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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Higher regulators and values of L-functions} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak \noindent\hyperlink{notation}{Notation}\dotfill \pageref*{notation} \linebreak \noindent\hyperlink{main_constructions_and_conjectures}{Main constructions and conjectures}\dotfill \pageref*{main_constructions_and_conjectures} \linebreak \noindent\hyperlink{deligne_cohomology_of_analytic_spaces}{Deligne cohomology of analytic spaces}\dotfill \pageref*{deligne_cohomology_of_analytic_spaces} \linebreak \noindent\hyperlink{multiplicative_structure}{Multiplicative structure}\dotfill \pageref*{multiplicative_structure} \linebreak \noindent\hyperlink{the_bloch_regulator}{The Bloch regulator}\dotfill \pageref*{the_bloch_regulator} \linebreak \noindent\hyperlink{relative_cohomology}{Relative cohomology}\dotfill \pageref*{relative_cohomology} \linebreak \noindent\hyperlink{complexes_with_logarithmic_singularities}{Complexes with logarithmic singularities}\dotfill \pageref*{complexes_with_logarithmic_singularities} \linebreak \noindent\hyperlink{deligne_cohomology_for_algebraic_varieties}{Deligne cohomology for algebraic varieties}\dotfill \pageref*{deligne_cohomology_for_algebraic_varieties} \linebreak \noindent\hyperlink{chern_classes_of_vector_bundles}{Chern classes of vector bundles}\dotfill \pageref*{chern_classes_of_vector_bundles} \linebreak \noindent\hyperlink{homologies}{Homologies}\dotfill \pageref*{homologies} \linebreak \noindent\hyperlink{for_smooth_analytic_spaces}{for smooth analytic spaces}\dotfill \pageref*{for_smooth_analytic_spaces} \linebreak \noindent\hyperlink{for_pairs_logarithmic_singularity}{for pairs (logarithmic singularity)}\dotfill \pageref*{for_pairs_logarithmic_singularity} \linebreak \noindent\hyperlink{for_schemes}{for schemes}\dotfill \pageref*{for_schemes} \linebreak \noindent\hyperlink{cycles}{Cycles}\dotfill \pageref*{cycles} \linebreak \noindent\hyperlink{hodge_conjecture_for_deligne_cohomology}{Hodge conjecture for Deligne cohomology}\dotfill \pageref*{hodge_conjecture_for_deligne_cohomology} \linebreak \noindent\hyperlink{regulators}{Regulators}\dotfill \pageref*{regulators} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak This page is supposed to be a review of the seminal article \begin{itemize}% \item [[Alexander Beilinson]] \emph{Higher regulators and values of L-functions}, Journal of Soviet Mathematics 30 (1985), 2036-2070, (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng}{mathnet (Russian)}, \href{http://dx.doi.org/10.1007%2FBF02105861}{DOI}) \end{itemize} 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]]. \hypertarget{overview}{}\subsection*{{Overview}}\label{overview} \hypertarget{notation}{}\subsubsection*{{Notation}}\label{notation} By [[analytic space]] we will mean [[real analytic space]]. Let $An$ denote the real [[analytic site]]. Consider the category $Ab(Sh(An))$ of [[abelian sheaves]] on $An$. Beilinson denotes by $D^+(An)$ its bounded above [[derived category]], i.e. the category of [[connective]] [[cochain complexes]] up to [[quasi-isomorphism]]. Given an analytic space $X \in An$, we will also consider its [[petit topos]] $X^\sim$ of sheaves on the site $Ouv(X)$ of open subsets. In $D^+(An)$ we have the complex $\Omega^\bullet$, of [[de Rham complex]]es of [[holomorphic forms]]. Let $\Omega^{\ge i}$ denote the ``stupid'' [[filtration]]. Fix a subring $A \subset \mathbf{R}$ and let $A(p) = A \cdot (2\pi i)^p \subset \mathbf{C}$ for $p \in \mathbf{Z}$. We will abuse notation and write $\mathbf{C} \in D^+(An)$ also for the constant sheaf valued in the complex concentrated in degree zero. Similarly we will view $A(p)$ as an object of $D^+(An)$. We will write $H^*_B(X, \mathbf{C})$ for the [[Betti cohomology]] of $X \in An$. \hypertarget{main_constructions_and_conjectures}{}\subsubsection*{{Main constructions and conjectures}}\label{main_constructions_and_conjectures} \hypertarget{deligne_cohomology_of_analytic_spaces}{}\paragraph*{{Deligne cohomology of analytic spaces}}\label{deligne_cohomology_of_analytic_spaces} For each $p \in \mathbf{Z}$, the inclusions $\Omega^{\ge p} \hookrightarrow \Omega^\bullet$ and $A(p) \hookrightarrow \mathbf{C} \hookrightarrow \Omega^\bullet$ induce a canonical morphism \begin{displaymath} \Omega^{\ge p} \oplus A(p) \longrightarrow \Omega^\bullet \end{displaymath} (given by the difference of the two inclusions). \begin{defn} \label{}\hypertarget{}{} For $p \in \mathbf{Z}$, the \textbf{Deligne complex of weight $p$}, denoted $A(p)_\D$, is defined as the [[mapping cone]] of the above morphism shifted by -1, hence fitting in the [[distinguished triangle]] \begin{displaymath} (\Omega^{\ge p} \oplus A(p))[-1] \longrightarrow \Omega^\bullet[-1] \longrightarrow A(p)_\D \longrightarrow \Omega^{\ge p} \oplus A(p). \end{displaymath} \end{defn} The Deligne cohomology is just the [[hypercohomology]] of this complex. That is, consider the [[right derived functor]] $R\Gamma(-, A(p)_D)$ of the functor of [[global sections]] on $An$ with values in $D^+(A-mod)$. \begin{defn} \label{}\hypertarget{}{} The \textbf{Deligne cohomology} of $X \in An$ of weight $p$ and degree $q$ with coefficients in $A$ is \begin{displaymath} H_D^q(X, A(p)) = H^q R\Gamma(X, A(p)_D). \end{displaymath} \end{defn} \begin{prop} \label{}\hypertarget{}{} There is a canonical [[long exact sequence]] \begin{displaymath} \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 \end{displaymath} \end{prop} This follows by applying the [[cohomological functor]] $R\Gamma(X, -)$ to the above [[distinguished triangle]]. \begin{lemma} \label{}\hypertarget{}{} For $p \le 0$ there is a canonical [[quasi-isomorphism]] \begin{displaymath} A(p)_D \stackrel{\sim}{\longrightarrow} \Omega^{\ge p}. \end{displaymath} For $p \gt 0$ there are canonical [[quasi-isomorphisms]] \begin{displaymath} \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} \end{displaymath} \end{lemma} \hypertarget{multiplicative_structure}{}\paragraph*{{Multiplicative structure}}\label{multiplicative_structure} \begin{prop} \label{}\hypertarget{}{} There exists a canonical morphism in $D^+(An)$ \begin{displaymath} - \cup - : A(i)_D \otimes^L A(j)_D \longrightarrow A(i+j)_D \end{displaymath} inducing the structure of a [[graded commutative ring]] on \begin{displaymath} H^*_D(X, A(*)) = \bigoplus_{p,q} H^p_D(X, A(p)) \end{displaymath} for all $X \in An$. \end{prop} \begin{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 $\Omega^\bullet$, $\Omega^{\ge *}$, $A(*)$, 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 $A(*) \to \Omega \leftarrow \Omega^{\ge *}$; Beilinson claims that the underlying complex of this [[dg-algebra]] has in each degree $p$ the Deligne complex $A(p)_D$ of weight $p$. This point is expanded on in \hyperlink{HopkinsQuick}{Hopkins-Quick}. Here we will simply give a formula for the quasi-isomorphic complex $(*)$ of Lemma 1 (assuming $i,j \gt 0$), which we will denote for the moment by $A(i)_E$. For $X \in An$, take $x \in \Gamma(X, A(i)_E)$, $y \in \Gamma(X, A(j)_E)$, and define \begin{displaymath} 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} \end{displaymath} 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 $R\Gamma(X, -)$ preserves monoids, so that one gets the structure of a graded commutative ring on the [[hypercohomology]] groups $H^*_D(X, A(*))$. \end{proof} \hypertarget{the_bloch_regulator}{}\paragraph*{{The Bloch regulator}}\label{the_bloch_regulator} Beilinson uses this cup product for $i=j=1$ to recover the [[Bloch regulator]]. \begin{theorem} \label{}\hypertarget{}{} \textbf{(Bloch)}. For each [[algebraic curve]] $X$ over $\mathbf{R}$, there is a canonical functorial homomorphism \begin{displaymath} r_X : K_2(X) \to H^1_B(X, \mathbf{C}^*) \end{displaymath} from the second [[algebraic K-theory]] group to the first [[Betti cohomology]] group with coefficients in $\mathbf{C}^*$. \end{theorem} \begin{proof} By Lemma 1, there are quasi-isomorphisms \begin{displaymath} \mathbf{Z}(1)_D \stackrel{\sim}{\to} \mathcal{O}^*[-1] \end{displaymath} induced by the [[exponential map]], and \begin{displaymath} \mathbf{Z}(2)_D \stackrel{\sim}{\to} (\mathcal{O}^* \stackrel{d \log}{\to} \Omega^1)[-1] \end{displaymath} induced by $x \mapsto \exp(x/2\pi i)$. It follows that the cup product \begin{displaymath} \cup : H^1_D(X, \mathbf{Z}(1)) \otimes H^1_D(X, \mathbf{Z}(1)) \longrightarrow H^2_D(X, \mathbf{Z}(2)) \end{displaymath} corresponds to a canonical homomorphism \begin{displaymath} \mathcal{O}^*(X) \otimes \mathcal{O}^*(X) \to H^1(X, \mathcal{O}^* \to \Omega^1). \end{displaymath} According to Deligne, the RHS classifies isomorphism classes of [[line bundles]] with [[holomorphic connection]]. Since $\dim(X) = 1$, all [[connections]] are [[integrable connection|integrable]] and this group is identified with $H^1_B(X, \mathbf{C}^*)$. Now by Matsumoto's theorem, giving a presentation of the $K_2$ group of a [[field]] by two generators and certain relations, one has \begin{displaymath} K_2(\mathcal{O}(X)) = (\mathcal{O}(X)^* \otimes \mathcal{O}(X)^*)/\lt t \otimes (1-t) \gt_{t \ne 0,1}. \end{displaymath} On the other hand, one has the Steinberg identity $t \cup (1-t) = 0$ for $t \in \mathcal{O}^*(X)$. It follows that the homomorphism above factors through $K_2(\mathcal{O}(X)) = K_2(\eta)$ for $\eta$ a generic point. To extend it to all of $X$, one uses the commutative diagram of localization [[exact sequences]] \begin{displaymath} \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} \end{displaymath} The first row comes from the [[Gersten-Quillen resolution]] for K-theory. \end{proof} \hypertarget{relative_cohomology}{}\paragraph*{{Relative cohomology}}\label{relative_cohomology} If $S$ and $T$ are [[toposes]] and $u^* : S \rightleftarrows T : u_*$ is a [[geometric morphism]], consider the [[Artin gluing]], i.e. the [[topos]] $(id_S/u_*) = (u^*/id_T)$ whose objects are morphisms $u^*(F) \to G$ for $F \in S$, $G \in T$, and morphisms are commutative diagrams. Write $(S, T)$ for this topos. \begin{defn} \label{}\hypertarget{}{} The functor of [[global sections]] on $(S, T)$ is the [[left exact functor]] \begin{displaymath} \Gamma(S, T, -) : Ch^+(Ab(S, T)) \longrightarrow Ch^+(Ab) \end{displaymath} defined by \begin{displaymath} \Gamma(S, T, F) = Cone(\Gamma(T, F_T) \to \Gamma(S, F_S))[-1] \end{displaymath} for each sheaf $F \in (S, T)$ given by $u^*(F_S) \to F_T$ with $F_S \in S$ and $F_T \in T$. Let $R\Gamma(S, T, -) : D^+(Ab(S, T)) \to D^+(Ab)$ denote its [[right derived functor]]. \end{defn} The rest of this section is just defining a [[monoidal product]] on $Ch^+(Ab((S, T)))$, and explaining that a [[monoid]] in $D^+(Ab((S,T)))$ will induce a [[monoid]] in $D^+(Ab)$, i.e. a [[dg-algebra]], after taking [[cohomology]]. \hypertarget{complexes_with_logarithmic_singularities}{}\paragraph*{{Complexes with logarithmic singularities}}\label{complexes_with_logarithmic_singularities} Let $\Pi$ denote the category of pairs $(X, \overline{X})$ with $\overline{X} \in An$ [[smooth manifold|smooth]] and $j : X \hookrightarrow \overline{X}$ an [[open subspace]] such that the complement $\overline{X} - X$ is a [[normal crossing divisor]]. The [[open immersion]] $j$ induces a functor $\Ouv(X) \to \Ouv(\overline{X})$ on the [[petit sites]] by mapping an open subspace $U \subset X$ to $U \subset X \subset \overline{X}$. This induces a [[geometric morphism]] $j^* : \overline{X}^{\sim} \rightleftarrows X^{\sim} : j_*$ on the [[petit toposes]]. Following the discussion of the previous section, we make the \begin{defn} \label{}\hypertarget{}{} The \textbf{topos of the pair} $(X, \overline{X})$ is defined to be the [[Artin gluing]] $(j^*/id)$, and will be denoted $(X, \overline{X})^{\sim}$. Hence a \textbf{sheaf on the pair} $(X, \overline{X})$ is a sheaf $F$ on $X$, a sheaf $G$ on $\overline{X}$, and a connecting morphism $j^*(G) \to F$. \end{defn} Let $\Omega^\bullet_{X,\overline{X}} \in \Ch^+(Ab((X, \overline{X})^{\sim}))$ denote the [[de Rham complex]] of [[holomorphic forms]] on $C$ with [[logarithmic singularities]] along $\overline{X} - X$. That is, $\Omega^\bullet_{X,\overline{X}}$ is an object of $Ch^+(Ab(\overline{X}^\sim))$ and we view it as a complex on $(X, \overline{X})^\sim$ by taking the part on $X^\sim$ to be trivial. Let $\Omega^{\ge p}_{X, \overline{X}}$ denote the stupid [[filtration]]. Now we define a complex of abelian sheaves $A(p)_D$ in $Ch^+(Ab((X, \overline{X})^{\sim}))$ as follows. \begin{defn} \label{}\hypertarget{}{} The \textbf{Beilinson-Deligne complex with logarithmic singularities} of weight $p$ of the pair $(X, \overline{X})$ is a complex of abelian sheaves on $(X, \overline{X})^\sim$, denoted $A(p)_D \in Ch^+(Ab((X, \overline{X})^\sim))$ and defined as follows. In degree $n \in \mathbf{Z}$, take the sheaf \begin{displaymath} A(p)_{D,X} = Cone^n(A(p) \to \Omega^\bullet_X) \end{displaymath} in $X^\sim$ (where the [[mapping cone]] is taken in $Ch^+(Ab(X^\sim))$), and the sheaf \begin{displaymath} A(p)_{D,\overline{X}} = \Omega^{\ge p}_{X, \overline{X}} \end{displaymath} in $\overline{X}^\sim$, together with the connecting morphism induced by the inclusion $j^*(\Omega^{\ge p}) \hookrightarrow \Omega^\bullet_X$. \end{defn} \begin{defn} \label{}\hypertarget{}{} The \textbf{Beilinson-Deligne cohomology with logarithmic singularities} of the pair $(X, \overline{X})$ in weight $p$ and degree $q$ and with coefficients in $A$, is the [[hypercohomology]] \begin{displaymath} H^q_D(X, \overline{X}, A(p)) = H^q R\Gamma((X, \overline{X})^\sim, A(p)_D). \end{displaymath} \end{defn} 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. \begin{defn} \label{}\hypertarget{}{} \end{defn} \hypertarget{deligne_cohomology_for_algebraic_varieties}{}\paragraph*{{Deligne cohomology for algebraic varieties}}\label{deligne_cohomology_for_algebraic_varieties} Let $\Pi' \subset \Pi$ denote the full subcategory spanned by pairs $(X, \overline{X})$ for which $\overline{X}$ is a [[smooth variety|smooth]] [[projective variety|projective]] [[algebraic variety]]. Let $V = V_\mathbf{R}$ denote the category of [[smooth variety|smooth]] [[quasi-projective variety|quasi-projective]] [[schemes]] over $\mathbf{R}$. By the [[GAGA]] principle, we have a functor $\sigma : \Pi' \to V$ which sends a pair $(X, \overline{X})$ to $X$. Conversely given $X \in V$, by [[Hironaka]] there exists a pair $(X, \overline{X}) \in \Pi'$ (a compactification). \begin{defn} \label{}\hypertarget{}{} Let $X \in V$ be a smooth quasi-projective algebraic variety over $\mathbf{R}$. Let $(X, \overline{X}) \in \Pi'$ be a compactification and define the \textbf{Beilinson-Deligne cohomology} of $X$ as \begin{displaymath} H_D^q(X, A(p)) = H^q R\Gamma(\overline{X}, X, A(p)_D) \end{displaymath} and \begin{displaymath} H_B^q(X, A(p)) = H^q R\Gamma(X, A(p)). \end{displaymath} \end{defn} 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 $H_D$ can be defined as cohomologies of certain complexes of [[Zariski topology|Zariski sheaves]]. He notes that this is not necessary for the remainder of the paper, so we omit this here. \hypertarget{chern_classes_of_vector_bundles}{}\paragraph*{{Chern classes of vector bundles}}\label{chern_classes_of_vector_bundles} There is a canonical morphism \begin{displaymath} c_1 : R\Gamma(X, \mathcal{O}^*)[-1] \longrightarrow H_D(X, A(1)) \end{displaymath} for each $X \in V$. This induces a canonical homomorphism \begin{displaymath} Pic(X) = H^1(X, \mathcal{O}^*) \longrightarrow H^2(X, A(1)) \end{displaymath} from the [[Picard group]] of [[invertible sheaves]]. \begin{defn} \label{}\hypertarget{}{} For an [[invertible sheaf]] $\mathcal{L}$ on $X$, its \textbf{first Chern class} is defined to be the image of the class of $\mathcal{L}$ under the above homomorphism. \end{defn} One can show that for $A = \mathbf{Z}$, this homomorphism is injective, and further surjective if $X$ is [[compact]]. Next Beilinson shows the [[projective bundle formula]] for Beilinson-Deligne cohomology. \begin{prop} \label{}\hypertarget{}{} \textbf{(Projective bundle formula)}. Let $E$ be an [[vector bundle]] of rank $r$ on $X$, let $\pi : \mathbf{P}(E) \to X$ be the associated [[projective bundle]], and $\mathcal{O}(1)$ the [[tautological sheaf]] on $\mathbf{P}(E)$. The homomorphism \begin{displaymath} \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)) \end{displaymath} is invertible. \end{prop} \begin{proof} By definition we have the distinguished triangle \begin{displaymath} H_B(X)[-1] \to R\Gamma(X, A(i))_D) \to R\Gamma(X, F^i) \oplus R\Gamma(X, A(i)) \to. \end{displaymath} One checks that it is compatible with the cup product. Since the morphism $A(i)_D \to A(i)$ 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. \end{proof} After the projective bundle theorem, one can define [[Chern classes]] of [[vector bundles]] following [[Grothendieck]]. In particular one gets the [[Chern character]] \begin{displaymath} ch : K_0(X) \longrightarrow \bigoplus_i H^{2i}(X, A \otimes \mathbf{Q}(i)). \end{displaymath} \begin{lemma} \label{}\hypertarget{}{} For each $i$, there exists a unique assignment to a [[vector bundle]] $E$ over $X \in \mathcal{V}$ a class \begin{displaymath} c_i(E) \in H^{2i}_D(X, A(i)) \end{displaymath} that is [[functorial]] with respect to [[inverse images]] and for which $A(i)_D \to A(i)$ sends $c_i(E)$ to the usual Chern class in [[Betti cohomology]]. \end{lemma} \begin{proof} We omit the proof and just recall the construction due to Grothendieck of the Chern classes. Let $r = rk(E)$ and write $P = \mathbf{P}(E)$. By the projective bundle formula one has \begin{displaymath} H_D^{2r}(P, \mathbf{Z}(r)) = \bigoplus_{i=0}^{r-1} H^{2j+2r}_D(X, \mathbf{Z}(r-j)) \end{displaymath} There exist $\gamma_i$ such that \begin{displaymath} \Sigma_{i=0}^r \pi^* \gamma_i \cup c_1(\mathcal{O}_P(1))^{r-i} = 0 \end{displaymath} with $\gamma_i \in H_D^{2i}(X, \mathbf{Z}(i))$ and $\gamma_0 = 1$. We define $c_i(E) = \gamma_i$. \end{proof} \hypertarget{homologies}{}\paragraph*{{Homologies}}\label{homologies} In this paragraph, our goal is to define the [[homology theory]] dual to Deligne cohomology, for schemes over $\mathbf{R}$. To do this, we first define functorial complexes on $\Pi_*$. Then they extend, more or less formally, to the category of finite type schemes over $\mathbf{R}$ and [[proper maps]]. Then we establish [[Poincare duality]]. \hypertarget{for_smooth_analytic_spaces}{}\paragraph*{{for smooth analytic spaces}}\label{for_smooth_analytic_spaces} \begin{defn} \label{}\hypertarget{}{} Let $X \in An$ be smooth. Let \begin{displaymath} \mathcal{A}^\bullet_X \qquad \text{(resp.} \quad \mathcal{D}^\bullet_X \text{)} \end{displaymath} denote the complex of $C^\infty$ [[differential forms|forms]] (resp. with [[distribution]] coefficients on $\mathcal{A}^{-p,-q}_{X}$). This is the [[totalization]] of the [[double complex]] $\mathcal{A}^{*,*}_X$ (resp. $\mathcal{D}^{*,*}_X$) formed by sheaves of $(p,q)$-forms (resp. with [[distribution]] coefficients). Let $\mathcal{A}^{\ge *}_{X}$ and $\mathcal{D}^{\ge *}_{X}$ denote the respective induced [[filtrations]]. Let \begin{displaymath} C'^\bullet(X, A(p)) \end{displaymath} denote the complex of $C^\infty$ [[singular chains]] with coefficents in the [[constant sheaf]] $A(p)$. Let \begin{displaymath} \mathcal{D}^\bullet_X = \Gamma_c(X, \mathcal{D}_X) \end{displaymath} denote the complex of [[global sections]] with [[compact support]]. (Here we view $\mathcal{D}_X$ as a sheaf on $X$.) \end{defn} \hypertarget{for_pairs_logarithmic_singularity}{}\paragraph*{{for pairs (logarithmic singularity)}}\label{for_pairs_logarithmic_singularity} Let $\Pi_* \subset \Pi$ denote the subcategory with the same objects and only morphisms $f : (X, \overline{X}) \to (Y, \overline{Y})$ which satisfy $f(\overline{X} - X) \subset \overline{Y} - Y$. \begin{defn} \label{}\hypertarget{}{} Let $(X, \overline{X}) \in \Pi_*$. Define the complexes \begin{displaymath} \mathcal{A}^\bullet_{(X, \overline{X})} = \mathcal{A}^\bullet_{\overline{X}} \otimes_{\Omega^\bullet_{\overline{X}}} \Omega^\bullet_{(X, \overline{X})} \end{displaymath} and \begin{displaymath} \mathcal{D}^\bullet_{(X, \overline{X})} = \mathcal{D}^\bullet_{\overline{X}} \otimes_{\Omega^\bullet_{\overline{X}}} \Omega^\bullet_{(X, \overline{X})} \end{displaymath} with the induced filtrations $\mathcal{A}^{\ge *}_{(X, \overline{X})}$ and $\mathcal{D}^{\ge *}_{(X, \overline{X})}$. Define the complex of relative $C^\infty$ singular chains on $(X, \overline{X})$ as the [[quotient chain complex]] \begin{displaymath} C'^\bullet(X, \overline{X}, A(i)) := C'^\bullet(\overline{X}, A(i))/C'^\bullet(\overline{X}-X, A(i)). \end{displaymath} Let \begin{displaymath} \mathcal{D}^\bullet(X, \overline{X}) = \Gamma_c(\overline{X}, \mathcal{D}^\bullet_{(X, \overline{X})}) \end{displaymath} denote the complex of sections with [[compact support]], with the induced filtration $\mathcal{D}^{\ge *}(X, \overline{X})$. \end{defn} Finally, define the complex $C'_D(X, \overline{X}, A(p))$ (this is the complex that will give us the Deligne homology groups $H'_D^*(X, \overline{X}, A(p))$). \begin{defn} \label{}\hypertarget{}{} For $(X, \overline{X}) \in \Pi_*$, define the complex \begin{displaymath} 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})). \end{displaymath} \end{defn} This is functorial on $\Pi_*$. \hypertarget{for_schemes}{}\paragraph*{{for schemes}}\label{for_schemes} Let $Sch_*$ denote the category of [[finite type]] schemes over $\mathbf{R}$ and [[proper maps]]. Let $V_* \subset Sch_*$ denote the subcategory of smooth quasi-projective schemes. Let $\Pi'_* = \Pi' \cap \Pi_*$ denote the category of pairs $(X, \overline{X})$ with $\overline{X}$ smooth projective, $X \subset \overline{X}$ open with $\overline{X} - X$ a normal crossing divisor, and with morphisms $f : (X, \overline{X}) \to (Y, \overline{Y})$ such that $f(\overline{X} - X) \subset \overline{Y} - Y$. \begin{lemma} \label{}\hypertarget{}{} The functor $C'^\bullet_D(-, A(p))$ on the category $\Pi'_*$ extends uniquely to a functor on $Sch_*$. In particular one gets has a [[distinguished triangle]] in $D^+(A-mod)$ \begin{displaymath} \longrightarrow H'_{dR}(X) \longrightarrow H'_D(X, A(p)) \longrightarrow F^i H'_{dR}(X) \oplus H'_B(X, A(p)) \longrightarrow, \end{displaymath} where $H'_{dR}$ is [[de Rham homology]], $F^i$ is the [[Hodge filtration]], $H'_B$ is the [[Borel-Moore homology]] with coefficients in the [[constant sheaf]] $A(p)$, and $H'_D(X, A(p))$ is the [[Deligne homology]], defined by the complex $C'_D(X, A(p))$. \end{lemma} \begin{lemma} \label{}\hypertarget{}{} \textbf{(Poincare duality).} Let $X$ be a smooth scheme of dimension $n$. There is a canonical isomorphism \begin{displaymath} H'_D(X, A(p)) = H_D(X, A(p+n))[2n]. \end{displaymath} \end{lemma} \begin{proof} Let $(X, \overline{X}) \in \Pi'_*$ be a compactification of $X$. Consider the presheaf on $\overline{X}$ of complexes of abelian groups, defined by \begin{displaymath} U \mapsto C'(\overline{X}, A(p))/C'(\overline{X} - (X \cap U), A(p)). \end{displaymath} Take its associated sheaf, and consider it as a complex of abelian sheaves, $\overline{C}'_{X, \overline{X}}(A(p))$. First of all note that \begin{displaymath} \overline{C}'_{X, \overline{X}}(A(p)) = j_*j^* \overline{C}'_{X, \overline{X}}(A(p)) \end{displaymath} where $j : X \hookrightarrow \overline{X}$ is the open immersion. Note that $j^* \overline{C}'_{X, \overline{X}}(A(p))$ is a [[flasque resolution]] of the sheaf $A(p+n)[2n]$ on $X$. Since the embedding \begin{displaymath} \Omega_{X, \overline{X}} \hookrightarrow \mathcal{D}_{X, \overline{X}}[-2n] \end{displaymath} is a [[filtered quasi-isomorphism]], and the [[associated graded objects]] $gr^p \mathcal{D}_{X, \overline{X}}$ are [[soft sheaves]], one gets \begin{displaymath} 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] \end{displaymath} \end{proof} \hypertarget{cycles}{}\paragraph*{{Cycles}}\label{cycles} Let $X$ be a scheme and $Y \in Z_n(X)$ an [[irreducible topological space|irreducible]] subscheme of dimension $n$. Note that the canonical homomorphism \begin{displaymath} H'_D^{-2n}(Y, A(-n)) \longrightarrow H'_B^{-2n}(Y, A(-n)) = A \end{displaymath} is invertible. Let $cl_D(Y)$ denote the element of $H'_D^{-2n}(Y, A(-n))$ corresponding to the unit $1 \in A$. Hence one gets a homomorphism \begin{displaymath} cl_D : Z_n(X) \longrightarrow H'_D^{-2n}(X, A(-n)) \end{displaymath} given by $cl_D[Y] = i_*(\cl_D(Y))$. If $X$ is smooth, by Poincare duality this corresponds to a homomorphism \begin{displaymath} cl_D : Z^n(X) \longrightarrow H_D^{2n}(X, A(n)) \end{displaymath} on the group of [[algebraic cyles]] of [[codimension]] $n$. \begin{lemma} \label{}\hypertarget{}{} If $X$ is smooth and compact, then for each $Y \in Z_n(X)$, if $cl_B(Y) \in H'_B^{-2n}(X, \mathbf{Z}(-n))$ is equal to 0, then $cl_D(Y)$ coincides with the Abel-Jacobi-Griffiths periods of the cycle $Y$. \end{lemma} \begin{proof} The distinguished triangle defining $\mathbf{Z}(n)_D$ induces, after passing to the associated [[long exact sequence]], a [[short exact sequence]] \begin{displaymath} 0 \to \mathcal{I}^n(X) \to H_D^{2n}(X, \mathbf{Z}(n)) \to Hdg^n(X) \to 0 \end{displaymath} where $\mathcal{I}^n$ is the $n$th [[intermediate Jacobian]] of [[Griffiths]], defined as \begin{displaymath} \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})) \end{displaymath} and $\Hdn^n(X)$ is the group of integral [[Hodge cycles]] of type $(n, n)$. Using the usual explicit model for the [[mapping cone]], $cl_D(Y)$ is the homology class of the cycle \begin{displaymath} (cl_F(Y), i_* cl_B(Y), 0) \in C'_D^{-2n}(X, \mathbf{Z}(-n)) \end{displaymath} where $i : Y \hookrightarrow X$ denotes the closed immersion, and $cl_F(Y) \in F^{-n}\mathcal{D}^{-2n}(X)$ is a distribution defined by integration over $Y$. Since $cl_B(Y) = 0$, we can choose $s \in C'^{-2n-1}(X, \mathbf{Z}(-n))$ such that $d(s) = i_*(cl_B(Y))$. By subtracting from $cl_D(Y)$ the boundary $(0, s, 0)$, we see that $cl_D(Y) = (cl_F(Y), 0, s)$. But the latter is precisely the definition of the periods of the cycle $Y$. \end{proof} \hypertarget{hodge_conjecture_for_deligne_cohomology}{}\paragraph*{{Hodge conjecture for Deligne cohomology}}\label{hodge_conjecture_for_deligne_cohomology} \hypertarget{regulators}{}\subsubsection*{{Regulators}}\label{regulators} \begin{lemma} \label{}\hypertarget{}{} \end{lemma} \begin{lemma} \label{}\hypertarget{}{} \end{lemma} \ldots{} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Alexander Beilinson]] \emph{Higher regulators and values of L-functions}, Journal of Soviet Mathematics 30 (1985), 2036-2070, (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intd&paperid=73&option_lang=eng}{mathnet (Russian)}, \href{http://dx.doi.org/10.1007%2FBF02105861}{DOI}) (reviewed in \hyperlink{EsnaultViehweg88}{Esnault-Viehweg 88}) \item [[Hélène Esnault]], [[Eckart Viehweg]], \emph{Deligne-Beilinson cohomology} in Rapoport, Schappacher, Schneider (eds.) \emph{Beilinson's Conjectures on Special Values of L-Functions} . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (\href{http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf}{pdf}) \item [[Michael Hopkins]], [[Gereon Quick]], \emph{Hodge filtered complex bordism}, \href{http://arxiv.org/abs/1212.2173v3}{arXiv}. \end{itemize} \end{document}