Currents on Complex Manifolds Let X Be a Complex Manifold and Ep,Q(X)

Home , Current (mathematics), Dolbeault cohomology

(p, q)-currents on complex manifolds Let X be a complex manifold and

E p,q(X)= {smooth (p, q) − forms on X}

Dp,q(X)= {smooth (p, q) − forms on X with compact support} and the dual space of Dp,q(X):

′n−p,n−q ′ p,q ∗ D (X) := Dp,q(X) := D (X) . ′ An element Θ ∈Dp,q is called a current of bidegree (n − p, n − q), or bidimension (p, q), or simply (n − p, n − q)-current.

We’ll show in the next chapter: Let S ⊂ X be a pure p-dimensional complex variety in ′ a complex manifold of dimension n. Then [S] ∈Dp,p(X) given by

[S](φ)= φ, ∀φ ∈Dp,p(X). ZS [S] is a closed positive (n − p, n − p)-current.

On an n dimensional complex manifold X, we have d = ∂ + ∂ so that by the deﬁnition ′ of dΘ, for any Θ ∈D p,q, we have

(∂ + ∂)Θ(φ)=(−1)p+q+1Θ (∂ + ∂) (φ), ∀φ ∈D2n−p−q−1, which implies ∂Θ(φ)=(−1)p+q+1Θ(∂φ), ∀φ ∈D2n−p−q−1. Hence ∂Θ ∈Dp,q+1. Also, if ∂Θ = 0, we say that Θ is ∂-closed.

Cohomology groups of a sheaf ([GH78]) Recall that we have the cochain complexes of diﬀerential forms: d d d d 0 → R →D0 −→D1 −→D2 −→D3 −→ ..., (26)

p p,0 ∂ p,1 ∂ p,2 ∂ p,3 ∂ 0 → ΩX →D −→D −→D −→D −→ .... from which we can deﬁne de Rham cohomology group and Dolbeault cohomology.

Now we still have the cochain complexes de Rham and Dolbeault sheaf resolutions

′ d ′ d ′ d ′ d 0 → R →D 0 −→D 1 −→D 2 −→D 3 −→ ..., (27)

61 p ∂ ′p,0 ′p,1 ∂ ′p,2 ∂ ′p,3 ∂ 0 → ΩX −→D →D −→D −→D −→ .... from which we can also deﬁne “de Rham cohomology” and “Dolbeault cohomology.” We want to prove that the new cohomology groups are isomorphic the old ones.

In (26) and (27), if we regard R as a sheaf of groups of constant real-valued functions, it gives two diﬀerent resolutions. 23 The question is whether or not these diﬀerent resolutions give the same cohomology group.

There are several ways to deﬁne cohomology groups for a sheaf and such groups are eventually all isomorphic. Let us discuss it as follows.

• Ceckˇ cohomology For any sheaf F of abelian group over a topological space, we have defined Ceckˇ cohomology group Hˇ p(X, F). • Resolution of flasque sheaves We also have introduced the notion of a flasque (or flabby) sheaf. 24 Any sheaf F on X admits a resolution 0 →F 0 →F 1 →F 2 → ... such that all sheaves F i, i ≥ 0, are flasque. If F is a sheaf of abelian groups over a manifold X, then F admits a resolution of flasque sheaves F →F • so that the j-th cohomology group Hi(X, F) of a sheaf F is the j-th cohomology of the complex is defined by Ker(φj : F j(X) →F j+1(X)) Hj(X, F) := . Im φj−1 : F j−1(X) →F j(X) That definition of cohomology is really independent of the chosen flasque resolution. If the manifold X is paracompact 25, then the above two groups are isomorphic: Hˇ p(X, F) ≃ Hp(X, F). • Resolution of acyclic sheaves Let F • be a resolution of a sheaf F by sheaves F •. We say that the resolution F • is acyclic on X if Hs(X, F q)=0 for all q ≥ 0 and s ≥ 0. The following theorem said that resolution of flasque sheaves can be replaced by acyclic sheaves.

23Recall that given a sheaf F over a manifold X, a resolution of F is a complex 0 → F 0 → F 1 → ... together with a homomorphism F→F 0 such that

0 →F→F 0 →F 1 →F 2 → ...... is an exact complex of sheaves. 24 Recall that a sheaf F is called flasque if for any open subset U ⊂ X the restriction map rU,X : F(X) → F(U) is surjective, i.e., every section of F on U can be extended to X. 25a paracompact space is a topological space in which every open cover admits an open locally finite refinement.

62 Theorem 8.1 (de Rham - Weil isomorphism theorem, [Demailly09], (6.4)) If F • is a resolution of a sheaf F by sheaves F • such that F • is acyclic on X, then there is a functorial isomorphism ≃ Hp(F •(X)) −→ Hp(X, F,d).

• Resolution of soft sheaves To determine which sheaf is acyclic, we introduce the deﬁnition: A sheaf F is called soft if the restriction Γ(X, F) → Γ(K, F) is surjective for any closed subset K ⊂ X, i.e., every section of F on a closed subset K can be extended to X.

Proposition 8.2 Soft sheaves are acyclic. Any sheaf of modules over a soft sheaf is soft and hence acyclic.

This is frequently applied to the sheaf of continuous (or diﬀerentiable) functions on a manifold which is easily shown to be soft. Notice that the sheaf of holomorphic functions on a complex manifold is not soft.

Example Let X be a real smooth manifold of dimension n. We consider the resolution of sheaves

d d 0 → R →E 0 −→E 1 → ... →E p −→E p+1 → ... →E n → 0.

0 ∞ p p Since E = EX = C (X) is soft, all sheaves E are EX -modules so that E are acyclic. Then de Rham cohomology groups of X are precisely

p R p • HDR(X, ) ≃ H (X, E ,d).

• Resolution of fine sheaves A fine sheaf over X is one with “partitions of unity,” i.e., for any open cover of the space X we can find a family of homomorphisms from the sheaf to itself with sum 1 such that each homomorphism is 0 outside some element of the open cover. Fine sheaves are usually only used over paracompact Hausdorff spaces X. Typical examples are the sheaf of continuous real functions over such a space, or smooth functions over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings. Fine sheaves over paracompact Hausdorff spaces are soft and acyclic.

63 de Rham and Dolbeault Theorems for currents A basic observation is that the Poincar´eand Dolbeault-Grothendieck lemmas still hold for currents. Namely, if (Dq,d) and ′ (D p,q, ∂) denote the complex of sheaves of degree q currents (resp. of (p, q) - currents), we still have De Rham and Dolbeault sheaf resolutions

′ d ′ d ′ d ′ d 0 → R →D 0 −→D 1 −→D 2 −→D 3 −→ ...,

p ′p,0 ∂ ′p,1 ∂ ′p,2 ∂ ′p,3 ∂ 0 → ΩX →D −→D −→D −→D −→ .... ′p Since the sheaves D are all EX -modules, they are acyclic so that we have the canonical isomorphisms

′q ′q+1 q q ′• Ker(d : D (X) →D (X)) H (X, R)= H (X, D ,d)= ′ ′ , (28) DR Im(d : D q−1(X) →D q(X))

′p,q ′p,q+1 ′ Ker(∂ : D (X) →D (X)) Hp,q(X)= Hq(X, D p,•, ∂)= . (29) Im(∂ : D′p,q−1(X) →D′p,q(X)) q R In other words, we can attach a cohomology class {Θ} ∈ HDR(M, ) to any closed current Θ of degree q, resp., a cohomology class {Θ} ∈ Hp,q(X) to any ∂ - closed current of bidegree (p, q).

26 ′ Historical remarks A current T ∈ D1(M) (i.e., we take test functions, instead of test forms) is called a distribution, or a generalized function. . Generalized functions were introduced by Sergei Sobolev in 1935. They were re-introduced in the late 1940s by Laurent Schwartz, who developed a comprehensive theory of distributions.

The ﬁrst origins of sheaf theory are hard to pin down they may be co-extensive with the idea of analytic continuation. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.

• 1936 E. Cechˇ associated a simplicial complex to an open covering. • 1938 H. Whitney gives a “modern” definition of cohomology. • 1943 N. Steenrod publishes on homology with local coefficients. • 1945 J. Leray publishes work carried out as a prisoner of war, motivated by proving fixed point theorems for application to PDE theory; it is the start of sheaf theory and spectral sequences.

26From en.wikipedia/weki/sheaf, and generalized function.

64 • 1947 H. Cartan reproves the de Rham theorem by sheaf methods, in correspondence with AndréWeil. Leray gives a sheaf definition in his courses via closed sets. • 1948 The Cartan seminar writes up sheaf theory for the first time. • 1950 The “second edition” sheaf theory from the Cartan seminar: the sheaf space (espace étalé) definition is used, with stalkwise structure. Supports are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time Kiyoshi Oka introduces an idea of a sheaf of ideals, in several complex variables. • 1951 The Cartan seminar proves the Theorems A and B based on Oka’s work. • 1953 The finiteness theorem for coherent sheaves in the analytic theory is proved by Cartan and Jean-Pierre Serre, as is Serre duality. • 1954 Serre’s paper Faisceaux algébriques cohérents (published in 1955) introduces sheaves into algebraic geometry. These ideas are immediately exploited by Hirzebruch, who writes a major 1956 book on topological methods. • 1955 A. Grothendieck in lectures in Kansas defines abelian category and presheaf, and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors. • 1956 Oscar Zariski’s report Algebraic sheaf theory. • 1957 Grothendieck’s Tohoku paper rewrites homological algebra; he proves Grothendieck duality (i.e., Serre duality for possibly singularities). • 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: schemes and general sheaves on them, local cohomology, derived categories (with Verdier), and Grothendieck topologies. There emerges also his influential schematic idea of ’six operations’ in homological algebra. • 1958 Godement’s book on sheaf theory is published. At around this time Mikio Sato proposes his hyperfunctions, which will turn out to have sheaf-theoretic nature.

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to algebraic topology. It was later discovered that the logic in categories of sheaves is intuitionistic logic. This shows that some of the facets of sheaf theory can also be traced back as far as Leibniz.