nLab
Higher regulators and values of L-functions

Contents

Contents

This page is supposed to be a review of the seminal article

introducing Beilinson-Deligne cohomology, Beilinson regulators, higher regulators, and the Beilinson conjectures.

Please note that it is currently in a very preliminary state, having been prepared quickly as notes for a seminar talk on the first section of the paper. It follows the paper very closely, and an interested reader might like to rewrite from the nPOV.

Overview

Notation

By analytic space we will mean real analytic space. Let AnAn denote the real analytic site. Consider the category Ab(Sh(An))Ab(Sh(An)) of abelian sheaves on AnAn. Beilinson denotes by D +(An)D^+(An) its bounded above derived category, i.e. the category of connective cochain complexes up to quasi-isomorphism. Given an analytic space XAnX \in An, we will also consider its petit topos X X^\sim of sheaves on the site Ouv(X)Ouv(X) of open subsets.

In D +(An)D^+(An) we have the complex Ω \Omega^\bullet, of de Rham complexes of holomorphic forms. Let Ω i\Omega^{\ge i} denote the “stupid” filtration.

Fix a subring ARA \subset \mathbf{R} and let A(p)=A(2πi) pCA(p) = A \cdot (2\pi i)^p \subset \mathbf{C} for pZp \in \mathbf{Z}. We will abuse notation and write CD +(An)\mathbf{C} \in D^+(An) also for the constant sheaf valued in the complex concentrated in degree zero. Similarly we will view A(p)A(p) as an object of D +(An)D^+(An).

We will write H B *(X,C)H^*_B(X, \mathbf{C}) for the Betti cohomology of XAnX \in An.

Main constructions and conjectures

Deligne cohomology of analytic spaces

For each pZp \in \mathbf{Z}, the inclusions Ω pΩ \Omega^{\ge p} \hookrightarrow \Omega^\bullet and A(p)CΩ A(p) \hookrightarrow \mathbf{C} \hookrightarrow \Omega^\bullet induce a canonical morphism

Ω pA(p)Ω \Omega^{\ge p} \oplus A(p) \longrightarrow \Omega^\bullet

(given by the difference of the two inclusions).

Definition

For pZp \in \mathbf{Z}, the Deligne complex of weight pp, denoted A(p) DA(p)_\D, is defined as the mapping cone of the above morphism shifted by -1, hence fitting in the distinguished triangle

(Ω pA(p))[1]Ω [1]A(p) DΩ pA(p). (\Omega^{\ge p} \oplus A(p))[-1] \longrightarrow \Omega^\bullet[-1] \longrightarrow A(p)_\D \longrightarrow \Omega^{\ge p} \oplus A(p).

The Deligne cohomology is just the hypercohomology of this complex. That is, consider the right derived functor RΓ(,A(p) D)R\Gamma(-, A(p)_D) of the functor of global sections on AnAn with values in D +(Amod)D^+(A-mod).

Definition

The Deligne cohomology of XAnX \in An of weight pp and degree qq with coefficients in AA is

H D q(X,A(p))=H qRΓ(X,A(p) D). H_D^q(X, A(p)) = H^q R\Gamma(X, A(p)_D).
Proposition

There is a canonical long exact sequence

H B q1(X,C)H D q(X,A(p))Ω pH B q(X,C)H q(X,A(p)) \cdots \to H^{q-1}_B(X, \mathbf{C}) \to H^q_D(X, A(p)) \to \Omega^{\ge p} H^q_B(X, \mathbf{C}) \oplus H^q(X, A(p)) \to \cdots

This follows by applying the cohomological functor RΓ(X,)R\Gamma(X, -) to the above distinguished triangle.

Lemma

For p0p \le 0 there is a canonical quasi-isomorphism

A(p) DΩ p. A(p)_D \stackrel{\sim}{\longrightarrow} \Omega^{\ge p}.

For p>0p \gt 0 there are canonical quasi-isomorphisms

A(p) D (A(p)𝒪Ω 1Ω p1) (*) (0𝒪/A(p)Ω 1Ω p1) \begin{aligned} A(p)_D &\stackrel{\sim}{\longrightarrow} (A(p) \to \mathcal{O} \to \Omega^1 \to \cdots \to \Omega^{p-1}) & (\ast) \\ &\stackrel{\sim}{\longrightarrow} (0 \to \mathcal{O}/A(p) \to \Omega^1 \to \cdots \to \Omega^{p-1}) & \end{aligned}

Multiplicative structure

Proposition

There exists a canonical morphism in D +(An)D^+(An)

:A(i) D LA(j) DA(i+j) D - \cup - : A(i)_D \otimes^L A(j)_D \longrightarrow A(i+j)_D

inducing the structure of a graded commutative ring on

H D *(X,A(*))= p,qH D p(X,A(p)) H^*_D(X, A(*)) = \bigoplus_{p,q} H^p_D(X, A(p))

for all XAnX \in An.

Proof

Beilinson gives an explicit formula using the usual explicit model for the mapping cone. He also remarks shortly that the product can be defined by observing that the obvious multiplicative structures on Ω \Omega^\bullet, Ω *\Omega^{\ge *}, A(*)A(*), turn each into a monoid object in the symmetric monoidal category of cochain complexes (of abelian sheaves), that is, into dg-algebras (of complexes of sheaves). Consider then the homotopy pullback of the diagram A(*)ΩΩ *A(*) \to \Omega \leftarrow \Omega^{\ge *}; Beilinson claims that the underlying complex of this dg-algebra has in each degree pp the Deligne complex A(p) DA(p)_D of weight pp. This point is expanded on in Hopkins-Quick.

Here we will simply give a formula for the quasi-isomorphic complex (*)(*) of Lemma 1 (assuming i,j>0i,j \gt 0), which we will denote for the moment by A(i) EA(i)_E. For XAnX \in An, take xΓ(X,A(i) E)x \in \Gamma(X, A(i)_E), yΓ(X,A(j) E)y \in \Gamma(X, A(j)_E), and define

xy={xy deg(x)=0ordeg(y)=0 xdy deg(x)>0anddeg(y)=j>0 0 otherwise x \cup y = \begin{cases} x \cdot y & \deg(x) = 0\quad\text{or}\quad\deg(y) = 0 \\ x \wedge dy & \deg(x) \gt 0\quad\text{and}\quad\deg(y) = j \gt 0 \\ 0 & \text{otherwise} \end{cases}

We omit the various verifications, that this defines a morphism of complexes, is associative, commutative, etc. One gets a monoid object in the category of cochain complexes of abelian sheaves. It only remains to see that RΓ(X,)R\Gamma(X, -) preserves monoids, so that one gets the structure of a graded commutative ring on the hypercohomology groups H D *(X,A(*))H^*_D(X, A(*)).

The Bloch regulator

Beilinson uses this cup product for i=j=1i=j=1 to recover the Bloch regulator?.

Theorem

(Bloch). For each algebraic curve XX over R\mathbf{R}, there is a canonical functorial homomorphism

r X:K 2(X)H B 1(X,C *) r_X : K_2(X) \to H^1_B(X, \mathbf{C}^*)

from the second algebraic K-theory group to the first Betti cohomology group with coefficients in C *\mathbf{C}^*.

Sketch of proof

By Lemma 1, there are quasi-isomorphisms

Z(1) D𝒪 *[1] \mathbf{Z}(1)_D \stackrel{\sim}{\to} \mathcal{O}^*[-1]

induced by the exponential map, and

Z(2) D(𝒪 *dlogΩ 1)[1] \mathbf{Z}(2)_D \stackrel{\sim}{\to} (\mathcal{O}^* \stackrel{d \log}{\to} \Omega^1)[-1]

induced by xexp(x/2πi)x \mapsto \exp(x/2\pi i). It follows that the cup product

:H D 1(X,Z(1))H D 1(X,Z(1))H D 2(X,Z(2)) \cup : H^1_D(X, \mathbf{Z}(1)) \otimes H^1_D(X, \mathbf{Z}(1)) \longrightarrow H^2_D(X, \mathbf{Z}(2))

corresponds to a canonical homomorphism

𝒪 *(X)𝒪 *(X)H 1(X,𝒪 *Ω 1). \mathcal{O}^*(X) \otimes \mathcal{O}^*(X) \to H^1(X, \mathcal{O}^* \to \Omega^1).

According to Deligne, the RHS classifies isomorphism classes of line bundles with holomorphic connection?. Since dim(X)=1\dim(X) = 1, all connections are integrable and this group is identified with H B 1(X,C *)H^1_B(X, \mathbf{C}^*).

Now by Matsumoto’s theorem, giving a presentation of the K 2K_2 group of a field by two generators and certain relations, one has

K 2(𝒪(X))=(𝒪(X) *𝒪(X) *)/<t(1t)> t0,1. K_2(\mathcal{O}(X)) = (\mathcal{O}(X)^* \otimes \mathcal{O}(X)^*)/\lt t \otimes (1-t) \gt_{t \ne 0,1}.

On the other hand, one has the Steinberg identity t(1t)=0t \cup (1-t) = 0 for t𝒪 *(X)t \in \mathcal{O}^*(X). It follows that the homomorphism above factors through K 2(𝒪(X))=K 2(η)K_2(\mathcal{O}(X)) = K_2(\eta) for η\eta a generic point. To extend it to all of XX, one uses the commutative diagram of localization exact sequences

K 2(X) K 2(η) xX(C)C * H B 1(X,C) H 1(η,C *) xX(C)C * \begin{array}{ccccc} K_2(X) & \longrightarrow & K_2(\eta) & \longrightarrow & \oplus_{x \in X(\mathbf{C})} \mathbf{C}^* \\ \downarrow & & \downarrow & & \downarrow \\ H^1_B(X, \mathbf{C}) & \longrightarrow & H^1(\eta, \mathbf{C}^*) & \longrightarrow & \oplus_{x \in X(\mathbf{C})} \mathbf{C}^* \end{array}

The first row comes from the Gersten-Quillen resolution? for K-theory.

Relative cohomology

If SS and TT are toposes and u *:ST:u *u^* : S \rightleftarrows T : u_* is a geometric morphism, consider the Artin gluing, i.e. the topos (id S/u *)=(u */id T)(id_S/u_*) = (u^*/id_T) whose objects are morphisms u *(F)Gu^*(F) \to G for FSF \in S, GTG \in T, and morphisms are commutative diagrams. Write (S,T)(S, T) for this topos.

Definition

The functor of global sections on (S,T)(S, T) is the left exact functor

Γ(S,T,):Ch +(Ab(S,T))Ch +(Ab) \Gamma(S, T, -) : Ch^+(Ab(S, T)) \longrightarrow Ch^+(Ab)

defined by

Γ(S,T,F)=Cone(Γ(T,F T)Γ(S,F S))[1] \Gamma(S, T, F) = Cone(\Gamma(T, F_T) \to \Gamma(S, F_S))[-1]

for each sheaf F(S,T)F \in (S, T) given by u *(F S)F Tu^*(F_S) \to F_T with F SSF_S \in S and F TTF_T \in T. Let RΓ(S,T,):D +(Ab(S,T))D +(Ab)R\Gamma(S, T, -) : D^+(Ab(S, T)) \to D^+(Ab) denote its right derived functor.

The rest of this section is just defining a monoidal product on Ch +(Ab((S,T)))Ch^+(Ab((S, T))), and explaining that a monoid in D +(Ab((S,T)))D^+(Ab((S,T))) will induce a monoid in D +(Ab)D^+(Ab), i.e. a dg-algebra, after taking cohomology.

Complexes with logarithmic singularities

Let Π\Pi denote the category of pairs (X,X¯)(X, \overline{X}) with X¯An\overline{X} \in An smooth and j:XX¯j : X \hookrightarrow \overline{X} an open subspace such that the complement X¯X\overline{X} - X is a normal crossing divisor?. The open immersion jj induces a functor Ouv(X)Ouv(X¯)\Ouv(X) \to \Ouv(\overline{X}) on the petit sites by mapping an open subspace UXU \subset X to UXX¯U \subset X \subset \overline{X}. This induces a geometric morphism j *:X¯ X :j *j^* : \overline{X}^{\sim} \rightleftarrows X^{\sim} : j_* on the petit toposes. Following the discussion of the previous section, we make the

Definition

The topos of the pair (X,X¯)(X, \overline{X}) is defined to be the Artin gluing (j */id)(j^*/id), and will be denoted (X,X¯) (X, \overline{X})^{\sim}. Hence a sheaf on the pair (X,X¯)(X, \overline{X}) is a sheaf FF on XX, a sheaf GG on X¯\overline{X}, and a connecting morphism j *(G)Fj^*(G) \to F.

Let Ω X,X¯ Ch +(Ab((X,X¯) ))\Omega^\bullet_{X,\overline{X}} \in \Ch^+(Ab((X, \overline{X})^{\sim})) denote the de Rham complex of holomorphic forms on CC with logarithmic singularities? along X¯X\overline{X} - X. That is, Ω X,X¯ \Omega^\bullet_{X,\overline{X}} is an object of Ch +(Ab(X¯ ))Ch^+(Ab(\overline{X}^\sim)) and we view it as a complex on (X,X¯) (X, \overline{X})^\sim by taking the part on X X^\sim to be trivial. Let Ω X,X¯ p\Omega^{\ge p}_{X, \overline{X}} denote the stupid filtration.

Now we define a complex of abelian sheaves A(p) DA(p)_D in Ch +(Ab((X,X¯) ))Ch^+(Ab((X, \overline{X})^{\sim})) as follows.

Definition

The Beilinson-Deligne complex with logarithmic singularities of weight pp of the pair (X,X¯)(X, \overline{X}) is a complex of abelian sheaves on (X,X¯) (X, \overline{X})^\sim, denoted A(p) DCh +(Ab((X,X¯) ))A(p)_D \in Ch^+(Ab((X, \overline{X})^\sim)) and defined as follows. In degree nZn \in \mathbf{Z}, take the sheaf

A(p) D,X=Cone n(A(p)Ω X ) A(p)_{D,X} = Cone^n(A(p) \to \Omega^\bullet_X)

in X X^\sim (where the mapping cone is taken in Ch +(Ab(X ))Ch^+(Ab(X^\sim))), and the sheaf

A(p) D,X¯=Ω X,X¯ p A(p)_{D,\overline{X}} = \Omega^{\ge p}_{X, \overline{X}}

in X¯ \overline{X}^\sim, together with the connecting morphism induced by the inclusion j *(Ω p)Ω X j^*(\Omega^{\ge p}) \hookrightarrow \Omega^\bullet_X.

Definition

The Beilinson-Deligne cohomology with logarithmic singularities of the pair (X,X¯)(X, \overline{X}) in weight pp and degree qq and with coefficients in AA, is the hypercohomology

H D q(X,X¯,A(p))=H qRΓ((X,X¯) ,A(p) D). H^q_D(X, \overline{X}, A(p)) = H^q R\Gamma((X, \overline{X})^\sim, A(p)_D).

One defines a cup product on these complexes in the same way as above, and gets a graded commutative ring structure on Beilinson-Deligne cohomology with logarithmic singularities.

Definition

Deligne cohomology for algebraic varieties

Let ΠΠ\Pi' \subset \Pi denote the full subcategory spanned by pairs (X,X¯)(X, \overline{X}) for which X¯\overline{X} is a smooth projective algebraic variety. Let V=V RV = V_\mathbf{R} denote the category of smooth quasi-projective schemes over R\mathbf{R}. By the GAGA principle, we have a functor σ:ΠV\sigma : \Pi' \to V which sends a pair (X,X¯)(X, \overline{X}) to XX. Conversely given XVX \in V, by Hironaka? there exists a pair (X,X¯)Π(X, \overline{X}) \in \Pi' (a compactification).

Definition

Let XVX \in V be a smooth quasi-projective algebraic variety over R\mathbf{R}. Let (X,X¯)Π(X, \overline{X}) \in \Pi' be a compactification and define the Beilinson-Deligne cohomology of XX as

H D q(X,A(p))=H qRΓ(X¯,X,A(p) D) H_D^q(X, A(p)) = H^q R\Gamma(\overline{X}, X, A(p)_D)

and

H B q(X,A(p))=H qRΓ(X,A(p)). H_B^q(X, A(p)) = H^q R\Gamma(X, A(p)).

One shows that these definitions are independent of the chosen compactification. By the above, one gets a cup product also on these cohomology groups.

Next Beilinson shows that H DH_D can be defined as cohomologies of certain complexes of Zariski sheaves. He notes that this is not necessary for the remainder of the paper, so we omit this here.

Chern classes of vector bundles

There is a canonical morphism

c 1:RΓ(X,𝒪 *)[1]H D(X,A(1)) c_1 : R\Gamma(X, \mathcal{O}^*)[-1] \longrightarrow H_D(X, A(1))

for each XVX \in V. This induces a canonical homomorphism

Pic(X)=H 1(X,𝒪 *)H 2(X,A(1)) Pic(X) = H^1(X, \mathcal{O}^*) \longrightarrow H^2(X, A(1))

from the Picard group of invertible sheaves?.

Definition

For an invertible sheaf? \mathcal{L} on XX, its first Chern class is defined to be the image of the class of \mathcal{L} under the above homomorphism.

One can show that for A=ZA = \mathbf{Z}, this homomorphism is injective, and further surjective if XX is compact.

Next Beilinson shows the projective bundle formula? for Beilinson-Deligne cohomology.

Proposition

(Projective bundle formula). Let EE be an vector bundle of rank rr on XX, let π:P(E)X\pi : \mathbf{P}(E) \to X be the associated projective bundle, and 𝒪(1)\mathcal{O}(1) the tautological sheaf? on P(E)\mathbf{P}(E). The homomorphism

c 1(𝒪(1)) jπ *: j=0 r1H D(X,A(ij))[2j]H D(P(E),A(i)) \oplus c_1(\mathcal{O}(1))^j \cup \pi^* : \bigoplus_{j=0}^{r-1} H_D(X, A(i-j))[2j] \longrightarrow H_D(\mathbf{P}(E), A(i))

is invertible.

Proof

By definition we have the distinguished triangle

H B(X)[1]RΓ(X,A(i)) D)RΓ(X,F i)RΓ(X,A(i)). H_B(X)[-1] \to R\Gamma(X, A(i))_D) \to R\Gamma(X, F^i) \oplus R\Gamma(X, A(i)) \to.

One checks that it is compatible with the cup product. Since the morphism A(i) DA(i)A(i)_D \to A(i) sends first Chern classes in Deligne cohomology to first Chern classes in Betti cohomology, the projective bundle formula?s for Betti cohomology and de Rham cohomology show that the map in question induces an isomorphism on the leftmost and rightmost members of the triangle. Hence the result follows.

After the projective bundle theorem, one can define Chern classes of vector bundles following Grothendieck. In particular one gets the Chern character

ch:K 0(X) iH 2i(X,AQ(i)). ch : K_0(X) \longrightarrow \bigoplus_i H^{2i}(X, A \otimes \mathbf{Q}(i)).
Lemma

For each ii, there exists a unique assignment to a vector bundle EE over X𝒱X \in \mathcal{V} a class

c i(E)H D 2i(X,A(i)) c_i(E) \in H^{2i}_D(X, A(i))

that is functorial with respect to inverse images and for which A(i) DA(i)A(i)_D \to A(i) sends c i(E)c_i(E) to the usual Chern class in Betti cohomology.

Proof

We omit the proof and just recall the construction due to Grothendieck of the Chern classes. Let r=rk(E)r = rk(E) and write P=P(E)P = \mathbf{P}(E). By the projective bundle formula one has

H D 2r(P,Z(r))= i=0 r1H D 2j+2r(X,Z(rj)) H_D^{2r}(P, \mathbf{Z}(r)) = \bigoplus_{i=0}^{r-1} H^{2j+2r}_D(X, \mathbf{Z}(r-j))

There exist γ i\gamma_i such that

Σ i=0 rπ *γ ic 1(𝒪 P(1)) ri=0 \Sigma_{i=0}^r \pi^* \gamma_i \cup c_1(\mathcal{O}_P(1))^{r-i} = 0

with γ iH D 2i(X,Z(i))\gamma_i \in H_D^{2i}(X, \mathbf{Z}(i)) and γ 0=1\gamma_0 = 1. We define c i(E)=γ ic_i(E) = \gamma_i.

Homologies

In this paragraph, our goal is to define the homology theory dual to Deligne cohomology, for schemes over R\mathbf{R}. To do this, we first define functorial complexes on Π *\Pi_*. Then they extend, more or less formally, to the category of finite type schemes over R\mathbf{R} and proper maps. Then we establish Poincare duality.

for smooth analytic spaces
Definition

Let XAnX \in An be smooth. Let

𝒜 X (resp.𝒟 X ) \mathcal{A}^\bullet_X \qquad \text{(resp.} \quad \mathcal{D}^\bullet_X \text{)}

denote the complex of C C^\infty forms (resp. with distribution coefficients on 𝒜 X p,q\mathcal{A}^{-p,-q}_{X}). This is the totalization of the double complex 𝒜 X *,*\mathcal{A}^{*,*}_X (resp. 𝒟 X *,*\mathcal{D}^{*,*}_X) formed by sheaves of (p,q)(p,q)-forms (resp. with distribution coefficients). Let 𝒜 X *\mathcal{A}^{\ge *}_{X} and 𝒟 X *\mathcal{D}^{\ge *}_{X} denote the respective induced filtrations.

Let

C (X,A(p)) C'^\bullet(X, A(p))

denote the complex of C C^\infty singular chains with coefficents in the constant sheaf A(p)A(p).

Let

𝒟 X =Γ c(X,𝒟 X) \mathcal{D}^\bullet_X = \Gamma_c(X, \mathcal{D}_X)

denote the complex of global sections with compact support. (Here we view 𝒟 X\mathcal{D}_X as a sheaf on XX.)

for pairs (logarithmic singularity)

Let Π *Π\Pi_* \subset \Pi denote the subcategory with the same objects and only morphisms f:(X,X¯)(Y,Y¯)f : (X, \overline{X}) \to (Y, \overline{Y}) which satisfy f(X¯X)Y¯Yf(\overline{X} - X) \subset \overline{Y} - Y.

Definition

Let (X,X¯)Π *(X, \overline{X}) \in \Pi_*. Define the complexes

𝒜 (X,X¯) =𝒜 X¯ Ω X¯ Ω (X,X¯) \mathcal{A}^\bullet_{(X, \overline{X})} = \mathcal{A}^\bullet_{\overline{X}} \otimes_{\Omega^\bullet_{\overline{X}}} \Omega^\bullet_{(X, \overline{X})}

and

𝒟 (X,X¯) =𝒟 X¯ Ω X¯ Ω (X,X¯) \mathcal{D}^\bullet_{(X, \overline{X})} = \mathcal{D}^\bullet_{\overline{X}} \otimes_{\Omega^\bullet_{\overline{X}}} \Omega^\bullet_{(X, \overline{X})}

with the induced filtrations 𝒜 (X,X¯) *\mathcal{A}^{\ge *}_{(X, \overline{X})} and 𝒟 (X,X¯) *\mathcal{D}^{\ge *}_{(X, \overline{X})}.

Define the complex of relative C C^\infty singular chains on (X,X¯)(X, \overline{X}) as the quotient chain complex

C (X,X¯,A(i)):=C (X¯,A(i))/C (X¯X,A(i)). C'^\bullet(X, \overline{X}, A(i)) := C'^\bullet(\overline{X}, A(i))/C'^\bullet(\overline{X}-X, A(i)).

Let

𝒟 (X,X¯)=Γ c(X¯,𝒟 (X,X¯) ) \mathcal{D}^\bullet(X, \overline{X}) = \Gamma_c(\overline{X}, \mathcal{D}^\bullet_{(X, \overline{X})})

denote the complex of sections with compact support, with the induced filtration 𝒟 *(X,X¯)\mathcal{D}^{\ge *}(X, \overline{X}).

Finally, define the complex C D(X,X¯,A(p))C'_D(X, \overline{X}, A(p)) (this is the complex that will give us the Deligne homology groups H D *(X,X¯,A(p))H'_D^*(X, \overline{X}, A(p))).

Definition

For (X,X¯)Π *(X, \overline{X}) \in \Pi_*, define the complex

C D(X,X¯,A(p))=Cone(𝒟 p(X,X¯)C (X,X¯,A(i))𝒟 (X,X¯)). C'_D(X, \overline{X}, A(p)) = Cone(\mathcal{D}^{\ge p}(X, \overline{X}) \oplus C'^\bullet(X, \overline{X}, A(i)) \longrightarrow \mathcal{D}^\bullet(X, \overline{X})).

This is functorial on Π *\Pi_*.

for schemes

Let Sch *Sch_* denote the category of finite type schemes over R\mathbf{R} and proper maps. Let V *Sch *V_* \subset Sch_* denote the subcategory of smooth quasi-projective schemes. Let Π *=ΠΠ *\Pi'_* = \Pi' \cap \Pi_* denote the category of pairs (X,X¯)(X, \overline{X}) with X¯\overline{X} smooth projective, XX¯X \subset \overline{X} open with X¯X\overline{X} - X a normal crossing divisor, and with morphisms f:(X,X¯)(Y,Y¯)f : (X, \overline{X}) \to (Y, \overline{Y}) such that f(X¯X)Y¯Yf(\overline{X} - X) \subset \overline{Y} - Y.

Lemma

The functor C D (,A(p))C'^\bullet_D(-, A(p)) on the category Π *\Pi'_* extends uniquely to a functor on Sch *Sch_*. In particular one gets has a distinguished triangle in D +(Amod)D^+(A-mod)

H dR(X)H D(X,A(p))F iH dR(X)H B(X,A(p)), \longrightarrow H'_{dR}(X) \longrightarrow H'_D(X, A(p)) \longrightarrow F^i H'_{dR}(X) \oplus H'_B(X, A(p)) \longrightarrow,

where H dRH'_{dR} is de Rham homology?, F iF^i is the Hodge filtration, H BH'_B is the Borel-Moore homology with coefficients in the constant sheaf A(p)A(p), and H D(X,A(p))H'_D(X, A(p)) is the Deligne homology?, defined by the complex C D(X,A(p))C'_D(X, A(p)).

Lemma

(Poincare duality). Let XX be a smooth scheme of dimension nn. There is a canonical isomorphism

H D(X,A(p))=H D(X,A(p+n))[2n]. H'_D(X, A(p)) = H_D(X, A(p+n))[2n].
Proof

Let (X,X¯)Π *(X, \overline{X}) \in \Pi'_* be a compactification of XX. Consider the presheaf on X¯\overline{X} of complexes of abelian groups, defined by

UC(X¯,A(p))/C(X¯(XU),A(p)). U \mapsto C'(\overline{X}, A(p))/C'(\overline{X} - (X \cap U), A(p)).

Take its associated sheaf, and consider it as a complex of abelian sheaves, C¯ X,X¯(A(p))\overline{C}'_{X, \overline{X}}(A(p)). First of all note that

C¯ X,X¯(A(p))=j *j *C¯ X,X¯(A(p)) \overline{C}'_{X, \overline{X}}(A(p)) = j_*j^* \overline{C}'_{X, \overline{X}}(A(p))

where j:XX¯j : X \hookrightarrow \overline{X} is the open immersion. Note that j *C¯ X,X¯(A(p))j^* \overline{C}'_{X, \overline{X}}(A(p)) is a flasque resolution? of the sheaf A(p+n)[2n]A(p+n)[2n] on XX. Since the embedding

Ω X,X¯𝒟 X,X¯[2n] \Omega_{X, \overline{X}} \hookrightarrow \mathcal{D}_{X, \overline{X}}[-2n]

is a filtered quasi-isomorphism, and the associated graded objects gr p𝒟 X,X¯gr^p \mathcal{D}_{X, \overline{X}} are soft sheaves?, one gets

RΓ(Ω X,X¯,Ω X,X¯ p)=Γ(X¯,(𝒟 X,X¯,𝒟 X,X¯ p+n))[2n] R\Gamma(\Omega_{X, \overline{X}}, \Omega^{\ge p}_{X, \overline{X}}) = \Gamma(\overline{X}, (\mathcal{D}_{X, \overline{X}}, \mathcal{D}^{\ge p+n}_{X, \overline{X}}))[-2n]

Cycles

Let XX be a scheme and YZ n(X)Y \in Z_n(X) an irreducible subscheme of dimension nn. Note that the canonical homomorphism

H D 2n(Y,A(n))H B 2n(Y,A(n))=A H'_D^{-2n}(Y, A(-n)) \longrightarrow H'_B^{-2n}(Y, A(-n)) = A

is invertible. Let cl D(Y)cl_D(Y) denote the element of H D 2n(Y,A(n))H'_D^{-2n}(Y, A(-n)) corresponding to the unit 1A1 \in A. Hence one gets a homomorphism

cl D:Z n(X)H D 2n(X,A(n)) cl_D : Z_n(X) \longrightarrow H'_D^{-2n}(X, A(-n))

given by cl D[Y]=i *(cl D(Y))cl_D[Y] = i_*(\cl_D(Y)). If XX is smooth, by Poincare duality this corresponds to a homomorphism

cl D:Z n(X)H D 2n(X,A(n)) cl_D : Z^n(X) \longrightarrow H_D^{2n}(X, A(n))

on the group of algebraic cyles? of codimension nn.

Lemma

If XX is smooth and compact, then for each YZ n(X)Y \in Z_n(X), if cl B(Y)H B 2n(X,Z(n))cl_B(Y) \in H'_B^{-2n}(X, \mathbf{Z}(-n)) is equal to 0, then cl D(Y)cl_D(Y) coincides with the Abel-Jacobi-Griffiths periods of the cycle YY.

Proof

The distinguished triangle defining Z(n) D\mathbf{Z}(n)_D induces, after passing to the associated long exact sequence, a short exact sequence

0 n(X)H D 2n(X,Z(n))Hdg n(X)0 0 \to \mathcal{I}^n(X) \to H_D^{2n}(X, \mathbf{Z}(n)) \to Hdg^n(X) \to 0

where n\mathcal{I}^n is the nnth intermediate Jacobian of Griffiths?, defined as

n=H B 2n1(X,C)/(H B 2n1(X,Z(n))F nH B 2n1(X,C)) \mathcal{I}^n = H^{2n-1}_B(X, \mathbf{C})/(H^{2n-1}_B(X, \mathbf{Z}(n)) \oplus F^n H^{2n-1}_B(X, \mathbf{C}))

and Hdn n(X)\Hdn^n(X) is the group of integral Hodge cycles of type (n,n)(n, n).

Using the usual explicit model for the mapping cone, cl D(Y)cl_D(Y) is the homology class of the cycle

(cl F(Y),i *cl B(Y),0)C D 2n(X,Z(n)) (cl_F(Y), i_* cl_B(Y), 0) \in C'_D^{-2n}(X, \mathbf{Z}(-n))

where i:YXi : Y \hookrightarrow X denotes the closed immersion, and cl F(Y)F n𝒟 2n(X)cl_F(Y) \in F^{-n}\mathcal{D}^{-2n}(X) is a distribution defined by integration over YY. Since cl B(Y)=0cl_B(Y) = 0, we can choose sC 2n1(X,Z(n))s \in C'^{-2n-1}(X, \mathbf{Z}(-n)) such that d(s)=i *(cl B(Y))d(s) = i_*(cl_B(Y)). By subtracting from cl D(Y)cl_D(Y) the boundary (0,s,0)(0, s, 0), we see that cl D(Y)=(cl F(Y),0,s)cl_D(Y) = (cl_F(Y), 0, s). But the latter is precisely the definition of the periods of the cycle YY.

Hodge conjecture for Deligne cohomology

Regulators

Lemma
Lemma

References

Last revised on October 28, 2014 at 19:52:11. See the history of this page for a list of all contributions to it.