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Appendices 43

A Support of the projection 43

1 Introduction

Intersection in mathematics has a long history. But the systemically devel- oped theories only started appearing in the 20th century. They are centered around the quotient rings that are derived from freely generated Abelian groups of non-quotient objects. While the quotient rings fit into the known axiomatic system well, the non-quotient groups do not.

For instance in classical approach the topological intersection for a real compact manifold X in homology or/and cohomology (quotient groups) are obtained from the non quotient objects – singular chains. Once the theory is set up, quotient groups are well-adapted to the axiomatic environment, but the groups of singular chains are not. Using cohomology we can let Hi(X; Z) be the cohomology of degree i with integer coefficients. The product is defined in an elaborated method through the intersection of chains as the cup product, ∪ : Hp(X; Q) × Hq(X; Z) → Hp+q(X; Z). (1.1) The product gives a ring structure to the cohomology. Then the focus is shifted to the framework of rings which provides rich mathematical struc- tures. However such a ring structure does not exist on the groups of chains. So the groups of singular chains plays a supporting role behind the cohomol- ogy groups.

Another example is the intersection in algebraic geometry defined by William Fulton in [3]. Its non-quotient objects are algebraic cycles, which form an Abelian group Z. Let X be a smooth projective variety over a field. Let CH(X) be the Chow groups (quotient groups). Then there is an elaborated product map through the intersection of algebraic cycles, • : CH(X) × CH(X) → CH(X) (1.2) such that CH(X) is a ring. The study of algebraic cycles has been motivated by the structure of the Chow rings. So the non-quotient Z group plays a supporting role since it does not have the similar ring structure. 2 BASIC PROPERTIES 3

Both ring structures ∪, • are classically known to be compatible with each other and functorial between spaces. These quotioned intersections play im- portant roles in the more general cohomological theory that later generalizes both quotient rings in the derive category. They created rich algebraic struc- tures that flourish beyond the original transcendental geometry. However the flourishing has been one-sided focusing on quotient groups mainly through homological algebra. In this paper we try to look into the other direction which studies the non-quotient objects. Our direction shows that the global invariant – cohomology ring may contain non-trivial geometry without quo- tient in the local charts. More specifically, the intersection in cohomology which is considered to be a quotient object, is related to the geometric mea- sure in transcendental geometry, which is a non-quotient object. The study of non-quotient objects for quotient objects is the real intersection theory. On the technical side, we let X be a connected, oriented manifold of di- mension m. In [9], using local charts, we defined the subspace C(X ) called Lebesgue currents in the space of currents such that there is a bilinear ho- momorphism – the intersection of currents, [·∧·]: C(X ) × C(X ) → C(X ) (1.3) (T1, T2) → [T1 ∧ T2]. The intersection is extrinsic, for it depends on De Rham data consisting of a De Rham covering of X . But it satisfies an important intrinsic relation,

supp([T1 ∧ T2]) ⊂ supp(T1) ∩ supp(T2). (1.4) where the support is the focus of the theory. In this paper which is not self-contained, we continue [9] to establish sufficient properties for the appli- cations. It is a detailed demonstration that by adjusting the extrinsic data, the well-known operations on quotient objects can be carried over to non- quotient objects 1. As a conclusion we give a short but immediate application that proves the generalized Hodge conjecture on 3-folds.

2 Basic properties

Let X be a connected oriented manifold. Let T1, T2 be two arbitrary singular chains. Then T1 ∩ T2 may not be a singular chain. To catch the essence of 1Intrinsically it is impossible. 2 BASIC PROPERTIES 4 such an intersection we are going to use a tool — current, the notion created by G. de Rham in 1950’s, [2].

We denote the space of C∞ forms with a compact support by D(X ), and its topological dual – the space of currents by D ′(X ). If necessary we add a superscript(a subscript) to denote the codimension (the dimension) of currents. The functional of a current T is denoted by

(•). (2.1) ZT

Lemma 2.1. Let Z ⊂ X be a submanifold. Let

i Z ֒→ X be the inclusion map. Let

D(X , Z)= {φ ∈ D(X ): φ|Z =0}, (2.2)

∞ where φ|Z is the pullback of the C -differential form by the inclusion map. So D(X , Z) ⊂ D(X ). (2.3) Then the sequence of topological dual

i∗ R D ′(Z) → D ′(X ) → D ′(X , Z) (2.4) is exact, where ′ stands for the topological dual and R is the restriction map.

Proof. We may assume Z is compact. It is trivial that R ◦i∗ = 0. Let’s show

ker(R) ⊂ Im(i∗).

Let U be a tubular neighborhood of Z and j : U → Z be a projection induced from the normal bundle structure of U. Let h be a C∞ function on X such that it has a compact support in U and it is 1 on Z. For any T ∈ D ′(X ), we define a current T ′ on Z 2 BASIC PROPERTIES 5

(·)= hj∗(·). (2.5) ′ ZT ZT Let T ∈ ker(R). We would like to show

′ i∗(T )= T.

It suffices to show that for any testing form of φ on X

∗ hj (φ|Z)= φ, ZT ZT or ∗ hj (φ|Z) − φ =0. (2.6) ZT   ∗ Since hj (φ|Z) − φ vanishes on Z, the formula (2.6) holds. If Z is non- compact, we can use a partition of unity to have the same proof. We complete the proof.

In [9], we have developed the local and global calculations of intersection of Lebesgue currents. We refer the readers to [2] and [9] for the necessary background material. Let’s give a summary. It started with the De Rham’s regularization.

Theorem 2.2. (G. de Rham) Let X be a connected, oriented manifold. Let ǫ be a small positive number. ′ There are linear operators Rǫ and Aǫ on D (X ) satisfying (1) a homotopy formula

RǫT − T = bAǫT + AǫbT. (2.7)

where b is the boundary operator. (2) supp(RǫT ),supp(AǫT ) are contained in any given neighborhood of supp(T ) provided ǫ is sufficiently small. ∞ (3) RǫT is C ; r r (4) If T is C , AǫT is C . (5) If a smooth differential form φ varies in a bounded set and ǫ is bounded above, then Rǫφ, Aǫφ are bounded. 2 BASIC PROPERTIES 6

(6) As ǫ → 0, RǫT (φ) → T (φ), AǫT (φ) → 0 uniformly on each bounded set φ.

X X In this paper we use the notations Rǫ , Aǫ to replace De Rham’s notations Rǫ, Aǫ.

Definition 2.3. (for De Rham’s regularization) X (a) We call Rǫ from Theorem 2.2 the De Rham’s smoothing operator, X Aǫ from Theorem 2.2 the De Rham’s homotopy operator, and the associated regularization the De Rham’s regularization. (b) We define De Rham data to be all items in the construction

We continue to have

Theorem 2.4. (B. Wang 2019) Let X be a connected, oriented manifold of dimension m, equipped with a De Rham data. Let T1, T2 ∈ C(X ) such that dim(T1)+ dim(T2) ≥ m. ∞ (1) Then there is a family of C forms, the kernel ̺ǫ(x, y) on X × X where ǫ> 0 such that X Rǫ (T2)= ̺ǫ(x, y) (2.8) Zy∈T2 where the right hand side is defined by the fibre integral similar to the fibre integral of the projection X×X → X (1st copy) (x, y) → x.

Furthermore ̺ǫ(x, y) is a closed form, which in case with a compact X is Poincar´edual to the diagonal of X × X in the cohomology group.2 (2) There exists a subspace C(X ) of D ′(X ), defined in geometric measure theory, such that for T1, T2 ∈ C(X ) continuous functional

X φ → lim Rǫ (T2) ∧ φ ǫ→0 T1 ∈ Z ∈ (2.9) D(X ) R

2 In the language of [2], ̺ǫ(x, y) is homologous to the diagonal for each non-zero ǫ. 2 BASIC PROPERTIES 7 exists and lies in C(X ), denoted by

[T1 ∧ T2]. (2.10)

Proposition 2.5. Let X be a manifold equipped with a De Rham data. Let i : Z ֒→ X be a submanifold. Let T ∈ C(X ). Then there is a current denoted by [Z ∧ T ]Z in Z such that

i∗([Z ∧ T ]Z ) = [Z ∧ T ]. (2.11)

Proof. For any φ ∈ D(X , Z),

X φ = lim Rǫ (T ) ∧ φ =0. (2.12) ǫ→0 Z[Z∧T ] ZZ Then by Lemma 2.1, there is a current in Z satisfying (2.11).

2.1 Basic properties Property 2.6. Let X a connected, oriented C∞ manifold of dimension m. Assume it is equipped with a De Rham data. In [9], we defined the intersection of currents,

C(X ) × C(X ) → C(X ) (2.13) (T1, T2) → [T1 ∧ T2], on the subspace C(X ) ⊂ D ′(X ) consisting of Lebesgue currents. For all Lebesgue currents T1, T2, the intersection [T1 ∧ T2] has properties:

(1) (Supportivity)

supp([T1 ∧ T2]) ⊂ supp(T1) ∩ supp(T2). (2.14)

(2) (Closedness) The intersection current [T1 ∧ T2] is closed if T1, T2 are. 2 BASIC PROPERTIES 8

(3) (Graded commutativity) There is a graded-commutativity. If

deg(T1)= p,deg(T2)= q, then pq [T1 ∧ T2]=(−1) [T2 ∧ T1] (2.15) or equivalently

limlim Rǫ(T1) ∧ Rǫ′ T2 ∧ φ = limlim Rǫ(T1) ∧ Rǫ′ T2 ∧ φ. (2.16) ǫ′→0ǫ→0 ǫ→0ǫ′→0 ZX ZX for the test from φ ∈ D(X ).

(4) (Cohomologicity) Let X be compact. We use hT i to denote the co- homology class represented by a closed current T . If T1, T2 are closed, in H(X ; R), we have hT1i∪hT2i = h[T1 ∧ T2]i. (2.17)

Hence if the cohomology hT1i, hT2i are integral, so is h[T1 ∧ T2]i. (5) (Associativity) There is an associativity

[T1 ∧ T2] ∧ T3 = T1 ∧ [T2 ∧ T3] . (2.18)    

(6) (Leibniz rule) If bT1, bT2 are Lebesgue and deg(T1)= p, then

p d[T1 ∧ T2] = [dT1 ∧ T2]+(−1) [T1 ∧ dT2]. (2.19)

Proof. (1) The proof is in [9].

(2) Let φ be a test form. By the definition

φ Zb[T1∧T2]

= lim RǫT2 ∧ dφ (2.20) ǫ→0 ZT1

= ± dRǫT2 ∧ φ ZT1 2 BASIC PROPERTIES 9

According to the homotopy (2.7),

bRǫT2 − bT2 = bbAǫT2 − bAǫbT2 (2.21)

Because T2 is closed, bRǫT2 =0.

So [T1 ∧ T2] is closed.

(3) (Graded commutativity ). The proof is in [9].

∞ (4) Let φ be a closed C form of degree deg(T1) + deg(T2). Denote the intersection number in cohomology by int(·, ·). Hence the intersection number, int(h[T1 ∧ T2]i, hφi) (2.22) is a well-defined real number that equals to

lim Rǫ(T2) ∧ φ. (2.23) ǫ→0 ZT1 (by De Rham’s theorem) which is

2 deg (T1) (−1) lim Rǫ(T1) (2.24) ǫ→0 ZT2∧φ (by the community (3)). Then we use (2.24) as De Rham’s Kronecker index

T1 ∧ (T2 ∧ φ)[1] which is well-defined between the currents T1 and T2 ∧ φ. We obtain that

int(h[T1 ∧ T2]i, hφi)= T1 ∧ (T2 ∧ φ)[1] (2.25)

On the compact manifold, the Kronecker index only depends the cohomol- ogy classes (see section 20, chapter IV [2]). Hence (2.25) is the intersection number int(hT1i, hT2 ∧ φi) (2.26) which by the associtivity of the intersection in cohomology is the same as

int(hT1i∪hT2i, hφi) (2.27) 2 BASIC PROPERTIES 10

Since φ is any closed form,

hT1i∪hT2i = h[T1 ∧ T2]i. (2.28)

(5) Let degrees of Ti be pi. The intersection of Lebesgue currents is still Lebesgue. Therefore we have a triple intersection expressed as a functional on the test forms •. Then (•)

[T1∧T2]∧T3 R   (2.29)

= lim lim lim Rǫ1 (T1) ∧ Rǫ2 (T2) ∧ Rǫ3 (T3) ∧ (•). ǫ3→0ǫ2→0ǫ1→0 ZX By the commutitavity in Proposition 4.4 of [9],

(•)= (•) p (p +p ) T1∧[T2∧T3] (−1) 1 2 3 [T2∧T3]∧T1 R R     p1(p2+p3) =(−1) lim lim lim Rǫ2 (T2) ∧ Rǫ3 (T3) ∧ Rǫ1 (T1) ∧ (•) ǫ1→0ǫ3→0ǫ2→0 ZX (2.30)

= lim lim lim Rǫ1 (T1) ∧ Rǫ2 (T2) ∧ Rǫ3 (T3) ∧ (•) ǫ3→0ǫ2→0ǫ1→0 X Z = (•).

[T1∧T2]∧T3 R  

(6) (Leibniz Rule) Let φ ∈ D(X ) be a test form. Let

deg(T1)= p,deg(T2)= q. 2 BASIC PROPERTIES 11

Then

b[T1 ∧ T2](φ)

= lim RǫT2 ∧ dφ ǫ→0 ZT1 q q+1 = lim (−1) d(RǫT2 ∧ φ)+(−1) dRǫT2 ∧ φ ǫ→0 ZT1  

= lim RǫT2 ∧ φ + lim dRǫT2 ∧ φ ǫ→0 q ǫ→0 q+1 Z(−1) bT1 Z(−1) T1 ( bT1, bT2 are Lebesgue) = φ + φ q q+1 Z(−1) [bT1∧T2] Z(−1) [T1∧dT2] Hence

q q+1 b[T1 ∧ T2]=(−1) [bT1 ∧ T2]+(−1) [T1 ∧ dT2]. (2.31) After change the sign, we found (2.31) is the same as (2.19).

2.2 Product and Inclusion Intersection theory in algebraic geometry has functoriality. We study its extension to the real intersection theory.

Definition 2.7. (1) Let U1, U2 be the De Rham data for the manifolds X1, X2 respectively. 1 2 2 1 2 Let Bi ⊂ Ui, Bj ⊂ Uj be the De Rham covering from U1, U2 and fi , fj are the associated convolution functions. Then given an order of the De Rham’s 1 2 1 2 covering Bi × Bj ⊂ Ui × Uj of X1 × X2, we define the De Rham data on the product X1 × X2 by taking the direct product of given data in each manifold. 1 2 For instance, we define the convolution functions (fi , fj ) on each open set 1 2 Bi × Bj . We call it the product De Rham data. i j i j (2) Given (B1, B2) ⊂ (U1, U2 ) the product De Rham covering of X1 × X2 that has the given order denoted by

i1 j1 i2 j2 (U1 , U2 ), (U1 , U2 ), ··· , 2 BASIC PROPERTIES 12

we construct new De Rham data on X1 by setting ordered De Rham covering as i1 i2 i3 U1 , U1 , U1 , ··· and on X2 by setting the De Rham covering as

j1 j2 j3 U2 , U2 , U2 , ···

These new De Rham data are called projection De Rham data with respect to the product De Rham data.

The important implication of these new De Rham data is the following projection formula.

Proposition 2.8. (Projection formula) Let X1×X2 be two manifolds equipped with a product De Rham data, X2 equipped with a projection De Rham data. Let P2 : X1 ×X2 → X2 be the projection, and σ ∈ C(X2), and T ∈ C(X1 ×X2). Then (1) X1×X2 ∗ X2 Rǫ (X1 ⊗ σ)=(P2) (Rǫ (σ)). (2.32)

(2) Let X1 be compact. Then

[(P2)∗(T ) ∧ σ]=(P2)∗[T ∧ (X1 ⊗ σ)]. (2.33)

Proof. (1). Assume X1, X2 are equipped with De Rham data, U1, U2 respec- tively. Let’s give a product De Rham data to X1 × X2 and projection De Rham data to X1 and X2. Each chart in the product De Rham covering i j (U1, U2 ) gives a De Rham’s smoothing operators on X1 × X2, denoted by X1×X2,(i,j) j Rǫ . Each chart in the projection De Rham covering U2 gives a De X2,j Rham’s smoothing operator on X2, denoted by Rǫ . We claim for any σ ∈ D ′(X ), Claim 2.9. X1×X2,(i,j) X2,j Rǫ (1 ⊗ σ)=1 ⊗ Rǫ (σ) (2.34) on X1 × X2 − ∂(Bi × Bj). where 1 is the current X1. Let’s consider both sides of (2.34) restricted to a neighborhood of a point (qi, qj) (both sides globally are not continuous, ¯i ¯j but in a neighborhood they are). If (qi, qj) ∈/ B1 × B2. Then both sides 2 BASIC PROPERTIES 13

i j are restricted to the 1 ⊗ σ in the neighborhood. If (qi, qj) ∈ B1 × B2, let dim(X1 )+dim(σ) φ ∈ D (X1 × X2) be supported in the neighborhood.

φ X ×X 1 2,(i,j) ZRǫ (1⊗σ) X2,j = Rǫ (σ) ∧ φ(x1, x2) Zx1∈Bi Zx2∈Bj = φ X 2,j Z1⊗Rǫ (σ)

( we should note that the partial degree of φ(x1, x2) in x1 is maximal). Then we obtain that (2.34) holds on the

X1 × X2 − ∂(Bi × Bj).

i j Continuing from Claim 2.9, we glue each piece (U1, U2 ) by taking De Rham’s compositions of smoothing operators on both sides of (2.34) to obtain that

X1×X2 X2 Rǫ (1 ⊗ σ)=1 ⊗ Rǫ (σ) (2.35) on X1 × X2 − ∪i,j∂(Bi × Bj),

X2 where Rǫ is obtained from the projection De Rham data with respect to the product De Rham data. Now since both sides of (2.35) are C∞, by the continuity, (2.35) holds on the closure X1 × X2. This completes the proof of part (1). (2). Since X1 is compact, P2 is proper. Then the pushforward (P2)∗ of currents is well-defined. Let φ be a test form on X2. We use projection De Rham data on X2 and product De Rham data on X1 × X2 to find

X2 φ = lim Rǫ (σ) ∧ φ ǫ→0 Z[(P2)∗(T )∧σ] Z(P2)∗T ∗ X2 = lim P2 (Rǫ (σ) ∧ φ) ǫ→0 ZT X2 ∗ = lim (1 ⊗ Rǫ (σ)) ∧ P2 (φ) ǫ→0 ZT 2 BASIC PROPERTIES 14

(Use part (1))

X1×X2 ∗ = lim Rǫ (1 ⊗ σ) ∧ P2 (φ) ǫ→0 ZT ∗ = P2 (φ) Z[T ∧(X1×σ)] = φ Z(P2)∗[T ∧(X1×σ)] This completes the proof.

Proposition 2.10. ( reduction to the diagonal) Let X be a compact mani- fold. Let T1, T2 ∈ C(X ). Let X be equipped with a De Rham data U. We give the product De Rham data to X × X and the associated projection De Rham data to each copy X 3. With these De Rham data we have the reduction to diagonal, [T1 ∧ T2]=(P2)∗[(T1 ⊗ T2) ∧ ∆] (2.36) where P2 : X×X →X (2nd copy) is the projection, the left hand side of intersection occurs in the second copy of X , and ∆ is the diagonal.

Proof. With induced De Rham data there is the projection formula (Propo- sition 2.8), ′ [T1 ∧ T2]=(P2)∗[T1 ∧ (X ⊗ T2)] (2.37) ′ where T1 is a current in X × X such that

′ (P2)∗T1 = T1.

Now we claim Claim 2.11. with the projection De Rham data also on the first copy X ,

(P2)∗[(T1 ⊗ X ) ∧ ∆] = T1 (2.38)

(T1 ⊗ X ) ∧ (X ⊗ T2)= T1 ⊗ T2. (2.39) 3 Each copy X will have different De Rham data depending on the order of the product De Rham data. 2 BASIC PROPERTIES 15

For (2.38), we let φ ∈ D(X ). Then

φ Z(P2)∗[(T1⊗X )∧∆] ∗ = (P2) (φ) Z[(T1⊗X )∧∆] X ×X ∗ = lim Rǫ (T1 ⊗ X ) ∧ (P2) (φ) ǫ→0 Z∆ ( Use projection formula, Proposition 2.8)

∗ X ∗ = lim (P1) (Rǫ (T1)) ∧ (P2) (φ) ǫ→0 Z∆ ( where P1 : X×X→X (1st copy) is the projection then identify X ≃ ∆)

X = lim Rǫ (T1) ∧ φ ǫ→0 ZX = φ. ZT1 Hence (2.38) is true. For (2.39), we let φ ∈ D(X ). Then by the projection formula

X ×X Rǫ (X × T2)= RX (T2) (2.40) Then

φ Z(T1⊗X )∧(X⊗T2) X ×X = lim Rǫ (1 ⊗ T2) ∧ φ ǫ→0 ZT1⊗X X = lim 1 ⊗ Rǫ (T2) ∧ φ ǫ→0 ZT1⊗X (2.41) X = lim Rǫ (T2) ∧ ( φ(x1, x2) ǫ→0 ZX Zx1∈T1
= φ(x1, x2) ZT2 Zx1∈T1 = φ. ZT1⊗T2 2 BASIC PROPERTIES 16

Hence with projection De Rham data on both X and product De Rham data ′ on X × X , Claim 2.11 is true. Then in (2.37), we replace T by (T1 ⊗ X ) ∧ ∆ and apply the associativity and commutativity to have

[T1 ∧ T2]=(P2)∗ [(T1 ⊗ X ) ∧ ∆ ∧ (X ⊗ T2)]  

=(P2)∗ [(T1 ⊗ X ) ∧ (X ⊗ T2) ∧ ∆]   ( by (2.39))

=(P2)∗ [(T1 ⊗ T2) ∧ ∆] .   This completes the proof.

.Let i : Z ֒→ X be the inclusion of a submanifold of dimension n

Proposition 2.12. (associativity) There exist De Rham data UZ , UX on Z, X respectively such that for any cellular chain W ⊂ Z and σ ∈ C(X ),

i∗([W ∧ [σ ∧ Z]Z ]) = [i∗W ∧ σ] (2.42) where the notation for [σ∧Z]Z is defined in Proposition 2.5, and W, to abuse the notation, denotes the current ∈ D ′(Z).

Proof. Let j : E → Z be a tubular neighborhood of Z in X . Thus E is diffeomorphic to a vector bundle of rank r. We denoted the bundle also by E. Let U be a De Rham data for Z such that each De Rham chart Ui lies in the trivialization. So −1 r j (Ui) =: Ui × R (2.43) where =: denotes a fixed diffeomorphism. Let B ⊂ Rr be the De Rham chart in the step 3 of the construction of De Rham’s regularization in [9]. Then E is equipped with a De Rham data UE whose De Rham covering is 2 BASIC PROPERTIES 17

−1 j (Ui), all i. Then we extend UE (arbitrarily) to X to have a De Rham −1 data UX . We denote the smoothing operator associated to the chart j (Ui) X ,i D by Rǫ . Let ξ ∈ (X ) be a function that is 1 on W and has support in a neighborhood of Z inside of E. Let W′ = ξj−1(W) be the current ∈ D ′(X ). By the associativity, [W′ ∧ [Z ∧ σ]] = [[W′ ∧ Z] ∧ σ]. (2.44) Notice that with vector bundle structure, W′ in a neighborhood of W meets Z transversely at the open cells of W. By Proposition 3.1 below, ′ [W ∧ Z]= i∗W. so ′ [W ∧ [Z ∧ σ]] = [i∗W ∧ σ]. (2.45) Let φ ∈ D(Z). By using the partition of unity, we may assume W lies in a single trivialization −1 r j (Ui) =: Ui × R of E. Then we calculate

j∗(φ) ′ Z[[Z∧σ]∧W ] X X ′ ∗ = lim Rǫ (σ) ∧ Rǫ′ (W ) ∧ j (φ) (ǫ,ǫ′)→0 ZZ X X ,i1 X ,ih ′ ∗ = lim Rǫ (σ) ∧ Rǫ′ ◦···◦ Rǫ′ (W ) ∧ j (φ) (ǫ,ǫ′)→0 ZZ (where i1 < ··· < ih, and apply Claim 2.9 to the trivialization (2.43) )

X Z,i1 Z,ih ∗ = lim Rǫ (σ) ∧ (1 ⊗ Rǫ′ ◦···◦ Rǫ′ (W)) ∧ j (φ) (ǫ,ǫ′)→0 ZZ X Z ∗ = lim Rǫ (σ) ∧ (1 ⊗ Rǫ′ (W)) ∧ j (φ) (ǫ,ǫ′)→0 ZZ Z ∗ = lim (1 ⊗ Rǫ′ (W)) ∧ j (φ) ǫ′→0 (2.46) Z[Z∧σ] = j∗(φ) Z[[Z∧σ]∧i∗W] = φ Z[[Z∧σ]Z ∧W] 3 DEPENDENCE OF THE LOCAL DATA 18

Hence ′ [W ∧ [Z ∧ σ]] = i∗[W ∧ [Z ∧ σ]Z ]. Combining with (2.45), we complete the proof.

Definition 2.13. Any pair of De Rham data such as UZ , UX satisfying (2.42) for any W will be called the inclusion De Rham data.

3 Dependence of the local data

The intersection of currents is extrinsic because De Rham data is extrinsic. The real intersection theory stresses inseparable nature of “intrinsic” and “extrinsic”. Let’s look into the dependence of the De Rham data to see how much is intrinsic and how much is extrinsic. We begin with the real case.

3.1 Real case

It is well-known that on a manifold, if two sub manifolds meet transversally at another submanifold, then the intersection should be defined as the in- tersectional manifold. The more useful version is its extension in algebraic geometry. The following proposition says that the tranversal intersection is a particular case of the intersection of currents, where the dependence of De Rham data disappears.

Proposition 3.1. Let X be a manifold of dimension m. If T1, T2 are cells of real dimension p, q with p+q ≥ m, and the intersection T1 ∩T2 is transversal at a connected, manifold V of dimension p + q − m. We assume V at each point can be oriented concordant with T1, T2. Then [T1 ∧ T2] is independent of U. Furthermore it is the current of integration over V . 3 DEPENDENCE OF THE LOCAL DATA 19

Proof. Let’s set up the coordinates for the cells. Let X = Rm have linear basis e1, ··· , em and coordinates x1, ··· , xm. Set up the subspaces,

p R = span(e1, ··· , ep) q R = span(em−q+1, ··· , em) p+q−m R = span(em−q+1, ··· , ep)

p p Let T1 = ∆ ⊂ R be the polyhedron defined by p

{ |xi| < 1} (3.1) i=1 X q Similarly T2 = ∆ is defined by m

{ |xi| < 1}, (3.2) i=m−q+1 X V = ∆p ∩ ∆q is defined by p

{ |xi| < 1}. (3.3) i=m−q+1 X m Let πp+q−m : R → V be the projection. The proof has two steps.

1st step: Notice [T1 ∧ T2] has a compact support, hence [T1 ∧ T2] is also evaluated at the forms in C∞(Rm). Let φ ∈ C∞(Rm) have degree p + q − m. m p q Let ξ ∈ D0(R ) that is equal to 1 on a bounded neighborhood K of ∆ ∩ ∆ . We consider a C∞ form

∗ ξ(φ − πp+q−m(φ|V )) (3.4) ∞ which has a compact support in K and is a limit of C forms ψn ∈ D(K) compactly supported in K − V¯ as n →∞. Because [T1 ∧ T2] is supported on V , but ψn is supported outside of V ,

ψn =0 Z[T1∧T2] for all n. By the continuity of the functional [T1 ∧ T2],

∗ φ = πp+q−m(φ|V ). (3.5) Z[T1∧T2] Z[T1∧T2] 3 DEPENDENCE OF THE LOCAL DATA 20

∞ ∗ Recall that the C form πp+q−m(φ|V ) is called a local constant slicing in [9]. Therefore it is a closed form.

Now we apply the homotopy formula (2.7). Let φ ∈ D(Rm) such that supp(φ) ∩ ∂(∆p) ∪ ∂(∆q) = ∅.

′ ′ For arbitrary De Rham’s regularization Rǫ, Aǫ
with fixed sufficiently small real numbers ǫ1, ǫ2, we apply the homotopy formula (2.7) to have

∗ πp+q−m(φ|V ) (3.6) Z[T1∧T2] ∗ = πp+q−m(φ|V ) (3.7) [(bA′ T +A′ bT )∧(bA′ T +A′ bT )] Z ǫ1 1 ǫ1 1 ǫ2 2 ǫ2 2 ∗ + πp+q−m(φ|V ) (3.8) [R′ T ∧R′ T ] Z ǫ1 1 ǫ2 2 Now we calculate the first integral (3.7)

∗ πp+q−m(φ|V ) [(bA′ T +A′ bT )∧(bA′ T +A′ bT )] Z ǫ1 1 ǫ1 1 ǫ2 2 ǫ2 2 ∗ = lim πp+q−m(φ|V ) ǫ→0 ′ ′ ′ ′ [Rǫ(bA T +A bT )∧Rǫ(bA T +A bT )] Z ǫ1 1 ǫ1 1 ǫ2 2 ǫ2 2 (because supp(bTi) ∩ supp(φ)= ∅.)

∗ = lim πp+q−m(φ|V ) ǫ→0 ′ ′ [bRǫA T ∧bRǫA T ] Z ǫ1 1 ǫ2 2 ∗ = ±lim d(π − (φ|V )) → p+q m ǫ 0 ′ ′ [RǫAǫ T1∧RǫAǫ T2] Z 1 2 =0 This shows that

∗ ∗ πp+q−m(φ|V )= πp+q−m(φ|V ). (3.9) [T ∧T ] [R′ T ∧R′ T ] Z 1 2 Z ǫ1 1 ǫ2 2 We observe that the right hand side of (3.9) does not involve the De Rham’s smoothing operator Rǫ, thus the current [T1 ∧T2] is independent of the choice of De Rham data U. 4 4 ′ The technique of using the arbitrary Rǫ is due to De Rham §20, [2]. 3 DEPENDENCE OF THE LOCAL DATA 21

2nd step: To calculate the intersection [T1 ∧ T2]. By the 1st step, we can m choose a particular De Rham’s data U that has one chart x1, ··· , xm for R . Also we choose a C∞ convolution function f(x) supported in a neighborhood of a unit ball B satisfying

f(x)dµ =1. (3.10) m ZR x µ where dµ = dx1 ∧···∧ dxm. Let ϑǫ(x)= f( ǫ )d ǫ . Let κ : Rm × Rm → Rm (3.11) (x, y) → x − y,

Denote the coordinates (x1, ··· , xm−q) by x1, (xm−q+1, ··· , xp) by x2 and m xi+1, ··· , xm by x3. Similarly for the second copy of R in (3.11), the corre- sponding coordinates are denoted by y1, y2, y3 respectively. Let x x x y y y g( 1 , 2 , 3 , 1 , 2 , 3 )= κ∗(ϑ ). (3.12) ǫ ǫ ǫ ǫ ǫ ǫ ǫ Let φ ∈ D(Rm) be a test form. Then we calculate the current

φ = lim Rǫ(T2) ∧ φ → [T ∧T ] ǫ 0 T Z 1 2 Z 1 (3.13) x1 x2 y2 y3 x1 = lim g( , , 0, 0, , ) ∧ φ(ǫ , x2, 0) ǫ→0 ǫ ǫ ǫ ǫ ǫ ZT1 Z(y2,y3)∈T2

x1 ∞ where φ(ǫ ǫ , x2, 0) is a test form, i.e. C form on T1 with a compact support. Now applying the fibre integral to that over T1, we obtain

φ Z[T1∧T2] x1 x2 y2 y3 x1 = lim g( , , 0, 0, , ) ∧ φ(ǫ , x2, 0) ǫ→0 i+j−m m−j j ǫ ǫ ǫ ǫ ǫ Zx2∈R Zx1∈R Z(y2,y3)∈R (3.14)

Then we make a change of variables,

x1 ǫ → x1, y2 ǫ → y2 (3.15) y3 ǫ → y3. 3 DEPENDENCE OF THE LOCAL DATA 22

Then

φ Z[T1∧T2] x2 = ± lim g(x1, , 0, 0, y2, y3) ∧ φ(ǫx1, x2, 0) ǫ→0 i+j−m m−j j ǫ Zx2∈R Zx1∈R Z(y2,y3)∈R (3.16)

Then we notice for each fixed x2, the fibre integral

x2 g(x1, , 0, 0, y2, y3) (3.17) j m−j ǫ Zy2,y3∈R ,x1∈R by formula (3.10), is 1. Therefore we obtain that

[T1 ∧ T2](φ)= φ(0, x2, 0) (3.18) i+j−m ZR Thus

[T1 ∧ T2](φ)= V φ|V . (3.19) where φ|V is the restriction φ(0, x2, 0) of φR to V . We complete the proof.

For non transversal intersection, there is no classification in general sit- uation because the intersection may not have the notion of dimension. But for some specific examples, the notion of the dimension exists.

Proposition 3.2. (real excess intersection) The intersection of currents [T1 ∧ T2] depends on U.

Proof. We prove it by using an example. Let X = R2, and be equipped with the De Rham data consisting of single chart R2 with the convolution function f satisfying

f(x1, x2)dx1 ∧ dx2 = 1 (3.20) 2 ZR 2 where x1, x2 are Euclidean coordinates of R . Let T1 = T2 be the current of integration over the finite piece of the parabola

2 x1 = x2 (3.21) 3 DEPENDENCE OF THE LOCAL DATA 23

containing the origin 0. Since T1, T2 are singular chains, [T1 ∧ T2] exists. Let φ(x) be a test function with a compact support. Denote the second copy of 2 R for the De Rham’s regularization by y1,y2. Then we calculate

φ [T1∧T2] k R 1 x1 − y1 x2 − y2 lim f( , )φ(x1, x2)(dx1 − dy1) ∧ (dx2 − dy2) ǫ→0 ǫ2 ǫ ǫ Zx∈T1 Zy∈T2 (3.22) 2 2 substitute x1 = x2,y1 = y2 for T1, T2, we obtain that

φ [T1∧T2] k R 2 (x2 − y2)(x2 + y2) x2 − y2 lim f( , )φ(x1, x2)(x2 − y2)dy2 ∧ dx2. ǫ→0 ǫ2 ǫ ǫ Zx2∈R Zy2∈R (3.23) Next we make a change of the variables u = (x2−y2) ǫ (3.24) (v = x2 + y2. Then [T1 ∧ T2](φ) k ǫu + v ǫu + v lim uf(uv,u)φ(( )2, )dv ∧ du → (3.25) ǫ 0 u∈R v∈R 2 2 Z Z k v 2 v (u,v)∈R2 uf(uv,u)φ(( 2) , 2 )dv ∧ du. Then the functional R v v φ → uf(uv,u)φ(( )2, )dv ∧ du (3.26) 2 2 2 Z(u,v)∈R defines a current supported on T1. So the intersection current

[T1 ∧ T2]

(which is (3.26)) is supported on T1, depending on the convolution function f. 3 DEPENDENCE OF THE LOCAL DATA 24

Example 3.3. (real proper intersection) Let X = R2 be equipped with the De Rham data consisting of single chart R2 with the convolution function f satisfying

f(x1, x2)dx1 ∧ dx2 = 1 (3.27) 2 ZR 2 where x1, x2 are Euclidean coordinates of R .

Case 1: Let T1 be a line through the origin 0 and T2 is another line segment through the origin. Then it is known that

[T1 ∧ T2]= δ0 if the order matches with the orientation of R2. (for instance see Proposition 3.1).

Case 2: Continuing from the setting in case 1, let T2 be the line x1 =0. Let T1 be a pieces of parabola 2 x1 = x2, x2 ∈ (−1, 1). (3.28)

Let’s calculate [T1 ∧ T2]. Let φ(x) be a test function supported in a neigh- borhood of the origin. We denote the second copy of R2 for De Rhams’ regularization by (y1,y2). Then

φ T ∧T Z[ 1 2] (3.29) 1 x1 x2 y2 = lim f( , − )φ(x1, x2)dy2 ∧ dx1. ǫ→0 ǫ2 ǫ ǫ ǫ Zx1∈T1 Zy2∈R Let

f1(x1)= f(x1, −y2)dy2. Zy2∈R Now we continue (3. 29) to have φ [T1∧T2] k R 1 x1 lim f1( )φ(x1, x2)dx1 ǫ→0 ǫ ǫ (3.30) Z(x1,x2)∈T1 k 0 +∞ φ(0) +∞ f1(x1)dx + 0 f1(x1)dx1 =0,   R R 3 DEPENDENCE OF THE LOCAL DATA 25

So [T1 ∧ T2]=0 for all convolution function f in the De Rham data. This example shows the formula supp([T1 ∧ T2]) = supp(T1) ∩ supp(T2) does not hold for singular chains.

Case 3: Continuing from the setting in case 2, let T2 be the line x1 =0. Let T1 be a piece of the cubic curve 3 x1 = x2, x2 ∈ (−1, 1). (3.31)

The same calculation in case 2 shows if order of T1, T2 is concordant with orientation of R2, then φ = φ(0). (3.32) Z[T1∧T2] Hence

[T1 ∧ T2]= δ0 (3.33) where δ0 is the δ-function at the origin. So the intersection is independent choice of convolution function f in De Rham data.

Remark All three cases in Example 3.3 belong to the case of “real proper intersection” (not fully defined) which by the step 1 of Proposition 3.1 is independent of De Rham data. They also coincide with Kronecker index T1 ∧ T2[1] defined by De Rham. However the multiplicity associated to the “proper” components is different from that of the complex cases. Thus such an intrinsic problem of determination of the multiplicity has no satisfactory answer.

3.2 Complex case

Proposition 3.4. Let f : X → Y be a regular map between two smooth projective varieties. Let W be a p dimensional algebraic cycle of X. Then the current f∗[W ] is the current of integration over the cycle

f∗W, 3 DEPENDENCE OF THE LOCAL DATA 26 where [W ] stands for the current of integration over the algebraic set.

Proof. Let W = i aiWi where Wi are irreducible and ai are non-zero inte- gers. Let f∗W = biSi where bi are non-zero integers divisible by non-zero P i ai and Si are irreducible subvarities. Let |W0| be the open sets of the support P 0 0 |W | such that f is smooth. Then correspondingly f(|W0|)= ∪iSi , where Si are open sets of Si. By the definition of the push-forward of algebraic cycles, the map f : Wi → Si (3.34) is a finite to one morphism, and f is restricted to an e´tal morphism on −1 0 f (Si ). Then using currents, we have

−1 0 bi 0 f∗[f (Si )] = Si . (3.35) ai Taking the closure and the sum over i, we obtain that

f∗( ai[Wi]) = bi[Si]. (3.36) i i X X Since for algebraic cycles, we have

f∗( aiWi)= biSi, (3.37) i i X X we complete the proof.

Theorem 3.5. Let X be a smooth projective variety of dimension n over C. Let T1, T2 be irreducible subvarieties of X of dimension p, q. To abuse the notations, the currents of integration over them are also denoted by T1, T2 respectively. Assume T1 ∩ T2 is proper. Then with an arbitrary De Rham data U on X, the current [T1 ∧ T2] is independent of U, and equals to the current of integration over the algebraic cycle

T1 · T2, where T1·T2 is the Fulton’s intersection ([3]) defined as the linear combination of all irreducible components of the scheme

T1 ∩ T2. Furthermore the assertion extends to algebraic cycles linearly. 3 DEPENDENCE OF THE LOCAL DATA 27

Remark Theorem shows that [T1 ∧ T2] in this case is intrinsic. This is the principle for the real intersection theory: every intrinsically defined intersection in the past can be interpreted as an intersection of currents that are independent of De Rham data.

Proof. Let’s fix the cycle T2. By example 11.4.2, [3], there is an algebraic cycle E1 rationally equivalent to T1 such that A meets T2 transversely (at an open set of each irreducible support). Without losing the generality, let’s have a simplified setting as follows. Let V ⊂ X × P1 be an irreducible 1 subvariety, and P1 : V → X, P2 : V → P are the projections. Let T2 ⊂ X −1 be an irreducible subvariety. Assume the cycle of the scheme P2 (1) is E1 −1 1 and the cycle of the scheme P2 (0) is T1, where 0, 1 are two points of P . Let I be a real curve in P1 connecting 0, 1. Next we consider two objects: currents and algebraic cycles. Using the currents, according to Proposition 4.12 below in section 4, we have the formula

[T1 ∧ T2] − [E1 ∧ T2]= d (P1)∗(VI (T2)) (3.38)   where VI(T2) is the current

[V ∧ (T2 ⊗ I)]. (3.39)

Next we consider the algebraic cycles in intersection theory where two rationally equivalent algebraic cycles are homotopic. More precisely we have the equation in singular cycles

′ T1 · T2 − E1 · T2 = d (P1)∗(VI (T2)) , (3.40)   ′ where VI (T2) is the singular cycle defined by the semi-algebraic set in the 1 complex manifold X ×P , T1 ·T2, E1 ·T2 are singular cycles obtained from the triangulation of the intersectional algebraic cycles, and d is the differential operator on the singular chains ( the boundary operator with a sign). We ′ should note that the current of integration over VI (T2) is the current VI (T2). Next we convert the equation (3.40) to that in currents to have

[T1 · T2] − [E1 · T2]= d (P1)∗(VI (T2)) . (3.41)   3 DEPENDENCE OF THE LOCAL DATA 28

Hence [T1 ∧ T2] − [E1 ∧ T2] = [T1 · T2] − [E1 · T2] (3.42) (where the difference of two sides should be noticed). By Proposition 3.1, since E1 meets T2 transversely,

[E1 ∧ T2] = [E1 · T2]. (3.43) Thus [T1 ∧ T2] = [T1 · T2]. We complete the proof.

Theorem 3.6. Let X be a smooth projective variety of dimension n over C. Let T1, T2 be subvarieties of X of codimension p, q. The currents of integration over them are also denoted by T1, T2 respectively. Assume T1 ∩ T2 is an excess intersection. Then

[T1 ∧ T2] (3.44) in general depends on the De Rham data U.

Proof. Let’s give an example where [T1 ∧ T2] is dependent of De Rham data. 2 Let P be a projective space over C with affine coordinates (z1, z2). Let T1 be the hyperplane z2 = 0, and T2 = T1. First it is not zero because its reduction to cohomology group is non-zero. Choose two open sets as De Rham’s covering: U1, the finite affine plane, and a small neighborhood U2 of 1 2 the infinity P ⊂ P . Choose real Euclidean coordinates x1,y1, x2,y2 for U1 such that z1 = x1 + iy2, z2 = x2 + iy2. Use these open covering and Euclidean coordinates to have a De Rham data 2 for P with a convolution function h(x1, x2,y1,y2) of the unit ball B in U1. Then we see in U1, ′ ′ 1 1 x1 − x1 x2 y1 − y1 y2 ′ ′ Rǫ (T2)= − 4 h( , , , )dx1 ∧ dy1 ∧ dx2 ∧ dy2, ǫ (x′ ,y′ )∈R2 ǫ ǫ ǫ ǫ ZZ 1 1 (3.45) 3 DEPENDENCE OF THE LOCAL DATA 29

′ ′ where xi,yi are the Euclidean coordinates for the second factor in the smooth- ing operator. The composing with another local smoothing operator from 1 U2 will not change the smooth current Rǫ (T2) in B. Thus for a test form φ supported in B, the integral

B Rǫ (T2) ∧ φ = (··· )dx2 ∧ dy2 =0. (3.46) ZT1 ZZx2=y2=0 This shows with this type of De Rham data,

[T1 ∧ T2] (3.47) is zero on U ∩ T1. Hence [T1 ∩ T2] is a 0-dimensional current supported at 1 the infinity point of T1. Since the ∞ = P is arbitrary, [T1 ∧ T2] is supported on an arbitrary set determined by the De Rham data.

Example 3.7. The intersection of currents of integration over algebraic cycles always exists because algebraic cycles are Lebesgue. But its intersection depends on De Rham data. Theorem 3.5 asserts that in case of a proper intersection, it is actually independent of De Rham data. But for excess intersection, the situation still goes back to the dependence of De Rham data, and the interpretation is different from Fulton’s.

Table 1: Excess intersection in complex case Intersection Cycle Cycle class Support

algebraic T1 · T2 not well-defined well-defined in the Chow ring, T1 ∩ T2 and cohomology ring

current [T1 ∧ T2] well-defined, not well-defined in the Chow ring, T1 ∩ T2 but U-dependent well-defined in cohomology ring 4 TOOLS FOR APPLICATION 30

4 Tools for application

4.1 Correspondence of a current

Lemma 4.1. Let X , Y be two compact manifolds, and PX be the projection

X×Y→X .

Then the image of the projection

′ (PX )∗ : C(X ×Y) → D (X ) lies in C(X ).

Proof. Notice there is a coordinates chart of X ×Y satisfying that the coor- dinates planes of X are also the coordinates planes for X×Y. Thus the two conditions of Lebesgue currents for X are implied by that for X ×Y.

Definition 4.2. Let X , Y be two compact manifolds. Let F ∈ C(X ×Y) (4.1) be a Lebesgue current. Let U be a product De Rham data on X ×Y. Let PX ,PY be the projections

X×Y→X , X×Y→Y.

Define pull-back of currents

F ∗(T ) by F ∗ : C(Y) → C(X ) (4.2) T → (PX )∗[F ∧ ([X ⊗ T )].

Define the push-forward F∗(T ) of currents by

F∗ : C(X ) → C(Y) (4.3) T → (PY )∗[F ∧ (T ⊗ [Y])]. 4 TOOLS FOR APPLICATION 31

Proposition 4.3. Let X,Y,Z be three smooth projective varieties over C. Let FXY , FYZ be algebraic cycles on X × Y and Y × Z respectively, and represent finite correspondences ([6]). Then

(FYZ)∗ ◦ (FXY )∗ =(FYZ ◦ FXY )∗, (4.4) where FYZ ◦ FXY denotes the multiplication of algebraic correspondences. Similarly ∗ ∗ t t ∗ (FXY ) ◦ (FYZ ) =(FXY ◦ FYZ ) , (4.5) t t where FXY , FYZ are the transposes of the correspondences.

Proof. Because they are finite correspondences, hence

(FXY × Z) · (X × FYZ ) (4.6) is a well-defined algebraic cycle Γ in X × Y × Z. Let σ ∈ C(X). By Theorem 3.5, evaluations of both sides of (4.4) is equal to

(PZ)∗[(σ ⊗ Y ⊗ Z) ∧ Γ], (4.7) where PZ : X × Y × Z → Z is the projection. So (4.4) is proved. The (4.5) is the transpose of (4.4).

Proposition 4.4. Let X,Y be compact complex manifolds. The pull-back and push-forward of currents extend Gillet and Soul´e’s push-forward of cur- rents and smooth pull-back of currents.

Proof. In [4], Gillet and Soul´edefined operations on the currents on compact complex manifolds. They include push-forward for proper maps and pull- back for smooth maps. We verify that these operations coincide with ours.

Let f : X → Y (4.8) 4 TOOLS FOR APPLICATION 32 be a regular map. Let F be its graph. Let T be a Lebesgue current on X. Let φ be a C∞ form on Y . We use product De Rham data on X × Y and projection De Rham data on X and Y . Then

φ ZF∗(T ) X×Y ∗ = lim Rǫ (T × Y ) ∧ (PY ) (φ) ǫ→0 ZF (by Proposition 2.8, the projection formula )

∗ X ∗ (4.9) = lim (PX ) Rǫ (T ) ∧ (PY ) (φ) ǫ→0 ZF X ∗ = lim Rǫ (T ) ∧ f (φ) ǫ→0 ZX = f ∗(φ). ZT This shows F∗(T )= f∗(T ) where f∗ is defined as the dual of the pullback on forms in 1.1.4, ([4]).

Now let f : X → Y (4.10) be a smooth map. Let F ⊂ X × Y be its graph. Let φ be a test form on X.

φ = φ ∗ ZF (T ) Z(PX )∗[F ∧(X×T )] X×Y ∗ = lim Rǫ (X × T ) ∧ (PX ) (φ) ǫ→0 (4.11) ZF (by Proposition 2.8, the projection formula )

∗ Y ∗ = lim (PY ) (Rǫ (T )) ∧ (PX ) (φ). ǫ→0 ZF Notice PY : F → Y (4.12) 4 TOOLS FOR APPLICATION 33 is isomorphic to f which is also smooth. Then we apply the fibre integral to have

∗ Y ∗ lim (PY ) (Rǫ (T )) ∧ (PX ) (φ)= f∗(φ). (4.13) ǫ→0 ZF ZT Thus F ∗(T )= f ∗(T ). (4.14) We complete the proof.

Proposition 4.5. Let X , Y be two compact manifolds. Let F ∈ C(X ×Y) (4.15) be a homogeneous closed, Lebesgue current. (a) Let T be a Lebesgue current of X or Y. Then supp(F∗(T )) is contained in the set

PY supp(F ) ∩ (supp(T ) ×Y) ;   supp(F ∗(T )) is contained in the set

PX supp(F ) ∩ (X × supp(T ) .  

(b) If T1, T2 are Lebesgue and closed (rspt. homologous to zero) in X and Y respectively, then F∗(T1), F∗(T2) are also closed (rspt. homologous to zero).

Proof. (a) Let S be a Lebesgue current on X ×Y. Let a∈ / PY (supp(S)). Then there is a neighborhood Ba ⊂Y of a, such that

(X × Ba) ∩ supp(S)= ∅.

Then for any φ ∈ D(Y) supported in Ba,

∗ (PY ) (φ)=0. (4.16) ZS Then (4.16) says a∈ / supp((PY )∗(S)). 4 TOOLS FOR APPLICATION 34

Hence So supp((PY)∗(S)) ⊂ PY (supp(S)). (4.17) Similarly supp((PX )∗(S)) ⊂ PX (supp(S)). (4.18)

Now we consider our case. Applying the assertion (4.17) for

S = F ∧ (T ⊗Y), together with part (1), property 2.6,

supp((PY )∗[F ∧ (T ×Y)]) ∩

supp PY (supp([F ∧ (T ×Y)])) (4.19)   ∩

supp PY (supp(F ) ∩ (supp(T ) ×Y)) .   The proof of

∗ supp(F (T )) ⊂ PX supp(F ) ∩ (X × supp(T ) . (4.20)   is similar. (b) By property 2.6, the currents

[F ∧ (T1 ×Y)], [F ∧ (X × T2)]

∗ are closed. Therefore F T2, F∗T1 are closed. If they are homologous to zero, then by the property 2.6,

[F ∧ (T1 ×Y)], [F ∧ (X × T2)]

∗ are homologous to zero in X , Y. Thus F T2, F∗T1 are homologous to zero. We complete the proof 4 TOOLS FOR APPLICATION 35

Example 4.6. Let X,Y be two smooth projective varieties over C,

f : X 99K Y be a rational map. Then there is graph

F ⊂ X × Y. (4.21)

Once X × Y is equipped with De Rham data (which does not have any ∗ requirements for X,Y ), there are homomorphisms F∗, F

F∗ : C(X) → C(Y ) (4.22) F ∗ : C(Y ) → C(X).

∗ When C(X), C(Y ) are reduced to cohomology, F∗, F are reduced to the usual cohomological correspondences.

4.2 Functoriality

The real intersection theory is not functorial in the usual category of algebraic varieties. However if we attach De Rham data, then it has a functoriality.

Proposition and Definition 4.7. Let Cord(C) be the category whose objects are the pairs of a smooth projective variety over C, and a De Rham data on it, denoted by (X, U). The morphisms are finite correspondences of X × Y .

Proof. The verification of the category is done in [6].

Definition 4.8. Let k be a whole number. Let (X, U) ∈ Cord(C). Define NkC(X) to be the linear span of Lebesgue currents

T ∈ C(X) 4 TOOLS FOR APPLICATION 36 satisfying (I) dim(T ) ≤ k, OR dim(T ) ≥ 2n − k OR, (II) supp(T ) lies in an algebraic set A of dimension dim(T )+ k ≤ . 2

A current in NkC(X) will be called Nk leveled. The objects NkC(X) with usual group homomorphisms form a subcategory of the Abelian category, de- noted by NkC.

Remark De Rham data U plays no role in both categories Cord(C) and NkC. However the functoriality needs De Rham data.

Proposition and Definition 4.9. The maps

(X, U) → NkC(X) (4.23) and Hom in Cord(C) → Hom in NkC Γ → Γ∗, (covaraint) (4.24) Γ → Γ∗, (contrvariant) define a covariant functor and a contravariant functor.

Proof. Let (X, UX ), (Y, UY ) be two objects in Cord(C). Let F ∈ Z(X×Y ) be a homomorphism in Cord(C). The corresponding homomorphsim in category NkC is defined is defined to be F∗

σ → (PY )∗[F ∧ (σ ⊗ Y )] where σ ∈NkC(X). It suffices to show F∗ satisfies two conditions: (a) F∗ maps a Nk leveled current to a Nk leveled current. (b) The map satisfies the composition criterion, i.e. if X,Y,W are smooth projective varieties over C, and Z1,Z2 algebraic cycles are finite correspon- dences between X,Y and Y, W , then

(Z2 ◦ Z1)∗ =(Z2)∗ ◦ (Z1)∗ (4.25) 4 TOOLS FOR APPLICATION 37

where Z2 ◦ Z1 is the composition of finite correspondences. Proof of (a): By the definition of the intersection of currents, we obtain

deg(F∗(σ)) = deg(σ). Let A be an algebraic set containing σ such that the level of σ is

k = deg(σ) − 2degC(A). For any algebraic set A, since F is a finite correspondence,

deg(F∗(A)) = deg(A). (4.26)

The level of F∗(σ) is deg(F∗(σ)) − 2deg(F∗(A)) which is equal to k = deg(σ) − 2degC(A). The proof for contravariant F ∗ is similar. Proof of (b): This is Proposition 4.3.

Let κ : C(X) → D ′(X) (4.27) T → dT. Let B(X) = κ−1(C(X)) be the sublinear space that at least contain singu- lar chains and C∞ forms. By Leibniz rule, part (6) of Property 2.6, the intersection [·∧·] send B(X) × B(X) to B(X). So we let

NkB(X)= NkC(X) ∩ B(X).

Proposition 4.10. Then (a) N•B(X) forms a decreasing filtration of complex of B(X) (b) its spectral sequence E• converges to the R coefficiented, algebraically leveled filtration defined as

2dim(X)−k r 2r+k Nk(X)= N H (X),k =0, ··· (4.28) − r=dimX(X) k on the total real cohomology H(X; R) (See [8] for details on leveled filtration). 4 TOOLS FOR APPLICATION 38

Proof. (a). Let T ∈NkB(X). If dim(T ) ∈ [0,k + 1] ∪ [2n − k − 1, 2n], then by the definition T is Nk+1 leveled. If dim(T ) ∈ (k +1, 2n − k − 1), then dim(T ) ∈ (k, 2n − k). If dim(T ) is not in above cases, there is an algebraic set A such that dim(T )+ k supp(T ) ⊂ A, and dim(A) ≤ . 2 This implies that

dim(T )+ k +1 supp(T ) ⊂ A, and dim(A) ≤ . 2 So T ∈Nk+1B(X). This shows that

N0B(X) ⊂···NkB(X) ⊂Nk+1B(X) ⊂···⊂B(X) form a filtration. Since the differential d on the differential form preserves the support, d maps NkB(X) to NkB(X). Thus (N•B,d) is a decreasing filtration of the complex. (b) Because B(X) includes singular chains, the limit of the spectral se- quence, p p+q Gr (H (N•B(X))) p,q X is a filtration on the cohomology

H(X; R).

Notice that by the definition, this filtration is the algebraically leveled filtra- tion.

4.3 Family of currents 4 TOOLS FOR APPLICATION 39

Definition 4.11. Let S and X be manifolds equipped with De Rham data. Let S × X be equipped with the product De Rham data. Let I ∈ C(S×X) be a homogeneous Lebesgue current. Let PX be the projection S ×X →X . We denote (PX )∗[({s} ⊗ X ) ∧ I] (4.29) by Is. The set {Is} for all such s in S will be called a family of of currents parametrized by S.

Remark We should note that we have abused the notation to use PX for its restriction PX |• to a subset. The family Is depends on extrinsic De Rham data.

Proposition 4.12. Let X be a manifold. Let S be real one dimensional. Let Iǫ ⊂ S be diffeomorphic to a finite closed interval of R with two end points 0 and ǫ> 0. Let S be equipped with De Rham data, S × X be equipped with a product De Rham data and X be equipped with corresponding the projection De Rham data. Let J be a Lebesgue current S × X . (4.30) Then for a closed current T ∈ C(X ), there is a well-defined current

JIǫ (T ) on R × X such that (1)

[Jǫ ∧ T ] − [J0 ∧ T ] kp p (4.31) =(−1) d((PX )∗JIǫ (T ))+(−1) [(Iǫ ⊗ T ) ∧ dJ ] where k = deg(J ),p = deg(T ), PX : R ×X →X is the projection. (2)

(PX )∗JI (T )= ǫ(PX )∗JI (T )) ǫ 1 (4.32) [(Iǫ ⊗ T ) ∧ dJ ]= ǫ[(I1 ⊗ T ) ∧ dJ ],

5 and [Jǫ ∧ T ] is continuous in ǫ ( in the topology of currents) . 5 The continuity for higher dimensional parameter space S does not hold 4 TOOLS FOR APPLICATION 40

Proof. Let R × X be equipped with a product De Rham data. We defined

JIǫ (T ) = [J ∧ (Iǫ ⊗ T )]. (4.33) Then we calculate

dJIǫ (T ) ( by Leibniz Rule.) p+1 = [d(Iǫ ⊗ T ) ∧J ]+(−1) [(Iǫ ⊗ T ) ∧ dJ ] p+1 = [({ǫ} ⊗ T −{0} ⊗ T ) ∧J ]+(−1) [(Iǫ ⊗ T ) ∧ dJ ] ( by associativity and commutativity, Property 2.6.) kp p+1 =(−1) [(({ǫ}−{0}) ⊗ X ) ∧ J ∧ (R ⊗ T )]+(−1) [(Iǫ ⊗ T ) ∧ dJ ] (4.34) By Definition 4.11,

(PX )∗[ ({ǫ}−{0}) ⊗ X ∧J ] = [Jǫ] − [J0]. Thus applying the projection formula, Proposition
2.8, we obtain

(PX )∗[ ({ǫ}−{0}) ⊗ X ∧ J ∧ (R ⊗ T )] = [Jǫ ∧ T ] − [J0 ∧ T ]. (4.35) Combination of (4.34),
(4.35) is the assertion (4.31). We complete the proof. (2) We apply the construction (4.33). Then (4.32) follows from the equal- ity of currents, Iǫ = ǫI1.

Finally by part (1) and formula (4.32) the family [Jǫ ∧ T ] is continuous in ǫ.

Example 4.13. Let X be a smooth projective variety of dimension n over C. Let T be a closed Lebesgue current representing a non-zero primitive cohomology class in Hn(X; Q). Let V ⊂ P1 × X. (4.36) be a Lefschetz pencil in X. Assume P1 × X is equipped with a product De Rham data. Let I = [V ∧ (P1 × T )]. (4.37) be the intersection current. Then its fibre currents It are exact for all t. But I is not because T is not. 5 GENERALIZED HODGE CONJECTURE ON 3-FOLDS 41

Example 4.14. Let X be a manifold and T a non-zero homogeneous Lebesgue 1 current in X . Let S × X be equipped with a product De Rham data. Let t0 1 be a point of S . Then I = {t0} ⊗ T gives a family of currents by Definition 4.11. Notice It = 0 for all t including t0, but I is non-zero, and not closed provided T is not.

5 Generalized Hodge conjecture on 3-folds

We give an application in complex geometry.

Theorem 5.1. Generalized Hodge conjecture is correct on a 3-fold X.

Proof. Let X be a smooth projective variety over C. In the cohomology vector space with rational coefficients, we denote the coniveau filtration of coniveau i and degree 2i + k by

N iH2i+k(X)

(defined in [5] where the coniveau filtration is called the arithmetic filtration) and the linear span of sub-Hodge structures of Hodge coniveau i and degree 2i + k by M iH2i+k(X). The generalized Hodge conjecture asserts that

M iH2i+k(X)= N iH2i+k(X). (5.1) for all existing i, k. For a 3-fold X, the only non-trivial cases are

M 1H2(X)= N 1H2(X)? (5.2) M 1H3(X)= N 1H3(X)? (5.3)

The formula (5.2) is well-known as Lefschetz (1, 1) theorem. For (5.3), Delign’s lemma 8.2.8, [1] already implies that

N 1H3(X) ⊂ M 1H3(X). 5 GENERALIZED HODGE CONJECTURE ON 3-FOLDS 42

Thus it is sufficient to prove

M 1H3(X) ⊂ N 1H3(X). (5.4)

Let L ⊂ H3(X; Q) be a sub-Hodge structure of coniveau 1. So it is polar- ized. In [7], Voisin used the intermediate Jacobian to prove a cohomological result that says there is a smooth projective curve C, and a Hodge cycle

Ψ ∈ Hdg4(C × X) (5.5) such that 1 Ψ∗(H (C; Q)) = L. (5.6) where Ψ∗ is defined as the image

hP i∗ Ψ ∪ ((•) ⊗ 1)) , (5.7)   of the cohomological map hP i with the projection P : C × X → X. Next we convert the cohomological expression (5.7) to current’s expression as

P∗ T ∧ ((•) ⊗ 1)) , (5.8)   where T is any current representing Ψ. (The formula (5.8) is the key turning point of the proof that has no difference from (5.7) in cohomology. However it carries the information of the support which is absent in (5.7)). Next we analyze the current T . Notice hP i∗ is a Hodge morphism, so hP i∗(Ψ) is a Hodge cycle in X. By the Lefschetz (1, 1) theorem hP i∗(Ψ) is algebraic on X. So there is a singular cycle TΨ on C × X representing the class Ψ such that the projection in currents satisfies

P∗(TΨ)= S + bW (5.9) where S is a current of integration over the algebraic cycle S, and bW is an exact Lebesgue current of dimension 4 in X. (adjust Ψ so S is non-zero). Consider another current in C × X

T := TΨ − [e] ⊗ bW (5.10) 43 denoted by T , where [e] is a current of evaluation at a point e ∈ C. Note T is Lebesgue. By adjusting the singular chain W continuously, we can assume the projection of the support of T satisfies

P (supp(T )) = supp(P∗(T )), (5.11) i.e. the projection of the support is the support of the projection. (See appendix for the proof). Thus we have the projection of currents

P∗(T )= S. (5.12) Let Θ be the collection of closed Lebesgue currents on C representing the classes in H1(C; Q). Now we apply the real intersection theory to establish the correspondence of currents ( in section 4.1),

T∗(Θ) (5.13) defined as (5.8). It gives a family of currents (parametrized by Θ) supported on the support of the current

P∗(T )= S, which is the integration over an algebraic cycle S, i.e. the the family of currents are all supported on the algebraic set |S|. This is a criterion for coniveau filtration in terms of currents, i.e. for β ∈ T∗(Θ), the cohomology class hβi of β satisfies

hβi ∈ ker H3(X; Q) → H3(X −|S|; Q) (5.14)   By Proposition 4.5 and Voisin’s assertion (5.6), the collection of cosmo- logical classes of the currents in T∗(Θ) consists of all classes in L. This shows L ⊂ N 1H3(X). We complete the proof.

Appendices

A Support of the projection

In this Appendix, we study the supports of cellular cycles in a Cartesian product . A SUPPORT OF THE PROJECTION 44

Let X be a compact manifold of dimension n. We use the following setting in algebraic topology. A p-singular simplex S consists of three elements: a p dimensional polyhedron ∆p in Rv, an orientation of Rv, and a C∞ map f of Rv to X. A chain is a linear combination of singular simplexes. The support |S| of S is the image of S in X. A point in S is a point in |S|. Let Y be another compact manifold of dimension m. Let

P : Y×X → X (A.1) be the projection.

Definition A.1. Let σ be a C∞ p-singular simplex of Y × X . Let a be an interior point of σ. If P−1 ◦ P(a) ∩ σ is a finite set, we say σ is finite at a. If σ is finite at all interior points of σ, we say σ is finite to X . The chain is finite if each simplex in the chain is finite.

Proposition A.2. For any C∞ p-singular simplex σ in the coordinates chart of Y × X with p ≤ dim(X ), there is barycentric subdiviosn (multiple times) of σ

Sd(σ)= Ci, (A.2) finite i X such that each simplex Ci is homotopic to another simplex finite to X and the homotopy is a constant on the ∂Sd(σ)

Proof. Let Rm, Rn, Rm × Rn be the coordinate’s charts for Y, X , Y × X respectively such that p ≤ n. We would like to show that there is a multi barycentric subdivision to divide N σ to a chain i=0 σi (a sum of smaller regular cells σi) such that there are ′ n ′ homotopy σ for each σi that is finite to R , and boundary of σ is the same i P i as that of σi. We use a claim to construct such small simplex σi. A SUPPORT OF THE PROJECTION 45

Claim A.3. Let g : Rk → Rl be a C∞ map with l ≥ k. Let q ∈ Rk be a point. Then there is an open ball B of q and continuous map g′ : Rk → Rl. such that 1) g is homotopically deformed to g′ such that at all points on ∂B and the boundary D of the unit ball g is fixed under the homotopy, 2) g′ in B\D is C∞ and finite to one to its image in Rl.

k Proof. of Claim A.3: Let θ1, θ2 be two analytic functions on R such that θ1 = ǫ, θ2 = 0 define ∂B and D where ǫ is the radius of B. We consider the homotopy

(1 − t)g + t(g + θ1θ2h), t ∈ [0, 1]. (A.3) where h is some C∞ function. Thus g is homotopic to

g + θ1θ2h (A.4)   The determinant of a maximal minor of the differential J of

g + θ1θ2h (A.5)   ∞ is a polynomial in θ1, θ2 whose coefficients are C functions of h. Thus for a small ǫ, by choosing a suitable h, the determinant is non-zero for all points in B with θ2 =6 1 and θ1 =6 ǫ, i.e. the differential J has full rank. By mean value theorem

g + θ1θ2h   is 1-to-1 to its image when restricted to B\D. It satisfies required conditions in Claim A.3.

Now let f be the composition of

(∆p)′ →Y×X →X where (∆p)′ is a neighborhood of ∆p . Next we cover ∆¯ p with finitely many balls Bi, i = 1, ··· ,l and the homotopy in Claim A.3 for each Bi. Consider p ′ the first open set B1. Applying Claim A.3, f is homotopic to f1 : (∆ ) → X such that the homotopy fix the map f on ∂B1 and D and f is homotopic REFERENCES 46

to f1 which is finite-to-one on B1. Then we repeat the homotopy from f1 to f2, from f2 to f3, ··· , from fl−1 to fl. Finally, we obtain a continuous map fl which is finite-to-one in each Bi and is equal to f on D. Let Ci be the barycentric subdivisions obtained from the covering Bi, i =1, ··· ,l. Then fl is homotopy to f. We complete the proof.

Proposition A.4. For any cellular cycle S in Y × X, of dimension p < dim(X), S is homopotic to a cycle finite to X.

Proof. Let

S = Ci (A.6) i X and each cell Ci satisfies Proposition A.2 with a homotopy hi. Then there is synchronized homotopy with the same parameter t ∈ [0, 1] such that Ci is ′ homotopic to another cell Ci 1-to-1 to X , but the boundary is fixed. Since the boundaries are not changed, this synchronized homotopy are glued together to yield a homotopy of the cycle S,

′ ′ S = Ci. (A.7) i X

References

[1] P. Deligne, Th´eorie de Hodge: III, Publ. Math IHES 44 (1974), pp 5-77.

[2] G. de Rham, Differential manifold, English translation of “Vari´et´es diff´erentiables”, Springer-Verlag (1980).

[3] W. Fulton, Intersection theory, Springer-Verlag (1980).

[4] H. Gillet, C. Soule´, Arithmetic intersection theory, Publications Math´ematiques de l’IHES,´ Volume 72 (1990), p. 93-174. REFERENCES 47

[5] A. Grothendieck, Hodge’s general conjecture is false for trivial rea- sons, Topology, Vol 8 (1969), pp 299-303.

[6] C. Mazza, V. Voevodsky, C. Weibel, Lecture Notes on Motivic Cohomology, Clay Mathematics Monographs (2000).

[7] C. Voisin, Lectures on the Hodge and Grothendieck-Hodge conjectures, Rend.Sem., Univ. Politec. Torino, Vol. 69, 2(2011), pp 149-198.

[8] B. Wang, Leveled sub-cohomology , Preprint, 2018

[9] B. Wang, Real intersection theory (I) , Preprint, 2020

Department of Mathematics, Rhode Island college, Provi- dence, RI 02908 E-mail address: [email protected], Fax: 1-401-456-4695