Introduction of the -cohomology Pierre Dolbeault Abstract . We - PowerPoint PPT Presentation

Introduction of the ∂ -cohomology Pierre Dolbeault Abstract . We recall results, by Hodge during the thirties, the early forties and 1951, by A.Weil (1947 and 1952), on differential forms on complex projective algebraic and K¨ ahler manifolds; then we describe the appearance of the ∂ -cohomology in relation to the cohomology of holomorphic forms. 1

Contents 1. Preliminaries 2. First unpublished proof of the isomorphism 3. Usual proof of the isomorphism 4. Closed holomorphic differential forms 5. Remarks about Riemann surfaces, algebraic and K¨ ahler man- ifolds 6. Fr¨ olisher’s spectral sequence 2

1.1. In [H 41] and former papers, Hodge defined harmonic differ- ential forms on a Riemannian manifold X ; using the Riemannian metric, he defined, on differential forms, the dual δ of the exte- rior differential operator d , the Laplacian ∆ = dδ + δd , harmonic forms ψ satisfying ∆ ψ = 0 and proved the following decompo- sition theorem: every differential form ϕ = H ( ϕ ) + dα + δβ and, from de Rham’s theorem: H p ( X, C) ∼ = H p ( X ). [H 41] W.V.D. Hodge, The theory and applications of harmonic integrals, (1941), 2th edition 1950. 3

Then Hodge gave applications to smooth complex projective al- gebraic varieties (chapter 4), the ambient projective space being endowed with the Fubini-Study hermitian metric: Hodge the- ory mimics the results of Lefschetz [L 24], via the duality be- tween differential forms and singular chains. The complex lo- cal coordinates being ( z 1 , . . . , z n ), Hodge uses the coordinates ( z 1 , . . . , z n , z 1 , . . . , z n ) for the C ∞ , or C ω functions and the type (with a slight different definition) ( p, q ) for the differential forms homogeneous of degree p in the dz j , and q in the dz j . 4

1.2. In a letter to G. de Rham in 1946 [W 47], A. Weil states that the results of ([H 41], chapter 4) are true for a compact K¨ ahler manifold and studies the following situation for closed meromorphic differential forms of degree 1 on a compact K¨ ahler manifold V : Let r = ( U j ) be a locally finite covering of V by open sets U j such that U j and U j ∩ U k � = ∅ be homeomorphic to open balls. For every j , let θ j be a d -closed meromorphic 1-form on U j such that on every U j ∩ U k � = ∅ , θ j − θ k = θ jk is holomorphic. Remark that: θ lj + θ jk + θ kl = 0 and dθ jk = 0 [W 47] A. Weil, Sur la th´ eorie des formes diff´ erentielles attach´ ees ` a une vari´ et´ e analytique complexe, Comment. Math. Helv., 20 (1947), 110-116. 5

The problem is to find a closed meromorphic 1-form θ having the singular part θ j on U j for any j . Using a result of Whitney, we construct smooth 1-forms ζ j on U j such that ζ j − ζ k = θ jk in the following way: assume already defined the forms ζ 1 , . . . , ζ k − 1 , ζ k is a C ∞ extension of ζ k − 1 − θ ( k − 1) k from U k − 1 ∩ U k to U k . Then, there exists, on V a smooth 1-form σ = dζ j on U j ; using the existence theorem of harmonic forms, we show that σ is harmonic of type (1 , 1) . The existence of θ is equivalent to σ = 0. Moreover remark that the 1-cocycle { θ jk } defines a fibre bundle [Ca 50]. 6

1.3. More generally, let { u jk } , where u jk , p ≥ 0, is a d -closed holomorphic p -form, be a 1-cocycle of the nerve of the covering r , then u jk is a holomorphic p -form on U j ∩ U k and u lj + u jk + u kl = 0 on U j ∩ U k ∩ U l � = ∅ . As above, there exist C ∞ ( p, 0)-forms g j on U j such that g j − g k = u jk and a harmonic form L p, 1 on V such that dg j = L p, 1 | U j . Conversely, on U j (small enough), from the Poincar´ e lemma, there exists, a ( p, 0)-form g j such that dg j = L p, 1 | U j ,then { u jk } = { g j − g k } is a 1-cocycle with u jk holomorphic [Do 51]. 7

1.4. In [ H 51], Hodge defined the differential operator d ” = n ∂ dz j of type (0 , 1); let d ′ = � n ∂ � ∂z j dz j of type (1 , 0); then j =1 ∂z j j =1 d = d ′ + d ” and d ” 2 = 0 = d ′ 2 . After [Ca 51], the use of d ′ , d ” and, on K¨ ahler manifolds, the operators δ ′ and δ ” became usual. 8

2. First unpublished proof of the isomorphism . 2.1. Let X be a paracompact (in particular countable union of compact sets) complex analytic manifold of complex dimension n . Let r = ( U j ) be a locally finite covering of X by open sets � U j such that U j and U j ∩ U k � = ∅ , or more generally U j � = ∅ j ∈ J for J ⊂ (1 , 2 , . . . , n ) be homeomorphic to open balls. It is always possible to replace r by a covering r ′ = ( U ′ j ) s.t. U ′ j ⊂ U j . We will use ˇ C ech cochains, cocycles and cohomology. 9

As in section 1.3, let { u jk } be a 1-cocycle of the nerve N r of r where u jk is a holomorphic p -form on U jk , with p ≥ 0 and u lj + u jk + u kl = 0 on U jkl = U j ∩ U k ∩ U l � = ∅ .. Then the ( p, 0)-forms u jk satisfy d ” u jk = 0. As above, there exist g j C ∞ ( p, 0)-forms such that g j − g k = u jk : then there exists a global d ”-closed ( p, 1)-form h such that h | U j = d ” g j . Conversely, given h on X , such that d ” h = 0, then, on U j (small enough), from the d ”-lemma (see section 3), there exists, a ( p, 0)-form g j such that d ” g j = h | U j ,then { u jk } = { g j − g k } is a 1-cocycle with u jk holomorphic. 10

2.2. Let E p,q be the sheaf of differential forms (or currents) of type ( p, q ) a complex analytic manifold X . = Ker( E p,q ( X ) d ′ ′ Z p,q ( X, C) → E p,q +1 ( X )) = Im( E p,q − 1 ( X ) d ′ ′ B p,q ( X, C) → E p,q ( X )) We call d ′ ′ -cohomologie group of type ( p, q ) of X , the C-vector space quotient H p,q ( X, C) = Z p,q ( X, C) /B p,q ( X, C) 2.3. Let Ω p the sheaf of holomorphic differential p -forms. From H 1 (Ω p ) ∼ = H p, 1 ( X, C). section 2.2, we have the isomorphism: 11

2.4. Let now { u jkl } be a 2-cocycle of the nerve N r of the covering r , where u jkl is a holomorphic p -form, we have: u mjk + u jkl + u klm + u lmj = 0 on U jklm = U j ∩ U k ∩ U l ∩ U m � = ∅ . The ( p, 0)- As above, there exist g jk C ∞ forms u jkl satisfy d ” u jkl = 0. ( p, 0)-forms such that g lj + g jk + g kl = u jkl on U jkl � = ∅ , then d ” g lj + d ” g jk + d ” g kl = 0 on U jkl � = ∅ , and three other analogous equations, the four homogenous equations are valid on U jklm . Then: d ” g lj = 0; d ” g jk = 0; d ” g kl = 0; d ” g lm = 0 on U jklm . If U jklm is small enough, from the d ”-lemma (see section 3), there exists, h jk such that d ” h jk = g jk on U jklm . The form h jk can be extended to U jk such that h jk + h kl + h lj = 0 on U jkl ; by convenient extension, there exists a form µ j on U j such that µ j − µ k = d ” h jk on U jk , and a d ”-closed ( p, 2)-form λ on X such that d ” µ j = λ | U j . Adapting the last part of the proof in section 2.1, we get the isomorphism: H 2 (Ω p ) ∼ = H p, 2 ( X, C). 12

3. Usual proof of the isomorphism . 3.1. Let F be a sheaf of C-vector spaces on a topological space X , on call r´ esolution of F an exact sequence of morphisms of sheafs of C-vector spaces j → L 0 d → L 1 d → . . . d → L n d ( L ∗ ) 0 → F → . . . Following a demonstration of de Rham’s theorem by [W 52] A. Weil, Sur le th´ eor` eme de de Rham, Comment. Math. Helv., 26 (1952), 119-145, J.-P. Serre proved: 13

3.2. Abstract de Rham’s theorem .- On a topological space X , let ( L ∗ ) be a resolution of a sheaf F such that, for m ≥ 0 and q ≥ 1 , H q ( X, L m ) = 0 . Then there exists a canonical isomorphism H m ( L ∗ ( X )) → H m ( X, F ) where L ∗ ( X ) is the complex 0 → L 0 ( X ) → L 1 ( X ) → . . . → L m ( X ) → . . . of the sections of ( L ∗ ). 14

′ Lemma .- On an open coordinates neighborhood U (with 3.3. d ′ coordinates ( z 1 , . . . , z n )) of a complex analytic manifold, the ex- n ∂ terior differential d = d ′ + d ′ ′ where d ′ ′ = � dz j . We have ∂z j j =1 ′ 2 = 0; this definition is intrinsic. d ′ In the same way, on U , every differential form of degree r , ϕ = ϕ r, 0 + . . . , ϕ 0 ,r where ϕ u,v = � ϕ k 1 ··· k u l 1 ··· l v dz k 1 ∧ · · · dz k u ∧ dz l 1 ∧ · · · ∧ dz l v ; the form ϕ u,v of bidegree or type ( u, v ) is define intrinsically. Lemma .- If a germ of differential form C ∞ t is d ′ ′ -closed, of type ( p, q ) , q ≥ 1 , there exists a germ differential form C ∞ s of type ( p, q − 1) such that t = d ′ ′ s . The Lemma is also valid for currents (differential forms with coefficients distributions). 15

It is proved by P. Dolbeault in the C ω case, by homotopy, as can e lemma. H. Cartan brings the proof to the C ω been the Poincar´ case by a potential theoritical method [Do 53]. Simultanously, the lemma has been proved by A. Grothendieck, by induction on the dimension, from the case n = 1 a consequence of the non homogeneous Cauchy formula see [Ca 53], expos´ e 18). 16

3.4. A sheaf F on a paracompact space X is said to be fine if, for every open set U of a basis of open sets of X and for every closed set A ⊂ U , exists an endomorphism of F equal to the identity at every point of A and to 0 outside U . If F is fine, then H q ( X, F ) = 0 for every q ≥ 1. From d ” Lemma follows the following resolution of the sheaf Ω p of the holomorphic differential p-formes: → E p, 0 d ′ → E p, 1 d ′ ′ → . . . d ′ ′ → E p,q d ′ ′ ′ 0 → Ω p j → . . . 17