Regulator Maps for Higher Chow Groups via Current Transforms
Pedro F. dos Santos Robert M. Hardt Paulo Lima-Filho






We show how  to use equidimensional algebraic correspondences between complex algebraic varieties to construct pull-backs and transforms of  certain  classes of geometric currents. Using  this construction we produce explicit formulas at the level of complexes for a regulator map from the Higher Chow groups of smooth complex quasi-projective  algebraic varieties to Deligne-Beilinson cohomology with integral coefficients. A distinct aspect of our approach  is the use of Suslin's complex  n ↦^p_Δ, eq(X,n)  of equidimensional cycles over Δ^n to compute Bloch's higher Chow groups. We calculate explicit  examples involving the Mähler measure of Laurent polynomials.



A review of swarmalators and their potential in bio-inspired computing
    Christian Bettstetter
    December 30, 2023
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§ INTRODUCTION

Using general principles  S. Bloch shows in <cit.> the existence of  natural cycle maps c pnX→^2p-n(X, p), from the higher Chow groups of a smooth complex algebraic variety X into any bigraded cohomology theory ^*(-, ∙) that: 
[1)]

  * satisfies homotopy axiom,
 
  * admits functorial cycle classes [Υ] ∈^2b_Υ(X,b) for subvarieties Υ⊂ X of pure codimension b ,  and
 
  * satisfies a weak purity property. 
 
 
 In particular, this shows the existence of  a regulator map with values in Deligne-Beilinson cohomology.  



The primary goal of this paper is to provide a structured  and explicit construction - at the level of complexes - of a regulator   map

    pnX⟶2p-npX,

from the higher Chow groups of a smooth complex algebraic variety X,  in their simplicial formulation and   coefficients  into integral Deligne-Beilinson cohomology. 

A distinct aspect of our approach  is the use of the complex  n ↦^p_Δ, eq(X,n) consisting  of cycles equidimensional over Δ^n
to compute Bloch's higher Chow groups. Here we rely on  Suslin's generic equidimensionality results  <cit.>, which imply that the inclusion into  Bloch's higher Chow complex ^p_Δ, eq(X,*) ↪^p_Δ(X,*) is a quasi-isomorphism (under mild conditions).

To compute  the Deligne-Beilinson  cohomology, in the case where X is projective, we use cone complex 

    (p)^*_(X) := 𝖢𝗈𝗇𝖾{(p)^*(X)  ⊕ F^p*X*X}[-1],

where  (p)^*(X) denotes the group of  integral currents, with (p) coefficients, and F^p *X denotes the Hodge filtration on the de Rham currents in X (see Appendix <ref>).

An element  γ∈(p)^k_(X) is represented as a triple 


    γ = ( T, θ, ϖ) ∈ (p)^k(X)⊕ F^p'𝒟^k (X)   ⊕k-1X,

whose differential is then given by 
dγ = (d T,  d θ,  θ - ϵ(T) - d ϖ ),
where ϵ is the inclusion ϵ(p)^*(X) ↪*X.

The regulator map must therefore associate to a  cycle Υ∈^p_Δ, eq(X,n) a triple 

    ( Υ)  = Υ∈(p)^2p-n(X) ×  F^p2p-nX×2p-n-1X,

such that
( ∂Υ)  = ( d Υ_, d Υ_Θ,  Υ_ - Υ_Θ - d Υ_W ),
 where ∂Υ denotes the boundary in the higher Chow groups complex.


In order to define the triple ( Υ) we first  introduce  geometrical constructions with currents that have independent interest. When translated to the equidimensional complex these associate  to  a codimension p  cycle  Υ  on X×Δ^n,
which is equidimensional over the algebraic simplex Δ^n a transform homomorphism Υ^∨^k( ^n ) →^k+2(p-n)( X), between groups of  currents defined by integration. We apply this construction to define  

    ( Υ) = 
    ( (-1)^n2(-2π)^p  Υ^∨_n,     (- 2π)^p-n Υ^∨_Θ_n,   (- 2π)^p-n Υ^∨_W_n),

 
where n  denotes the  degree n current defined by integration on the topological simplex n⊂Δ^n ={ [ z_0 : … : z_n ] ∈^n | z_0 + … + z_n ≠ 0 } (with the standard orientation)
, Θ_n is the current represented by the meromorphic form  ∑_i=0^n (-1)^i dz_0/z_0…dz_i/z_i…dz_n/z_n  in ^n and W_n is a degree n-1 current relating n and Θ_n (see Section <ref>).

The properties of equidimensional cycles make the transform of currents into a seamless operation yielding the desired map of complexes.
In particular, contrasting with other constructions of the cycle maps from the higher Chow groups to Deligne Beilinson cohomology  (<cit.>, <cit.>),
no moving lemmas are needed in our construction and the resulting homomorphism  (<ref>) is defined with  coefficients.

Below we summarize the content of each section of the paper.


We start with a brief recollection in Section <ref> of  the notions of equidimensional and relative algebraic cycles, stating the results from <cit.> that are relevant in our constructions. Then we introduce the complexes we use to define Deligne-Beilinson cohomology, along with a glossary of the currents and forms that we use. For the reader's convenience, we provide in Appendix <ref> a brief review of geometric measure theory and detailed references. 


















The technical core of the paper lies in Section <ref>, where we use algebraic correspondences to construct pull-backs of  currents:
if X is a smooth connect variety, B is a smooth variety  and Υ⊂ X × B is a codimension p  subvariety, which  is dominant  over B, 
we show in Proposition <ref> the existence of pull-back maps Υ^#kB→k+2iX× B,
where, as above,  kB denotes the currents of degree k in B that are representable by integration (measure coefficients).

Furthermore, Υ^# sends currents of type (p,q) to currents of type (p+i, q+i). 

Under appropriate conditions (Proposition <ref>.<ref>) – e.g., Υ equidimensional over B – the pull-back Υ^# S of a locally normal current S coincides with the intersection of currents Υ∩{ X × S}; and d(Υ# S) = Υ^# (dS) (Corollary <ref>). 

When B  is proper one can define a transform Υ^∨kB→k+2(i-n)X by Υ^∨ S = pr_1# ( Υ^# S). where pr_1  X× B → X is the projection. 








As mentioned above, this transform with B = ^n is used in  the definition of the regulator map of a projective variety X. 

If  U is quasiprojective,  let U↪ X ↩ D = X-U be a projective compactification of U with a simple normal crossings divisor D.
If Υ⊂ U ×Δ^n is equidimensional over Δ^n = ^n - H_∞, we show that the constructions   above induce a transform 

    Υ^∨k^n→k-2nXlog D ,

and we study its behavior with respect to hyperplanes H ≠ H_∞ and boundaries. 
In particular we show the following.





















Corollary <ref>. 
Using the  notation in Definition <ref>, the following holds. 

      
  * Given a smooth hypersurface H⊂^n,   H≠ H_∞, denote H = H  ∩Δ^n. Then 
Υ_|H    and  (  Υ_|  H) induce the same transform
   
    Υ_|H^∨ = (  Υ_|  H)^ ∨  kH→k+2(i-n)XD .

   
  * If S is a current in ^n
   vanishing suitably at H_∞ (see Definition <ref>) then the identity
   d ( Υ^∨_S ) =  Υ^∨_dS   
   holds in k+2(i-n)+1XD.














In Section <ref> we introduce a fundamental triple of currents (n, Θ_n, W_n) in complex projective space ^n = ^n()^an, with the analytic topology. The construction starts with a special  nested sequence of closed semi-algebraic subsets R_n,0⊂ R_n,1⊂⋯ R_n, n = ^n, which are suitably oriented to define semi-algebraic chains R_n,j∈n+j^n. The current R_n,0  corresponds to the natural orientation of the topological simplex n := Δ^n(_≥ 0) ⊂Δ^n()^an≡^n - H_∞, where H_∞ = { [] |_n() := z_0 + ⋯ + z_n = 0 } is the hyperplane at infinity. 




Next, for 0≤  j≤ n, denote  θ^n_j := ∑_r=0^j (-1)^r 0∧⋯∧r∧⋯∧j ∈ Ω^j(^n)log D_j, where D_j is the divisor given by z_0⋯ z_j = 0,  and define 

    ω^n_j := (-1)^j log( 1 - _j()/z_j) ∧θ^n_j-1,

where _j() := z_0 + ⋯ + z_j. In Proposition <ref> and Corollary <ref> we 
exhibit formulas for the boundaries of both [θ^n_j] and R_n,jω_j^n, and show that they define normal currents in ^n
(i.e., both the currents and their boundaries are representable by integration). 

With these preliminaries in place, we define the fundamental triple 

    (n, Θ_n, W_n)  ∈ n^n⊕ F^n n^n⊕n-1^n,

where  Θ_n ∈ F^n n^n denotes the current in ^n represented 
by θ^n_n,

and
W_n is the normal current
W_n := ∑_j=1^n (-1)^j(j+1)/2 (-2 π)^n-j  R_n,jω_j.
For all n≥ 0, the fundamental triple satisfies the  following identity, shown in  Corollary <ref>):

    d W_n     =  (-1)^n(n+1)/2Θ_n - (2π)^n n -  ∑_r=0^n (-1)^r ι_r#( W_n-1) ,

where ι_r ^n-1↪^n denotes the inclusion of the r-th coordinate hyperplane. 

We conclude the section by establishing that the currents in the fundamental triple satisfy the conditions of Corollary <ref> with respect to algebraic cycles in a product X×^n which are equidimensional over Δ^n. This amounts essentially to having a controlled vanishing at infinity. 





In Section <ref> we use the constructions the previous sections to define our  map of complexes 

    𝒵^p_eq(U;n)    ⟶Γ( X; (p)_, (X,U)^2p-n),

where 𝒵^p_eq(U;n) is the Bloch-Suslin (chain) complex of equidimensional cycles and
(p)_, (X,U)^2p-n is a complex of acyclic sheaves computing the Deligne-Beilinson homology of U. More precisely, if Υ⊂ U×Δ^n lies in 𝒵^p_eq(U;n) then
(Υ) =  Υ, where

    Υ_  :=        (-1)^n2(-2π)^p  (Υn)∩U    ∈     ^2p-n_loc(U; (p))      
    Υ_Θ  :=       (- 2π)^p-n ΥΘ_n    ∈      F^p 2p-nXD   
    Υ_W  :=      (-2π)^p-n ( ΥW_n)∩ U    ∈     2p-n-1U .

Here, ^2p-n_loc denotes the sheaf of locally integral currents, and F^p *XD denotes the Hodge filtration on the currents in X with logarithmic pole along D. See Appendix <ref>.








In the last section we calculate the regulator map in a  family of examples,  first introduced as Example <ref> to illustrate equidimensional cycles. 
There, starting with a subfield ⊂, we consider  a (Laurent) polynomial  (  ) ∈[ ] of degree d in n-variables
and introduce  a correspondence Γ_ in U_×Δ^n+1, where  U_:= { ()^n - Z_}× with
Z_⊂ ()^n   the zero set of (  ).
We show  that Γ_ is an equidimensional correspondence in n+1n+1U_ which is, in fact,  a cycle in 
 n+1n+1U_, thus representing an element in CH^n+1_Δ(U_, n+1). 
 
We show that - under simple conditions on the polynomial  - the class [  (Γ_)] ∈n+1n+1U_ has a non-trivial “transcendental” component coming from H^n+1(U_, )/H^n+1(U_, )
in the exact sequence
 
    0→  H^n(U_, )/H^n(U_, (n+1))  →n+1n+1U_
    → H^n+1(U_, (n+1))⊕ F^n+1H^n+1(U_, ) →⋯ .


Evaluating the resulting homomorphism 
γ H_n(U_, ) →/(n+1)
(<ref>) on a particular homology class 
we obtain

    -  ^n() ∈/(n+1),

where () is the logarithmic Mähler measure of the polynomial .

The polynomial _α(x, y) = α  + x +1/x + y + 1/y, with α > 4 satisfies the conditions needed for the calculations. For example, when 
 α=8 one obtains
γ ( _2×{ 1 }) =  
- ^2 (_8) = 96   L(E_24, 2) ≠ 0 ∈/(3),
where L(E_24,z) is the L-series of the rational elliptic curve E_24  of conductor 24.










§ PRELIMINARIES



In this section we  recall the necessary properties of equidimensional cycles, and introduce the notation for the forms and currents that are used throughout the paper. We conclude the section recalling Deligne-Beilinson cohomology. 


 


 §.§ Complexes of equidimensional cycles


Equidimensional  cycles over a base space play a key role in the study and applications of algebraic cycles. Examples include the development of morphic cohomology <cit.>  and the alternative presentation of motivic complexes in <cit.>. Here we summarize  Suslin's generic equidimensionality results <cit.> and <cit.>, the key ingredients in our applications. 




Let → B be a scheme of finite type over a noetherian base scheme  B and assume that  is irreducible, with =d and B= n .
An algebraic cycle  W = ∑ n_i W_i  on  is said to be dominant over B if  each W_i is dominant over a component of B.  It  is called equidimensional of relative dimension r if for each s∈ B and each component W_i of W, the fiber  (W_i)_s  is either empty or each of its components has dimension r .

We denote by qB the group of algebraic cycles of codimension q in  that are dominant and equidimensional over B, of relative dimension r= d-n-q.



The following result summarizes the key properties of dominant equidimensional cycles that are relevant to this discussion



<cit.>

Let W be an equidimensional dominant cycle on  of relative dimension r over a base scheme B.

   
  * If B  is normal (or geometrically unibranch) then W is a relative cycle of relative dimension r.
   
  *     If B is regular, then W is a universally integral relative cycle. Hence,  for each map f T → B, there exists a unique  relative cycle (with integral coefficients) W_|T on ×_B T over T, such that for every point  t ∈  T, the pullback t^*(W_|T) to _t agrees with f(t)^*W.   
 







  * The cycle W_|T is called the pullback of W along f. 

  * It follows from the construction that whenever W_|T is non-zero, then it is  a dominant equidimensional cycle over T.

  *  If V  T  B are maps between regular schemes then W_|V = ( W_|T )_|V.

  * If T⊂ B is a closed immersion of regular schemes and W is an equidimensional cycle on X over B, then the pull-back cycle W_|T coincides with the image of W under the intersection-theoretic  pull-back homomorphism induced by  T×_B X→ X as explained in <cit.> and <cit.>.





  §.§.§ Simplicial groups of equidimensional cycles



Let  X be an equidimensional scheme of finite type over 𝕜,  and  denote  pnX := p{X×Δ^n}Δ^n. It follows from Theorem <ref>.<ref> that the assignment n ↦pnX defines a simplicial subgroup p∙X⊂p∙X, with associated chain complex   p*X.





  §.§.§ Generic equidimensionality


The next result can be seen as a general moving lemma that  has   geometric/measure-theoretic consequences in characteristic zero, yielding a natural  construction of the regulator maps. 

<cit.>

Let X be an equidimensional quasi-projective scheme of finite type over 𝕜. Then the inclusion map
p*X↪p*X
is a quasi-isomorphism whenever p≤X.



The condition p≤X imposes no restriction when addressing Higher Chow Groups in general. Indeed,  the homotopy property and general equidimensionality give natural quasi-isomorphisms:

    p*X[rr]_-htpy. inv. ^-≃      p*X×^p      @_(->[ll]_-≃^-Thm.<ref>  p*X×^p.







Let ⊂ be a field, and consider 
a polynomial  (  ) ∈[ ] of degree d in n-variables
 = (t_1, …, t_n),  and 
let Z_⊂ ()^n  be its zero set. 

Denote U_:= { ()^n - Z_}×, with coordinates (;λ), and  let 
 = (z_0, … ,z_n+1) ∈Δ^n+1 be coordinates satisfying 
 ∑_r=0^n+1 z_r =1.
 
Define a correspondence Γ_ in U_×Δ^n+1 by

    Γ_ := 
    
    z_n+1( λ + ( ) )     = λ
    
    z_0 -  t_1z_1      = 0 
    
    z_0 + z_1 - t_2 z_2      = 0 
    ⋮   ⋮
    
    z_0 + z_1 + ⋯ + z_n-1 - t_n z_n     = 0
    .

We claim that Γ_   is an equidimensional correspondence in n+1n+1U_ and a cycle in 
 n+1*U_, thus representing an element in CH^n+1_Δ(U_, n+1).

To prove this claim, consider the linear forms 

    _j() = z_0 + ⋯ + z_j ,    for    j≥ 0,
 
and define the auxiliary polynomial

    R_() = (z_1 ⋯ z_n)^d( _0()/z_1, _1()/z_2, …, _n-1()/z_n) ∈[] .

Now, let Y_⊂Δ^n+1 be the divisor given by (z_0 z_1 ⋯ z_n+1)(_0() ⋯_n()) R_() =0, 
and observe that Γ_∩ (U_× Y_) = ∅. 

It follows that, over Δ^n+1 - Y_,
the correspondence Γ_ is the graph of the map

    ψΔ^n+1 - Y_   ⟶  U_
       ⟼( (); z_n+1/1- z_n+1(()) ),

where () = 
( _0()/z_1, _1()/z_2, …, _n-1()/z_n) .
 
We conclude that Γ_ is an equidimensional correspondence in n+1n+1U_. By definition,  the faces ∂_i Δ^n+1, i=0,…, n+1, are contained in Y_, and this suffices to show that  Γ_ is a cycle in 
 n+1*U_. 






 §.§ Forms, currents and Deligne cohomology




  §.§.§ Glossary: Forms and currents
 

In Appendix <ref>  the reader will find the  definitions and relevant properties of the  objects listed below, along with detailed references. 

Here, M is a smooth oriented manifold of dimension m, X is a smooth proper algebraic variety, and D is a  divisor in X with simple normal crossings.


l c p6.0cm| l

Notation     :     Description     References 
 

kM     :     Complex-valued differential forms of degree  k  on M.        <ref> 


kM     :      Compactly supported differential forms of degree k  on M.       <ref>  


pqX     :     Smooth forms of type (p,q)  on X.      <ref> 


     :     Sheaf of holomorphic functions on complex manifolds.       


Ω^p(X)     :     Holomorphic forms of degree p  on X.      


pXD⊂Ω^p_X      :      Subsheaf consisting of the holomorphic  p-forms that vanish on D.     <ref>.<ref> 


pqXD     :     =  pXD⊗^0,q_X     Defn. <ref>.<ref> 


^p,q_c(X)𝗇𝗎𝗅𝗅  D     :     = Γ_c(X,pqXD ) 


kM     :     Bounded Baire forms  of degree  k.      Defn. <ref>.<ref>  


kM     :     Continuous  forms of degree  k  on M.      Defn. <ref>.<ref> 


kM     :     Compactly supported continuous  forms of degree  k  on M.      XXXXXX  



kM     :     DeRham currents of degree k  on M.      <ref>


pqX    :     Currents of type (p,q) on X.    <ref> 


kM     :      Currents representable by integration.    <ref>.<ref>  


T     :      Measure associated to T ∈kM.    <ref>.<ref>  


kXD     :     Currents of degree k on X with log poles along D.     Defn. <ref> 


pqXD     :     = p+qXD∩pqX.    


F^p^k_X     :     = ⊕_p ≤ r ≤ k_X^r,k-r, Hodge filtration on sheaves of smooth forms.     


F^p '_X^k     :     = ⊕_p ≤ r ≤ k'_X^r,k-r, Hodge filtration on sheaves of currents on X.     


kM, kM     :     Integral and locally integral currents of degree k  on M, respectively.     <ref>.<ref>   


_loc^k(M;G)     :     Locally integral chains of degree k on  M with coefficients in G.     <ref>.<ref>



^k_loc(p)(M)     :     =  _loc^k(M;(p))      


kX     :     L^1_loc-forms of degree k on M.     Defn. <ref>.<ref>    


kM, kM     :     Normal and locally normal currents of degree k  on M.      <ref>.<ref> 


T, f, y     :      Slice of a normal current T on M by a map f  M→ N at the point y ∈ N.     <ref> 
 







  §.§.§ Deligne-Beilinson cohomology


Given a subring 
A⊂ and p∈, denote A(p) := ^p · A ⊂. For a topological space X, let A(p)_X denote the locally constant sheaf on X with values in A(p).

Let U = X-D @^(->[r]^-    X    @_(->[l]^- D,  be a good compactification of a smooth complex algebraic variety U, where X is a smooth proper variety and D is a DNC.
Denote by ϵ A(p)_U →Ω_U^* and 
ι : F^pΩ_X^*log D →_*Ω_U^* the natural inclusions. 






The Deligne-Beilinson complex   of (X, U, ) is defined as

    A(p)_ := A(p)_, (X,U) := 𝖢𝗈𝗇𝖾(  R_* A(p)_U ⊕ F^pΩ_X^*log D   R_*Ω^*_U )[-1].

The hypercohomology of this complex is independent of the good compactification  (up to canonical isomorphisms), and one defines the Deligne-Beilinson cohomology of U as 
H^k_(U, A(p)) := ^k( X, A(p)_, (X,U)),  k ≥ 0.
We refer the reader to <cit.> for further details.



We work primarily with A = in this paper, but  the arguments hold  for arbitrary subrings A ⊂.
Since the sheaves
'𝒟_U^k, '𝒟_X^k log D and  (p)^k_U,loc are acyclic,
we use the quasi-isomorphisms
Ω^*_U '𝒟_U^* and 
(p)_U (p)^*_U,loc ,
and the filtered quasi-isomorphism
( Ω^*_Xlog D , F^*) ( '𝒟_X^* log D ,   F^* ),
to obtain our preferred acyclic resolution of the Deligne-Beilinson complex. In particular, we 
use the identification


    H^k_(U; (p)) 
    = H^k(
    𝖢𝗈𝗇𝖾{^*_loc(p)(U)  ⊕ F^p*XD*U}[-1]
    )

to represent  an element γ∈(p)^k_(U) as a triple 


    γ = ( T, θ, ϖ) ∈ (p)^k_loc(U)⊕ F^p'𝒟^k (X)log D ⊕k-1U.

The differential d  (p)^k_(U) →(p)^k+1_(U) 
is then given by 

    dγ = (d T,  d θ,  θ_|U - ϵ(T) - d ϖ ),

where ϵ is the inclusion ϵ(p)^*_loc(U) ↪*U,
and θ_|U the restriction of  θ to U.




For simplicity, we often use T instead of  ϵ(T) when considering a locally integral current T simply as a De Rham current. 














§ CORRESPONDENCES AND TRANSFORMS



In this section,  algebraic correspondences are used to construct pull-back homomorphisms and transforms on certain classes of currents. We first present the main results and  applications before exhibiting their proofs, starting with  the key result.



Let X be a connected, smooth projective variety and let B be a smooth variety, with X =m and B = n. 
An  irreducible subvariety Υ⊂ X × B  of codimension i which is dominant over B  induces  a pull-back homomorphism on currents represented by integration

    Υ^#  kB⟶k+2iX× B,     k ≥ 0.

Furthermore, if S is a current of type (r,s) then Υ^#S has type (r+i, s+i).


This theorem is proven in Section <ref>. 
Next, we explain that the  pull-back homomorphisms Υ^# behave particularly well when the currents satisfy appropriate conditions with respect to Υ.




Fix a Riemannian metric on B. Given a smooth subvariety H ⊂ B,  let H⊂ W^τ be a tubular          τ-neighborhood with smooth boundary ∂ W^τ,  τ >0.  We say that a normal current S  on B vanishes suitably along H  when

   
  * S(H) = dS(H) = 0; 
   
  * The intersection S∩ [∂ W^τ] exists for all τ sufficiently small, and 
 S∩ [∂ W^τ] converges weakly to zero as τ goes to zero. 




If  Υ is a correspondence as in Theorem <ref>, then  there exists a closed subset F ⊂ B with F≥ 2 such that Υ_| B -F := Υ∩{ X× (B-F) } is equidimensional and dominant over B-F. The next result  is proven in Section <ref>. 



Assume that  Υ⊂ X × B satisfies the conditions of   Theorem <ref>.

 
  *  Let
S be a locally normal current in B whose support is contained in B - F, the domain over which Υ is equidimensional. Then the intersection of  currents Υ∩(  X × S ) –  in the sense of <cit.> – exists and satisfies

    Υ^#S  =  Υ∩(  X × S ) .



  *  Assume that B is also proper and  that the exceptional set F is contained in a smooth subvariety 
H ⊊  B.  If  S is  a normal current on B that vanishes suitably along H
then

    d( Υ^# S ) =  Υ^#(dS).

In particular, Υ^#(S)  is a normal current. 




When Υ is not dominant over B we set  Υ^# to be the zero map, so that  the pull-back operation gives rise to a  pairing

    𝒵^i(X× B) ⊗kB⟶k+2iX× B,   σ⊗ S ⟼∑_r n_r Υ^#_r S,

where σ = ∑_r n_r Υ_r is an algebraic cycle of codimension i in X× B.  




Let B be   a proper variety. The  projections X  X × B  B  induce   transform homomorphisms
associated to an algebraic cycle σ∈^i(X× B):


    σ  kB   ⟶k+2(i-n)X,    
    S ⟼σS := π_1#( σ^#S).














If S is locally integral (respec. sub-analytic, semi-algebraic) and S⊂ B-F, then Υ^#S  and  Υ^∨_S  are also locally integral (respec.  sub-analytic,  semi-algebraic).



We now study the pull-back and transform homomorphisms on quasiprojective varieties over algebraic simplices.  More precisely, we start with a smooth quasiprojective variety   U  of dimension m and let Υ⊂ U ×Δ^n be an irreducible subvariety of codimension i in U×Δ^n, whose projection onto Δ^n is dominant and has equidimensional fibers.

Consider  a projective compactification 
U↪ X ↩ D, where D = X-U is a divisor with simple normal crossings, and let Υ⊂ X ×^n be the closure of Υ in X×^n.
The next result is proven in Section <ref>. 





Let Υ⊂ U×Δ^n be as above. Then Υ induces a well-defined pull-back  homomorphism

    Υ^#k^n⟶k+2iX×^n{D×^n},    S ↦ Υ^#S,

satisfying the following properties.

   
  *  Let H⊂^n be a smooth hypersurface H≠ H_∞. Then 
   the correspondences Υ_|H and (  Υ_|  H) in X× H induce the same pull-back homomorphisms
   
    ( Υ_|H)^#   =    (  Υ_|  H)^ #  kH→k+2iX× H{D × H} .

   
  *  If S is a current in ^n vanishing suitably at H_∞ (see Definition <ref>) then the identity
   d ( Υ^# S )  =  Υ^#  (dS)   
   holds in k+2i+1X×^n{D ×^n}.




Using the proposition above we can define transforms in the quasiprojective case. 




Let Υ⊂ U×Δ^n be equidimensional over Δ^n,  and let U↪ X be a compactification of U with NCD D = X - U.  If  π_1  X×^n → X is the projection, define the transform 

    Υ^∨k^n→k+2(i-n)Xlog D

as the homomorphism S ↦Υ_S^∨ := π_1#Υ^#(S).





Using the  notation in Definition <ref>, the following holds. 

      
  * Given a smooth hypersurface H⊂^n,   H≠ H_∞, denote H = H  ∩Δ^n. Then 
Υ_|H    and  (  Υ_|  H) induce the same transform
   
    Υ_|H^∨ = (  Υ_|  H)^ ∨  kH→k+2(i-n)XD .

   
  * If S is a current in ^n
   vanishing suitably at H_∞ (see Definition <ref>) then the identity
   d ( Υ^∨_S ) =  Υ^∨_dS   
   holds in k+2(i-n)+1XD.

 


The rest of this section is devoted to constructing  the pull-backs and transforms discussed above, and to proving these and other results  that may have an  independent interest. 

The reader mostly interested in the regulator maps can skip the rest of this section and proceed to Section <ref> with no loss of continuity in the narrative.



 §.§ Integration along the fiber



We first recall the main features of the classical construction on an oriented fiber bundle (E, π, B, F) over an oriented n-manifold B with compact r-dimensional fiber F.  The integration along the fiber 
homomorphisms
πr+pE→pB, p≥ 0
are completely characterized by the following. 

Given α∈*B and ϕ∈*E one has:

   
  Projection formula π( π^*α∧ϕ)   =  α∧πϕ
   
  Fubini theorem ∫_E π^*α∧ϕ =  ∫_B α∧πϕ


These homomorphisms can be used to define a pull-back map on currents, as the adjoint operation π^#  kB→kE that sends S ∈kB to the current π^# S defined by π^# S ϕ↦ S(π_!ϕ). Using this pull-back  one can extend classical synthetic constructions in algebraic geometry to currents. As an example we have the algebraic join of algebraic cycles, a key ingredient in the homotopy-theoretic applications  studied in <cit.>, <cit.>, <cit.> and related work. 


In the study of cycle maps for Lawson homology <cit.>,  the algebraic join operation on algebraic cycles was extended to  a complex join  of currents. The main idea is to write  ^n+1 as a direct sum ^n+1 = V ⊕ W and let 𝐁 denote the blow-up of ^n at
(V⊕ 0 ) ⨿(0⊕ W). Since  𝐁 is a ^1-bundle over (V)×(W), it comes with a blow-down map 
b 𝐁→^n and a bundle projection 
π𝐁→(V) ×(W).
The pull-back map π^# '^k((V)×(W)) →'^k(𝐁)   preserves algebraic cycles, semi-algebraic chains, normal and integral currents.  We can now define a pairing

    #_  r(V)×s(W)⟶r+s(V⊕ W)

that sends (R,S) to R#_ S := b_#π^# ( R × S). 





 §.§ Generalized integration along the fibers and current pull-backs



Our next step is to extend integration along the fibers to a broader context, at the expense of restricting the domain of the corresponding pull-back homomorphism of currents. 


Start with  a smooth projective variety X,  and let B be an arbitrary smooth variety, with X = m and B=n. Consider an irreducible subvariety Υ⊂ X × B of codimension i which is  dominant over B. 

Assume that X and B are connected, 
and consider the Zariski closed subset 

    F= F_Υ := { b ∈ B |π^-1_2(b) > m-i }⊂ B.
 
It is clear that the codimension of F is greater or equal than 2. 




Given Υ⊂ X × B and F as above
one can define for all k≥ 0 an integration along the fiber homomorphism 

    Υ_!   kX× B⟶k-2(m-i)B,

sending continuous forms on X× B to bounded Baire forms on B,
so that for all ϕ∈*X× B  the following holds.

   
  * Υ_!ϕ is continuous on B-F.

   
  * If ϕ lies in kX× B, then there is a dense Zariski open subset V ⊂ B-F on which Υ_!ϕ is a smooth form.

   
  *  
   (Fubini) The form Υ_!ϕ represents the current π_2#( Υϕ). In other words, for all β∈*B one has

    ∫_B Υ_!ϕ∧β =  Υ( ϕ∧π_2^* β).

In particular, Υ_!ϕ represents a normal current when ϕ is smooth.

   
  *  
   (Projection formula) For any α∈*B one has Υ_!(ϕ∧π_2^* α) = Υ_!(ϕ)∧α.

   
  * 
   If ϕ  is a form of type (p,q) then Υ_!ϕ has type (p+i-m, q+i-m ).
   
  *  There is a constant λ >0, depending on Υ, such that for a compact subset K ⊂ B  the following inequality holds:

    Υ_! ϕ_K ≤λ 𝐌_K(Υ) ϕ_X× K,

where 𝐌_K(Υ) := 𝐌(χ_X× KΥ), and χ_X× K is the characteristic function of X× K.








Let π_1  X × B → X and π_2  X × B → B denote the projections and let π'_1 Υ→ X and π'_2 Υ→ B 
denote the compositions π_1∘ and π_2∘, respectively. 


Step I: Smooth forms


Start with a smooth form ϕ∈kX× B. 
Given a resolution of singularities p Υ→Υ,  let ρΥ→ B be the composition ρ = π_2 ∘ (∘ p) =  π_2'∘ p. The constructions that follow are summarized in the following commutative diagram.

    Υ[dl]_-p[ddl]^(.3)ρ|![rd];[ld]      Υ_|V = ρ^-1(V)  @_(->[ll]^-[ddl]^(.3)ρ@..>[dl]  
    
     
    X × B [dr]_π_2   Υ@_(->[l]_-[d]_(0.35)π'_2      Υ_|V  = π_2'^-1(V) [d]_(0.35)π_2'@_(->[ll]_
       B       V@_(->[ll]^-


By generic flatness and  smoothness, one can find a Zariski open (dense) V ⊂ B such that ρΥ_|V→ V is smooth and π'_2  Υ_|V→ V is flat. 





Since algebraic-geometric smooth maps are submersions, it follows from Ehresman's fibration theorem that - in the analytic topology - they are smooth fiber bundles, in the differential geometric sense. 




Therefore, given ϕ∈*X× B, we can define a form ρ̂_!(ϕ) on V by

    ρ̂_!(ϕ) =
    ρ_!{  (∘ p)^* ϕ}_|ρ^-1V .

where ρ_!{  (∘ p)^* ϕ}_|ρ^-1V is obtained as integration along the fiber of a smooth fiber bundle projection.




The form ρ̂_!(ϕ) gives a well defined germ at the generic point of B. In other words,  given any two resolutions of singularities the resulting forms agree on a non-empty Zariski open.  

To prove the claim, recall that  any two resolutions are dominated  by a third one, and hence it suffices to assume that the resolution p' Υ' →Υ factors through pΥ→Υ.  Hence we have a proper birational isomorphism πΥ' →Υ so that 
p' = p ∘π.

    Υ'_|V' @^(->[r]^-[dd]_ρ'   Υ'  [ddr]_ρ'[rr]^π[dr]^-p'      Υ[dl]_-p[ddl]^ρ   Υ_|V@_(->[l]^-[dd]^ρ
          Υ[d]|(0.35)π'_2   
    V' @^(->[rr]^-      B       V@_(->[ll]^-


Let ∅≠V' ⊂ B be a Zariski open so that Υ'_|V' V' is a smooth map.  Then π sends Υ'_|V∩ V' to Υ_|V∩ V'. Given ϕ∈qX× B denote ϕ' := (∘ p')^*ϕ∈qΥ' and  ϕ:= (∘ p)^*ϕ∈qΥ, hence ϕ' = π^* ϕ. Then for a form α∈*V∩ V' one has

    ∫_V∩ V'α∧ρ̂'     (ϕ)
    := 
    ∫_Υ'_|V∩ V' (ρ'^*α) ∧ϕ' 
    =
    ∫_Υ'_|V∩ V' (π^* ρ^*α) ∧π^*ϕ
        =
    ∫_π(Υ'_|V∩ V') (ρ^*α) ∧ϕ=
    ∫_Υ_|V∩ V' (ρ^*α) ∧ϕ =  ∫_V∩ V'α∧ρ̂(ϕ).

Since the identity above holds for every α∈Υ_c(V∩ V', ^*), one derives an equality of smooth forms
ρ̂'( ϕ)_|V∩ V' = ρ̂( ϕ)_|V∩ V',
thus proving the claim. 




Now, fix a resolution of singularities p Υ→Υ,  and let V ⊂ B be the domain of ρ̂_!(ϕ), as above. 
Define a preliminary form  ψ_o on B by 

    ψ_o :=
    ρ̂_!(ϕ)    ,  on  V 
    
     0    ,  on  G = B - V.


Write G = ⋂_n∈ U_n where G ⊂ U_n+1⊂U_n+1⊂ U_n ⊂⋯    is a nested family of  neighborhoods of F. For each n∈ choose a smooth function σ_n  B → [0,1] satisfying σ_n(x) =  1    ,  if  x ∉ U_n 
 0    ,  if  x ∈U_n+1, and define 
θ_n = σ_n   ρ̂ ( ϕ ) on V 
and θ_n ≡ 0 on U_n+1. 
Then each θ_n is a smooth form on B and θ_n(x) ψ_o (x) for all x∈ B. This shows that ψ_o is a Baire form that is smooth on V.

Now, observe that  the slicing map t∈ V ↦Υ, ρ , t is continuous and  that Υ, ρ , t  = Υ_t is the current given by algebraic cycle associated to the scheme theoretic fiber Υ_t = ρ^-1(t). Therefore,  we have a continuous family of effective algebraic cycles on a projective variety; see <cit.>. Since V is connected, the degree of each cycle is constant and the mass (associated to the metric induced by the Fubini-Study metric of some projective embedding of  X) of the fibers Υ_t is uniformly bounded. As a result we conclude that   ψ_o is a bounded Baire form.

It follows that ψ_o represents a unique class [ψ_o ] in k-2(m-i)B. Furthermore, given   α∈2(m+n-i) - kB
one has 

    ∫_Bα∧ψ_o    =  
    ∫_Vα∧ρ̂(ϕ)      (B - V    has measure  0)
        = ∫_Υ_|Vρ^*α∧ϕ_|ρ^-1V   (Fubini)
       = ∫_Υρ^*α∧ϕ   (measure 0 argument again)
       =
    ∫_Υ (∘ p)^*π_2^*α∧   (∘ p)^* ϕ   (by definition)
        =
    Υ(  (∘ p)^* ( π_2^*α∧ϕ)  ) 
        =
    (∘ p)_#Υ ( π_2^*α∧ϕ)    
        =
    Υ ( π_2^*α∧ϕ).     (p is birational isomorphism )


It follows that

    ∫_B ψ_o∧α = (-1)^k∫_B α∧ψ_o = (-1)^kΥ ( π_2^*α∧ϕ) = Υ ( ϕ∧π_2^*α) .

In conclusion,  the form ψ_o represents the current π_2#(  Υϕ).


In order to prove the first three statements in the proposition, we first show that ρ̂_! (ϕ) has a  continuous (and hence unique) extension to U:=B-F, the domain over which Υ is equidimensional.






Fix p_0 ∈ U. 
Using partitions of unity, it suffices to assume that ϕ  is supported on X× W, where W is the domain of a coordinate chart Ψ W → W'⊂^n with coordinates = (z_1, …, z_n) so that Ψ(p_0) = 0 , and that  ϕ has the form 

    ϕ = g π_1^*α∧π_2^* β ,

where g = g(, ) is a smooth function on X× B,  α∈^*(X)
 and  β∈^*(B)  is given in coordinates by β = h()dz_I∧ dz̅_J.
 
 
If α≠ 2(m-i) define Υ_!(ϕ) = 0. 

Now, assume α = 2(m-i),  |I|+|J| = k -2(m-i) and let
σ_I,J be the sign of the shuffle so that  Ω =  σ_I,J  dz_I∧ dz̅_J∧ dz_I^C∧ dz̅_J^C,
 is the volume form in ^n.


Define γ^ϵ in local coordinates by 

    γ^ϵ = σ_I,J  f^ϵ()   dz_I^C∧ dz̅_J^C ,

where 
f^ϵ() is a “bump” function whose support is contained in the ϵ-ball D_ϵ⊂ W' around 0 in ^n, with ∫ f^ϵ() d_2n() = 1.
Hence, β∧γ^ϵ =  h() f^ϵ() ∧Ω. 

It follows from <cit.> that
   
    ∫_U      h() f^ϵ() Υ, π_2, ( g  π_1^*α) d_2n()  = 
    ( Υπ_2^* ( h f^ϵ∧Ω) ) ( g π_1^* α)  
        =
     Υ( g π_1^* α∧π_2^*( β∧γ^ϵ ) 
        )    =
     Υ( 
    { g π_1^* α∧π_2^* β}∧π_2^*γ^ϵ) 
        =
     Υ( 
    ϕ∧π_2^*γ^ϵ) = π_2#( Υϕ) (γ^ϵ)   (<ref>)=∫_U  ψ_o∧γ^ϵ

Therefore

    lim_ϵ→ 0 ∫_U  ψ_o∧γ^ϵ  =  lim_ϵ→ 0∫_U        = 
     lim_ϵ→ 0∫_U   h() f^ϵ() Υ, π_2, ( g  π_1^*α) d_2n() 
        =
      h(p_0)  Υ, π_2, p_0 ( g  π_1^*α)
 
Using the fact that the slicing function p_0 ↦Υ, π_2, p_0 is a continuous function on U <cit.> one concludes that the last term in (<ref>)  is a continuous function at p_0. We conclude that the first term shows how to (re)define ψ_o at p_0 to make it continuous on U. In other words, we can extend  ρ̂_!(ϕ) (<ref>) to a bounded Baire form ψ_o which is continuous on U = B -F. In particular, this extension does not depend on the resolution of singularities and is generically smooth. 
Finally, define

    Υ_!(ϕ) =
    ψ_o    on   U= B-F 
    
    0     on  F.

This is the desired form, satisfying the first three statements of the proposition.


Step II: Continuous forms


Now, let  ϕ be a continuous form. One can also assume that ϕ has the form (<ref>), with g continuous. 

The essential ingredient here is the fact that the slices Υ, π_2, are normal currents, which are in fact represented by effective algebraic cycles, and hence they can be applied to continuous forms. Now, the  arguments in (<ref>) and (<ref>) apply, and show that Υ_!ϕ can be defined and is a continuous form on U= B-F, and the same continuity arguments still hold in the continuous case to show that the extension of Υ_!ϕ by zero on F defines a bounded Baire form on B.


To prove the last assertion of the proposition,   note that for each compact  K ⊂ B,  the current 
π_2#{[ Υ∩ (X× K) ] ϕ_|X× K}
is represented by the form χ_K ·Υ_! ϕ. 
The last assertion now follows directly from definitions.








  * When Υ is not dominant over B, we define Υ_! as the zero map,   to have a homomorphism
σ_! kX× B→k-2(m-i)B
associated to  any algebraic cycle σ∈^i(X× B).


  * We also use Υ_!ϕ    to denote the class in *B represented by Υ_!ϕ. Restricting ^1_loc⊗_^0^*_B to the Zariski topology on B, the proposition above states that the germ  (Υ_!ϕ)_ is continuous whenever  is a point of codimension 1. Furthermore, this germ is ^∞ at the generic point η when ϕ is a smooth form.


The continuity in codimension 1 described in the remark above is essential in the  construction of the desired chain maps from higher Chow groups.  The following result is a first  step in that direction. 




Consider a smooth embedding H ↪ B and let Υ⊂ X× B be irreducible and dominant over B, as in Proposition <ref>, with its structure of reduced closed subscheme of X× B. Let Υ_|H denote the algebraic cycle in X× H associated to the closed subscheme Υ×_B H. Given ϕ∈*X× B, the following holds. 



  * If H is a hypersurface in B then the forms
 ( Υ_|H)_! (1×)^*(ϕ) and  ^* Υ_!(ϕ) coincide on a dense Zariski open subset of H. 


  * If the codimension of H  is arbitrary but Υ is equidimensional and dominant over B, in the sense of Definition <ref>,  then 
 
    ( Υ_|H)_! (1×)^*(ϕ) =  ^* Υ_!(ϕ) .







Let F ⊂ B be the closed subvariety so that Υ_!ϕ is continuous on B-F, and let F' ⊂ H denote the corresponding subvariety for ( Υ_|H)_! (1×)^*(ϕ). Recall that  F and F' have codimension at least 2, and hence F” := F' ∪ (F∩ H) is a proper Zariski closed subset  of H. It follows that  both
( Υ_|H)_! { (1×)^*(ϕ)  } and  ^* Υ_! ϕ are continuous on B- F”.

Let 
Φ_H^ϵ∈2B be a Thom form for the normal bundle of H whose support is contained in an ϵ-neighborhood of H in B.
Given α∈*H pick some α̂∈*B such that ^*α̂= α. Then


    ∫_H    ^*Υ_!(ϕ) ∧α = ∫_H ^*{Υ_!(ϕ) ∧α̂}  = lim_ϵ→ 0∫_B Φ_H^ϵ∧Υ_!(ϕ) ∧α̂
        =
     lim_ϵ→ 0∫_B Υ_!(ϕ) ∧Φ_H^ϵ∧α̂=  lim_ϵ→ 0  Υ(  ϕ∧π_2^* ( Φ_H^ϵ∧α̂) )       
        =  lim_ϵ→ 0  ∫_Υϕ∧π_2^* ( Φ_H^ϵ∧α̂) = 
     lim_ϵ→ 0∫_Υπ_2^*Φ_H^ϵ∧ϕ∧π_2^*(α̂).

Since π_2^*Φ^ϵ_H is a Thom form for the normal bundle of X× H in X× B, one concludes from the identities above that
∫_H  ^*Υ_!(ϕ) ∧α =  ( [X× H] ∩ [Υ ]) (ϕ∧π_2^*α̂) .
Here we use  [Υ] ∩ [X× B] to denote both the intersection of algebraic cycles and its associated current.

Applying the identity
 (1×)_* ( Υ_|H) = [ Υ ] ∩ [ X × H] <cit.>
of algebraic cycles on X× B one concludes that 

    ∫_H  ^*Υ_!(ϕ)  ∧α    = 
    (1×)_#( Υ_|H) 
    (ϕ∧π_2^*α̂)  = ( Υ_|H) 
    (1×)^* (ϕ∧π_2^*α̂) 
        =
    ( Υ_|H) ( 
    (1×)^* (ϕ) ∧π_2^* ^*α̂) =
    Υ_|H( 
    (1×)^* (ϕ) ∧π_2^* α) 
        =
    ∫_H  (Υ_|H)_!( (1×)^*ϕ) ∧α.

Since α is arbitrary, this identity shows that 
^*Υ_!(ϕ) and 
(Υ_|H)_!( (1×)^*ϕ)
coincide in the domain B- F” where both forms are continuous. This proves the first statement. 

For the second statement, observe that since Υ→ B is a proper, dominant map, it is surjective and hence the algebraic cycle Υ_|H is  equidimensional and surjective over H, as well. In particular both forms  ( Υ_|H)_! (1×)^*(ϕ) and  ^* Υ_!(ϕ) are continuous on H. Now, one can use the same arguments as in the proof of the first statement and the result follows. 






 §.§ Correspondences and transforms of  currents



In this section we show how to construct the pull-back homomorphisms on currents that are represented by integration, induced by correspondences, and prove the main properties of these constructions. . 



  §.§.§ The projective case






Let X be a connected, smooth projective variety and let B be a smooth variety, with X =m and B = n. 
Given an  irreducible subvariety Υ⊂ X × B  of codimension i, define a pull-back homomorphism

    Υ^#  kB⟶k+2iX× B,     k ≥ 0,

by sending S ∈kB to the current Υ^#S defined  on  ϕ∈2(m+n-i)-kX× B 
as S( Υ_!(ϕ) ). In other words, the pull-back is the adjoint of the generalized integration along the fiber. 


In the proof  below we  show that this pull-back operation above  is well-defined. 
(of Theorem <ref>)
The assignment ϕ↦ S(Υ_!ϕ) = S(ρ̂_!ϕ ) is well-defined since Υ_!ϕ is  a compactly supported bounded Baire form, as shown in Proposition <ref>,  and S is represented by integration.  Now,  let  K ⊂ B be a compact set such that (ϕ) ⊂ X × K and, for a current T ∈*X× B,  denote 𝐌_K(T) := 𝐌(χ_X× K T), where χ_X× K is the characteristic function of X× K.
Then 

    |S(Υ_!ϕ )| ≤𝐌_K( S)  Υ_! ϕ≤𝐌_K(S)  𝐌_K(Υ ) ϕ,
  where the last inequality follows from Proposition <ref>.<ref>.
This suffices to show that Υ^#S is indeed a current representable by integration.

Finally, the fact that Υ^# sends a current of type (r,s) to a current of type (r+i, s+i) follows directly from Proposition <ref>.<ref>.






If the current S ∈kB is given by a form ω∈kB then it follows from Proposition <ref>.<ref>  that Υ^#S = Υπ_2^* ω .





 

  §.§.§ Properties of the pull-back and transform operations



We now proceed to prove Proposition <ref> and discuss a few more properties of the pull-back maps. 

(of Proposition <ref>)
Consider  Υ⊂ X × B  as in Theorem <ref> and let
S be a locally normal current in B whose support is contained in B - F, the domain over which Υ is equidimensional. 

To prove the first assertion in the proposition, it suffices to assume that Υ⊂ X × B is equidimensional  and dominant over B and that S is a normal current in B. 
The following commutative diagram summarizes the notation for the indicated projections. 

    X × B × B [dd]^π_2@/_2.2pc/[ddll]_π_1@/^2.2pc/[ddrr]^π_3[dl]_π_12[dr]^π_23      
        X× B [dl]^ρ_1[dr]_ρ_2       B× B  [dl]^q_1[dr]_q_2   
    
    X        B           B






Let Δ B → B× B denote the diagonal inclusion and let Δ_B  ⊂ B× B denote the diagonal. With Υ and S as above then Υ× B is equidimensional over B× B and 


  * 
    (Υ× B)^#( B × S)  =  Υ× S.


  * Given φ∈^*(X× B× B) one has

    Δ^*( {Υ× B}_!(φ) )  =  Υ_!( (1×Δ)^* φ).






(of Lemma)
First we show the following


If φ is a form on X× B× B of the form π_12^*α∧π_3^* β, with α∈rX× B and β∈sB then

    (Υ× B)_!( φ)  =  q_1^*( Υ_!α) ∧ q_2^*(β) .


To prove the claim, pick a form θ in B× B of the form θ = q_1^*τ∧ q_2^*η, with τ∈2(m+n-i) -rB and η∈2n-sB. Then


    ∫_B× B     (Υ× B)_!(φ) ∧θ =
    Υ× B (  φ∧π_23^*θ)  
        = 
    Υ× B (  π_12^*α∧π_3^*β∧π_23^*(q_1^*τ∧ q_2^* η)   ) 
        =
    (Υ× B )(  π_12^*α∧π_3^*β∧π_2^*τ∧π_3^* η) 
        =
    (-1)^r+s(Υ× B )(  π_12^*α∧π_2^*τ∧π_3^*β∧π_3^* η) 
        =
    (-1)^r+s(Υ× B )(  π_12^*{α∧ρ_2^*τ}∧π_3^*{β∧η}) 
       = 
    (-1)^r+sΥ(  α∧ρ_2^*τ) · B  ( β∧η)  
        =
    (-1)^r+s( ∫_B Υ_!α∧τ)·( ∫_B β∧η) 
       = (-1)^r+s∫_B× B q_1^* (Υ_!α∧τ )∧ q_2^*( β∧η  ) 
       =     ∫_B× B q_1^* (Υ_!α ) ∧ q_2^*( β) ∧ q_1^*( τ ) ∧ q_2^*( η  ) 
        =    ∫_B× B q_1^* (Υ_!α ) ∧ q_2^*( β) ∧θ.

The claim now follows from the identities above and the continuity of the forms (Υ× B)_! (φ) and q_1^*(Υ_!α)∧ q_2^*(β). 


From the definitions and the Claim, it follows that for a form φ as above one has

    (Υ× B)^#(  B × S )(φ)      := (  B × S )( (Υ× B)_! φ) 
        Claim <ref>=(  B × S ) ( q_1^* (Υ_!α ) ∧ q_2^*( β)) 
    =
     B  (Υ_!α) · S(β)  
         = Υ(α) · S(β)
     = 
     ( Υ× S )(φ).

This proves the first assertion of the Lemma. 

The second assertion follows directly from Proposition <ref>.<ref>. Indeed, one just  needs to replace B by B× B and H ↪ B by Δ B ↪ B × B in the statement, and use the fact that Υ≡ B×_Δ (Υ× B)  = (Υ× B)_| Δ_B.


Now,  we conclude the proof of the first assertion in Proposition  <ref>. Using partitions of the unity, it suffices to consider a form φ∈*X× B× B whose support is contained in X× K × K, where K ⊂ V is a compact contained in the domain  of a coordinate chart ψ V  V'⊂^n.

Then

    (1×Δ)_#  (Υ^#S) (φ)     def=Υ^#S ( (1×Δ)^*φ )    def= S ( Υ_!( (1×Δ)^*φ) ) 
       (<ref>)=
    S(Δ^*( {Υ× B}_!(φ) ) )  = (Δ_# S) (  {Υ× B }_!(φ)  ).


Given a form β in B× B, with β⊂ K × K, let δ V× V →^n be the “difference map” δ(u,v) = ψ(u)-ψ(v). Then 

    (Δ_# S) (  β)     =  B× S, δ, 0  (β)  =  lim_ϵ→ 0∫_B f^ϵ(b)  B× S, δ, b (β) d ℒ_2n(b) 
        =
      lim_ϵ→ 0 ( B× S) ( β∧δ^*(f^ϵΩ) )

On the other hand, taking β = {Υ× B}_!(φ), one has

    ( B× S) ( β∧δ^*(f^ϵΩ) )     =
    ( B× S) ({Υ× B}_!(φ) ∧δ^*(f^ϵΩ) )  
        def={Υ× B}^# (B× S) (φ∧π_3^*δ^*(f^ϵΩ) )  
       = 
    {Υ× B}^# (B× S) (φ∧δ'^*(f^ϵΩ) ) 
       (<ref>)=
    (Υ× S) (φ∧δ'^*(f^ϵΩ) ) 
        = 
    ∫_B f^ϵ(b) Υ× S, δ', b  (φ) dℒ_2n(b).

Taking the limits when → 0 one gets:

    Υ× S, δ', 0  (φ)  =  (1×Δ)_#( Υ^# S ) (φ).

One concludes that  {Υ∩  (  X × S)}(φ)  =  ( Υ^# S ) (φ).


To prove the second assertion of Proposition <ref>, assume that S is normal and vanishes suitable along H. Given φ∈kX× B, the condition 
S(H) = 0 gives 
Υ^#S(dφ) := S (Υ_!(dφ)) = {S∩ (B-H)}(Υ_!(dφ)) .
Since S is normal one has

    { S ∩ (B-H)}(Υ_!(dφ)) = lim_τ→ 0{ S ∩ (B-W^τ)}(Υ_!(dφ)).
 

Now, for each τ > 0 sufficiently small the current S∩ (B-W^τ) is normal, by hypothesis, and its support is contained in B-W^τ⊂ B-F. The first assertion of the proposition then gives

    { S ∩ (B-W^τ)} (Υ_!(dφ))   := 
    Υ^#( S ∩ (B-W^τ))(dφ) 
         =
    ( Υ∩{ S ∩ (B-W^τ)})(dφ) =
    ∂( Υ∩{ S ∩ (B-W^τ)})(φ)
        =
    ( ∂Υ∩{ S ∩ (B-W^τ)})(φ)   +
    ( Υ∩∂{ S ∩ (B-W^τ)})(φ)  
        =
    ( Υ∩∂{ S ∩ (B-W^τ)})(φ)  
        =
    ( Υ∩{∂ S ∩ (B-W^τ)})(φ)  +
    ( Υ∩{ S ∩∂(B-W^τ)})(φ)  
        =
     {∂ S ∩ (B-W^τ)}(Υ_!φ)  +
     { S ∩∂(W^τ)}(Υ_!φ) .


The hypothesis on S and the identities above give

    S(Υ(dφ))     = lim_τ→ 0{ S ∩ (B-W^τ)} (Υ_!(dφ)) 
        =
    lim_τ→ 0{∂ S ∩ (B-W^τ)}(Υ_!φ)  +
    lim_τ→ 0{ S ∩∂(W^τ)}(Υ_!φ) 
        =
    (∂ S ∩ (B-F)) (Υ_!φ) = (∂ S ) (Υ_!φ)  = Υ^#(∂ S) (φ).

This concludes the proof of the proposition. 



The next result is a direct consequence of  Proposition <ref>.
Recall that the transform 
Υ  kB⟶k+2(i-n)X is given by S ↦ΥS := π_1#( σ^#S).
See Definition <ref>.



Let Υ⊂ X× B be dominant over B, as in Proposition <ref>. 
Given a smooth hypersurface H ↪ B and a  current S ∈*H  representable by integration,
the identity 

    Υ^∨__# S =  ( Υ_|H)^∨_S

holds whenever one of the following conditions occur.


  * 
 Υ is equidimensional over B;

  * For each proper Zariski closed subset Z ⊊ H one has S(Z) = 0. 





In the first case, consider  φ∈*X. Then

    Υ^∨__# S (φ)     :=  Υ^#__# S(π^*_1φ)   = _#S( Υ_!(π^*_1φ)) = S( ^* Υ_!(π^*_1φ)) 
       (<ref>)= 
      S (  {Υ_|H}_! (1×)^*(π^*_1φ )  ) =   S (  {Υ_|H}_! φ) :=  ( Υ_|H)^∨_S (φ).


Now,  observe that if F  ⊂ B is the set over which  Υ is not equidimensional, then 
F∩ H ⊊ H, since F≥ 2.  The second case now follows from Proposition <ref>.i.  and  the identities (<ref>). 








 


  §.§.§ The quasi-projective case
 

Let Υ⊂ U ×Δ^n be an irreducible subvariety of codimension i in U×Δ^n, whose projection onto Δ^n is dominant and has equidimensional fibers.

Let U↪ X ↩ D be  a projective compactification 
where D = X-U is a divisor with simple normal crossings, and let Υ⊂ X ×^n be the closure of Υ in X×^n.


We know that Proposition <ref> gives a homomorphism 
Υ_! kX×^n→k^n so that Υ_! ϕ is continuous on ^n - F, for any continuous form ϕ on X×^n.  Unfortunately, the exceptional set F may intersect Δ^n, and this prevents us from directly using the arguments from the projective case. The redeeming factor is that we are interested in currents with log poles.



Using the notation above, let ϕ∈kX×^n be a continuous form that vanishes at D ×^n. Then Υ_!ϕ extends continuously to Δ^n.




First assume that   ϕ∩ (D ×^n) = ∅, and denote Υ' = Υ∩ (U×^n). 

Using partitions of unity, we may assume that ϕ = g π_1^*α∧π_2^*β (<ref>), as in the proof of Proposition <ref>. 

It follows from the properties of the slicing of analytic chains <cit.> that the identities (<ref>) and (<ref>) still hold for Υ', since ϕ is compact and lies in U×^n.
As in the proof of Proposition <ref>, it follows that we can define Υ'_!ϕ continuously on Δ^n and this clearly coincides with Υ_!ϕ on Δ^n - F.

In the general case, when  ϕ∩ (D ×^n) ≠∅, we may still assume that ϕ has the form  ϕ = g π_1^*α∧π_2^*β. Fix a product Riemannian metric on X ×^n and let { U_n, ρ_n }_n∈ be a system of neighborhoods of D in X, with ρ_n  X → [0,1] smooth and satisfying 

  
  * D ⊂ U_n+1⊂U_n+1⊂ U_n, for all n∈; 
  
  * ⋂_n≥ 0 U_n = D, and 
  
  * ρ_n ≡ 1    ,   on U_n+1, 
 
  0    ,   on   X - U_n.




Since D  is compact, for each ϵ >0  there is an n_0 ≫ 0 such that 

    ρ_n ϕ≤ϵ,    for  n ≥ n_0.


Now, denote ϕ_n := (1-ρ_n)ϕ. It is clear from the definition and (<ref>) that ϕ_n converges uniformly to ϕ on X×^n and (ϕ)∩ (D ×^n) = ∅. In particular,  {Υ_!(ϕ_n) }_n∈ is a sequence of bounded Baire forms that are continuous on Δ^n. We proceed to show that this sequence converges uniformly to Υ_!(ϕ).

Denote G_r,k(z) :=   Υ' , π_2, z  ( (ρ_r-ρ_r+k) g π_1^*α) and observe that this is a continuous function on Δ^n. Since Υ' = Υ∩ (U×^n) and (ρ_r-ρ_r+k) g π_1^*α⊂ U×^n,   for each z_0 ∈Δ^n-F one has 
G_r,k(z_0) =   Υ, π_2, z_0  ( (ρ_r-ρ_r+k) g π_1^*α) .


As explained in  the proof of Proposition <ref>, for z_0 ∈Δ^n - F, the mass of the slice 
Υ, π_2, z_0 is bounded by a constant d, regardless of z_0, since this is a continuous family of effective algebraic cycles in a projective variety.
Therefore, 

    | G_r,k(z_0) | ≤ d  (ρ_r-ρ_r+k) g_∞ α_X  ≤ d ρ_r g_∞ α_X  .
 
For r fixed, this inequality holds for all k and z_0∈Δ^n - F. Since Δ^n - F is dense in ^n and G_r,k(z) is continuous on Δ^n, it follows that (<ref>) holds for every z_0∈Δ^n.


Now, fix z_0 ∈Δ^n and let γ^ϵ =f^ϵ dz_I∧ dz̅_J, as in (<ref>). Following (<ref>)
we get

    ∫_^n(Υ'_!(ϕ_r+k)- Υ'_!(ϕ_r) )∧γ^ϵ    = ∫_^n(Υ'_!(ϕ_r+k- ϕ_r) )∧γ^ϵ
        = 
    ∫_^n h(z)f^ϵ(z) G_r,k(z_0) d _2n(z).

Therefore, 

    | ∫_^n(Υ'_!(ϕ_r+k)- Υ'_!(ϕ_r) )∧γ^ϵ|    ≤∫_^n |h(z)| f^ϵ(z) | G_r,k(z_0)|  d _2n(z) 
       ≤( d ρ_r g_∞ α_X   ) ∫_^n |h(z)| f^ϵ(z)  d _2n(z) .

Taking the limit as ϵ→ 0 gives

    | lim_ϵ→ 0∫_^n(Υ'_!(ϕ_r+k)- Υ'_!(ϕ_r) )∧γ^ϵ|  = lim_ϵ→ 0|∫_^n(Υ'_!(ϕ_r+k)- Υ'_!(ϕ_r) )∧γ^ϵ|   
       ≤( d ρ_r g_∞ α_X   ) lim_ϵ→ 0 ∫_^n |h(z)| f^ϵ(z)  d _2n(z) 
        =
    ( d ρ_r g_∞ α_X   ) |h(z_0)| ≤( d h_∞ α_X   ) ρ_r g_∞ .

The inequalities above imply that Υ'_!(ϕ_r+k) - Υ'_!(ϕ_r)_K ≤( d h_∞ α_X   ) ρ_r g_∞ 
on each compact K ⊂Δ^n.

It follows that the sequence  {Υ_!(ϕ_r) }_r∈ converges uniformly on each compact subset of Δ^n,  and that the limit coincides with Υ_!(ϕ) on Δ^n - F. The result follows. 



We now prove Proposition <ref>.

(of Proposition <ref>)

Consider Υ⊂ U×Δ^n as in Lemma <ref>, and let H⊂^n be a smooth hypersurface H≠ H_∞. It follows that Υ intersects X× H properly and we denote the algebraic cycle [Υ]∩ [ X× H]  by  Υ_|H. 
Now, write Υ_|H= A + B + C, where B⊂ D×^n, C⊂ X×{ H ∩ H_∞}, and no component of A is contained in D×^n ⋃   X×{ H∩ H_∞}.

By definition, given ϕ∈*X×^n one has (  Υ_|H)_!(ϕ) = A_!(ϕ) + B_!(ϕ), since no component of C is dominant over H. Furthermore, if ϕ vanishes at D×^n then (  Υ_|H)_!(ϕ) = A_!(ϕ). On the other hand, intersection theory shows that the restriction of  an intersection of two algebraic cycles to an open set coincides with the intersection of their respective restrictions. Therefore, 
Υ_|H∩ ( U×Δ^n) = A∩ (U×Δ^n)
coincides with 

    (  Υ∩{U×Δ^n}) ∩(  [X× H]∩{U×Δ^n}) =  [ Υ] ∩ [U× H] =: Υ_|  H,
 
where H = H ∩Δ^n = H- H_∞.

Let ( Υ_|  H) the algebraic cycle in X× H obtained by taking the closure of each component of Υ_|  H while keeping their multiplicities. The arguments above show that ( Υ_|  H)  = A, and hence

    ( Υ_|  H)_!(ϕ)  =  (  Υ_|H)_!(ϕ),

for all ϕ∈*X×^n satisfying ϕ_|Y×^n=0 .




























It follows from Theorem <ref> that the homomorphisms 
( Υ_|H)^# and (  Υ_|  H)^ #
are well defined and we conclude that 
Proposition <ref>.<ref>  follows from (<ref>).

To prove Proposition <ref>.<ref> we use the same arguments   in the proof of Proposition <ref>.<ref> along Lemma <ref>.







We conclude this section observing that the transform operation does not depend on the compactification chosen, up to canonical isomorphism.



Let Υ⊂ U×Δ^m be as above and let  U↪ X' ↩ D'  be another compactification.  Assume there is a map of pairs f  (X', D') → (X, D) which is the identity on U, and 
let Υ and Υ' denote the closures  of Υ in X×^n and X' ×^n, respectively. Then  the following diagram commutes.
 
    k^n[rr]^-Υ'[rrd]_-Υ      *X'(log D') [d]_≅^f_#
          *X(log D) ,

where Υ and Υ' are the current transforms associated to Υ and Υ'.





This result follows directly from the description of the integration along the fibers in the proof of Proposition <ref> and from the behavior of the slicing operation under orientation-preserving diffeomorphisms  <cit.> .
















§ GEOMETRIC CURRENTS ON .



This section  introduces  the currents on complex projective spaces that  play a key role in subsequent constructions. The final outcome is the triple   ( Θ_n,  n, W_n) that we call the fundamental triple of currents in ^n. 




 §.§ The basic semi-algebraic currents

Let us start with the topological and algebraic simplices, seen as semi-algebraic subsets of  ^n. Write = (z_0, …, z_n) and let  [] = [ z_0 : ⋯ : z_n] denote the corresponding homogeneous coordinates on ^n.  


  


For 0≤ j ≤ n, consider the linear form _j() := z_0 + ⋯ + z_j.



  * Denote by H_j ⊂^n the j-th coordinate hyperplane given by z_j=0, and let L_j⊂^n denote  the  hyperplane given by _j() = 0.  


  * Let ι_r ^n-1↪^n  denote the natural inclusion identifying ^n-1 with H_r.  Given 0≤ k ≤ n define the operation 

    τ_k   m^n-1   ⟶m+	2^n
    
       T    ⟼τ_k(T) := ∑_r=0^k (-1)^r ι_r#( T ).



  * When j=n we denote H_∞ := L_n ⊂^n, and call it the hyperplane at infinity. The algebraic n-simplex
 Δ^n := ^n - H_∞,
is canonically identified with the complex affine space 


Δ^n  = { (u_0, …, u_n) | u_0+ ⋯ + u_n = 1 }⊂^n+1.



  * Via this identification, the standard topological simplex n⊂^n+1 sits inside Δ^n as the semi-algebraic set

    n = { (x_0,…, x_n) | ∑_r=0^n x_r =1,  and  0≤ x_r ≤ 1  for all  r =0,…, n . }.

For simplicity, we also denote by n  the current n associated to the canonical orientation of the topological simplex.

 
 











Given 1≤ j ≤ n, consider the semi-algebraic set S_j  ⊂^n given by

    S_j := { [z_0:⋯:z_n]  | z_j = t _j() ,  for some  t ∈ [0, 1] },


and let S_j ∈1^n denote the corresponding semi-algebraic chain 
oriented so that


   d  S_j  =  L_j-1  -  H_j .


For 0≤ j < n ≤ n define


R_n,j  = S_n∩ S_n-1∩⋯∩   S_j+1⊂^n.

We show in the next proposition that this intersection 

is proper, and hence, we can suitably orient R_n,j to have

    R_n,j =   S_n∩ S_n-1∩⋯∩ S_j+1.

Note that R_n,n-1  = S_n, and for completeness define R_n,n  = ^n and R_n,j = ∅, when n<j.

In order to perform calculations, it is useful to parametrize  R_n,j as follows. Let   [ :λ] be  homogeneous coordinates on ^j  with = (u_0, …, u_j-1 ). Write = (s_0, …, s_n-j) ∈Δ^n-j,  with ∑_r=0^n-j s_r = 1, and define the algebraic map

    Φ_n,j  ^j×Δ^n-j   ⟶^n 
    
    ([  :λ ],  )    ⟼ [  : s_0 λ - () : s_1 λ : ⋯ : s_n-jλ ].










Using the notation above, the following holds:
 

 
  *  The map Φ_n,j induces an isomorphism between affine spaces
 
    Φ_n,j  ( ^j - H_j) ×Δ^n-j^n - L_n = ^n - H_∞ ,

where  H_j  ⊂^j is the hyperplane given by λ = 0.  


  *  The image of  ^j×n-j under Φ_n,j is precisely R_n,j. In particular, Φ_n,j induces an isomorphism between the semi-algebraic set ^j×n-j and R_n,j - B, where
we are identifying ^j = ^j - H_j and  B = L_n ∩ R_n,j = H_∞∩ R_n,j  =  L_j∩ H_j+1∩⋯∩ H_n.


  *  The collection { S_j,  …, S_n } intersects properly in the sense of Definition <ref>. In particular, the real codimension of R_n,j is n-j.


  *  (Boundary formula)

    d  R_n,j     =   (-1)^n-j-1 R _n,j+1∩ L_j   
     - (-1)^n ∑_r=j+1^n(-1)^r ι_r#( R_n-1,j)


 
  *  One has an identity of currents:
 
    Φ_n,j #^j×n-j  =   R_n,j.














































First observe that the identity  _n ∘Φ_n,j((, λ), ) = λ  shows that  Φ_n,j sends ^j ×Δ^n-1  = (^j- H_j) ×Δ^n-j into Δ^n. On the other hand, the map

    ΨΔ^n    ⟶^j ×Δ^n-j
    
     ( z_0, …, z_n)    ⟼( [z_0 : ⋯ : z_j-1 : 1 ] , ( 1- z_j+1 - ⋯ - z_n, z_j+1, …, z_n ) )

gives an inverse to the restriction of Φ_n,j to ^j ×Δ^n-j. 
This concludes the proof of assertion <ref>). 




Given 0≤ j < n, denote = (t_j+1, …, t_n) and define polynomials 

    ^n_j,r() = 
    
    (1-t_n)(1-t_n-1) ⋯ (1-t_j+1)    ,  if  r = j 
    
    (1-t_n)(1-t_n-1)⋯ (1- t_r+1) t_r   ,  for  j< r< n 
    
    t_n     ,  if  r =n.

It is easy to see that  

    ∑_r=j^n ^n_j,r() = 1,

and that the induced map   [0,1]^n-j→n-j  sending ↦  (s_0(), …, s_n-j()), with s_k() = ^n_j,j+k(), k=0, …, n-j,  is a parametrization of the  simplex . 


By definition, the set R_n,j is described by the following conditions:

    z_j+1    = t_j+1_j+1(),   for some  t_j+1∈ [0,1] 
    ⋮         ⋮
    
    
    z_n       = t_n  _n(),     for some  t_n  ∈ [0,1]


Performing successive substitutions of the type 
_n-1()  = _n () - z_n   = _n() - t_n _n() = (1-t_n)_n()
one concludes that [] = [z_0  ⋯  z_n]  ∈ R_n,j if and only if there are t_j+1, …, t_n  ∈ [0,1] such that


    z_j+1        = _n()  (1-t_n) ⋯(1-t_j+2)t_j+1        = _n() ^n_j+1,j()        = _n()s_0()        

   ⋮           ⋮          ⋮    m          


     z_n-1        = _n() (1- t_n)t_n-1           = _n() ^n_n-1,j()        = _n()s_n-j-1()             

    z_n          = _n() t_n          = _n() ^n_n,j()        = _n()s_n-j().  m        





 Hence,  it follows   from (<ref>) that
the image of  ^j ×n-j under  Φ_n,j is R_n,j,   and that Φ_n,j induces an isomorphism 
 
    (  ^j-H_j ) ×n-j ≅  R_n,j   - B,

with B = R_n,j∩ H_∞. This proves assertion <ref>). 




The proof that the  family { S_j, …, S_n} intersects properly  follows  from the equations (<ref>). Since 
this is a family of codimension 1 semi-algebraic sets, the isomorphism (<ref>) implies that R_n,j =  n+j. Assertion <ref>) follows.





By definition,

    d    (   S_n∩ S_n-1∩⋯∩ S_j+1)  =∑_r=j+1^n (-1)^n-r S_n∩⋯∩ d S_r ∩⋯∩ S_j+1
       =
    ∑_r=j+1^n (-1)^n-r S_n ∩⋯∩(  L_r-1  -  H_r)  ∩⋯∩ S_j+1
        =  (-1)^n-j-1 S_n∩⋯∩ S_j+2∩ L_j
    -
    ∑_r=j+1^n (-1)^n-r S_n∩⋯∩ H_r∩⋯∩ S_j+1
        =  (-1)^n-j-1 R_n,j+1∩ L_j 
    -
    ∑_r=j+1^n (-1)^n-rι_r#( 
     S_n-1∩⋯∩ S_j+1) 
        =  (-1)^n-j-1 R_n,j+1∩ L_j  
    - (-1)^n
    ∑_r=j+1^n (-1)^rι_r# R_n-1,j

where the third identity follows from the fact that S_k ∩ L_k  = 0 , for all k. This proves the boundary formula in assertion <ref>).





Let ι_r ^m-1→^m denote the inclusion as the r-th coordinate hyperplane and note that  
Φ_n,j∘ (1 ×ι_r) = ι_j+r∘Φ_n-1,j,
for 1 ≤ r ≤ j.
Furthermore,  when r=0, it is easy to see  that 
{Φ_n,j∘ (1 ×ι_0)}_#^j×n-j-1 =   R_n,j+1∩ L_j   .
Therefore, 

    d   Φ_n,j#^j×n-j 
    =   
    (-1)^n-j+1∂Φ_n,j#( ^j×n-j)   
       = 
    (-1)^n-j+1Φ_n,j#(  ^j×∂n-j) 
       = 
     (-1)^n-j+1Φ_n,j#( ^j×∑_r=0^n-j(-1)^r ι_r#( n-j-1) )
       =
     (-1)^n-j+1∑_r=0^n-j (-1)^r 
     {Φ_n,j#∘ ( 1×ι_r) }_#(
     ^j+n -n×n-j-1) 
        =
     (-1)^n-j+1{Φ^n_n,j#∘ ( 1×ι_0)}_#^j+n -n×n-j-1
           +(-1)^n-j+1∑_r=1^n-j (-1)^r ι_j+r#∘Φ^n-1_n-1,j#^j×n-j-1
        =
    (-1)^n-j-1 R_n,j+1∩ L_j    -  (-1)^n ∑_k=j+1^n (-1)^k ι_k#∘Φ^n-1_n-1,j#^j×n-j-1
       =
    d  R_n,j.






Using the identity above and assertion <ref>) we conclude the proof of the proposition. 










The given orientation on R_n,0 identifies the current
R_n,0 with n .
Furthermore,  if  R_n,j is the measure associated to the integral current R_n,j
for 0≤ j ≤ n, then every proper Zariski closed subset Z⊂^n   has R_n,j-measure zero. 




The first assertion follows directly from the proof of the proposition. To prove the second assertion, first note that each R_n,j is Zariski dense in ^n, since it contains n. Therefore, Z∩ R_n,j must be a semi-algebraic subset of R_n,j of dimension strictly less than n+j = _(R_n,j), and hence R_n,j(Z) =0.






 §.§ The canonical -forms



 Fix  integers 0≤ j≤ n.

   

   
  * 

For j≤ n define meromorphic j-forms   jn in ^n as 0n  =1 and for j>0
   
    jn  := ∑_r=0^j (-1)^r dz_0/z_0∧⋯∧dz_r/z_r∧⋯∧dz_j/z_j.

This is a form with log-poles along the divisor D_j  := H_0∪⋯∪ H_j. 
 

  * 
Define forms jn on ^n by setting
0n   = 0, and  jn  := (-1)^jħ_j j-1n
for 1 ≤ j ≤ n,
where

    ħ_j[] =
    log( 1 - _j()/z_j)  
       ,  if  [] ∉ S_j 
    
    
    0    ,  if   [] ∈ S_j,

and _j() = z_0 + ⋯ + z_j, as in Definition <ref>. 






The function  ħ_j is holomorphic on ^n - S_j, see (<ref>),  and  lies in (^n). The form jn is also in j-1^n, 
thus yielding a current [jn ] ∈j-1^n represented by integration. 

Similarly, the forms θ^n_j  define  [θ^n_j ] ∈2n-j^n.













To simplify the statement and proof of the next result, we denote 

    β_k,j^n:= τ_k( R_n-1,jω^n-1_j   ) := ∑_r=0^k (-1)^r   ι_r#( R_n-1,jω^n-1_j   )  ∈ n^n.

  
  


Given 1≤ j≤ n the following holds:

 

  *  Boundary formula for [jn]: 
    d[jn]  =   - ∑_r=0^j (-1)^r _r#  [j-1n-1]  =  
    -  τ_j(  [j-1n-1]) .



  * 
The  form j-1n is S_j-summable. Hence,
   S_n j-1n
is represented by integration. 

  *  The  form jn is R_n,j-summable. Hence,   R_n,jjn is represented by integration. 


  * Boundary formula for [jn]:
 

    d[jn]      =  [jn]  - (-1)^j   S_j j-1n  + ∑_r=0^j-1(-1)^r ι_r#([j-1n-1])  
        =  [jn]  - (-1)^j   S_j j-1n  + τ_j-1( [j-1n-1] )



  * Boundary formula for R_n,j [jn]: (See (<ref>) for notation.)

    d  (  R_n,jω^n_j ) 
         =
     (-1)^n {β_j,j^n   -  β_n,j^n  + (-1)^j β_j-1,j-1^n  }
        +
    (-1)^n{  (-1)^j(  R_n,jθ^n_j )   
     -   (  R_n,j-1θ_j-1^n)    } .




 The proof of statement (<ref>) is a standard residue calculation.
The proof of statements (<ref>) and (<ref>) are given in Proposition <ref>, 
Appendix <ref>. 

Since R_n,j∩ S_j =  R_n,j-1, then S_j has R_n,j-measure zero. It follows from equations (<ref>)  that Z_n,j := H_0 ∪⋯∪ H_j-1∪ S_j ∪ L_n  also satisfies R_n,j (Z_n,j) = 0.  Thus, for any smooth form β on ^n one has

    R_n,j (ω_j^n ∧β)  = { R_n,j∩ ( ^n - Z_n,j )}(ω_j^n ∧β)   =lim_δ→ 0(  R_n,j∩ U^δ_n,j)  (ω_j^n ∧β),

where U^δ_n,j⊂^n is the  set described below. 

For 0≤ r ≤ n  let N^δ_r denote the complement of a tubular δ-neighborhood of the coordinate hyperplane H_r, thus having smooth real-analytic boundary and satisfying  ⋂_δ > 0( ^n - N^δ_r)= H_r.

Finally, consider the map
σ_j ^n - S_j → given by
σ_j([]) = 1 - _j()/z_j and define
W^δ_j := σ^-1_j (O_δ) where 
O_δ⊂ is the complement of the usual tubular δ-neighborhood of (-∞, 0] in . 
Now, define

    U^δ_n,j := N^δ_0 ∩⋯∩ N^δ_j-1∩ W^δ_j.

It is easy to see that when δ is small enough, the  closed sets in this intersection  intersect properly and have  semi-analytic boundaries. Let us write

    R_n,j∩ U^δ_n,j =  R_n,j∩ N^δ_0 ∩⋯∩ N^δ_j-1∩ W^δ_j .

Then




 




    ∂(  R_n,j∩ U^δ_n,j)  =  (∂ R_n,j) ∩ U^δ_n,j +   R_n,j∩∂ U^δ_n,j
        =
      (∂ R_n,j) ∩ U^δ_n,j +   R_n,j∩∂(  N^δ_0 ∩⋯∩ N^δ_j-1∩ W^δ_j ) 
        = (
    ∂ R_n,j) ∩ U^δ_n,j
          +
     R_n,j∩(  ∑_r=0^j-1 N^δ_0 ∩⋯∩∂ N^δ_r ∩⋯∩ N^δ_j-1) ∩ W^δ_j 
          +
       R_n,j∩ N^δ_0 ∩⋯∩ N^δ_j-1∩∂ W^δ_j


Notice that ω^n_j is smooth on U^δ_n,j for all δ>0. Hence, given a smooth form ϕ on ^n one has

    d  (  R_n,jω^n_j )( ϕ)  = 
    (-1)^n   R_n,j ( ω^n_j ∧ dϕ)      = (-1)^n lim_δ→ 0(  R_n,j∩ U^δ_n,j)  (ω_j^n ∧ dϕ) 
         = (-1)^n lim_δ→ 0(  R_n,j∩ U^δ_n,j)  ( (-1)^j-1{
    d( ω_j^n ∧ϕ)  -  dω^n_j ∧ϕ}) 
        =
    (-1)^n+j-1lim_δ→ 0 ∂(  R_n,j∩ U^δ_n,j)  
     (  ω_j^n ∧ϕ)   
    - (-1)^n+j-1lim_δ→ 0(  R_n,j∩ U^δ_n,j)  (θ^n_j ∧ϕ) 
        =
    (-1)^n+j-1lim_δ→ 0 ∂(  R_n,j∩ U^δ_n,j)  
     (  ω_j^n ∧ϕ)  
    - (-1)^n+j-1(  R_n,jθ^n_j ) (ϕ)


We now use  the terms (<ref>) – (<ref>) to write down the limit  (<ref>).  



First,  apply Proposition <ref>.<ref> to (<ref>) and get

    (    ∂ R_n,j∩ U^δ_n,j) ( ω^n_j ∧ϕ) 
        =
    (  { R_n,j+1∩ L_j   
     + (-1)^j ∑_r=j+1^n(-1)^r ι_r#( R_n-1,j)
     }∩ U^δ_n,j) ( ω^n_j ∧ϕ)  
        =
    (   
      (-1)^j ∑_r=j+1^n(-1)^r ι_r#( R_n-1,j)
      ∩ U^δ_n,j) ( ω^n_j ∧ϕ) ,

where the last identity follows from the fact that the restriction of ω^n_j to R_n,j+1∩ L_j ∩ U^δ_n,j is equal to zero.



We use  the notation (<ref>) in what follows. 
A direct inspection shows that for j+1≤ r≤ n 
one has ι_r^*ω_j^n = ω_j^n-1. 
Therefore,

    lim_δ→ 0 (    ∂ R_n,j∩ U^δ_n,j) ( ω^n_j ∧ϕ) 
        = 
    lim_δ→ 0(     
      (-1)^j ∑_r=j+1^n(-1)^r ι_r#( R_n-1,j)
       ∩ U^δ_n,j) ( ω^n_j ∧ϕ) 
        = 
      (-1)^j ∑_r=j+1^n(-1)^r ι_r#( R_n-1,j)    ( ω^n_j ∧ϕ) 
        
    = 
     (-1)^j ∑_r=j+1^n(-1)^r ι_r#( R_n-1,jω^n-1_j   )    ( ϕ)  
     
     = 
     (-1)^j {β^n_n,j - β^n_j,j}  ( ϕ).




Now, for 0≤ r≤ j-1 one has 


    lim_δ→ 0 (  R_n,j∩ N^δ_0 ∩⋯∩∂ N^δ_r ∩⋯∩ N^δ_j-1∩ W^δ_j ∩ V^δ_n )  (ω^n_j ∧ϕ) 
        = (-1)^r+1 ι_r#(  R_n-1, j-1ω^n-1_j-1) (ϕ) .
 


Indeed, noting that ι_r^*ω_j^n = -ω_j-1^n-1,  denoting by  E_r,δ the total space of the circle bundle ϱ^δ_r ∂  N^δ_r → H_r  and using integration along the fiber of ϱ_r^δ, we obtain

    lim_δ→ 0 (  R_n,j∩ N^δ_0 ∩⋯∩∂ N^δ_r ∩⋯∩ N^δ_j-1∩ W^δ_j )  (ω^n_j ∧ϕ) 
        =  lim_δ→ 0 (  R_n,j∩E_r,δ|_ N^δ_0  ∩⋯∩N^δ_r∩⋯∩  N^δ_j-1∩  W^δ_j  ∩ H_r ω^δ_j  )  ( ϕ)
        = 
     lim_δ→ 0 R_n, j∩ι_r#(  H_r∩  N^δ_0  ∩⋯∩ N^δ_r ∩⋯∩  N^δ_j-1∩  W^δ_j  ϱ_r^δ_!ω^n_j) (ϕ)
        = 
     ι_r#(  R_n-1, j-1  (-1)^r+1ω^n-1_j-1) (ϕ).
 
 
 












Therefore, the term in (<ref>) gives

    lim_δ→ 0 { R_n,j∩(  ∑_r=0^j-1 N^δ_0 ∩⋯∩∂ N^δ_r ∩⋯∩ N^δ_j-1) ∩ W^δ_j } ( ω^n_j ∧ϕ)  
        =  -  ∑_r=0^j-1 (-1)^rι_r#(  R_n-1, j-1ω^n-1_j-1) (ϕ) 
        = -β_j-1,j-1^n(ϕ).




Now a direct calculation gives:



    lim_δ→ 0 { R_n,j∩ N^δ_0 ∩⋯∩ N^δ_j-1∩∂ W^δ_j } ( ω^n_j ∧ϕ) 
        =
      (   R_n,j∩ S_j ) ((-1)^jθ_j-1^n∧ϕ) =
      (-1)^j   R_n,j-1  (θ_j-1^n∧ϕ) 
        = 
    (-1)^j  (  R_n,j-1θ_j-1^n) (ϕ).


































Putting all together in (<ref>) one obtains


    d    (  R_n,jω^n_j )( ϕ)  =
    (-1)^n+j-1lim_δ→ 0 ∂(  R_n,j∩ U^δ_n,j)  
     (  ω_j^n ∧ϕ)
    
           
    - (-1)^n+j-1(  R_n,jθ^n_j ) (ϕ)  
        =
    (-1)^n+j-1 (-1)^j {β_n,j^n(ϕ)   -  β_j,j^n (ϕ)  }
        +
    (-1)^n+j-1{  -β_j-1,j-1^n(ϕ)  }
        +
    (-1)^n+j-1{ 
     (-1)^j (  R_n,j-1θ_j-1^n) (ϕ)  }
         - 
     (-1)^n+j-1(  R_n,jθ^n_j ) (ϕ) 
        =
     (-1)^n {β_j,j^n (ϕ) -  β_n,j^n(ϕ) + (-1)^j β_j-1,j-1^n(ϕ) }
        +
    (-1)^n{  (-1)^j(  R_n,jθ^n_j ) (ϕ) 
     -   (  R_n,j-1θ_j-1^n) (ϕ)  }









The currents R_n,j [jn] and [jn] are normal, for all 0≤ j ≤ n.


































































































 §.§ The fundamental triple of currents on 


We  now have all the ingredients to build the desired triple. 



 
 Set
W_0  = 0 and for n ≥ 1 define

    W_n   := (-1)^n2∑_j = 1^n (-1)^j2^n-j  R_n,jjn .

Denote Θ_n := [nn] and call

    ( Θ_n,  n, W_n)  ∈ F^n n^n⊕^n(^n) ⊕n-1^n

the fundamental triple of currents on ^n.






The fundamental triple ( Θ_n,  n, W_n) satisfies the following identity

    d W_n     =  Θ_n -  (-1)^n+12 (2π)^n n + ∑_r=0^n (-1)^r ι_r#( W_n-1) 
        =  Θ_n - (-1)^n2 (-2π)^n n + τ_n ( W_n-1).





It follows from the proposition that

    dW_n = 
    (-1)^n2∑_j=1^n (-1)^j2 (2π)^n-j 
    d (  R_n,jω_j )  
        =
    (-1)^n2∑_j=1^n (-1)^j2 (2π)^n-j 
    (-1)^n {β_j,j^n  -  β_n,j^n + (-1)^j (2π)  β_j-1,j-1^n }
          + 
      (-1)^n2∑_j=1^n (-1)^j2 (2π)^n-j
     (-1)^n+j{(  R_n,jθ^n_j )  }
          - 
     (-1)^n2∑_j=1^n (-1)^j2 (2π)^n-j
    (-1)^n  { (  R_n,j-1θ_j-1^n )   }.



Using the fact that β^n_n,n= β^n_0,0 = 0, R_n,n =  ^n  and R_n-1,n = 0, along with a simple reindexing, the above sum gives

    dW_n =  
     - (-1)^n2 + n  ∑_j=1^n (-1)^j2 (2π)^n-jβ_n,j^n        + 
         R_n,nθ^n_n      - 
     (-1)^n2+n  (2π)^n R_n,0
        =  
    (-1)^n-12∑_j=1^n-1 (-1)^j2^n-jτ_n( R_n-1,jω^n-1_j   )  
              + 
        R_n,nθ^n_n      - 
     (-1)^n+12  (2π)^n R_n,0
        =
        Θ_n   -   (-1)^n+12  (2π)^n n
           +    τ_n((-1)^n-12∑_j=1^n-1 (-1)^j2^n-1-j R_n-1,jω^n-1_j   )      
         
        =
        Θ_n    -  (-1)^n+12  (2π)^n n +    τ_n( W_n-1).












§ THE REGULATOR MAP



Let U be a smooth quasiprojective variety and let U↪ X ↩ D be a projective compactification of U, with D:= X-U a divisor with normal crossings.




Let Υ⊂ U ×Δ^n be an irreducible subvariety of codimension p ≤U = m which is equidimensional and dominant over Δ^n. Using  Propositions <ref> and <ref>, along with  (<ref>) and Remark <ref>, one can define a triple of currents
Υ  by

    Υ_  :=        (-1)^n2(-2π)^p  (Υn)∩U    ∈     ^2p-n_loc(U; (p))      
    Υ_Θ  :=       (- 2π)^p-n ΥΘ_n    ∈      F^p 2p-nXD   
    Υ_W  :=      (-2π)^p-n ( ΥW_n)∩ U    ∈     2p-n-1U .



Define group homomorphisms

    𝒵^p_eq(U;n)    ⟶Γ( X; (p)_, (X,U)^2p-n) 
    Υ   ⟼Υ,

where (p)_, (X,U)^2p-n is the complex giving Deligne cohomology, as in  Section <ref>.






Let U be a smooth quasiprojective variety  and let  U ↪ X  ↩ Y=X-U be a projective compactification of U, with Y a divisor with normal crossings. The assignment Υ↦(Υ) defines a map of complexes 

    𝒵_Δ, eq^p(U;*) ⟶(p)_^2p-*(U)

which gives the regulator maps 

    CH^p(U;n) ⟶ H^2p-n_(U;(p)),

for all 0≤ n≤ 2p, whenever (U) ≥ p.



The theorem follows directly from Lemmas <ref> and <ref> below. 







The currents Θ_n,  n and W_n in ^n vanish suitably along the hyperplane at infinity H_∞, in the sense of Definition <ref>. 




This is clear for n and  Θ_n.  For  W_n  it suffices to show that R_n,jjn
vanishes suitably at H_∞, for all 0≤ j ≤ n.

We use the parametrization    Φ=Φ_n,j  ^j×n-j→^n  of   R_n,j given in (<ref>). Consider   the neighborhood of R_n,j given by
W^τ = Φ (N_j^τ×n-j ), where N_j^τ denotes the τ-tubular neighborhood of the hyperplane H_j⊂^j in the Fubini-Study metric.

Given β∈n^n we need to show that

    lim_τ→ 0S∩∂ W^τ (β) = lim_τ→ 0∫_R_n,j∩∂ W^τjn∧β=0.


Now we can assume, with no loss of generality,  that
(Φ^*( jn∧β)) ⊂  U_0 ×n-j = (^j - H_0)×n-j. Note that 

    U_0∩∂ W^τ = {[1::λ] ∈^j |∈^j-1,  λ∈ and  |λ| = √(1+||^2)δ(τ) } ,

with δ(τ)0.
 

Using  an appropriate parametrization of 
U_0∩∂ W^τ≅^j× S^1×n-j
we can write S∩∂ W^τ (β) as the integral

    ∫_^j-1× [0,2π]× [0,1]^n-jlog( 1+()/1+()+δ(τ)√(1+|  |^2) e^iθ t_1 ) 
    φ(,λ,)/x_1… x_j-1δ(τ)dV,

where ()=x_1+… +x_j-1,  dV denotes the volume form on ^j-1× [0,2π]× [0,1]^n-j and  φ is a smooth function with compact support. The result now follows from the dominated convergence theorem.






With Υ as above, denote ∂_k Υ := Υ_|  H_k, where H_k is the k-th coordinate hyperplane in ^n, image of the inclusion ι_k ^n-1↪^n and  H_k =  H_k ∩Δ^n. Then

   
   
  * (∂Υ)_:= ∑_k=0^n (-1)^k ( ∂_k Υ)_ =  d( Υ_)
   
   
  * ( ∂Υ)_Θ := ∑_k=0^n (-1)^k ( ∂_k Υ)_Θ =  
   d( Υ_Θ)
   
   
  * ( ∂Υ)_W :=    ∑_k=0^n (-1)^k ( ∂_k Υ)_W = Υ_Θ- Υ_ -   d( Υ_W )
 



By definition

    dΥ_    := (-1)^n2 (-2π)^p d Υn  Cor. <ref>=  
    
    (-1)^n2(-2π)^p  Υd n
        = (-1)^n2 (-2π)^p (-1)^n+1Υ∂n
        =   (-1)^n-12 (-2π)^p     ∑_k=0^n (-1)^k Υι_k#(n-1)
        =   (-1)^n-12 (-2π)^p    ∑_k=0^n (-1)^k  (Υ_|H_k)^∨_n-1,

where the last identity follows from Proposition <ref> and Corollary <ref>.

On the other hand,  Corollary <ref> shows that
(Υ_|H_k)^∨_n-1 = (Υ_|∘H_k)^∨_n-1, and one concludes from (<ref>) that 

    dΥ_    =  
       (-1)^n-12 (-2π)^p      ∑_k=0^n (-1)^k  (Υ_|∘H_k)^∨_n-1
       def=
     (-1)^n-12 (-2π)^p     ∑_k=0^n (-1)^k  (∂_k Υ)^∨_n-1
        
    =  (-1)^n-12 (-2π)^p      ( ∂Υ)^∨_n-1 def= ( ∂Υ)_ .

This proves identity p1.

Similarly, Corollaries  <ref> and <ref>.II,  Lemma <ref> and Proposition <ref>  give

    dΥ_Θ    := (-2π)^p-n d  ΥΘ_n = (-2π)^p-n ΥdΘ_n
       (<ref>)= (-2π)^p-n (-2π) ∑_k=0^n (-1)^k Υι_k#(Θ_n-1)
        = (-2π)^p-(n-1)∑_k=0^n (-1)^k  (Υ_|H_k)^∨_Θ_n-1.

Then, once again Corollary  <ref>.I shows that
(Υ_|H_k)^∨_Θ_n-1 = (Υ_|∘H_k)^∨_Θ_n-1, and one concludes from (<ref>) that 

    dΥ_Θ    =  
    (-2π)^p-(n-1)∑_k=0^n (-1)^k  (Υ_|∘H_k)^∨_Θ_n-1def=
    (-2π)^p-(n-1)∑_k=0^n (-1)^k  (∂_k Υ)^∨_Θ_n-1
        =  (-2π)^p-(n-1) ( ∂Υ)^∨_Θ_n-1 def= ( ∂Υ)_Θ ,

proving identity p2.

Finally, 

    d Υ_W  =
     (-2π)^p-n d  ΥW_n = (-2π)^p-n ΥdW_n
       Cor.<ref>= (-2π)^p-n{ΥΘ_n - (-1)^n2(-2π)^nΥn + ∑_k=0^n (-1)^k Υι_k#W_n-1}
       def=Υ_Θ - Υ_ - (-2π)^p-(n-1)∑_k=0^n (-1)^k Υι_k#W_n-1
        =
    Υ_Θ - Υ_ - (-2π)^p-(n-1)(∂Υ)^∨_W_n-1 def=Υ_Θ - Υ_ - (∂Υ)_W,

where the second to last identity follows from
Corollaries  <ref> and <ref>.II, along with  Proposition <ref>
in the same fashion as the previous cases. 










§ EXAMPLES AND BEYOND





 §.§ On the Mahler measure of  polynomials


In Example <ref> we use a polynomial () ∈[] = [ t_1, …, t_n ] with  ⊂, to construct a cycle Γ_∈n+1n+1U_, where U_ = ( ^n - Z_) × and Z_⊂^n is the zero set of ,  given by

    Γ_ = 
    
    z_n+1( λ + ( ) )     = λ
    
    z_0 -  t_1z_1      = 0 
    
    z_0 + z_1 - t_2 z_2      = 0 
    ⋮   ⋮
    
    z_0 + z_1 + ⋯ + z_n-1 - t_n z_n     = 0
    .

It is easy to see that the same construction works when  is a Laurent polynomial. 

In order to calculate (Γ_) we discuss a few preliminaries. 



  §.§.§ General conditions and calculations




Consider () ∈[] with () = d,  and recall that we introduced 

    R_() = (z_1 ⋯ z_n)^d( _0()/z_1, _1()/z_2, …, _n-1()/z_n) ∈[]

to define a divisor 
Y_⊂Δ^n+1 given by  (z_0 z_1 ⋯ z_n+1)(_0() ⋯_n()) R_() =0 
that satisfies Γ_∩ (U_× Y_) = ∅. 


   
  * Define 
_ := {  (, λ) ∈ U_| (1+t_1)⋯ (1+t_n)(λ + ()) =  0 }⊂ U_.
This is the union of the graph of - with ⋃_r=1^n{( ×⋯×{- 1 }×⋯×) ×}∩ U_. Note that  ( _×Δ^n+1) ∩Γ_ = ∅. 
   
  * Define

    E   Δ^n+1 - Y_    ⟶   U_,   ⟼( _0()/z_1, …, _n-1()/z_n, _n()/z_n+1)  
    ℓ_ U_   ⟶ U_,    ( , λ) ⟼( , ()/λ) 
    ϕ U_ - _   ⟶Δ^n+1,    ( ; λ) ⟼(z_0(;λ), …, z_n+1(;λ) ),

where

    ϕ:
    
    z_0(;λ)     = (  ∏_r=1^n  t_r/1+t_r)()/() + λ
    
    z_j(;λ)     = 1/1+t_j(  ∏_r=j+1^n  t_r/1+t_r)()/() +λ,   for   j=1, …, n; 
    
    z_n+1(;λ)     = λ/() + λ.






Using the notation above along with Definition <ref>, the following holds. 



  * Let ω_j := ω_j^n+1 be as in Definition <ref>.<ref> and denote
β_j := -  log(-  t_j)  dt_1/t_1∧⋯∧dt_j-1/t_j-1  .
Then

    ϕ^* ω_j     =   β_j ,   for  j=1,… n, 
    ϕ^* ω_n+1   =  - log{- ()/λ}dt_1/t_1∧⋯∧dt_n/t_n,    and
    ϕ^*θ_n+1     = (-1)^ndt_1/t_1∧⋯∧dt_n/t_n∧d λ/λ



   
  * The  map  ψΔ^n+1  - Y_→ U_, 
introduced in   (<ref>), can be factored as


    ψ :  (Δ^n+1 - Y_)  U_  U_,

and its image is U_ - _ .


  * The map ϕ U_ - _→Δ^n+1 is the inverse of ψ.





The proof follows from straightforward calculations. 


Next we introduce some integral currents that appear in subsequent calculations.



Let _n denote the compact torus _n := (S^1)^n ⊂ ()^n. For j= 0, …, n define

    T_j :=   ()^ j× (0, ∞)^n+1 - j ⊂ ()^n+1 .
 
Then T_j defines an analytic chain in ()^n+1, and we use the same symbol to denote its restriction to U_.






From now on,  assume that the polynomial  satisfies the following:

   
  * () has no zeros on the torus _n= (S^1)^n,  i.e. (S^1)^n ×⊂ U_;
   
  * ( (0, ∞)^n ) ⊂ (0,∞); 
   
  * If Gr_-⊂ U_ denotes the graph of the restriction of - to _n, then Gr_- intersects T_j properly,   for each j=1,…, n. 






Condition P2. implies  that ℓ_# (0, ∞)^n+1   =   -  (0, ∞)^n+1, see (<ref>). 









The following polynomials satisfy the conditions above. 

   
  *  Constant polynomials () = k ∈, with k>0.
   
  *  The Laurent polynomial 


    _α(x, y) = α +x + 1/x + y + 1/y

with α∈, whenever  α > 4.




Assume that one has a fixed NCD compactification U_↪ X_↩ D.




For all j = 0,…, n the restriction of Γ^∨_,  R_n+1, j to U_p  (see Defn. <ref>) is the semi-algebraic chain  
Γ^∨_,  R_n+1,j :=    ℓ_# T_j  .



A direct calculation shows that the map E  (<ref>) induces an  orientation-preserving diffeomorphism between  
R_n+1, j∩ (Δ^n+1 - Y_) and  T_j  ∩ (U_ - _ ) .  By definition, the restriction of  Γ^∨_,  R_n+1, j to U_p - _
is given by 

    ψ_#{ R_n+1,j∩  (Δ^n+1 - Y_) }    =    ℓ_#{ E_# R_n+1,j∩  (Δ^n+1 - Y_)}
        =    ℓ_#{ T_j ∩ (U_ - _ ) } .

The result follows. 





For j=1, …, n, the restriction of   Γ^∨_,  R_n+1,jω_j to U_ is given by 

    Γ^∨_,  R_n+1,jω_j := 
    ℓ_#( 
     T_j β_j
    ).

and

    Γ^∨_,  R_n+1,n+1ω_n+1 = 
    Γ^∨_, [ ω_n+1]  = -  [ log{-()/λ}dt_1/t_1∧⋯∧dt_n/t_n] .




It follows from Lemma <ref> and Corollary <ref>, together with Remark <ref>, that  (Γ_)=( Γ_, , Γ_, Θ, Γ_, W) – see Definition <ref> – is completely determined by







































    Γ_,     =  (-1)^n2 (2π)^n+1 (0, ∞) ^n+1
    Γ_, Θ    = (-1)^n[ dt_1/t_1∧⋯∧dt_n/t_n∧d λ/λ] 
    
    Γ_, W    =   (-1)^n+12∑_j=1^n (-1)^j2^n+1 - j ℓ_#(  T_j β_j )   
    
          -  [ log{-()/λ}dt_1/t_1∧⋯∧dt_n/t_n]





  §.§.§ Explicit periods


Let ∈[] be an arbitrary (Laurent) polynomial satisfying Conditions <ref>, and let ∈[] be the constant polynomial. By definition, U_⊂ U_ = ^n+1 and we can restrict Γ_ to U_ and  obtain a cycle in

    Γ := Γ_ -  Γ_ ∈ n+1n+1U_.

It follows from (<ref>), (<ref>) and (<ref>) that

    (Γ )=( 0, 0, Γ_, W - Γ_, W)

and, as a result,  we see that [Γ]  
comes from  H^n(U_; )/ H^n(U_, (n+1)) in the  exact sequence
 
    0→  H^n(U_, )/H^n(U_, (n+1))  →n+1n+1U_
    → H^n+1(U_, (n+1))⊕ F^n+1H^n+1(U_, ) →⋯ .


In particular,  [Γ]    induces a homomorphism


    γ H_n(U_, ) →/(n+1).



Condition P1 on  shows  that _n×{ -1 } represents a non-trivial class in H_n(U_,), and we use the following lemma to calculate γ (_n ×{ -1 }).



Let () ∈[] satisfy the three conditions above. Then

   
  * For all j=1, …, n one has T_j   ∩ _n ×{ -1 } = 0.
   
  * The intersection 
      𝕏_n,j :=   ℓ_p#(  T_j   ) ∩ _n ×{- 1 } is well-defined and satisfies 
      
    𝕏_n,j (β_j) = 0 ,   for all      j=1, …, n.

      
  * One has 
      
   
  * ∫__n×{ -1 }β_n+1 = 0
   
  * ∫__n×{ -1 }log{-()/λ}dt_1/t_1∧⋯∧dt_n/t_n   =  ^n (), where () is the (logarithmic) Mahler measure of the polynomial .

   





The first statement follows from the fact the the supports of the respective currents do not intersect.

Now, observe that 

    ℓ_p^-1 (_n ×{ -1 } ) =   { (, λ) ∈ U_|∈_n   and  λ = - ()} = Gr_-.

The identity above together with Conditions <ref>.P3 show that the intersection 

    T_j ∩Gr_- = ± T_j ∩ℓ_#^-1_n ×{-1 }

exists. Applying ℓ_p# to (<ref>) shows that 𝕏_n,j exists. 

Now, let σ_j  U_→ U_ denote complex conjugation on the j-th coordinate, j=1, …, n.
Then, σ_j reverses the orientation of both T_j and _n×{ -1 }. If follows that 
σ_j#𝕏_n,j = 𝕏_n,j. On the other hand, restricted to _n one has σ_j^* β_j = - β_j. It follows that 𝕏_n,j(β_j) = ( σ_j #𝕏_n,j)(β_j) = 𝕏_n,j(σ_j^*β_j) = - 𝕏_n,j(β_j). This concludes the proof of statement b. 

To prove statement c.i) just note that the restriction of β_n+1 to _n×{ - 1 } is zero. 

Finally, since  has real coefficients,  the integral of {()}dt_1/t_1∧⋯∧dt_n/t_n over _n is zero. This proves statement c.ii). 






Let () be a Laurent polynomial with real coefficients, satisfying Conditions <ref>. Then the homomorphism
γ H_n(U_, ) →/(n+1)  (<ref>)
satisfies 

    γ ( _n×{ - 1 }) =   -  ^n() ∈/(n+1).





This follows directly from the definition of Γ and from Lemma <ref>. 





For the polynomial _α(x, y) = α + x + 1/x + y + 1/y in Example <ref>, it is shown in
 <cit.> that for any α∈  one can describe (_α) in terms of hypergeometric functions: 
 
    (_α) = 𝐑𝐞{logα -  2/α^2433/2, 3/2, 1, 12,2,216/α^2}.

For α > 0, one also has

    (_α) = α/4𝐑𝐞{321/2, 1/2, 1/21, 3/2α^2/16},

see <cit.> and <cit.>. In particular, for α= 8, Corollary <ref> shows that  the homomorphism γ H_2(U__8, ) →/(3) satisfies

    γ  ( _2×{ -1 }) = - ^2 (_8) = 96   L(E_24, 2) ≠ 0 ∈/(3),

where L(E_24,z) is the L-series of the rational elliptic curve E_24  of conductor 24.





 §.§ Beyond the equidimensional framework



Equidimensional cycles provide a simple conceptual framework that yields an explicit construction of the regulator map at the level of complexes. However, the fundamental triple (Θ_*, *, W_*) can still be used in “suitably transversal” situations where equidimensionality is not satisfied. 



Given a smooth quasiprojective variety U, let  _Δ,⋔^p(U;*) ⊂_Δ^p(U;*) denote the subcomplex
generated by those irreducible correspondences Υ⊂ U×Δ^n  satisfying the following. Given a proper face f Δ^r ↪Δ^n then

   
  * Υ intersects  U × f(Δ^r) properly.
   
  * Given a NCD compactification X of U, with divisor D = X-U,  then
   Υ∩ (  X × f_#Θ_r) induces a current in *XlogD, and
      Υ∩ (  U × f_# W_r)  induces a current in *U and
       Υ∩ (  X × f_#r) induces a current in _loc^*(U ) .



The arguments  in the equidimensional case can be used to define a map of complexes
 _Δ,⋔^p(U;*) →(p)^*_(U), and this  suffices to provide another approach to an explicit regulator map, along the lines of <cit.> once the following is proven. 
 
 Claim:  The inclusion of complexes _Δ,⋔^p(U;*) ↪_Δ^p(U;*) 
is a quasi-isomorphism when U is a smooth quasiprojective variey.

This will be addressed in a forthcoming note. 













§ BACKGROUND ON CURRENTS





 §.§ Forms and Currents





Let  denote one of the following sites: smooth manifolds with the ^∞ topology; real or complex manifolds with the analytic topology; or algebraic varieties with the Zariski topology. If  is a sheaf on  and M ∈𝖮𝖻𝗃(  ), we denote by _M the restriction of  to the small site M_ of M.





  §.§.§ Forms on smooth manifolds



Let ^k be  the sheaf of -valued smooth differential forms of degree k on smooth manifolds,  and let  
kM := Γ(M,  ^k) and kM := Γ_c(M,^k)  respectively denote the spaces of   k-forms and k-forms with compact support  on a manifold M.  

The ^∞ topology on kM is defined  by uniform convergence on compact subsets of the derivatives of all orders of the coefficients of the forms in local coordinates. One topologizes the compactly supported forms kM by saying that β_i β∈kM if there is a compact set K ⊂ M such that (β_i)⊂ K for all i∈, and the sequence converges to β in the ^∞ topology on kM. See <cit.> or <cit.>.




Consider the  sheaves  and  of continous, respec. locally integrable,  complex-valued functions on manifolds.

   
  *  
   Since  is a sheaf of ^0-modules, one can define  ⊗_^0^k and let
   kM := Γ(M, ⊗_^0^k) denote the space of forms of degree k on M with coefficients in , or the “ k-forms on M”.
   
  * 
   Similarly,  kM := Γ(M, 𝒞⊗_^0^k) denotes the  continuous k-forms on M.  kM  is given the topology of uniform convergence on compact sets of the component functions of forms in any local coordinate system.
   
  * The continuous forms with compact support, kM := Γ_c(M, ^k), can be topologized by saying that a sequence {β_i }_i∈ converges to β∈kM if there is a compact set K ⊂ M such that (β_i)⊂ K for all i∈, and the sequence converges to β in the ^∞ topology on kM.

   





The bundles Λ^k TM ⊗ and Λ^k T^*M⊗ inherit  canonical Hermitian metrics induced by the Riemannian metric on M. In particular, given γ either in Λ^k T_x M or in Λ^k T^*_xM for some x∈ M,  we denote by |γ|_x the length of γ in the induced metric.  


(<cit.>, <cit.>)

 Let φ  be a (possibly discontinuous) k-form on M.
 
  
   
  * Given x ∈ M, define
   
    φ(x) := 
    sup{ | φ (γ) |   | γ∈Λ^kT_xM   is a decomposable  r-vector and |γ |_x  ≤ 1  }.

When φ  is a continuous form, the assignment x ↦φ(x) defines a continuous function φ M →. 

  
  * Given any K⊆ M, define  the comass of φ on K as

    ν_K(φ) := sup{φ(x) | x ∈ K }.


The form  φ is called  bounded when  ν_M(φ) < ∞.
   
  * 
   We say that φ is a Baire form if φ is the pointwise limit of a sequence {ϕ_r , r ∈} of continuous forms ϕ_r ∈kM.  Denote by kM the space of bounded Baire k-forms on M.   





  §.§.§ Forms on complex manifolds
(See <cit.>)


For  p≥ 0, let Ω^p be the sheaf of holomorphic p-forms on complex manifolds and denote by  the sheaf of holomorphic functions. 

On complex manifolds,   one has a decomposition  ^k = ⊕_p+q=k^p,q, where ^p,q is the sheaf of smooth forms of type (p,q). 

The exterior derivative d ^k →^k+1 is canonically written as d = d' + d” with d' ^p,q→^p+1,q and d”^p,q→^p,q+1.






Defining
F^p ^k := ⊕_p≤ r ≤ k^r,k-r, the exterior derivative makes (F^p^*, d) into a subcomplex of (^*,d) so that one gets the Hodge filtration  ^* = F^0 ^* ⊃⋯⊃ F^p ^* ⊃ F^p+1^* ⊃⋯ on forms. 



Consider  a smooth proper complex manifold  X, and
let D= ⋃_i D_i be a simple normal crossing divisor (DNC) on X.  This means that the irreducible components D_i are smooth, and each x ∈ X has a coordinate neighborhood U where x ≡  (0,…, 0) and D∩ U = { (z_1, …, z_d) | z_1 ⋯ z_k = 0 }, for some 0 ≤ k ≤ d.

Denote U:= X- D ↪ X and let 1XD⊂_*Ω^1_X be the (locally free) sheaf of meromorphic 1-forms on X generated over _X by the forms df/f where f is a holomorphic function whose zero set is contained in D.
Define pXD := 1XD∧⋯∧1XD. Since D is a DNC,  *XD⊂_* Ω^*_U is a locally free subsheaf of graded _X-algebras generated locally in the neighborhoods described above by Ω^*_X and the forms i, i=1,…, k.
The Hodge filtration F of *XD is the decreasing filtration F^p*XD := ⊕_r ≥ ppXD.




Let D @^(->[r]     X    @_(->[l]_- U be as above.

   
  * 
    pXD⊂Ω^p_X:  the subsheaf consisting of the holomorphic p-forms that vanish on D. 
   
  * 
   pqXD :=  pXD⊗^0,q_X.
   
  *  
   pqXD :=  pXD⊗^0,q_X






  §.§.§ Currents on smooth manifolds
(<cit.>, <cit.>, <cit.>)


Let M be a smooth m-dimensional manifold. The sheaf kM of (deRham) currents of degree k on M associates to each open subset U ⊂ M the vector space kU consisting of the continuous linear functionals on  m-kU. 
In this context, elements in  m-kU are called
 test forms.



Let M be a smooth m-dimensional manifold, and consider T ∈kM, S ∈p,qX, ω∈rM, and test forms φ∈*M.

   
  * If N ⊂ M is an oriented submanifold of codimension r denote by N ∈rM the current φ↦ N (φ) :=  ∫_N ω, defined by integration along N. 
    
  * We define [ ω ]  ∈rM  by  [ω](φ) = ∫_M ω∧φ. Hence, the assignment ω↦ [ω] induces an inclusion kM↪kM. More generally, a form β in kM  defines a current [β] ∈kM in a similar fashion. 
   
  * Define Tω∈k+rM by (Tω)(φ) = T(ω∧φ). Note that this is a generalization of the exterior product of forms, namely, [β] ω = [ β∧ω ].
The current Tω is often denoted by T∧ω in the literature. 
   
  * The boundary bT of T is the adjoint of the exterior derivative on forms: 
    bT(φ) = T(dφ) .

   
  * The exterior derivative d kM→k+1M is defined as 
    dT := (-1)^k+1 bT,
 so that the inclusion *M↪*M becomes a map of complexes. 
   
  * Denote the restriction of T to an open set W⊂ M by T_|W and define the support (T) of T as the intersection of all closed sets F such that T_|M-F =  0.
   
  * If f  M → N is a smooth map such that f_| T is proper, one defines the push-forward f_#T ∈k+m-nN by f_# T(ψ) := T(f^* ψ). Note that d∘ f_# = f_#∘ d.








  §.§.§ Currents on complex manifolds
(<cit.>)


When X is a complex manifold of dimension d,  the sheaf p,qX of currents of type (p,q)  consists of those currents that vanish on all test forms of type (r,s) ≠ (d-p, d-q).




Consider  D @^(->[r]     X    @_(->[l]_- U as before. 
Define the currents with log poles pqXD as the continuous linear functionals on 
   
    ^d-p,d-q_c(X)𝗇𝗎𝗅𝗅  D  := Γ_c(X,d-pd-qXD ),
 in the relative topology from p,qX.
Define 

    kXD := ⊕_p+q=kpqXD
 
and

    F^p kXD := ⊕_r≥ prk-rXD  .

    This gives a filtered complex 
    *XD = F^0*XD⊃⋯⊃ F^p
        *XD⊃⋯ .






  §.§.§ Special currents
 




  *  We say that a current T ∈kM  is representable by integration, or  a current of  order 0, or a current with measure coefficients if it extends to a continuous linear functional on m-kM. Denote the space of all such currents by kM.


  * If T is represented by integration, it follows from Riesz representation theorem that there is  a Radon measure T on M, and a T-measurable (m-k)-vector field ξ_T such that T is given by

    T(φ) = ∫_M φ, ξ_T  d T,

on test forms φ. By the dominated convergence theorem, one can define T(β)  on bounded Baire forms β∈_c^k(M). See <cit.>.


  * Define
(T) := sup{ T(φ) |φ∈kM and ν_K(φ) ≤ 1 }. The current T is said to have finite mass when (T)< ∞.


  *  A current T is called locally normal if  both T and dT are represented by integration. It is called normal if it is locally normal and (T) is compact. 
We denote by kM and kM the spaces of locally normal and normal currents of degree k on M, respectively.  See <cit.> or <cit.>.



  * The rectifiable currents are defined as the completion in the mass norm  of the group of Lipschitz push-forwards of finite polyhedral chains in some Euclidean space, and a current T is locally rectifiable if for each x∈ M there is a rectifiable current T_x such that x∉(T-T_x).


  *  A current T is called  locally integral if both T and dT are locally rectifiable. It is called integral if it is locally integral and (T) is compact. 
We denote by kM and kM the spaces of locally integral and integral currents of degree k on M, respectively.  See <cit.>, <cit.> and <cit.>

  * Standard arguments show that kM⊂kM.

  *  If G is a finitely generated abelian group, we denote ^k_loc(M;G) := kM⊗ G,
and
^k(M;G) := kM⊗ G for the groups of locally integral and integral G chains.








 §.§ Slicing and intersection of locally normal currents




  §.§.§ Slicing locally normal currents


Here we summarize material from <cit.> and <cit.>. 
Let M be a smooth m-dimensional Riemannian manifold,  and let N be an oriented n-dimensional Riemannian manifold
with (unit) orientation form n. Denote by _n the Hausdorff measure on N induced by the metric. 
Given a locally Lipschitz map f  M → N, a normal current T ∈kM and a bounded Baire form ϕ∈iN one can define a current 

    T, f, ϕ∈k+iM ,

which coincides with T f^*ϕ whenever f is a smooth map. 



If f  M → N is a locally Lipschitzian map and  T  ∈kM is a normal current of degree k≤ m-n, then for _n-almost all y∈ N there exists a unique current T, f, y ∈k+nM which can be defined as follows.
Denote by B_ρ(y) the ball  of radius ρ centered on y∈ N, and let χ_B_ρ(y) denote its characteristic function. Then

    T, f, y  (ψ) = (-1)^n(m-n-k) lim_ρ→ 0 1/_n(B_ρ(y))  T, f,   χ_B_ρ(y)n (ψ).




Properties of the slicing function:



   
  * T, f, y ⊂(T)∩ f^-1{ y }

   
  * Whenever k < m-n and T, f, y exists, so does   dT, f, y  = d  T, f, y.

   
  * Whenever ψ∈qM, with q ≤ m-n-k, and T, f, y exists, so does 
   
    Tψ, f, y  = (-1)^nq T, f, y ψ .


   
  * For every bounded Baire form ϕ∈k-nM one has
   
    T, f, n(ϕ) = ∫_N T, f, y (ϕ) d_n(y).


   
  * If  N → is a  bounded Baire function and ψ∈k-nM then
   
    T, f,   n(ψ) = ∫_N T, f, y (ψ) (y)  d_n(y)

   and
   
    ( T, f,   n) =  ∫_N(  T, f, y  )   |(y)|  d_n(y).




Using slicing of currents, Federer introduces the notion of intersection of (locally) normal  currents.[This can be defined, more generally, for locally flat currents]  Given S ∈kM, T∈rM, with k+r ≤ m, one says that the intersection of S and T exists provided there is a current S∩ T ∈k+rM characterized by the condition:

Let γ M → M× M be the diagonal map. If h U  U' is an orientation-preserving diffeomorphism from an open subset of U to U'⊂^m, and δ^m×^m →^m is the “difference" map δ(, ) = -, then

    (γ_|U)_#{ (S∩ T)_|U} = (-1)^k(m-r) (S× T)_|U× U, δ∘ (h× h), 0

In other words, when the intersection exists, it is determined locally by the slice of the product of the currents under the 
difference map, over the origin 0∈^m.




 When the currents S, T are locally integral (repec. semi-algebraic chains, sub-analytic chains) and the intersection exists, then S∩ T is integral (respec. semi-algebraic chains, sub-analytic chain). 




  §.§.§ Slicing Analytic Chains


Here we summarize material from <cit.>. 



Let X be an oriented real analytic manifold.


  * A k-dimensional locally integral current T in X is called a k-dimensional analytic
chain if X can be covered by open sets U for which there exist k and (k - 1)-dimensional real analytic subvarieties Z and W of U with U ∩T⊂ Z and U ∩(b T)⊂  W. 

  * It
follows from <cit.> that an analytic chain T is a locally finite sum of chains corresponding to integration over certain k dimensional oriented analytic submanifolds of X.

  * If X is a real analytic manifold, we denote by kX⊂_k^loc(X) the group of k-dimensional real analytic chains on X.




(<cit.>)
Let f  M → N be an analytic map between oriented real analytic manifolds of dimensions m and n, respectively.   Given an analytic chain T in M of dimension k, denote

    Y =
    { y ∈ N | 
    	[   { f^-1(y) ∩(T)}      ≤ k-n,   and; { f^-1(y) ∩(bT) }         ≤ k -n -1 ]. 
    }.

Then the slicing function y  ↦ T, f, y, maps Y into the k-n dimensional analytic chains in M, and is continuous in the flat norm topology in ^loc_k-n(M).



It follows directly from this theorem that when M is an oriented real analytic manifold and the currents S, T are analytic chains, then S∩ T is an analytic chain, whenever this intersection exists. 

There are simple support criteria that ensure  the existence of the intersection of analytic chains. 


(<cit.>)

Given analytic chains S, T in M, with S = s, T = t, one says that { S, T } intersect properly (or intersect suitably) iff

   
  * s+t ≥ m ,
   
  * [ (S) ∩(T)] ≤ s+t -m,
   
  * [ {( bS) ∩(T) }∪{(S) ∩( bT) }] ≤ s+t -m -1






If { S, T} intersect properly, then the intersection of S and T exists and S∩ T is an analytic chain in M of dimension s+t-m,  and 
    b (S∩ T) = (-1)^m-tbS ∩ T + S ∩ bT.






 §.§ Main quasi-isomorphisms


See <cit.>,   <cit.>,   <cit.>. 



Let M be a smooth manifold. 
Then the sheaves kM, '^k_M and ^k_M,loc are acyclic, for all k≥ 0 and the following maps of complexes are quasi-isomorphisms:
   
    _M *M'^*_M    and  _M ^*_M,loc .








Let D @^(->[r]     X    @_(->[l]_- U be an NCD compactification of the smooth complex variety U. Then one has a commuting diagram of quasi-isomorphisms of complexes on X, where the rightmost arrows are filtered quasi-isomorphisms. 

    R_* Ω_U^* = _* Ω_U^*   [d]^≃   ( *XD, F^*) [l]_≃[d]_≃^filtered
    
       R_*_U [ur]^≃[r]^≃[dr]^≃   _**U[d]^≃   ( ^*_Xlog D, F^*) [l]_≃[d]_≃^filtered
       _*'^*_U   ( '^*_Xlog D , F^*) [l]_≃

















§ FURTHER TECHNICAL PROOFS


 



Let S, T be normal currents of dimensions s and t, respectively, on an m-dimensional oriented manifold M. If the intersection S∩ T exists, and  ω is a form on M which is both T-summable and S∩ T-summable, then
the intersection S ∩ (T ω) exists and satisfies

    S ∩ (T ω)  =  (S∩ T) ω.




One may assume that M = ^m. Denote δ^m →^m ×^m the diagonal map and let f ^m ×^m →^m denote the difference map f(, ) =  - .

Then, by definition, S∩ T is the unique normal current on M that satisfies

    δ_# ( S∩ T) = (-1)^(m-s)t S× T,  f, 0.

Now, if π_2 ^m ×^m →^m is the projection onto the second factor, one has:
δ_#(  (S∩ T) ω)
= 
δ_#(  (S∩ T) δ^* π_2^* ω)  
= 
δ_#( S∩ T) π_2^*ω.

Given a test form φ on M one then has 

    δ_#( S∩ T) π_2^*ω (φ)     =
    δ_#( S∩ T)( π_2^*ω∧φ) = 
    (-1)^(m-s)t S× T,  f, 0 ( π_2^*ω∧φ) 
        =
    (-1)^(m-s)tlim_ϵ→ 01/V_ϵ (S× T) ( f^* Ω_ϵ∧π_2^* ω∧φ) 
        = 
    (-1)^(m-s)t + |ω| |φ|lim_ϵ→ 01/V_ϵ (S× T) ( f^* Ω_ϵ∧φ∧π_2^* ω)

where Ω_ϵ is the restriction of the volume form on ^m the ball B_ϵ(0)  of radius ϵ around 0,  V_ϵ is the volume of B_ϵ(0), and |φ|, |ω| are the degrees of the forms. 

On the other hand, one can approximate f^*Ω_ϵ∧φ by a sequence of sums of forms of the type π_1^* α∧π_2^* β, see <cit.>. We have

    (S× T) ( π_1^*α∧π_2^* β∧π_2^* ω)
        =(-1)^|β| |ω|  
    (S× T) ( π_1^*α∧π_2^*(ω∧β) ) 
     
       =(-1)^|β| |ω|  S(α) T(ω∧β) 
        = (-1)^|β| |ω| { S × ( T ω) }( π_1^* α∧π_2^*β).


Now, note that 
|α| + | β| = m + |φ| and we can assume |α| = s, hence 
|β| = m-s +  |φ |. 
This gives

    (S× T) ( f^* Ω_ϵ∧φ∧π_2^* ω) = 
    (-1)^(m-s+| φ |) | ω |{ S × ( T ω) }( f^*Ω_ϵ∧φ)


Taking ϵ→ 0 in (<ref>) and using (<ref>) one gets 

    δ_#( S∩ T) π_2^*ω  (φ)  = 
    (-1)^(m-s)t + |ω| |φ| + (m-s+| φ |) | ω |  S × (T ω), f , 0 (φ) 
    
     = 
    (-1)^(m-s)t + |ω| |φ| + (m-s+| φ |) | ω | + (m-s) (t - |ω|)δ_#( S ∩ (Tω) )(φ)
    
    = (-1)^ 2(m-s)t + 2 |ω| |φ|δ_#( S ∩ (Tω) )(φ).








Given 0≤ j <n ≤ the following holds.

   
  *  For all j+1 ≤ k ≤ n one has L_k ∩ R_n,j =  0.
   
  * If R_n-1,j() and R_n-1,j(-1) denote the current R_n-1,j in the projective spaces ^ and ^-1, respectively, and ι_n^-1↪^ is the inclusion of ^-1 as the coordinate hyperplane H_n, then

    ι_n # R_n-1,j(-1) 
    =
     H_n ∩ R_n-1,j() .






It follows from (<ref>) and definitions that  L_k ∩ R_k,j = L_k ∩ S_j+1∩⋯∩ S_k  = L_k ∩ H_j+1∩⋯∩ H_k = L_j ∩ H_j+1∩⋯∩ H_k, and the latter is an intersection of k-j+1 linearly independent hyperplanes. It follows that

    L_k    ∩ R_n,j()  = L_k ∩ R_k,j() ∩ R_n,k() = 
     L_j ∩ H_j+1∩⋯∩ H_k  ∩ R_n,k() 
       ≅ R_n-(k-j+1),k -(k-j+1)(  -(k-j+1)) = R_n-k+j-1,j-1(-k+j-1).

Now, if L_k ∩ R_n,j≠ 0  then (L_k ∩ R_n,j()) must be 2+j-n-2. On the other hand  R_n-k+j-1,   j-1(-k+j-1)= (2 + j -n-2) - (k-j), which is strictly less than what it should be. This proves the first assertion. 

The second assertion follows directly from the definitions.






Given 1 ≤   j ≤ n, the following holds:


  * The differential form θ_j-1 is S_j-summable and hence one can define the flat current S_j ∩θ_j-1 =   S_j θ_j-1∈^j(^n).
 

  * The differential form ω^n_j is R_n,j-summable and hence one can define the flat current R_n,j∩ω^n_j =  R_n,jω^n_j   ∈^n-1(^n).





In order to prove the first statement we show that the restriction of θ_j-1 to S_j lies in Γ( S_j, ⊗_^0^j-1), using the parametrization of S_j,
Φ^n -1 ×1→^n given by Φ([:λ:],) =[:s_0λ-():s_1λ:],
where ()=u_0+…+u_j-2, λ∈ and ∈^n-j (we assume that 
j>1). 
In the case j=n, Φ is the parametrization Φ_n-n-1 of Proposition <ref>. 

Denote β := s_0λ - (). Then, as a form on ^n-1×1 one has
Φ^*θ_j-1 = θ_j-2∧dβ/β +  (-1)^j-1∧dlog(),
where dlog() := du_0/u_0∧⋯∧du_j-2/u_j-2. Write

dβ/β =  
s_0λ/β·dλ/λ  +  λ/β· ds_0   -  d()/β 

and observe that


θ_j-2() ∧d()/() =  (-1)^j-2dlog().

Therefore, 

    Φ^*θ_j-1    = s_0λ/β·θ_j-2∧dλ/λ +  λ/β·θ_j-2∧ ds_0 
          -
    (-1)^j-2()/β·dlog()
    +(-1)^j-1dlog() 
        = 
    s_0λ/β·θ_j-2∧dλ/λ +  λ/β·θ_j-2∧ ds_0  + (-1)^j-1{()/β + 1    }dlog() 
       =
    s_0λ/β·θ_j-1 +  λ/β·θ_j-2∧ ds_0 
        =  
    s_0λ/s_0λ - ()·θ_j-1 +  λ/s_0λ - ()·θ_j-2∧ ds_0.

To show this form is in Γ(^n-1×1, ⊗_^0^j-1) one simply needs to restrict it to standard coordinate charts and observe that  the coefficients are in .



For the proof of the second  statement we consider the parametrization Φ=Φ_n,j of R_n,j
given  in (<ref>). Locally Φ^*ω^n_j is a sum of terms of the form log y dlog() with (y,)∈×^j-1. One easily checks that 
log y dlog()∈j-1^j hence the result follows.





amsalpha



