∈ =  5. Orthogonal group: ( ) ( ){ }nO n , A GL n , A A I .= ∈ =  T 6. Special orthogonal group: ( ) ( ){ }SO n , A O n , det A 1 .= ∈ =  7. Unitary group: ( ) ( ){ }nU n A GL n , A*A I ,= ∈ = where A* A .= T 8. Special unitary group: ( ) ( ){ }SU n A U n det A 1 .= ∈ = 9. Simplectic group: ( ) ( ){ }Sp n , A GL 2n , A JA J ,= ∈ =  T where n n O I J -I O   =     (10) If G and H are two Lie groups, then the direct product G×H has a structure of a Lie group. (11) If G is a Lie group and H is a closed normal subgroup of G , then G H∕ has the structure of a Lie group. Readers who want more examples and desire to know more about Lie groups, can consult (Warner,1983, Knapp, 2001, Brocker and Diek or Helgason, 1978). Remark 8. In all that follows we will assume, without loss of generality, that manifolds are connected. Submanifolds Definition 9. Let M be a manifold of dimension N . A subset N of M is a submanifold of dimension p if for every point x n∈ , there exists a local chart ( )U , ,ϕ with m U∈ such that (1) ( ) 1 2U V V ,ϕ ≅ × where 1V (resp. 2V ) is an open subset of p (resp. n-p ). (2) ( ) { }1U N V 0 .ϕ ∩ ≅ × . The definition above can be reformulated as follows: For each x n∈ , there exists a local chart ( )( )1U, x ,..., x ,n with x U∈ such that ( ) ( ){ }jU N x U x x 0 for j p 1,..., n .ϕ ∩ ≅ ∈ = = + 4 The torus / nR Z is isomorphic to ( ) 1S n , where { }1S 1 .z z= ∈  = KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE .n+1 fold of-is a submani ( ){ }n 1n n 1 21 n+1 ii 1S x ,..., x x 1++ == ∈ | =∑R ) 1( .Example 10 (2) The set ( ) ( ){ }kp, q p.qM A M rank A k= ∈  =  . ( ) ( ) I p,q I k p,qM U M< ‚ is a submanifold of (3) Let G be a Lie group and let H be an abstract subgroup of G . It is well known that H is a submanifold of G (and hence a Lie group) H⇔ is closed in G . Tangent Bundle The aim of this section is to define the tangent bundle TM of a differentiable manifold M . Let m M∈ , and let f (resp. g ) be a differentiable function defined in a neighborhood U (resp. V ) of m . We say that f and g are equivalent, and we write f g< , if there exists a neighborhood W U V, m W,⊂ ∩ ∈ such that f in Wg≡ . It is easy to see that R is an equivalence relation. The equivalence class of a function f defined in a neighborhood of m , called a germ of f at m , will be denoted by f . The set of germs at m will be denoted by mE It can be shown that mE is an − algebra. Definition 11. A tangent vector at m M∈ is linear map mX :E , satisfying ( ) ( ) ( )ˆ ˆ ˆˆ ˆ ˆX f . fX X f ,g g g= + . ., Xi e is an − derivation of the − algebra mE Definition 12. (1) The set of tangent vectors to M at a point m M∈ is called the tangent space to M at m and is denoted by mT M . (2) The dual of mT M , denoted by * mT M , is called the cotangent space of M at m . (3) The disjoint union m M mT M∈C (resp. * m M mT M∈C ), denoted TM (resp. *T M ), is called the tangent (resp. cotangent) bundle of M . 2. Vector bundles Definition 13. Let M be a C ∞ -differentiable manifold of dimension n . A vector bundLe of rank r is a differentiable manifold E together with: (1) A surjective C ∞ -differentiable map : E Mπ → , such that for every x M∈ , ( ) 1 xE Xπ −= has a structure of a real r- dimensional vector space, where r is independent of x . (2) An open covering ( ) IUα α ∈ of M and diffeomorphisms αφ (called trivializations) ( )1U: E U U , r αα α α ϕ π −= → ×  , such that for each x U ,α∈ the map BOUDJEMAA ANCHOUCHE { }E r rxϕαχ → × →  is a  -linear isomorphism. The vector bundle is denoted by ( )E, M,π or simply by E . Let Uα and U β are two open subsets of M such that ( ) ( ) ( ){ }1 i i 1 1 1 1U U m, ,..., m U , ,..., , i ,r i i i i riπ ζ ζ ζ ζ α β− ≅→ × = ∈ ∈ =  . For each couple ( ),α β , consider the functions ( ) ( ) ( )( ) ' 1 , i, k 1 i, j : U U G r x x r g x g g β α β α α β β β α α ϕ ϕ − ≤ ≤ = ∩ → = o a L where ( ), i, k U U .g C β α α β ∞∈ ∩ For m U Uα β∈ ∩ , the points ( )1 1m , ,..., r α αζ ζ and ( )2 1m , , ..., r β βζ ζ represent the same point on ( ) ( )1 U U U U rE α β α βπ − ≅ ∩ → ∩ ×  , if and only if ( )1 2 i j 1 m m m and m , i=1, ..., r. r k gβ β ααζ ζ = = = = ∑ Example 14. (1) If M is a differentiable manifold, then M r×  is a vector bundle called the trivial bundle of rank r . (2) Let M be a differentiable manifold and consider the projection given by ( ) m : TM M X x m X T Mif π π → = ∈a Then it can easily be seen that ( )TM , M,π is a vector bundle of rank dim Mn = R . Similarly for T * M . (3) A vector bundle of rank 1r = is called a line bundle. (4) Since a vector bundle is a family of vector spaces of a given dimension parameterized by a C ∞ - manifold M , we can perform on vector bundles the same operations performed on vector spaces, such as the direct sum, the tensor product, the dualization, and so on. Definition 15. A Lie aLgebra is vector space V endowed with a bilinear map [ ] ( ) [ ] , : V V V , , ,ζ η ζ η × → a called the Lie bracket, such that: (1) [ ] [ ], , , Vfor allζ η ζ η ζ η= − ∈ , (2) [ ] [ ], , , , 0 for all , Vζ η γ γ ζ η ζ η   + = ∈    (Jacobi Identity). Let G be a Lie group. It can be shown that the tangent space eT G to G=g at the identity e has the structure of a Lie algebra and the tangent bundle TG of G is trivial, more precisely TG G g≅ × . KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE Definition 16. (1) A section s of the C ∞ -vector bundle ( )E , M ,π over an (open) subset U of M is a C ∞ -map s: U E→ such that Us idπ =o . The space of C ∞ -sections of E over U is denoted by ( )U , EΓ . If U M= , then we denote ( )U , EΓ simply by ( )EΓ . (2) A section of the vector bundle TM (resp. T * M ) over an open set U of M is called a vector field (resp. a differential form) defined over U . The space C ∞ -vector fields over U will be denoted by ( )U, TMΓ . Let M be C ∞ differentiable manifold of dimension , m Mn ∈ and let ( )1 nx ,..., x be a local coordinate system defined in an open set U , with m U∈ . A C ∞ -vector field X over U may be written ( ) ( ) n i i=1 i X x x x ζ ∂ = ∂ ∑ where ( )i U,Cζ ∞∈  . Definition 17. The differential mdf of a function ( ) 1f U,C∈  at m M∈ is a linear form on mT M defined as follows: for each n i m i=1 i X T M x m ζ ∂ = ∈ ∂ ∑ , ( ) ( ) ( ) ( )m i i=1 i f df X X f m m n x ζ ∂ = = ∂ ∑ . In particular, for if x= , we have ( ) ( )i imd x X ζ= . Hence we can write n m i i=1 i f df dx x ∂ = ∂ ∑ If ( )1 nx ,..., x is a local coordinate system defined in an open subset U of M , with m U∈ , then 1 n ,..., x x  ∂ ∂   ∂ ∂  (resp. ( )1 ndx ,..., dx ) is a C ∞ -frame of TM (resp. T * M ) over U . The frame ( )1 ndx ,..., dx is the dual of the frame 1 n ,..., x x  ∂ ∂   ∂ ∂  , i.e. i i, j, j dx x δ  ∂ =  ∂  where i, j, δ is the Kronecker symbol. Definition 18. Let X and Y be two vector fields defined in an open subset U of M . The Lie bracket of X and Y is the vector field [ ]X , Y defined by [ ]( ) ( )( ) ( )( )X, Y f X Y f Y X f ,= − where ( ) 2f U, TMC∈ . If ( )1 nU, x ,..., x is a local chart with m U∈ and if BOUDJEMAA ANCHOUCHE n n i i=1 i=1i i X and U x xi Y inζ η ∂ ∂ = = ∂ ∂ ∑ ∑ , then [ ] n n i i=1 i=1 X, Y x x x i j j j j i ζη ζ η  ∂∂ ∂ −  ∂ ∂ ∂  ∑ ∑ Embeddings Definition 19. Let M and N be two C ∞ -manifolds. A C ∞ -map f : M N→ is a continuous map such that ( ) ( )-1ofo : U V mapis a Cϕ ϕ ∞Ψ → Ψ − , for all charts ( ),U ϕ of M and ( )V, Ψ of N such that ( ) 1U Vϕ −⊂ . Definition 20. Let M and N be two C ∞ -manifolds and ( )m m f m df : T M T N→ , C ∞ -map. The differential is defined by ( ) ( ) ( )m ˆdf X X ofg g= , where g is a function defined in a neighborhood of ( )f m representing ĝ . It can be proved that the definition is independent of the choice of a representative of ĝ . If we put N =  in Definition 20, then we get the differential of a function which was already defined in Definition 17. Definition 21. Let M and N be two C ∞ -manifolds and f : M N→ a C ∞ -map. (1) The map f is said to be an immersion if : ( )m m f mdf : T M T N→ is injective for all m M∈ . (2) The map f is said to be an embedding if : (a) f is an injective immersion, and (b) f is a homeomorphism between M and ( )f M , where ( )f M is endowed with the induced topology of N . Isomorphism of Vector Bundles Definition 22. Let E and F be two vector bundles over a differentiable manifold M. A map between the vector bundles E and F is a C ∞ map : E F,ν → such that: (1) ( )x xE F , for all x Mν ⊆ ∈ , and (2) xx x xE : E Fν ν= → is linear for all x M∈ . The vector bundles E and F are said to be isomorphic if there exists a map : E Fν → such that x x x: E Fν → is an ismorphism for all x M∈ . We denote this by E F≅ . A vector bundle (resp. a complex vector bundle) E over M is said to be a trivial bundle if KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE Pull-Back of Vector Bundles Definition 23. Let be a C ∞ -map between two C ∞ manifolds M and N and E a vector bundle over N . The pull-back bundle f * E is a vector bundle over M defined by ( ) ( )f mmf * E E= for all m M∈ . The transition functions of f * E are given by the pull-back of the transition functions of E . Hermitian Metrics on Vector Bundles Definition 24. A C ∞ -frame over an open subset U of M is an r -tuple ( )1 ,..., rσ σ , where such ( )U, Eiσ ∈ Γ , such that ( ) ( )( )1 x ,..., r xσ σ is a basis for xE for all x U∈ . Let U be an open subset of M such that ( )1U: E U U rϕ π ϕ−= → ×% , and let ( )1 i rei ≤ ≤ be the canonical basis of r To the trivialization ϕ , we associate the C ∞ -frame ( )1 i rei ≤ ≤% of UE , where ei% is defined by ( ) ( )1e x : x, e x Ui iϕ −= ∈% Then any section ( )s U, E∈ Γ can be written as ( ) ( ) ( ) 1 s x x e x r i i i a = = ∑ % with ( )a U,i C ∞∈ R . Conversely, to C ∞ − frame ( )1 ,..., rσ σ , ( )U, Eiσ ∈ Γ we can associate a trivialization of E over U as follows ( )( ) U 1 : E U x, a ,...., a r r ϕ → ×  a where ( )1a ,..., a r are defined as follows: ( ) ( ) 1 x a x r i i i σ σ = = ∑ , The definition of a C ∞ − frame for a complex vector bundle is obvious and is left to the reader. Definition 25. Let E be C ∞ − complex vector bundle of rank r over a C ∞ manifold M of dimension n . A Hermitian metric h on E is a family ( )x Mh x ∈ , where h x , is a Hermitian inner product on E x , such that if ( ) ( )1 i re ,i C U ∞ ≤ ≤ ∈% C is a C ∞ − frame for the complex vector bundle E defined in an open set U , then, the functions ( ) ( )( ) i, j x i j h : U x h e , e ,x x →  % %a are smooth, i.e., ( )i, jh : U, i, j 1,..., r.C for all ∞∈ =C BOUDJEMAA ANCHOUCHE 3. Tensor and differential forms In this section we will assume some knowledge of the properties of tensor and exterior algebras. Let M be a differentiable manifold of dimension n and consider the following vector bundles ( )( )p,q m M m p,qT M:= T M ,∈t ( ) ( )( )p *m M mM : T MpA ∈= ∧t , ( ) ( )p *m M mp=0M : T MA ∞ ∈  = ⊗ ∧    t , where ( ) q-timesp-times * * m m m m mp,q T M T M .. T M T M .. T M= ⊗ ⊗ ⊗ ⊗ ⊗ 64474486447448 and ( ) p-times p * * * m m mT M T M ... T M∧ = ∧ ∧ 6447448 . Definition 26. (1) The vector bundle ( )p,qT M (resp., ( ) ( )M , M pA A ) is called the tensor bundle of type ( ),p q (resp., the exterior p-bundle, exterior algebra bundle) over M . (2) A C ∞ − section of the vector bundle ( )p,qT M (resp. ( ) ( )M , M pA A ) is called a smooth tensor field of type ( ),p q (resp. a smooth differential p − form, a smooth differential form) on M . The set of all smooth sections of ( )p,q MT (resp., ( ) ( )M , M pA A ) will be denoted by ( )p,q MT (resp., ( ) ( )M , Mpε ε ). If ( )1 xx ,..., n is a local coordinate system defined in an open subset U of M , with m U∈ , n 1 xi i =  ∂   ∂  (resp. { }n 1dxi i = ) is a C ∞ − frame for TM (resp. * mT M ) over U . Therefore we get C ∞ − frames for ( ) ( )p,TM , T*Mp q ∧ and ( ) p p 0 T * M ∞ =⊗ ∧ over U . For example, a differential p − form ω in U will be written ( ) ( ) 1 1 1 i 1 i ... i n x ,..., i x dx ... x , p p p i ia dω ≤ < < ≤ = ∧ ∧∑ where 1i , ..., i pa are C ∞ − functions in U . Definition 27. Let (resp. β ) be a p − form (resp. q − form). The exterior product of α and β , denoted by α β∧ , is a ( )p q+ -form defined by ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) p+q 1 p+q 1 p p 1 p+q 1 ,..., : ,..., ,..., pq s σ σ σ σ σ α β ξ ξ ε σ α ξ ξ β ξ ξ+ ∈ ∧ = ∑ , where 1 p+q ,...,ξ ξ are p q+ vector fields, p+q S is the group of permutations of the set { }1,..., p q+ and ( )ε σ is the signature of the permutation σ . KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE The exterior product is associative and satisfies ( ) ( ) ( )pq p q1 for M and Mα β β α α ε β ε∧ = − ∧ ∈ ∈ . Theorem 28. There exists a unique operator ( ) ( )d : M M ,ε ε→ such that (1) ( ) ( ) p p+1M Mdε ε⊂ . (2) ( ) ( ) ( ) ( ) p pd d 1 d , M , Mα β α β α β α ε β ε∧ = ∧ + − ∧ ∈ ∈ . (3) d d 0=o (4) If f C ∞∈ , then df is the differential of f . If ( ) p Mα ε∈ , then ( ) p 1d Mα ε +∈ is defined by ( ) ( ) ( ) ( ) ( ) p 1 1 p+1 p+1 i 1 i+jp 1 1 p+1 1 j p 1 ˆ,..., -1 ,..., , ˆ ˆ1 , , ,..., ,..., ,..., i i i i j i j dα ξ ξ ξ α ξ ξ ξ α ξ ξ ξ ξ ξ ξ + = + ≤ ≤ +  = +    −   ∑ ∑ where 1 p+1 ˆ , ...,ξ ξ are vector fields and îξ means that the vector field iξ is omitted. 4. Riemannian manifolds Definition 29. A Riemannian metric on a C ∞ differentiable manifold M is a tensor field g of type ( )0, 2 , i.e., ( )0,g 2 M∈ T , such that at each mm M, g∈ is a positive definite symmetric bilinear form, i. e., ( ) { }mgm X, X 0, X T M 0 ,for all> ∈  ( ) ( ) mgm X,Y gm Y,X X,Y T Mfor all= ∈ Definition 30. A Riemannian manifold is a pair ( )M,g , where M is a C ∞ differentiable manifold and g is a Riemannian metric. Let X and Y be two elements of mT M , and let ( )1 nx ,...,x be a local coordinate system defined in an open subset U of M , with m U∈ , and suppose that n i=1 X x mi ξ ∂ = ∂ ∑ and n i=1 Y x mi i ξ ∂ = ∂ ∑ Then ( ) ( ) n ij i,j=1 gm X,Y g m ,i jξ ζ= ∑ where ( )i,j mg m g ,x m x mi j  ∂ ∂ =   ∂ ∂  We write n i,j i,j=1 g g dx xi jd= ⊗∑ BOUDJEMAA ANCHOUCHE Let ( )M, g be a Riemannian manifold. Using the metric g , it is possible to compute the length of any curve on M joining any two points x and y of M . Definition 31. Let ( )M, g and ( )N, h be two Riemannian manifolds. A C ∞ − map f : M N→ , is said to be an isometry if f is a C ∞ -diffeomorphism and if for every m M∈ , and for every m, T Mξ η ∈ , we have ( ) ( ) ( )( ) ( )m mf mh df , df gm ,ξ η ξ η= . We write g f * h= Example 32. (1) Since the tangent bundle of n is trivial, i. e., n n nT ≅ ×  , the Euclidean metric on n denoted by ecg is defined by ( ) ( ) n 1 1 1 1m ,v , m ,v T∈  ( ) ( )( )ec 1 1 1 2 1 2g m ,v . m ,v v .v .= (2) Consider the unit sphere ( ){ }n n+1 2 21 n+1 1 n 1S x ,..., x x ... x 1+= ∈ + + = . The Euclidean metric on n+1 induces a metric on nS , i.e., if n n 1S i +→  is the obvious embedding of nS in n 1+ , then Sn ecg i * g= is the "canonical" metric on nS , where ecg is the Euclidean metric on n 1+ . (3) Let ( ){ }2n n 2 21 n 1 nD x x ,..., x x x ... x 1 ,= = ∈ = + + < be the unit disc. Since n n nTD D ,≅ ×  then a metric nD g on nD can be defined as follows ( ) ( )( ) ( ) D, x 22 g x, , x, : 1 x ξ η ξ η ⋅ = − (4) Every compact connected Lie group admits a bi-invariant metric, i.e., a metric for which left and right translations are isometries. For details see (Gallot and Hulin, 1990). Partition of unity The aim of this section is to introduce a very powerful tool called" partition of unity" which allows one to construct global objects such as metrics, differential forms, vector fields,..., by gluing local ones. Partition of unity plays also a fundamental role in the definition of integration on manifolds. Definition 33. Let M be a C ∞ -manifold and let ( )i IUi ∈ be a covering of M . A C ∞ - partition of unity subordinate to the covering ( )i IUi ∈ is a collection of maps ( )i Iiθ ∈ , where ( )M,i Cθ ∞∈  satisfying: (1) supp ( ) Ui iθ ⊆ for all i I∈ , where supp ( ) ( ){ }i xx M 0 .iϕ ϕ= ∈ ≠ (2) ( ) [ ]U 0,1 i Ii i for allθ ⊆ ∈ KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE (3) For every x M∈ , there exists an open subset x xV , x V∈ such that xi V 0θ ≡ except for a finite set of iθ . (4) For every ( )i Ix M , x 1iθ∈∈ =∑ ((3). implies that the sum is finite). Definition 34. A Hausdorff space M is said to be paracompact if every open covering { } IUi i∈ of M there exists an open covering { }j j JV ∈ of M such that: (1) For every j J∈ , there exists j i I∈ such that ji V Uj ⊂ (we say that { }j j JV ∈ is a refinement of { }i i IU ∈ ). (2) Each point m M∈ has a neighborhood mW such that m j W V φ∩ ≠ only for finitely many values j of J (we say that { }j j JV ∈ is locally finite). Paracompactness is a generalization of compactness (every Hausdorff compact topological space is paracompact). Theorem 35. Let M be a paracompact C ∞ -manifold and let ( )i i IU ∈ be a covering of M . Then there exists a C ∞ - partition of unity subordinated to ( )i i IU ∈ . Theorem 36. There exists at least one Riemannian metric on any paracompact C ∞ -manifold. The construction goes as follows: Let ( )i I,i iU ϕ ∈ be an atlas of M and let ( )i Iiθ ∈ be a partition of unity subordinate to the covering ( )i i IU ∈ . Fix a scalar product ( )Q .,. on n where dim Mn = . Then it is easy to see that ( ) i I * i ig θ ϕ ∈  =  ∑ Q is a metric on M . For more details, see (Aubin, 2001). 5. The levi-Civita connection Definition 37. Let M be a differentiable manifold. A Linear connection ∇ on M is a map ( ): TM TM TM∇ ×  → ( ), Y Yξξ ∇a , satisfying the following conditions (1) If mT Mξ ∈ and ( )Y ∈Γ ΤΜ , then mY T Mξ∇ ∈ . (2) The restriction of ∇ to ( )m MΤ ×  ΤΜ is bilinear. (3) ( ) ( ) ( )mfY = f Y f m Yξ ξξ∇ ⋅ + ∇ for all ( )mT M, Y TMξ ∈ ∈  and for all f differentiable function on M . (4) If ( )X TM∈ Γ and ( )Y TM∈Γ are such that X is of class rC and Y is of class 1rC + , then ( )x Y TM∇ ∈ Γ is of class rC . Definition 38. (1) The torsion of a linear connection ∇ is a map T defined by ( ) ( ) ( )T : TM TM TMΓ × Γ → Γ ( ) ( ) [ ]x YX,Y T X,Y Y- X- X,Y= ∇ ∇a BOUDJEMAA ANCHOUCHE where [ ]X,Y is the Lie bmcket of the vector fields X and Y . (2) The curvature R of a linear connection ∇ is a two form with values in Hom ( ) ( )( )TM , TMΓ Γ , defined by ( ) ( ) ( ) ( )( )R : TM TM Hom TM TMΓ × Γ → Γ , Γ ( ) ( ) [ ]x Y Y x x,YX,Y R X,Y = ∇ ∇ − ∇ ∇ − ∇a For simplicity we write ( ) [ ] [ ]x Yx Y ,R X,Y , ∇ ∇= ∇ ∇ − ∇ . Remark 39. One can easily check that: (1) The curvature R is a (1, 3) tensor. (2) ( ) ( )R X,Y R X,Y= − . (3) If T 0= , then ( ) ( ) ( )R X , Y R Y, Z X R Z , X Y 0Z + + = (Bianchi's Identity). (4) The value of ( )R X , Y Z at a point m M∈ depends only on the values X , Y and Z at m . Theorem 40. Let ( )M , g be a Riemannian manifold. Then there exists a unique linear connection which is torsion free and compatible with the metric, i.e., a connection g∇ satisfying the following two conditions: (1) ( )T X , Y 0= for all ( )X , Y T M∈ Γ . (2) ( ) ( ) ( )g gx xX g Y, Z g Y, Z g X, Z⋅ = ∇ + ∇ for all ( )X,Y,Z TM∈ Γ . where T is the torsion of the connection g∇ . The connection g∇ defined above is called the Levi-civita a connection of the metric g and will simply be denoted by ∇ . For a proof, see (Aubin, 2001). The vector field x Y∇ is called the covariant derivative of the vector field Y in the direction of the vector field X . The covariant differentiation can be extended to tensors of type ( )p,q , see (Kobayashi and Nomizu, 1969) . In a local coordinate system ( )1 nx ,..., x defined in an open subset U of M , we write i n k ij k=1j kx x x ∂ ∂ ∂ ∂ ∇ = Γ ∂ ∂ ∑ . Definition 41. The functions k ijΓ defined in U are called the ChristoffeL SymboLs of the Levi-Cevita connection ∇ . Easy computations in a local coordinate system ( )1 nx ,..., x defined in an open subset U of M show that the Christoffel symbols can be expressed in terms of the metric as follows nk k ij ijj i i j 1 g g g g 2 x x x=1      ∂ ∂ ∂ Γ = + −   ∂ ∂ ∂  ∑ where ( )k 1 k, n g   ≤ ≤ is the inverse of the matrix ( )ij 1 i,j ng ≤ ≤ . KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE 6. Sectional and ricci curvatures Curvature is a fundamental and central concept in Riemannian geometry. In this section we will introduce two types of curvatures, namely the sectional curvature and the Ricci curvature. Let ( )M , g be a Riemannian manifold and R its curvature. Using the metric g , we can view R as a ( )0, 4 tensor (denoted also by R ) as follows ( ) ( )( )R X,Y,Z,W : g R X,Y Z,W= . The properties of R can be summarized in the following Proposition 42. The Riemannian curvature R satisfies the following properties (1) ( ) ( ) ( ) ( )R X,Y,Z,W R Y,X,Z,W R X,Y,Z,W R Y,X,W,Z= − = = . (2) ( ) ( )R X,Y,Z,W R Z,W,X,Y= . Let ( )m2,T MG be the Grassmannian of 2-dimensional vector subspaces of mT M and let ( ) ( )m M m2,TM U 2,T M∈=G G be the corresponding Grassman bundle. Let mu be a two dimensional vector subspace of mT M with basis ( ),ξ η . Then it can be shown that the real number ( ) ( )( ) ( ) ( ) ( ) m m 2 m m m g R , , g , g , g , ξ η ξ η ξ ξ η η ξ η = − T u is independent of the choice of the basis ( ),ξ η of mu . Definition 43. Let ( )M , g be a Riemannian manifold. The sectional curvature of ( )M , g at m M∈ , denoted by ( )sect m , is the function ( ) ( ) ( ) ( ) ( ) m m m m sect m : 2, T M sect m → =  a G u u T u Remark 44. If ,ξ η are two linearly independent vectors in mT M , then we put ( ) ( ) ( )msect m ,ξ η = T u . where mu is the plane spanned by the vectors ,ξ η . We will say that ( )sect resp. , , cc≥ ≤ > < if ( ) ( ) ( ) ( ) ( ) ( )( )m msect m , cg , resp.sect m , , , cg , ,ξ ξ ξ ξ ξ ξ ξ ξ≥ ≤ > < for all vectors, mT Mξ ∈ and for all m M∈ . Definition 45. Let ( )M,g be a Riemannian manifold. For each m M∈ , the Ricci curvature tensor of ( )M,g at m , denoted by ( )gRi mc is defined by ( ) ( ) ( ) g m mRi m : T M T M , Ric , c ξ η ξ η × →  a BOUDJEMAA ANCHOUCHE where ( ) ( )( ) ( )ni=1 Ric , Trace Ric , Ric , , e ,i ie ξ η ζ ξ ζ η ξ η = = ∑ a and ( )1i i ne ≤ ≤ is a mg -orthonormal basis of mT M . As defined, g Ric is a bilinear form which will sometimes simply be denoted by Ric if the metric is clear from the context, and ( )gRi mc will simply be denoted by mRic . As in the case of sectional curvature, we will say that ( )Ri . , , cc c resp≥ ≤ > < if ( ) ( ) ( ) ( )( )m m m mRic , cg , resp. Ric , , , cg , ,ξ ξ ξ ξ ξ ξ ξ ξ≥ ≤ > < for all vectors mT Mξ ∈ and for all m M∈ . 7. A volume comparison theorem The positivity of the Ricci curvature imposes strong topological and geometric constraints on Riemannian Manifolds as shown in Theorem 46, Theorem 57 and Theorem 58 below. Theorem 46. ( Myers, 1941) Let ( )M,g be a complete Riemannian manifold and suppose that gRic ,c≥ where c is a positive constant. Then M is compact and its fundamental group is .finite. Remark 47. Negative Ricci curvature has no topological implications on Riemannian manifolds of dimension 3≥ as is shown by the following result. Theorem 48. (Lohkamp, 1994) Any manifold of dimension 3≥ admits a metric with negative Ricci curvature. Definition 49. Let ( )M,g be a Riemannian manifold and let ( ): a,b Mγ → , be a differentiable map (called a smooth curve). The length of the curve denoted by ( )ι γ is given by ( ) ( ) ( ) ( )( )tg t , t dt , b a γ ι γ γ γ= ∫ & & where ( ) ( ) ( ) t d d t t d , t a,b , dt dt γ γ γ    = = ∈      & , dγ is the differential of the map γ and d dt is the unit vector on  . The length of a piecewise smooth curve is the sum of the lengths of its smooth pieces. Definition 50. (1) The distance between two points x and y , denoted by ( )gd x , y , is the infimum of the lengths (With respect to the metric g ) of all piecewise 1C -curves from x to y . KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE (2) The metric g induces a distance function g d on M , giving a metric space ( )gM , d . We say that the metric g is complete if ( )gM , d is a complete metric space. Remark 51. It can be proved that the topology of M induced by the metric gd is the same as the original topology of M . Definition 52. The geodesic ball ( )gB m , r centered at m and with radius r is defined by ( ) ( ){ }g gB m , r x M d m , x r= ∈ < , where g d is the distance function induced by the Riemannian metric. Integration of differential forms on manifolds Let M be an C ∞ -manifold of dimension n and let ϖ be a smooth n -form defined on M . The aim of this section is to give a meaning to the following expression M ρϖ∫ , where ρ is a compactly supported function on M , i.e., supp ( ) Kρ ⊆ , where K is a compact subset of M . Let us start first with the following Definition 53. Let M and N be two C ∞ -manifolds and f : M N→ a C ∞ -map. If Φ is a smooth n -form on N , then f * Φ is the smooth N -form defined on M by ( ) ( ) ( ) ( ) ( )( )1 n m 1 m nf mmf * ,..., : df , ..., df ,ξ ξ ξ ξΦ = Φ , for all m M∈ and for all 1 n m,..., T Mξ ξ ∈ Let M, ρ and ϖ w be as above. Suppose first that supp ( ) iUρ ⊆ , where ( )i iU ,ϕ is a local chart and ( )1 nx ,...,x are the corresponding local coordinates. Then ( ) ( )1i 1 n 1 n* x ,..., x dx ... dx ,iϕ ϖ ϑ− = ∧ ∧ where iϑ is a smooth function defined in ( )i iUϕ with real values. We put (1) ( ) ( )i i 1 iM U * ϕ ρϖ ϕ ϖ−=∫ ∫ (2) ( ) ( ) ( ) ( )i i 1 i 1 n 1 n 1 nU o x ,..., x x ,..., x dx ... dxiϕ ρ ϕ ϑ − ∧ ∧∫ We have to prove that the definition is independent of the choice of the local chart containing the support of ρ . For this, suppose that ( )j jU ,ϕ is a local chart and ( )1 ny ,..., y are the corresponding local coordinates with supp ( ) jUρ ⊆ , and put (3) ( )( )j j 1 jM Uϕ ρϖ ϕ ϖ−=∫ ∫ (4) ( )( ) ( ) ( )j 1 j 1 n j 1 n 1 nU o y ,....,y y ,...,y dy ... dy ϕ ρ ϕ ϑ− ∧ ∧∫ where j ϑ is defined by ( ) ( )1j j 1 n 1 n* y ,...,y dy ... dyϕ ϖ ϑ− = ∧ ∧ BOUDJEMAA ANCHOUCHE In general the right hand side of the expressions (1) and (3) are not necessarily equal, but if we restrict ourselves to "orientable manifolds" then we can choose an atlas of local charts such that (1) and (3) are equal. To be more precise, let us introduce the following Definition 54. (1) A C ∞ -manifold M is said to be orientable if there exists a C ∞ -atlas ( )i i i IU ,ϕ ∈ such that the Jacobian Jac ( ) 1 i joϕ ϕ − of the C ∞ -diffeomorphism ( ) ( ) ( )1i j j i j i i jo : U U U Uϕ ϕ ϕ ϕ− ∩ → ∩ is positive for all i , j I∈ , i.e., ( ) ( )( ) 1 1 1 n 1 i j j n n 1 n y y x x Jac o x : det 0 , y y x x ϕ ϕ ϕ− ∂ ∂  ⋅ ⋅ ⋅ ∂ ∂   ⋅ ⋅ ⋅ ⋅ ⋅   = >⋅ ⋅ ⋅ ⋅ ⋅   ⋅ ⋅ ⋅ ⋅ ⋅   ∂ ∂ ⋅ ⋅ ⋅ ∂ ∂  for all i , j I∈ and for all i jx U U∈ ∩ , where ( ) ( )( )1 n 1 nx ,..., x . y ,..., yresp is the system of local coordinates corresponding to the chart ( ) ( )( )j j i iU , . U ,respϕ ϕ . (2) The manifold M is said to be oriented if such an atlas has been chosen. (3) A chart ( )U,ϕ is said to be compatible with the orientation if ( )1iJac o 0, ifor allϕ ϕ − > . It can be shown that Theorem 55. A C ∞ -man(fold M of dimension n is orientable if and only (f there exists a differential n -form ω such that m 0ω ≠ for all m M∈ . If M is oriented and the atlas ( )i IU ,i iϕ ∈ is compatible with the orientation, then ( ) ( ) ( )( )i i j j 1 1 i jU U * * , ϕ ϕ ϕ ϖ ϕ ϖ− −=∫ ∫ i.e., the integral M ρϖ∫ is independent of the choice of the local chart. Suppose now that ρ is a smooth function defined on M with compact support. Then we define M ρϖ∫ as follows iM M i I Xρϖ ρϖ ∈ = ∑∫ ∫ (5) where ( )i i IX ∈ is a partition of unity subordinate to an atlas ( )i IU ,i iϕ ∈ which is compatible with the orientation. The right hand side of (5) is a finite sum and it can be proved that it is independent of the choice of the partition of unity. KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE Integration over riemannian manifolds. Let ( )M, g be a Riemannian manifold of dimension n , let ( ) ( )( )i 1 i nU, U, xϕ ≤ ≤= be a local chart, and suppose that f is a measurable function on M with compact support included in U . Then the integral gM fdV∫ is defined as follows ( )( ) ( ) ( )( )( )1 1g ij 1 nM MfdV f x det g x dx ...dxϕ ϕ ϕ − −=∫ ∫ If f is a measurable function with compact support in M , then we define gM fdV∫ as follows i g i gM M fdV X fdV ∈ = ∑∫ ∫ where ( ) i I iX ∈ is a partition of unity subordinate to an atlas ( ) i I U ,i iϕ ∈ .It can be proved that the sum is well defined, i.e., only a finite number of terms in the sum are nonzero, the sum is independent of the choice of the atlas and of the partition of unity subordinated to it. Suppose now that M is oriented and let ( ) i I U ,i iϕ ∈ =A be an atlas which is compatible with the orientation. Let ( )U,ϕ be a local chart belonging to A and let ( )1 nx ..., x be the local coordinate system associated to ϕ and consider the following n -form: ( )( )x ij x1 ndet g x d ... dx ,ϖ = ∧ ∧ where det ( )ijg is the determinant of the metric n ij i ji=1 g g dx dx= ⊗∑ . Consider another local chart ( ),V υ ∈/ A and let ( )1 ny ,..., y the corresponding local coordinates, and put ( )( )y ij 1 ndet g y dy ... dyϖ = ∧ ∧ An easy computation shows that x y in U Vϖ ϖ= ∩ . Therefore ϖ is a global n -form, called the volume form associated to the Riemannian metric g and it will be denoted by g dV . Definition 56. The volume of the geodesic ball ( )gB m , r denoted by ( )( )g gvol B m , r is given by ( )( ) ( )g g g B m,r vol B m , r : dVg= ∫ The following two results are very important and will be needed in part 4. Theorem 57. (Bishop and Crittenden, 1964) Let ( )M , g be a complete noncompact Riemannian manifold of positive Ricci curvature, dim M n=R . Then ( )( ) ng gvol B m , r cr r 0 ,for all≤ > where c is a positive constant independent of r . There is a generalization of Bishop's Theorem by Gromov, but for our purposes, Bishop's Theorem is enough. The proof of Bishop's (resp. Gromov's) Theorem (Theorem 57) uses special local coordinates coming from the exponential map and some properties of Jacobi fields, for more details, see (Gallot and Hulin, 1990). BOUDJEMAA ANCHOUCHE Theorem 58. (Calabi, Yau, see [14]) Let ( )M,g be a complete noncompact Riemannian manifold of nonnegative Ricci curvature. Then ( )( )g gvol B m,r cr r 0for all≥ > , large enough. de Rham isomorphism Definition 59. (1) A p - form w is called closed, or d - closed, if d 0ω = . (2) A p -form ω is called exact if dω β= for some ( )p 1− -form β . Since 2d 0= , every exact form is closed. Hence we see that the vector space of exact p -forms is a subspace of the space of closed p-forms. Definition 60. The thp de Rham cohomology group of M , denoted by ( ) p DRH M , is given by ( ) { } { }pDRH M p / pclosed forms exact forms= − − de Rham's Theorem says that ( ) p DRH M is isomorphic to the thp singular cohomology5 of M . More precisely consider the following map ( ) ( ) [ ] [ ] [ ]( ) *p DR p Z X : H M H M , Zω ω ω → = ∫  a where ( )pH M ,  is the singular thp homology group with real coefficients, ( ) * pH M,  its dual, W is a representative of its de Rham cohomology class [ ]ω , and Z is a thp cycle representing its real differentiable singular homology class [ ]Z . The fact that the map X is well defined is an easy consequence of Stokes theorem. For more details, see (Warner, 1983). Theorem 61. (de Rham) The map X is an isomorphism. If M is a compact oriented manifold, then Poincare duality implies that ( ) ( )* ppH M , H M ,≅  Combining de Rham isomorphism and Poincare isomorphism, we obtain ( ) ( )* pp DRH M, H M≅ Definition 62. Let M be a compact manifold. The nonnegative integer ( )ib M defined by ( ) ( )ii DRb M dim H M ,=  is called the i th− Betti number of the manifold M . Holonomy groups The aim of this section is to introduce the "holonomy group" of a Riemannian manifold and to state two of the fundamental results in Riemannian Geometry, namely de Rham's and Berger's Theorems. The material of this section will not be used elsewhere in this paper, so it may be skipped if the reader whishes to do so. To any Riemannian manifold ( )M,g of dimension n n, we can associate a closed subgroup of ( )SO n , called the holonomy group of the metric; lots of information about a Riemannian metric is encoded in its holonomy group. Let us first define the holonomy group. 5 For more details on singular (co)-homology, the readers can consult [20]. KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE Let ( ): a,b Mγ → be a smooth curve. The tangent vector d dt γ to the curve γ is defined by ( ) 00 0 tt d d , t a,b dtdt d γ γ    = ∈      , where dγ is the differential of the map γ and d dt is the unit vector on  . For a function f defined in a neighborhood of' ( )0tγ , we get ( ) ( ) ( ) 0t 0 d fd f t dt dt γγ   =     o Let ( )M , g be a Riemannian manifold of dimension n and let ∇ be the Levi-Cevita on M . Definition 63. (1) A vector field X along a curve ( ): a,b TMγ → is a map ( )X : a , b TM→ such that ( ) ( ) ( )tX t T M t a,bfor allγ∈ ∈ . (2) A vector field X is said to be parallel along a differentiable curve γ if d dt X 0.γ∇ = Let ( ) n i 1 n i=1 i X X ,..., , x and γ γ γ ∂ = = ∂ ∑ be the expression of the vector field X and the curve γ in a local chart ( )( )1 nU, x ,..., x . Then ( ) ( )( ) ( ) ( ) j in n j k d ik j 1 i , k 1 jdt dX t d t X 0 t X t 0 dt dt xγ γ γ = =   ∂ ∇ = ⇔ +  =  ∂   ∑ ∑ . (6) Let 1m and 2m be two points of M and let [ ]: a , b Mγ → be a continuous curve which is smooth in ( )a , b , and joining 1m and 2m , i.e., ( ) 1a mγ = and ( ) 2b mγ = and let 1X be a vector in m1T M . Since the equation (6) is linear, by Cauchy's theorem, it has a unique solution ( )X t for all [ ]t a , b∈ satisfying ( ) 1X a X= . Definition 64. The vector ( )2X X b= is called the parallel transport of 1X from 1m to 2m along the curve γ . The parallel transport along a piecewise smooth curve γ is defined in an obvious way. Then to each piecewise curve γ is associated an isometry (with respect to the scalar product on m1T M and m2T M induced by g ) 1 2m m : T M T Mγ →T This map is both linear and invertible and so defines an element of ( )mGL T M . Let m M∈ and consider ( ) [ ] ( ) ( ){ }piecewise and smooth,o m : a,b Ml a b mγγ γ γ= → = = . BOUDJEMAA ANCHOUCHE It can be shown that ( )o ml is a group under composition of paths. To a composition 1 2γ γo , we associate the linear transformation 1 2 1 2 m m o : T M T Ml l lγ γ γ γ= →o Definition 65. The holonomy of ( ) g or g∇ based at m is defined as ( ) ( ) ( ){ }gm mh o l l GL T M o mlγ γ∇ = ∈ ∈ The holonomy ( ) g mh o l ∇ is a subgroup of the orthogonal group ( )O n , moreover, it is a Lie group. If ( ) 1 g mh o l ∇ and ( )2 g mh o l ∇ are the holonomy groups of the connection g∇ at the points 1 2m , m , then there exists an element ( )A O n∈ such that ( ) ( ) 2 1 g g 1 m mh o l Ah o l A −∇ = ∇ This allows us to speak of "the" holorromy group of a Riemannian manifold ( )M , g and denote it simply by ( )gh o l ∇ . If we consider in the definition of the holonomy group only contractible closed curves, then we obtain what is called the restricted holonomy group, and it can be proved that the two coincide in case the manifold M is simply connected. Remark 66. The following set ( ) ( ) ( ){ }0 gm mh o l l GL T M l o m , is a contractible curveγ γ∇ = ∈ ∈ is a connected normal subgroup of ( ) g mh o l ∇ , and is called the restricted holonomy group. Definition 67. (1) The representations of ( ) g mh o l ∇ in mT M are isomorphic and therefore called the holonomy representation. (2) A Riemannian manifold ( )M , g is irreducible if its holonomy representation is irreducible. Theorem 68. (de Rham 1952) Let ( )M , g be a connected, simply connected complete Riemannian manifold. Then: (1) there exists a canonical decomposition ( ) ( ) ( ) ( ) isometric 0 0 1 1 k kM , g M , g M , g ... M , g≅ × × × where ( )0 0M , g is a Euclidean space (possibly reduced to a point) and ( )i iM , g , i 1,...,k= are irreducible simply connected complete Riemannian manifolds. (2) For ( )1 km m ,...,m M= ∈ , let ( ) ( )i g m m ih o l O T Mi ∇ ⊂ be the holonomy group of iM at im and let ( )gmh o l ∇ be then the holonomy of ( )M , g at m . Then ( ) ( ) ( ) ( )0 1 k 2 0 1 k g g gg m m m mh o l h o l h o l ... h o l ,∇ ≅ ∇ × ∇ × × ∇ where the action of ( ) ( ) ( ) 0 1 k 0 1 k g g g m m mh o l h o l ... h o l∇ × ∇ × × ∇ on 0 1 km m 0 m 1 m k T M T M T M ... T M ,≅ × × × is through the product representation. Such a decomposition is unique up to an order. KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE The symmetric spaces were classified by E . Cartan around 1920 and their holonomy groups are well understood. For example, a compact symmetric space is a homogeneous space G/H , where G is a compact Lie group and H is the identity component of the fixed locus of an involution of G and the holonomy G/H is H 6. So we can exclude symmetric spaces from our study of the holonomy of Riemannian manifolds. Theorem 69. (Berger, 1953) Let ( )M , g be an irreducible simply connected Riemannian manifold which is not isomorphic to a symmetric space. Then the holonomy ( ) gh o l ∇ of ( )M , g is one of the following groups ( )gh o l ∇ dim M Type of the metric SO (n) n Generic Riemannian U (r) 2r Kähler SU (r) , r 3≥ 2r Calabi-Yau Sp (r) 4r Hyper Kähler Sp (r) Sp (1) 4r Quaternion - Kähler 2G 7 Ricci Flat Spin(7) 8 Ricci Flat Remark 70. (1) Initially it was thought that there is no manifold with holonomy 2G or Spin (7), but recently, D. Joyce was able to construct a compact 7-dimensional manifold with holonomy 2G (see [28] ) , and a compact 8- dimensional manifold with holonomy Spin (7), (see Joyce, 1996). (2) Since ( ) ( ) ( ) ( )Sp m SU 2m U 2m SO 4m⊂ ⊂ ⊂ we deduce that every hyperkahler manifold is a Calabi- Yau manifold, every Calabi- Yau manifold is a Kähler manifold, and every Kähler manifold is orientable. Kähler manifolds will be introduced in Part 2, but we will not study the other types of manifolds in this paper. (3) It is very well known that the groups appearing in Berger's list (Theorem 69) are connected to the three division algebras  (real numbers), c (complex numbers), H (quaternions). The connection is stated as follows - ( )SO n is a group of automorphism of n . - ( )U n and ( )SU n are groups of automorphisms of nc . - ( )Sp n and ( ) ( )Sp Sp 1n are groups of automorphisms of nH , where H is the space of quaternions. - 2G and Spin (7) can also be realized as automorphism groups of some other structures. Kähler manifolds will be the subject of Part 2. As for the other classes of manifolds, excellent accounts can be found in (Salamon, 1989, Joyce, 2000, Beauville, 2006, Bryant, ). Part 2. Kähler manifolds We have seen in the previous part that if the holomony ( ) ghol ∇ of a simply connected irreducible complete Riemannian manifold ( )M,g of dimension 2n is icluded in ( )U n , then the manifold is "Kähler". 6 For more details on symmetric spaces, see [23]. BOUDJEMAA ANCHOUCHE The aim in this part is to give a short introduction to" Kähler manifolds". Since we mentioned the Hodge decomposition theorem without proof, skipped some other fundamental results in Klihler geometry, and since most of the proofs are either sketchy or missing, we suggest the following references (Kobayashi and nomizu, 1969, Griffiths and Harris, 1978 and Demailly, ) for more detail. 9. Holomorphic functions Definition 71. Let Ω be a nonempty connected open subset of nC . A complex valued function f on Ω is called holomorphic if for each ( ) 0 0 0 1 n, ...,z z z= ∈ Ω , there exists a neighborhood U and a power series ( ) ( )1 n 1 n 1 n 0 0 0 ,..., 1 1 n n ,..., c z z ... z z α α α α α α ≥∈ − −∑  that converges to ( )f z for z U∈ , where 1 n,...,cα α are complex numbers and 0≥ is the set of nonnegative integers. We denote by ( )H Ω the set of holomorphic functions on Ω . Let ( )1 nz z ,..., z= be the coordinate system of n . Write 2 j-1 2 jz x 1xj = + − and identify n with 2 n via the correspondence ( ) ( )1 n 1 2 2nz ,..., z x ,x ,...,xa , and consider the following notation 2 1 2 2 1 2 1 1 : 1 , : 1 z 2 x x 2 x xz jj j j j j− −    ∂ ∂ ∂ ∂ ∂ ∂ = − − = + −      ∂ ∂ ∂ ∂ ∂∂    . (7) The following characterization of holomorphic functions is very useful Theorem 72. ( ) ( )2oc j f f L 0 j 1, 2,..., n , z lH for  ∂ Ω = ∈ Ω  = =  ∂  where ( ) 2 ocLl Ω is the set of locally square integrable functions on Ω and the derivative is taken in the sense of distributions. For a proof of Theorem 72 see (Ohsawa, 2002). Definition 73. Let Ω (resp. ′Ω ) be a nonempty connected open subset of n (resp. m ). (1) A map ( ) m1 k,..., :ϕ ϕ ϕ= Ω →  is called holomorphic if each j :ϕ Ω →  is holomorphic, 1 j m≤ ≤ . If ( )ϕ ′Ω ⊆ Ω , then we write ( )1 k,..., :ϕ ϕ ϕ ′= Ω → Ω Suppose that m n= . A holomorphic map : ′ψ Ω → Ω is called a biholomorphism if there is a holomorphic map :φ ′Ω → Ω KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE such that o id o id ,and φ′Ω Ωψ ψ = Ψ = where idΩ (resp. id ′Ω ) is the identity map of Ω (resp. ′Ω ). In case ′Ω = Ω , a bihlomorphism :ψ Ω → Ω is called an automorphism of Ω . The set of automorphisms of Ω denoted by Aut ( )Ω is a group under composition of mappings. A beautiful theorem of H. Carlan says that if Ω is a bounded domain in nc , then Aut ( )Ω is a Lie group. Bedford & Dadok and Saerens and Zame proved independently that any compact Lie group can be realized as the automorphism group of a pseudoconvex domain in some nC 7 . 10. Complex manifolds If in Definition 2 we consider open subsets of n instead of open subsets of n , and biholomorphisms instead of C ∞ -diffeomorphism, then we get what we call" a complex structure" on M . Definition 74. A complex manifold of dimension n is a pair ( )M,ψ , where M is a locally Euclidean space of dimension n and Ψ is a complex structure on M. The manifold ( )M, ψ will be denoted simply by M. The pair ( ) ( )( )i i i 1 nU , or U , z ,..., zϕ , is called a local complex chart and ( )1 nz ,..., z is called a local complex coordinate system defined in iU . We have already defined in the category of C ∞ - manifolds what. we mean by a submanifold, an embedding, an immersion, a C ∞ -map between manifolds, isomorphism of vector bundles, pull back of vector bundles,...etc. With obvious modifications, we get the notions of complex submanifolds, holomorphic embeddings,...etc. For more details, the interested reader can consult (Kobayashi and Nomizu (1969) or other textbooks dealing with complex manifolds. If ( )i iU ,ϕ , is called a local complex chart and ( )1 nz ,..., z is a local complex coordinate system defined in iU , then ( ) n 2 ni iU .ϕ ⊆ ≅  Put j 2 j-1 2jz x 1x i i= + − , j 1,..., ,n= and consider the following maps ( ) ( )( ) 2 n i 1 2 n : U m x m ,..., x m , i i i →  a ñ The maps ( ) ( )1 : U U U U ,α β β α β α α β− ∩ → ∩oñ ñ ñ ñ are C ∞ - diffeomorphisms. The collection ( ) i I U ,i i ∈ ñ is a C ∞ -atlas on M. This C ∞ -structure on M is called the underlying C ∞ -structure of the complex manifold M. An example of a complex manifold, which will play a fundamental role in what follows, is given by: Example 75. The projective space nP The projective space deserves a special attention. Since it plays a very important role in what follows, we will try to describe it in some detail. As a set, the projective space is defined by BOUDJEMAA ANCHOUCHE { }( )n n+1 \ 0 / ,=  P where the equivalence  is defined by ( ) ( ) *0 n 0 nz ,..., z w ,..., w ⇔ ∃ ∈ Ch such that ( ) ( )0 n 0 nz ,..., z w ,...,w= h . Denote by π the projection { } ( ) [ ] n+1 n 0 n 0 n : \ 0 z ,..., z z ,...,z , π → a P where [ ]0 nz ,...,z is the equivalence class of ( )0 nz ,..., z . The projective space nP can be equipped with a complex atlas ( )i 0 i nU ,i ϕ ≤ ≤ as follows: Let ( ){ }n+10 nU z ,...,z z 0 , i 0,..., n,i i= ∈  ≠ =% C and consider the following map ( ) n 0 i-1 i+1 n 0 n i i : U z z z z z ,...,z ,..., , ,..., , z z z z i i i i ψ →       %  a The maps iψ are continuous and it is clear that ( ) ( ) [ ] [ ]0 n 0 n 0 n 0 nz ,..., z y ,...,y z ,..., z y ,...,yi iψ = ψ ⇔ = Let ( )U Ui iπ= % .Then the map [ ] n 0 i-1 i+1 n 0 n i i : U z z z z z ,..., z ,..., , ,..., z z z z i i i i ϕ →        a is a well defined bijective map. The maps iϕ are open, for, if V is an open subset of Ui , then ( ) ( )( )1V Vi iϕ π −= ψ is open. Therefore the map iϕ realizes a homeomorphic between Ui and n , moreover, the maps ( ) ( ) ( ) 1 j j 0 i-1 i+1 n-1 0 n-1 j j o : U U U U z z z z1 z ,...,z ,..., , , ,..., , z z z z z j i i i j i j j j ϕ ϕ ϕ ϕ− ∩ → ∩        a are biholomorphic. Thus ( )i 1 i nU ,i ϕ ≤ ≤ as defined above is a complex atlas, which gives nP the structure of a complex manifold. The projective space nP is a compact complex manifold, since ( )n 2n+1S ,π=P where ( ){ }2 22n 1 n+10 n 0 nS z ,...,z z ... z 1+ = ∈  + + = . Example 76. The following are complex manifolds: KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE (1) n . (2) Any open subset of n . (3) Let ( )r, nG be the set of all r -dimensional vector subspaces of n , called the Grassmannian manifold of r - linear subspaces of n . It can be proved that ( )r, nG is a compact complex manifold of dimension ( )r n - r . 1f r 1= , then we get the projective space nP . (4) If 1M and 2M are complex manifolds, then 1 2M M× is a complex manifold. More examples of complex manifolds will be given below. Definition 77. A complex Lie group is a complex manifold G endowed with a group structure such that the map ( ) ( ) 1 G G G g, h g, h gh , ν ν − × → =a is holomorphic. Example 78. The following are complex Lie groups: (1) The linear group ( )GL n, : is the set of all n n× invertible matrices with complex entries. (2) Special linear group: ( ) ( ){ }SL n, A GL n, det A 1= ∈  =  . (3) The special orthogonal group ( ) ( ){ }nSO n, A SL n, A A I= ∈  =  T (4) The complex symplectic group: ( ) ( ){ }Sp n, A GL 2n, A JA J ,= ∈  =  T where n n 0 I J , I 0 −  =     and nI is the identity matrix of ( )GL n, . For more details, the interested reader can consult (Warner,1983, Knapp,2001 or Helgason,1970). Quotient Manifolds A very important procedure to construct new complex manifolds from known ones is given as follows. Let G be a subgroup of the group of automorphisms of a complex manifold M. It is easy to see that the relation  defined in M by x y if there exists an element g G∈ such that ( )y g x= is an equivalence relation. The set of equivalence classes is denoted by M / G .In general, the quotient space M / G of a complex manifold M is not necessarily a complex manifold. But by imposing some constraints on G, the complex structure on M will induce a complex structure on the quotient M / G . Definition79. (1) A subgroup G of the automorphism group of a complex manifold M is said to be properly discontinuous if for every compact sets 1K and 2K of M, ( ){ }1 2g G g K K ,φ∈  ∩ ≠ < + ∞# i.e., the number of elements g G∈ such that ( )1 2g K K φ∩ ≠ is finite. BOUDJEMAA ANCHOUCHE (2) The subgroup G is said to be fixed point free if no element of { }MG \ id has a .fixed point, where MId is the identity automorphism of M. Theorem 80. Let G be a subgroup of the group of automorphisms of a complex manifold M. If G is fixed point free and properly discontinuous, then the quotient space M / G has a canonical structure of a complex manifold induced from that of M. Example 81. Torus. Consider the vector space n (which has the structure of a complex Lie group) and take 2n vectors ( )1 ,..., , j 1,..., 2nj j j nυ υ υ= = which are  -linearly independent. The vectors { }j 1 j 2n υ ≤ ≤ generate a discrete subgroup 2 n j j j j 1 G r r ,υ =   =  ∈    ∑  of n . Then it can be shown that n n / G= T is a compact complex manifold (actually a compact complex commutative Lie group) called a Torus. The vectors { } 1 j 2nj υ ≤ ≤ are called the periods of nT and the matrix 1 n 1 1 1 1 2n 2 n ... . ... . . ... . . ... . ... υ υ υ υ        Λ =         is called the period matrix. Example 82. Hopf Manifold. Let 1 n,...,ϕ ϕ an be n complex numbers such that j 1α > for j 1,..., ,n= and let G g= be the cyclic group generated by the automorphism ( ){ } ( ){ } ( ) ( ) n n 1 n 1 1 n n g : \ 0,..., 0 \ 0, ..., 0 z ,...,z z ,..., z .α α →  a The group G is fixed point free and properly discontinuous. Therefore the quotient space ( ){ }( )1 n n,...,M \ 0,..., 0 / Gα α =  , called the Hopf manifold, is a compact complex manifold. It can be proved that 1 n,..., Mα α is diffeomorphic to 2n-1 1S S× , where jS is the j -dimensional sphere. Example 83. Iwasawa manifold. Let M be the subgroup of ( )3GL  defined by ( ) 1 3 3 2 1 2 3 1 z z M 0 1 z z ,z ,z . 0 0 1      =  ∈          KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE Denote by G the discrete group of matrices with entries in the ring of Gaussian integers [ ] { }i a bi a,b= +  ∈  , i.e., ( ) [ ]( ) 1 3 3 2 1 2 3 1 z z G 0 1 z z ,z ,z i , 0 0 1      =  ∈          It can be shown G acts by biholomorphisms on M such that the quotient M / G , known as the Iwasawa manifold, is a compact complex manifold, with ( )dim M / G 3= . Blow-Up of Manifolds A second procedure to construct new complex manifolds from given ones is the blow-up. In this section we will explain what is meant by blowing up a complex manifold at a point. Let us first start with a special case: nM = ∆ where n∆ is the unit polydisc, i.e., ( ){ }n 1 n 1 nz ,...,z 1,..., 1 ,z z∆ =  < < and consider the following set  ( ) [ ]( ){ }n n n-10 1 n 1 n i j j i: z ,...,z , w ,...,w z w - z w 0∆ = ∈ ∆ ×  =P It can be shown that n 0∆ is a complex manifold and that { }  ( ) biholomorphic n n 1 0\ 0 \ 0 ,π −∆ ≅ ∆ and ( )1 n-10 ,π − ≅ P where π is the projection n n 0:π ∆ → ∆ ( ) [ ]( ) ( )1 n 1 n 1 nz ,...,z , w ,..., w z ,...,za The complex manifold n 0∆ is called the blow-up of n∆ and ( ) 1 0E π −= is called the exceptional divisor. Consider now the general case, i.e., let M be a complex manifold, dim M n= and let m be a point of M. The "blow up" of M at m is defined as follows: Let ( )U,ϕ be a local chart, with ( )m U, m 0ϕ∈ = and ( ) nU ,ϕ = ∆ the unit polydisc in nc . Then Definition 84. The blow up of M , denoted by  mM is defined as follows  { }( ) nm 0M : M \ m ,π= ∪ ∆ where { }( ) n0M \ m π∪ ∆ means that U is replaced with n 0∆ . It can be shown that the definition of  mM is independent of the choice of the local coordinates. From the definition, we see that  mM is a complex manifold which comes equipped with a projection  m: M M ,π → such that ( ) { } biholomorphism 1 mM \ m M \ m ,π − ≅ BOUDJEMAA ANCHOUCHE and ( ) 1 n-1mπ − ≅ P is called the exceptional divisor. 11. Holomorphic vector bundles Definition 85. Let M be a complex manifold of dimension n . A holomorphic vector bundle of rank rover M is a complex manifold E together with (1) A surjective holomorphic map MEπ →: , such that ( ) 1 x xE π −= has a structure of a complex r - dimensional vector space, where r is independent of x . (2) An open covering ( )Uα α ∈ of M and biholomorphisms α' called trivializations ( )1U: U U rE α α α αϕ π −  = → ×  such that for each x Uα∈ , the map { } r rx x ,E ϕ α→ × →  is a  -linear isomorphism. Example 86. (1) If M is a complex manifold, then M r×  is a vector bundle of rank r , called the trivial bundle. (2) The tangent bundle 1,0 MT of a complex manifold M (see 9 below for a definition). (3) Let M be a compact complex manifold and denote by M its universal cover, ( )1 Mπ its fundamental group acting on M on the right, and let ( ) ( )1: M GL Vρ π → be a complex linear representation of ( )1 Mπ . Then V M Vρ ρ= × is a holomorphic vector bundle over M , where  M V M V/ ,ρ× = ×  and ( ) ( ) ( ) ( ) 1 1 2 2 1 1 2 1 2 1 m ,v m ,v g M m m .g v v such that and π ρ − ⇔ ∃ ∈ = =  g Almost Complex Manifolds Definition 87. An almost complex structure J on a C ∞ -differentiable manifold M is a section of ( )d MnE T , such that at each m 2 m Mm M , J id∈ = − T , where m M idT is the identity transformation of m MT . An almost complex manifold is a pair ( )M, J where M is a C ∞ -differentiable manifold and J is an almost complex structure. Remark 88. (1) It can be shown that every almost complex manifold is orientable and of even dimension. (2) Not every orientable manifold of even dimension can carry an almost complex structure, for example Borel and Serre (1951) proved that the spheres nS do not admit an almost complex structures if 2, 6n ≠ . Proposition 89. Every complex manifold cames an almost complex structure. Let ( ) n 1 nz ,...,z ∈  , and put j 2 j-1 2 j z x 1x= + − With respect to the coordinate system ( )1 n n+1 2nx ,...,x ,x ,...,x , we define an almost complex structure as follows 2 j-1 2 j 2 j 2 j-1 J J , j 1, 2, ..., n. x x x x and    ∂ ∂ ∂ ∂ = = − =      ∂ ∂ ∂ ∂    (8) KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE Therefore, every complex manifold is an almost complex manifold. The converse is not true in general, i.e., an almost complex manifold need not be a complex manifold. But if we impose some restrictions on the almost complex structure then the converse becomes true. Define the torsion JN of an almost complex structure J by ( ) [ ] [ ] [ ] [ ]J X, Y JX, JY X, Y J X, JY J JX, Y ,N = − − − where X and Y are vector fields. The torsion JN is a tensor, called the Nijenhuis tensor. Definition 90. An almost complex structure is said to be integrable if has no torsion, i. e., if ( )J X, Y 0N = for all vector fields X and Y . Theorem 91. (Newlander and Nirenberg, 1957) An almost complex structure is a complex structure if and only if it has no torsion. Let M be a complex manifold and ( )J Mnd∈ E T be the almost complex structure induced from the complex structure of M see Birkenhake and Lange (1992), where MT tangent bundle of the underlying C ∞ -manifold M . The endomorphism J extends to the complex vector bundle MT , where MT is the complexification of the real tangent space of the underlying C ∞ -manifold M , i.e., M M= ⊗  T T . From 2 Mid= − TJ , we deduce that 1,0 1,0M M M ,= ⊕  T T T { } { }1,0 1,0M X M JX 1X and M X M JX 1X= ∈  = − = ∈  = − − T T T T (9) Then 1,0 MT is a holomorphic vector bundle of rank n and 1,0 MT is the conjugate of 1,0 MT , i.e., 1,0 1,0M M≅T T . We can identify MT with 1,0 MT via the correspondence ( ) 1,01M X X 1 JX M2 − −∋ ∈aT T (10) Consider a local complex chart ( )( )1 nU, ,...,z z where 2 1 2x 1xj j jz −= + − . Then 1 2 n ,..., x x  ∂ ∂   ∂ ∂  constitutes a basis for MT over  . (also a basis for MT over c ), i.e., 1 2n 1 2n M span ,..., , M span ,..., , x x x x    ∂ ∂ ∂ ∂ = =    ∂ ∂ ∂ ∂       T T and 1 2n ,..., z z  ∂ ∂   ∂ ∂  (resp. 1 2n ,..., z z  ∂ ∂   ∂ ∂  ) a basis for 1,0 MT (resp. 1,0 MT ), where 1z ∂ ∂ and 1z ∂ ∂ are as in 7, in other words, 1 n 1 n M span ,..., , ,..., , z z z z  ∂ ∂ ∂ ∂ =   ∂ ∂ ∂ ∂    T and 1,0 1,0 1 n 1 n M span ,..., , M span , ..., z z z z    ∂ ∂ ∂ ∂ = =    ∂ ∂ ∂ ∂      T T The dual of the holomorphic vector bundle 1,0 MT will be denoted by 1 MΩ . Let M and N be two complex manifolds. Recall that a C ∞ -map f : M N→ between the underlying C ∞ - manifolds is said to be holomorphic if BOUDJEMAA ANCHOUCHE ( ) ( )1o f o : U Vϕ ϕ−ψ → ψ is a holomorphic map, for all local complex charts ( )U,ϕ of M and ( )V,ψ of N such that ( ) 1U Vϕ −⊂ . At each point m M∈ , the differential ( )m f mdf : M N ,→T T is an  -linear map. By complexification, we get a linear map (denoted also by df ) ( ) ( ) ( )m f mdf : M N→ T T In general ( )( ) ( ) ( )1,0 1,0m f mdf : M N⊄T T It can be shown that Theorem 92. Let M and N be two complex manifolds and f : M N→ a C ∞ -map between the underlying C ∞ - manifolds. Then f is holomorphic if and only if ( )( ) ( ) ( )1,0 1,0m m f mdf M N m Mfor all⊆ ∈T T 12. Differential forms on complex manifolds Let M be a complex manifold of dimension ,n ω a differential p -form, ( )U,ϕ a local chart and let ( )1 nz ,...,z be the system of complex coordinates associated to ϕ . Put j 2 j-1 2 j z x 1x= + − . Then ( )1 n n+1 2nx ,...,x ,x ,...,x are the local coordinates of the underlying C ∞ -manifold in U. In ( )( )1 n n+1 2nU, x ,...,x ,x ,...,x we write ( ) ( ) 1i i i 1 i<... T T J for some alternating 2 2n n× integral matrix J , where the condition 2) above means that the matrix 11 −− ∧ ∧> J is positive definite. Definition 110. A torus nT which is a projective variety is called an Abelian variety. Readers who want to know more on Abelian varieties can consult (Birkenhake and Lang, 1992, Griffiths and Harris, 1978 or Mumford,1974) . Remark 111. Every projective manifold M is necessarily Kähler, since the restriction of the Fubini-Study metric of nP to M is Kähler. The converse is not true, i.e., not every Kähler manifolds is projective. For example if we take any lattice Λ not satisfying the conditions in Theorem 109, then n n / G= T is Kähler but not projective. Therefore, we have {Projective Manifolds} ≠ ⊂ {Compact Kähler Manifolds} Definition 112. A quasi-projective variety is a an open subset (in the Zariski topology) of a projective variety. The readers who want to know more about complex algebraic geometry, can consult the following references: Demailly, , Griffiths and harris, 1978 or Shafarevich, 1977. Part 3. Compact Kähler manifolds of positive ricci curvature (Fano Manifolds) Compact Kähler manifolds of positive Ricci curvature playa very important role in the classification of complex algebraic varieties. The aim of this section is to give an overview of some of the main results in this area. In all what follows, we will assume some knowledge of sheaf theory, a very nice introduction can be found for example in Kodaira, 1986 or Griffiths and Harris, 1978. 15. Chern curvature of a hermitian line bundle Let M be a compact complex manifold and consider the following exact sequence of sheaves * exp0 0 ,M M→ → → → O O (15) where  is the constant sheaf, MO the sheaf of holomorphic functions and * MO the sheaf of nonvanishing holomorphic functions. The exact sequence (15) induces an exact sequence of cohomology groups ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 * 1 1 1 * 2 0 H M, H M, H M, H M, H M, H M, H M, ... M M M M δ → → → → → → → →    O O O O (16) KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE The set of holomorphic line bundles on M constitute a group under the tensor product called the Picard group and denoted by ( )Pi Mc .It can be proved that Proposition 113. ( )Pi Mc is isomorphic to ( ) 1 *H M, MO . In what follows, we will identify ( )Pi Mc with ( )1 *H M, MO . Definition 114. The image ( ) ( ) 2H M,Lδ ∈  of a line bundle ( ) 1 *H M, ML ∈ O is called the first Chern class of L and is denoted by ( )1c L . In what follows, we will identify ( )1c L to its image in ( ) 2 DRH M using the following ( ) ( ) ( ) ( ) de Rham Isomorphism 2 2 2 1 DRH M, H M, H Mc L ∈ ⊂ ≅  Definition 115. Let M be a complex manifold of dimension n. The first Chern class of M denoted by ( )1 Mc is defined by ( ) ( )11 1 MM : kc c −= where 1 n 1,0 MK M , − = Λ T is the anticanonicalline bundle of a complex manifold M . Let ( ), hL be a Hermitian line bundle over M , and let ( )iUi ∈ be an open covering of M such that iU i UiL → × ñ ׀ is an isomorphism. The point ( )M, Uiζ ∈ ×  is identified with the point ( ) ( )( )1o M, m, g m Uj i ji jζ ζ− = ∈ × ñ ñ . The functions g ji are nonvanishing holomorphic functions defined in U Ui j∩ . It is very easy to check that the functions ( ) , g ji i j , satisfy the following relation g g 1 in U U g g g U U U ji ij i j ji ik jk i j kin = ∩  = ∩ ∩ (17) Conversely, given a set of nonvanishing holomorphic functions g : U U ,ji i j∩ →  satisfying (17), we can define a line bundle with transition functions ( ) , g ji i j , as follows: We put ( )( )i iU /L = ×C   where C is the disjoint union, and ( ) ( )i jU z, z , Uξ ξ′ ′× ∈ ×  ∋ if and only if z z′= and ( ),g z .i jξ ξ′ = For each m M∈ , and mv L∈ we define the norm h v with respect to the metric h as follows BOUDJEMAA ANCHOUCHE ( )2 m2 hv , je ϕζ −= where it j ϕ is C ∞ function defined in U ,j and ( ) ( )m, vjζ = ñ . The definition of i' implies that ,1n g in U U .i i i j i jϕ ϕ= + ∩ Since ,i j g is a non vanishing holomorphic function, it follows that , 1n gi j is pluriharmonic, i.e., ,1 1n g 0.i j− ∂∂ = Therefore 1 1 in U U ,i j i j− ∂∂ = − ∂∂ ∩' ' and hence defines a global (1,1) form on M . Definition 116. The Chern curvature of the Hermitian line bundle ( ), hL , is the real closed (1,1) form given by h i1 ϕ= − ∂∂Θ Remark 117. Let { }iUi ∈ and { }, i,ji jg ∈ be as above, and assume that all the intersections U Ui j∩ are simply connected (if nonempty). Then { } ( )1 *,g H M,i j ML  = ∈  O . Since the holomo1phic functions ,i jg are non vanishing on U Ui j∩ , then choose a branch of log ,i j g , which is one valued holomo1phic function, and put ( )1 log log log 2ijk jk ik ij σ π − = − − +g g g Then it can be shown that the Chern class ( )1c L is given by ( ) { } ( )21 H M, .ijkc L σ = ∈   To state the next theorem we need the following definitions. Definition 118. Let M be a complex manifold. (1) A subset D of M is called a hypersurface if for every m M∈ , there exists a neighborhood U of m and a nonzero holomo1phic function f on U such that ( ){ }D U x U | f x 0 .∩ = ∈ = (2) A divisor D is a formal linear combination of a finite number of irreducible analytic hypersurfaces Di with integer coefficients i.e., D D , .i i i ν ν ν= ∈∑  (a) The divisor D is said to be effective if 0iν ≥ , and it is called a reduced divisor if 1iν = for all i . (b) An effective divisor D Diν= ∑ is called a simple normal crossing divisor if D is reduced, each component Di is smooth, and for x D∈ , there exist ( )U,z , x U∈ , a local complex chart, such that { }1 kU D z U | z ...z 0∩ = ∈ = for some k n≤ . To each divisor D Di iν ν= ∑ , we can associate a line bundle, denoted [ ]D as follows: Let ( )jU ,z and j kx U U∈ ∩ , be two local coordinate systems and suppose that ( ){ }II i I iU D z U | f z 0 ; I j,k,∩ = ∈ = = KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE where |fi is a holomorphic function in IU , I = j,k . Then j i i f f k is a nonvanishing function in j k U U∩ Put ( ) ( ) ( ) i j,k i i f z f z f z ij k ν   = Π      (18) Then it is easy to see that j,k f satisfies the condition (17), and hence are the transition functions of a line bundle. Definition 119. The line bundle with transition functions { }j,kf defined in 18 is called the line bundle associated to the divisor D and denoted by [ ]D . Remark 120. Over projective algebraic manifolds, every line bundle L is of the form [ ]DL for some divisor D L , i.e., [ ]DLL ≅ Theorem 121. Let ( ), hL be a Hermitian line bundle over M and let hΘ be its curvature. Then ( ) ( )21 h DR 1 c H M 2 L π  − = Θ ∈    , where h 1 2π  − Θ    is the cohomology class of h 1 2π − Θ in ( ) 2 DRH M . Moreover, h 1 2π  − Θ    is independent of the choice of the metric h. For a proof, see Griffiths and Harris (1978). Definition 122. Let ML → be a line bundle over a compact complex manifold M, dim M n= and let ( ) ( )21 H M,c L ∈  be the first Chern class of L . We say that ( )1c L is positive (resp. negative) and write ( )1 0c L > (resp. ( )1 0c L < ) if the cohomology class ( )1c L can be represented by a (1,1) closed real form n i j i,j 1 1 dz d , 2 i j z π = − Ω = ∧∑ ' where the matrix ( )1 i n, 1 j n i jϕ ≤ ≤ ≤ ≤ is positive definite (resp. negative definite) at each point m M∈ . The line bundle L is said to be positive (resp. negative) if ( )1 0 ,c L > (resp. ( )1 0c L < ). Remark 123. It can be shown that a line bundle ML → over a compact Kähler manifold M is positive if and only if there exists a Hermitian metric on L with positive curvature. The expression of the Ricci curvature in the Kähler case is very simple, more precisely we have Proposition 124. Let (M,g) be a compact Kähler manifold, where ( ) ( ) n i, j i,j 1 g z 1 g z dz dzi j = = − ⊗∑ BOUDJEMAA ANCHOUCHE Then ( )g i, jRic 1 1 det g ,n  = − − ∂∂   and ( )g 1 1 Ric c M , 2π   =   where g 1 Ric 2π      is the de Rham cohomology class of g 1 Ri 2 c π 16. Line bundles on projective spaces The aim of this section is to sketch a proof of the fact that the Picard group of ( ) nPic P is an infinite cyclic group and then construct a generator of ( ) nPic P . Proposition 125. ( )nPic isomorph ≅ P Proof. (Sketch of a proof): We start from the following well known fact ( ) ( )n nr 0 if r odd b dim H , 1 if r even r = =    P P (19) From (103) and (19), we deduce that ( )np n 0 if p qdim H , 1 if p q q ≠Ω =  =  P P In particular ( )n n np n 0dim H , 0 if p 1 (since ).= ≥ = Ω P P PP O O (20) From the exact sequence (16), and (20), we deduce that ( ) ( ) ( )nn 1 n * 2 nPic H , H , .≅ ≅ P P PO The Proposition follows from the well known fact that ( )2 nH , ≅ P A construction of a generator of ( ) nPic P , called the tautological line bundle, goes as follows Definition 126. The tautological line bundle over nP , denoted by ( )n 1−PO , is the line subbundle of the trivial bundle n n+1× P defined as follows: ( ) [ ]( ){ }n n n+11 , | .zz ξ ξ− = ∈ × ∈ P PO The dual of the line bundle ( )n 1−PO will be denoted by ( )n 1PO , i.e., ( ) ( )n n *1 : 1− = P P O O For a positive integer k , we put KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n n n n n n k-times k-times k 1 1 ... 1 -k 1 1 ... 1 k k = ⊗ = ⊗ ⊗ = − ⊗ = − ⊗ ⊗ − 644474448 64444744448 P P P P P P P P O O O O O O O O A definition of ( )n 1−PO ( )( )nresp. 1PO in terms of its transition functions is left to the reader. It can be shown that Proposition 127. The Picard group of nP is generated by the line bundle ( )n 1PO , i.e., ( ) ( )nnPic 1= PP O For a proof Proposition 127, see Kodaira (1986). Remark 128. (1) (2) If n K P is the canonical line bundle of nP , then ( )n nK n 1≅ − −P PO (3) If D is a smooth hypersurface of degree d in nP , i.e., [ ] ( ){ }n0 n 0 nD z ,...,z | f z ,...,z 0 ,= ∈ =P where f is a homogeneous polynomial of degree d , then [ ] ( )nD d≅ PO Proposition 129. The Fubini-Study (1,1) form FSω on nP represents the first Chern class of the line bundle ( )n 1PO , i.e., ( )( ) [ ]n1 FS1c ω=PO Therefore, ( )n 1OP is positive. Proof. (Sketch of the proof) We will construct a Hermitian metric on the line bundle ( )n 1PO , and we will show that its curvature coincides with the Fubini-Study metric. Since the fiber of the line bundle ( )n 1−PO at a point [ ] nz ∈ P is given by [ ] ( )n , z 1 .z.− = PO then, we define a Hermitian metric h on ( )n 1−PO as follows ( ) n 22 0 n h i i 0 z ,...,z z = = ∑ If { } ( ) ( ) n n+1 1 n 1 n : U \ 0 z ,...,z 1, z ,....,z s ⊃ →  a P is a local section, then ( ) n 22 0 n h i i 1 z ,....,z 1 zs = = + ∑ BOUDJEMAA ANCHOUCHE The curvature of the metric h is given by ( ) 2h 1 n h1 1n s z ,...,z= − − ∂∂Θ Hence the the curvature of the induced Hermitian metric *h on the dual bundle ( )n 1OP is given by ( ) ( ) * * 2 1 nh h 2 1 n h n 2 i i 1 1 1n s z ,...,z 1 1n s z ,...,z 1 1n 1 z = = − − ∂∂ = − ∂∂   = − ∂∂ +    ∑ Ä It can be shown (see Griffiths and Harris, page 30) that hÄ ( therefore *h Ä ) is independent of the local lifting (21) .By Theorem 121, the (1,1) form *h 1 2π − Ä represents the first Chern class of the line bundle ( )n 1PO . Since the Fubini-Study metric FSω on nP is equal to *h 1 2π − Ä , we deduce that FSω represents the first Chern class of the line bundle ( )n 1PO . 17. Kodaira embedding theorem As was said above, every projective manifolds is Kähler, but the converse is not true. Kodaira's embedding Theorem gives a characterization of projective manifolds among compact Kähler manifolds as those admitting positive line bundles, more precisely we have Theorem 130. (Kodaira, 1954) (Kodaira Embdding Theorem) Let ( )M, g be a compact Kähler manifold and let ML → be a holomorphic line bundle. Then L is positive if and only if there exists a holomorphic embedding n: Mψ → P ( )m m ,ψa of M into some projective space such that ( )( )n* 1 L ν⊗ψ =PO for some positive integer ν . Proof. (Sketch of a Proof) Suppose that such an embedding ψ exists. Since FSω represents ( )( ) ( )n *1 FSc 1 , ωψPO represents ( )( )n1c 1  ψ  PO and obviousely it is a positive (1,1) form. But ( )( ) ( )( )( ) ( ) ( ) n n * 1 1 1 1 c 1 c 1 , c c .L Lν υ ν⊗  ψ = /  = = P P O O Hence ( )* FS 1 ω ν ψ represents ( )1c L , i.e., L is positive. The converse is hard. For a proof, see (Kodaira, 1954), or (Griffths and Harris, 1978). Corollary 131. A compact Kähler manifold ( )M, g with positive Ricci curvature is projective. Proof. By Proposition 124, KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE ( )1 1 Ric c M 2 gπ   =   Hence -1 MK is positive. The corollary from the Kodaira's Embedding Theorem by taking -1 MKL = . Definition 132. A holomorphic line bundle ML → over a compact Kähler manifold M is said to be ample (resp. very ample) if there exists an embedding nM υ/→ P for some positive integer n such that ( )( )n 1L ν ψ⊗ = PO for some positive integer ( )( )( )n *. 1 .resp Lν π= ψ P O Theorem 133. A compact Kähler manifold ( )M, g is projective if and only if there exists a Kähler metric h (not necessarily equal to g ) such that the cohomology class [ ]hω is rational, i.e., [ ] ( ) 2 h H M, ,ω ∈  where hω is the (1,1) form associated to the Kähler metric h . Proof. (Sketch of the proof) Suppose that M is projective, i.e., i: embedding nM → P . Then the restriction * FSi ω of the Fubini-Study metric on nP to M is a Kähler metric on M . Since the Fubini-Study metric is the first Chern class of the autological line bundle ( )n 1PO on nP , it has integral cohomology class, i.e., ( ) * 2 FSi H M,ω  ∈   . For the converse, see Griffiths and Harris (1978). As a consequence of Hodge decomposition Theorem, we have the following Corollary 134. Let ( )M, g be a compact Kähler manifold satisfying ( )2 MH M, 0.=O (22) Then M is projective. Proof. (Sketch of the Proof) The assumption (22) is equivalent to ( ) ( )2,0 0, 2M, M, 0.H H= =  Then Hodge Decompostion Theorem implies that ( ) ( )2 1,1H M, H M, .≅  Then there exists d -closed (1,1) -forms 1 ,..., kθ θ on M such that their cohomology classes [ ] [ ] ( )21 ,..., M,k Hθ θ ∈  and [ ] [ ]{ }1 ,..., kθ θ constitute a basis for ( ) 2H M, .Hence, there exists k real numbers 1 ,..., kα α suh that [ ] 1 k g i i i ω α θ = = ∑ . Choose k rational numbers 1 ,..., kβ β suh that iβ is very close to iα and consider the following (1,1) form [ ] 1 . k i i i ϖ β θ = = ∑ Then ϖ is d -closed, positive, and satisfies [ ] ( ) 2H M,ϖ ∈  .Theorem 133 implies that M is projective. BOUDJEMAA ANCHOUCHE 18. Fano manifolds Definition 135. A projective manifold is said to be Fano if 1 MK − is ample, where 1 MK − is the anticanonical line bundle. Example 136. (1) There is only one Fano manifold of dimension one, it is 1P . (2) Fano manifolds of dimension two are called Del-Pezzo surfaces and are completely classified. They are 1 1 2× ×P P P , and ( ) 2 1p ,..., p , 1 k 8 ,k ≤ ≤P where ( ) 2 1p ,..., pkP is obtained by blowing up 2P at 1p ,..., pk in general position. (3) The projective space nP is a Fano manifold. (4) Let M be a smooth complete intersection of ν hypersurfaces of degree 1d ,...,dν in nP . Then n 1 M i=1 K n 1 - d .i ν −  = +    ∑OP In particular, M is Fano if and only if ii=1 d 1.n ν < +∑ (5) Let G be a complex reductive group and P a parabolic subgroup of G . Then the homogeneous space M G / P= is a Fano manifold. From the algebro-geometric point of view, a Fano manifold M is a smooth projective variety with ample anticanonical bundle 1 MK − , and from the differential geo- metric point of view, a Fano manifold M is a compact complex manifold admitting a Kähler metric with positive Ricci curvature. The two definitions are equivalent by Yau's solution of the Calabi conjecture Theorem 137. (Yau, 1977) A smooth projective variety M is Fano if and only if M admits a Kähler metric with positive Ricci curvature. As was mentioned before, there is close interplay between curvature and topology of a manifold. In the case of a compact Riemannian manifold, we have seen (Theorem 46) that the fundamental group is finite. In the Kähler case we can say more. Theorem 138. (Kobayashi, 1961) A compact Kähler manifold M with positive Ricci curvature is simply connected, i.e., ( )1 M 0.π = The proof uses the Riemann-Roch-Hirzubruch Theorem, Myer's Theorem, and the following result of Bochner Proposition 139. A compact Kähler manifold of positive Ricci curvature admits no non-zero holomorphic p - forms, for p 1≥ .. For a proof see Kobayashi (1987). Part 4. Einstein-Kähler metrics on fano manifolds Definition 140. (1) A Kähler metric h on a compact complex manifold M is said to be Einstein-kähler if h hRic c ,ω= where c is a constant, hω is the closed (1,1) form associated to the Kähler metric h and hRic is the Ricci (1,1) form associated to the Kähler metric h . (2) A compact Kähler manifold M is said to be Einstein-Kähler if it admits an Einstein-Kähler metric. One of the main problems related to Fano manifolds and not solved yet is to decide when a compact Kähler manifolds of positive Ricci curvature has an Einstein- Kähler metric. Obstructions to the existence of Einstein- KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE Kähler metrics on compact Kähler manifolds of positive Ricci curvature were discovered by several mathematicians. Below, we will introduce two obstructions discovered by Matsushima and Futaki and an invariant constructed by Tian. 19. Matsushima's obstruction Let M be a complex manifold. An automorphism (biholomorphism) F : M M→ of M is a holomorphic map from M to M such that there exists a holomorphic map G : M M→ with MG o F F o G id= = , where Mid is the identity map of M . The set of automorphisms of M , denoted by ( )Aut M , has a structure of a group under the composition of maps. In case the complex manifold M is compact, we can say more, Theorem 141. Let M be a compact complex manifold. Then ( )Aut M is a complex Lie group and its Lie algebra ( ) ( )( )M ie Aut M= Ly consists of holomorphic vector fields of M . For a proof, see (Kobayashi and Nomizu, 1969). Definition 142. A complex Lie group G is called reductive if the Lie algebra g of G is isomorphic to the complexification of the Lie algebra k of a compact Lie group K , i.e., ≅ ⊗  g k The Lie algebra k is called a real. form of the Lie algebm g . Example 143. (1) If ( )G G n,= L , then G K=  , where ( )K U n= . Moreover ( ) ( )nl n, M ,≅ g g= and ( ) ( ){ }*n A l n, A A 0 .u= = ∈  + =k g (2) If ( )G SO n,=  , then G K=  , where ( )SO nK = . Moreover ( ) ( ){ }*o n, A l n, A A 0 ,= = ∈  + = g s g and ( ){ }A l n, A A 0 .∈  + =k = g T . (3) If ( )G Sp n,=  , then G K=  , where ( )K Sp n= . Moreover ( ){ }A l 2n, A J JA 0 ,∈  + =g = g T and ( ){ }A l 2n, A J JA 0 .∈  =k = g T + More details about Lie groups (resp. algebraic groups) can be found in (Knapp, 2001) or (Brocker, et al., 1985) (resp. (Borel, 1991) or (Humphreys, 1975). Theorem 144. (Matsushima, 1957) The automorphism group ( )Aut M of a compact Einstein-Kähler manifold of positive Ricci curvature is reductive. At the level of Lie algebras, Matsushima's theorem says that if M be a compact Einstein-Kähler manifold with non zero Ricci curvature then the Lie algebra ( )Mi of infinitesimal isometries (or Killing vector fields) is a real form of the Lie algebra ( )h M of holomorphic vector fields, i.e., ( ) ( ) ( )h M M 1 M .i i= ⊕ − BOUDJEMAA ANCHOUCHE Remark 145. If ( )1 M 0 ,c < then ( )Aut M is finite, and consequently ( )h M 0.= . Therefore Matsushima's Theorem is not obvious only when ( )1 M 0.c > . Example 146. (Futaki, 1988 ) Let ( )2 1 2p ,p%P be the projective plane 2P blown up at two points 1 2p , p . It can be shown that ( )2 1 2p ,pP is a compact Kähler manifold with positive Ricci curvature whose automorphism group is not reductive, therefore it cannot carry an Einstein-Kähler metric. In dimension two, the reductivity of ( )h M is the only obstruction to the existence of Einstein-Kiihler metrics as shown by the following result Theorem 147. (Tian, 1990) Any compact complex surface M with positive Ricci curvature admits an Einstein- Kähler metric if ( )h M is reductive. In higher dimension, there are other obstructions to the existence of Einstein-Kähler metrics besides Matsushima's obstruction. 20. Futaki's invariant For the construction of Futaki's invariant, we need the following Lemma 148. Let θ be a real ( )p,p form on a compact Kähler manifold which is cohomologous to zero, i.e., dθ β= . Then there exists a ( )p -1,p -1 form η such that 1 .θ η= − ∂∂ The proof uses the Hodge Theorem for the ∂ operator (see [33], Proposition 7.24, for the details). Let g ω be a (1,1) form associated to a Kähler metric g and suppose that ( )1 Mg cω ∈ .Then g 1 Ri 2 c π represents also ( )1 Mc , and by Lemma 148, we have g 1 Ri 1 F , 2 gg c ωωπ − = − ∂∂ where F gω is a global function on M which is defined up to a constant. In (Joyce, 2000), Futaki introduced the following linear functional ( ) ( ) ( ) F n F gM h M: 1 X X X F . 2 gω ω π → − = ∫a CL L Theorem 149. (Futaki, 1983) The functional FL is independent of the choice of the Kähler form ( )1 Mcω ∈ . In particular F L is invariant under the automorphism group ( )Aut M of M and FL is a Lie algebra homomorphism. Moreover, if M admits an Einstein-Kähler metric, then FL vanishes identically. KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE The following example exhibits an example of a compact Kähler manifold M with positive Ricci curvature such that ( )Mh is reductive, but F L is not identically zero. Example 150. (Futaki, [18]) Let r and s be two positive integers, 1 : r s rπ × →P P P the projection to the first factor, and 2 : r s sπ × →P P P the projection to the second factor. Let ( ) ( )* *r,s 1 21 1 ,r s r sπ π= ⊕ → ×O OP P P PE and let , M r s be the total space of the projective bundle ( ),r sEP . It can be shown that (1) ( )1 r,sM 0 ,c > , (2) ( )r, sh M is reductive, (3) If r 1= and s 2 ,= , then F 0.≠L , Therefore 1,2 M is a Fano manifold which cannot carry an Einstein-Kähler metric with positive Ricci curvature. There are other generalizations of Futaki's invariant which we will not consider in this paper. 21. Tian's invariant The vanishing of the Futaki invariant is a necessary condition for the existence of a Einstein-Kähler metric but not a sufficient one. In Tian (1997), Tian constructed a compact Kähler manifold which does not admit any Einstein-Kähler metric even though F 0=L . Let us introduce Tian's invariant: let ( )M, g be a compact Kähler manifold of dimension n , and let ( ) ( )2 g M 1 P M,g C M, 0, sup 0 , 2 ω ω π  −  = ∈  = + ∂∂ =      φφ φ φ> where ( ) n g i, i jj i,j 1 1 g z dz dz ,ω = = − ∧∑ is the associated (1,1) form expressed in local holomorphic coordinates, and where " 0"ω >φ means that ω' is a positive definite (1,1) form. Proposition 151. Let ( )M, g be as above. Then there exists two positive constants c and a such that c and α such that ( )z n gM e c ,α ω− ≤∫ ' for all ( )P M, g∈φ . Consider the following invariant defined by Tian (1997) : ( ) ( ) ( ){ }z ngMt M, g sup 0 c > 0, e c for all P M, g .αα ω−= >  ∃ ≤ ∈∫ φ φ BOUDJEMAA ANCHOUCHE Theorem 152. (Tian, 1997) Let ( )M, g be a Fano manifold of dimension n , where gω represents the first Chern class of M , i.e., ( )g 1 Mcω  =  . If ( ) nt M, g , n 1 > + then M admits an Einstein-Kähler metric. Estimates of ( )t M, g were used by several authors to prove the existence of Einstein-Kähler metrics on certain Fano manifolds. 22. Uniqueness of einstenin-kähler metrics with positive ricci curvature Let M be a Fano manifold, and let ( )MEK be the set of all Einstein-Kähler metrics on M . The automorphism ( )Aut M acts on ( )MEK via pull-back. Let ( )0Aut M be the identity component of ( )Aut M . Theorem 153. (Bando and Mabuchi, 1985) Suppose that ( )MEK φ≠ and let go be an Einstein-Kähler metric. Then ( ) ( )0 0M Aut M . ,EK g= , i.e., any Einstein-Kähler metric g is of the form * 0gψ for some ( )0Aut M .ψ ∈ Remark 154. If ( )h M 0= , then the Einstein-Kähler metric is unique if it exists. Part 5. Complete noncompact Kähler manifolds of positive ricci curvature The structure of noncompact manifolds is richer and at the same time harder to explore than the structure of compact manifolds. Complete noncompact Kähler manifolds of positive ricci curvature are less studied, and therefore less understood compared to Fano manifolds. Nonetheless, lots of partial results have bee proved although a general theory is still missing. In this part, we will survey some of these results, close to the author's interests, obtained on this topic. 23. Construction of complete kähler metrics of positive ricci curvature The problem of existence of complete Kähler metrics of positive Ricci curvature on smooth quasi-projective varieties was considered by several authors, among them S. T. Yau, G. Tian, R. Kobayashi, S. K. Yeung, and others. By solving certain complex Monge-Ampere equations, where the ideas are close to the ones introduced by Yau in his solution of the Calabi conjecture, G. Tian and S. T. Yau, proved the following Theorem 155. (Tian, Yau 1990) Suppose that M is a smooth complex projective variety and that D M⊂ is a smooth ample divisor. Let Ω be any smooth closed (1,1) form in the cohomology class [ ]( )( )1MK D −⊗ . Then M \ D admits a complete Kähler metric whose Ricci curvature form is equal to the restriction of Ω to M \ D . As a consequence, we get KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE Corollary 156. Let M and D M⊂ be as above. If [ ]( ) 1MK D − ⊗ is ample, then M \ D admits a complete Kähler metric with positive Ricci curvature. To state the next Theorem, we need the following Definition 157. A complete noncompact Kähler manifold ( )M, g of dimension 2n ≥ with positive Ricci curvature is said to be of standard type if it satisfies the following three conditions (1) n gM Ric ,< +∞∫ (2) there exists a constant 1c such that for all r 0 ,> ( )( )g 0vol B x , rg (3) there exists a constant 2c such that for all m M ,∈ ( ) ( )( ) 2 2 c | sect M | , 1 r m ≤ + where ( )( )g 0vol B x , rg is the volume of the geodesic ball ( )g 0B x , r with respect to the metric g , Ricg is the Ricci (1, 1) form associated to the Kähler metric g , and ( )sect m is the sectional curvature of g at the point m . Simultaneously with Tian and Yau, using the continuity method for the Monge-Ampere equation, Yeung obtained the following Theorem 158. (Yeung 1990) If M is a smooth projective variety with dim M 2≥ , and D is a smooth hypersurface such that the associated line bundles [ ]D and [ ]( ) 1 M K D − ⊗ are positive, then the affine variety M \ D admits a complete Kähler metric of positive Ricci curvature and of standard type. Example 159. Let nM , 2n= ≥P and let D be a smooth hypersurface in nP of degree, where 1 d n≤ ≤ . Since [ ] ( )nD d= PO and [ ]( ) ( )n 1 MK D n 1 d − ⊗ = + − P O are positive, Yeung's Theorem, (Theorem 158) implies that nM \ D= P admits a complete Kähler metric of positive Ricci curvature and of standard type. An extension of Kobayashi's result (Theorem 138) to the noncompact case was obtained by Tsuji Theorem 160. (Tsuji, 1988) Let M be a smooth projective variety and let D be a simple normal crossing divisor. If [ ]( ) 1MK D − ⊗ is ample then M \ D is simply connected, i.e., ( )1 M \ D 0π = 24. Compactification of complete Kähler manifolds of positive ricci curvature One way to approach noncompact complete Kähler manifolds of positive Ricci curvature is to try first to compactify them (if it is possible to do that), i.e., realize them as open subsets, in the Zariski topology, of some projective manifolds, via some "noncompact" version of Kodaira embedding. This was achieved by Mok (1990) for complete Kähler manifolds of positive Ricci curvature and of standard type. More precisely, Mok's result says that BOUDJEMAA ANCHOUCHE Theorem 161. (Mok, 1990) Let ( )M, g be a complete noncompact Kähler manifold of dimension 2n ≥ with positive Ricci curvature and of standard type. Then M is biholomoryhic to a quasi-projective variety M \ D , where M is a smooth projective variety and 1 jj 0 D D = = ∑ is a normal crossing divisor. More precisely, there exists a finite set of sections { } ( )n 0 qj Mjs H M, K −⊂ of class 2L for some positive integer q , where -1 MK is the anticanonical line bundle, such that ( ) ( ) n q 0 n : M M P m s m ,..., s m , ψ → ⊂   a (23) is an embedding, and q : M M \ Dυ =/ . Mok's embedding Theorem says more or less that -1 MK is "ample" and can therefore be seen as a noncompact analogue of Kodaira's embedding Theorem. Remark 162. It can be proved, under the assumption of Theorem 161, (see Borel and Serre, 1951) that ( ) ( )1 1M M \ D ,π π= < +∞# # and therefore 0 1H M, 0 M   Ω =    Remark 163. (1) The completeness of the metric in the noncompact case is crucial. All metrics are complete on compact manifolds. (2) The constmint on the volume growth is natural, since by combining the Calabi -Yau's theorem (Theorem 58) and Bishop's Theorem (Theorem 57), on a complete Riemannian manifold with positive Ricci curvature we have ( )( ) 2n1 g g 0 2c r vol B x , r c r≤ ≤ (3) An extension of Mok's embedding theorem to the case of a volume growth slower than Euclidean was obtained by To (1991). 25. Logarithmic kodaira dimension of complete Kähler manifolds of positive ricci curvature To introduce the logarithmic Kodaira dimension of ( )M, D , we need some definitions. Definition 164. A closed subset V of a complex manifold M, dim M ,n =  is called an analytic subvariety if for every m M∈ , there exists a neighborhood U of m and a finite number of holomorphic functions ( ) m m 1 mf ,...,fν on U , called defining functions, such that ( ) ( ) ( ){ }m m1 mV U x U | f x ... f x 0 .ν∩ = ∈ = = = A point m V∈ is called a smooth point, if there exists an open subset U′ of m and defining functions m m 1 kf ,...,f in U′ such that KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE ( ) ( ) m m 1 1 1 n m m 1 k 1 n m m k k 1 n f f . . . z z . . . . . f ,...,f rank rank k .. . . . . z ,...,z . . . . . f f . . . z z  ∂ ∂   ∂ ∂    ∂   = =  ∂      ∂ ∂   ∂ ∂  The integer k is called the codimension of V at m . The set of smooth points is denoted by smV . It can be proved that if smV is connected, then it is a complex submanifold of M of dimension - kn , in which case we write odim V kc = . Remark 165. The defining functions of an analytic subvariety in a neighborhood o.t a point are not unique. Definition 166. Let M and N be two projective algebraic manifolds. (1) A rational map Ψ from M to N , denoted by : M ---- NΨ > , is given by a holomorphic map : M \ V N ,ψ → where V is a subvariety of M , such that odim V 2c ≥ . (2) We say that a rational map : M ---- NΨ > is birational, if there exists a rational map : N ---- Mφ > such that φΨ o is the identity as a rational map. If such a birational map : M ---- NΨ > exists, then we say that the algebraic manifolds M and n are birational. Let n: M ----φ > P be a rational map defined by a holomorphic map n: M \ V ----φ > P , where V is a subvariety of M , such that odim V 2c ≥ , and let ( )( ){ }nψ m, m M \ V .φ = ∈ × P Denote by ψΓ the closure (in the Zariski topology) of ψ Γ in nM × P and by n ψ2 ,π :Γ → P the projection to the second factor. The image ( )Mφ of M is given by ( ) ( )ψ2M .φ π= Γ Let M be a smooth projective manifold, D be a normal crossing divisor on M and let [ ]( )( ) [ ]( )( ){ }0M MM, K D : | H M, K D 0 .νν⊗ = ∈ ⊗ ≠NS Let us assume that [ ]( )( )MM, K D 0⊗ ≠S , and for [ ]( )( )MM, K Dν ∈ ⊗S , consider the rational map ( ) ( ) ( ) N 0 N : M --- m m m ,..., m , ν ν ν ν ν ν φ φ >  =  a P φ φ where BOUDJEMAA ANCHOUCHE [ ]( )( )0 MN dim H M, K D 1,νν = ⊗ − and { }0 N,..., νν νφ φ is a basis for [ ]( )( )0 MH M, K D ν⊗ Definition 167. The Logarithmic Kodaira dimension of M M \ D= , denoted by ( )k M , is given by ( ) ( ) [ ]( )( ) [ ]( )( ) ( ) [ ]( )( ) M max M M M, K D k M k M \ D dim M M, K D . M, K D if ifν φ φ φ ν −∞ ⊗ =  = = ⊗ ≠  ∈ ⊗   S S S Remark 168. The Logarithmic Kodaira dimension has the following properties: (1) It is independent of the compactification, i.e., if biholomorphic biholomorphic 1 1 2 2M M \ D M \ D ,≅ ≅ where ( )1 2M . Mresp is a smooth projective variety and ( )1 2D . Dresp is a normal crossing divisor on ( )1 2M . Mresp and if the isomorphism biholomorphic 1 21 2M \ D M \ D≅ extends to a birational map birational 1 2M ---- M> then ( ) ( )1 21 2k M \ D k M \ D= (24) For a proof of (24) , see Litaka (1977). (2) It is not a biholomorphism invariant. For a proof, see Hartshorne (1970). Theorem 169. (Anchouche, 1998) Let ( )M, g be a complete noncompact Kähler manifold ( )M, g of dimension 2n ≥ with positive Ricci curvature and of standard type and let ( )M, D be the compactification obtained in Theorem 161. Then ( )k M = −∞ Proof. (Sketch of the Proof) In all what follows, we identify M with M \ D . Remark first that [ ]( )( ) ( )0 , DM MH M, K D M, K ,ν ν⊗ ≅ ϒ where ( ),D MM, Kνϒ is the set of meromorphic sections of MK which are holomorphic in M and admitting poles along the divisor D of order at most ν . Step1: We establish the following inequality KÄHLER MANIFOLDS POSITIVE RICCI CURVATURE ( ) ( ) ( )g g n 1 2 n 3 4 n 3 n n g g2g n B x,r B x,r 1 D 1 s c 1n , s ii i ν ω ω + + + =      ≤    Π    ∫ ∫ where [ ]( )( )0 MH M, K Ds ν∈ ⊗ is the section of the line bundle [ ]Di defining the divisor Di , i.e., ( )1Ds 0 , si ii D iD − = is the norm of the section SD, defined by a fixed Hermitian metric hi on the line bundle [Di], and 115111 is the norm of the section s with respect to the metric on MK induced from the metric g . Step 2: Let n 1 2n i 0 1 D 1 1n c qu 1n s s i i i i = =    ψ = − + +     Π ∑ n 2 i 0 c qu 1n si =    ψ = +      ∑ where u is a solution of the equation ( )g u Ricgr∆ = T (see Mok, 1984) , g∆ is the Laplacian associated to the Kähler metric g,c is chosen in such a way that 1ψ and 2ψ are subharmonic functions, and si are the sections of q MK − appearing in Mok's embedding Theorem 161. Therefore, we get the Riesz Representation ( ) ( ) ( )( ) ( ) ( ) ( )0 0 x n 0 0 g g B x ,r x x 1im G x , y y y , g j j jρ ρ ω → +∞ ψ = ψ + ∆ ψ∫ where ( ) 0xG .,.ρ is the Green kernel of the geodesic ball ( )0B x , rg for the operator g∆ . Step 3: There exist a constant c such that i n 2 DM Mi 1 c 1 s c , i ω ω = − ≤ − ∂∂ ≤∏ where M ω is the Fubini-Study metric on M . Step 4: Using the previous steps, we get ( ) ( ) ( )( ) ( ) n 2n gg B x,r y y cr 1n r x 2 for 1, 2. g j iω∇ψ ≤ + =∫ The estimate above is the hardest pat of the proof. Step 5: Using the inequality obtained in Step 4 and following a similar scheme developed by John-Nirenberg, we get ( ) ( ) ( )( ) ( ) n 2n 2 2 ,r gg B x,r y y cr 1n r x r 2 for 1, 2. g j j iω +ψ − ≤ + + =∫ Θ where ,ri Θ is the solution of the following Dirichlet's problem ( ) ( )0 in B x,rg , r g | B x,r, r g j j j  ∆ =  ∂ = ψ Θ Θ BOUDJEMAA ANCHOUCHE Step 6: using the previous steps, we get ( ) ( ) ( )( )( ) n 1 2n 2 2 n 34 x 3 2n r c 1n r r x 2 s x c r n ν + + + +  + + + ≤ where the constants c and xc are independent of r . Since n 2≥ , the theorem is obtained by letting r → + ∞ . For more details, see Anchouche (1998). The vanishing theorem above puts a lot of restriction on the structure of the manifold M , for example if we assume that dim M 2=C then, as a consequence of Theorem 169 and Miyanishi's classification of open algebraic surfaces [40], we obtain the following Theorem 170. (Anchouche, 1998) Let M be a complete Kähler surface of positive Ricci curvature and of standard type, and let ( )M, D be the compactification obtained in Theorem 161. Then (1) M is a rational surface. (2) Each component j D of D is a smooth rational curve, i.e., jD is isomoryhic to 1P . (3) D is a tree. (4) None of the components j D is exceptional. The Theorem above is a consequence of Theorem 169 and Miyanishi's classifi- cation of quasi projective surfaces of logarithmic Kodaira dimension − ∞ . The fact that the Kähler manifold M is of infinite volume implies part (4) of the Theorem 170. In [2], it has been shown that the 2L sections of the line bundle q XK − , and the volume form of the metric g have no essential singularities near the divisor at infinity D . As a consequence we obtain a comparison between the form forms of the complete Kähler metric g and the Fubini-Study metric of the compactification. 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