Holmstrom Absolute Hodge cohomology

Absolute Hodge cohomology

Wildeshaus on the Eisenstein symbol

Saito: Bloch’s conjecture, Deligne cohomology, and Higher Chow groups

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Absolute Hodge cohomology

MR0862628 (87m:14019) Be\u\i linson, A. A. Notes on absolute Hodge cohomology. Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), 35–68, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986. 14F25 (11G45 11R70 14C30 19F27) PDF Doc Del Clipboard Journal Article Make Link

The author gives a new construction of Deligne cohomology, which puts it into a more conceptual framework. He also explains his regulator map (from $K$-theory) using this new point of view. The main point is as follows: Consider a $C$-scheme $X$. By a result of \n P. Deligne\en [Inst. Hautes Études Sci. Publ. Math. No. 40 (1971), 557; MR0498551 (58 #16653a)], the Betti cohomology of $X$ supports a mixed Hodge structure. Even better, this construction gives an object in a derived category of Hodge structures. Now the Deligne cohomology of $X$ is simply given by $\roman{Ext}^i(Z (m),H^*(X))$, where Ext is computed in this derived category, $Z(m)$ denotes the Tate-Hodge structure, and $i$ and $m$ have to be suitably chosen.

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Absolute Hodge cohomology

Articles on regulators:

Beilinson: Higher regulators and values of L-functions (in Russian). (1984)

Some work by Rob de Jeu on curves over number fields.

MR0406981 (53 #10765) 12A70 Lichtenbaum, Stephen Values of zeta-functions, ´etale cohomology, and algebraicK-theory. Algebraic K-theory, II: ‘‘Classical’’ algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 489–501. Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973. The author expounds the Birch-Tate conjecture about the value of the zeta-function ?(F, s) of an algebraic number field F at s = −1 in terms of cohomology, as well as in terms of K2 [see the author, Ann. of Math. (2) 96 (1972), 338–360; MR0360527 (50 #12975)], and then he offers a conjecture in connection with Quillen’s conjecture, extending the latter to a conjecture for any odd negative number s. In the last section he defines the mth regulator of F and again proposes a conjecture about this and ?(F, s). {For the entire collection see MR0325308 (48 #3656b).}

MR721399 (85d:14017) 14C35 (13D15 18F25) Coombes, K. R. (1-OK); Srinivas,V. Srinivas,Vasudevan Aremark onK1 of an algebraic surface. Math. Ann. 265 (1983), no. 3, 335–342. The authors prove two interesting theorems, which are part of a general program initiated by Bloch and Beilinson whose goal is to relate higher algebraic K-groups of algebraic varieties to their real cohomology groups via a regulator map. Let X be a nonsingular projective surface over an uncountable algebraically closed field k, such that CH2(X), the group of codimension 2 cycles modulo linear equivalence, is finite-dimensional in the sense of Mumford. They prove that the cokernel of the product map PicX Z k? ! H1(X,K2) isN-torsion for some integerN. If, in addition, AlbX = 0, then the same result holds for Kn(k)!H0(X,Kn), for any n. They give the following application to commutative algebra. LetR = kx, y, z/(zpn −f(x, y)) be a normal domain of characteristic p. Let G be the Grothendieck group of R-modules of finite length and finite projective dimension. The theorem is that G is a direct sum of Z and an abelian group which is N-torsion for some N.

MR0700583 (84k:12005) Chinburg, T.(1-WA) Derivatives of $L$-functions at $s=0$. Compositio Math. 48 (1983), no. 1, 119–127. 12A70 PDF Doc Del Clipboard Journal Article Make Link

In a series of 4 papers on “$L$-functions at $s=1$” H. Stark generalized the known class number formula about the residue of the $\zeta$-function at $s=1$ to a conjecture about values of Artin $L$-functions at $s=1$. In the form emphasized by Tate this conjecture states that the leading coefficient of an Artin $L$-function $L(s,\chi)$ at $s=0$ is the product of the regulator $R(\chi)$ (to be defined for nonabelian characters) and an algebraic number $A(\chi)$. More precisely: $A(\chi^\alpha)=A(\chi)^\alpha$ for all automorphisms $\alpha$ of $C$.

Tate proved this to be true for characters $\chi$ with values in $Q$. On the other hand, Lichtenbaum showed in 1975 that for rational representations the leading coefficient of its $L$-function is the product of a regulator (defined by Lichtenbaum) and an Euler characteristic in étale cohomology. Starting from this result, the author gives another proof of Tate’s theorem. To do this he has to compare the different concepts of regulators (Lichtenbaum’s and Tate’s) and close the small gap between rational and rational-valued characters.

MR0760999 (86h:11103) Be\u\i linson, A. A. Higher regulators and values of $L$-functions. (Russian) Current problems in mathematics, Vol. 24, 181–238, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984. 11R42 (11G40 11G45 11R70 14C35 18F25 19F27) PDF Doc Del Clipboard Journal Article Make Link

In this fundamental paper, the author sets forth his theory of higher regulators for algebraic $K$-theory, presents conjectures\break about them and their relationships with values of $L$-functions, and proves some of the conjectures in special cases. The program for the future described in this paper (and subsequent ones by the same author) is a natural and satisfying generalization to arithmetic varieties $X$ of the Lichtenbaum conjectures for $K$-groups of number rings, the Hodge conjecture, the Tate conjecture about algebraic cycles, the Birch and Swinnerton-Dyer conjecture about elliptic curves, and Bloch’s conjecture about $K_2$ of elliptic curves. Resolution of these conjectures is at the head of the agenda for algebraic $K$-theory, and, considering the intractibility of the conjectures which it generalizes, it may remain there for decades.

Define $H^k_{{\scr A}}(X_{Z},Q(i))$ to be the subspace of $K_m(X_{Z})\otimesQ$ on which the Adams operations act with weight $i$, and where $m=2i-k$. The ${\scr A}$ (in later papers changed to ${\scr M})$ signifies that this is the “absolute motivic cohomology”. Here $X_{Z}$ denotes a regular model for $X$ proper over $Z$. The higher regulator appears as a map $r\colon H^k_{{\scr A}}(X,Q(i))\to H^k_{{\scr D}}(X,R(i))$. The target of this map is Deligne-Be\u\i linson cohomology, and is a finite-dimensional real vector space whose dimension is determined by the Hodge numbers of $X\otimesR$. The map $r$ is constructed by the author as a Chern character, and is an instance of the universality of $H_{{\scr A}}$.

The author produces a suitable rational structure on the highest exterior power of $H_{{\scr D}}$ (not necessarily trivial when $H_{{\scr D}}=0$), allowing one to define a number $\det(r)$ up to a rational factor (here the reviewer deviates from the author’s notation slightly). One also considers the $L$-function $L(s)=L_{k-1}(X,s)$ associated to the representation of $\text{Gal} (\overline{Q}/Q)$ on $H^{k-1}(X\otimes _{Q}\overline{Q},Q_l)$: it conjecturally admits analytic continuation and a functional equation $s\leftrightarrow k-s$. The critical strip is of width 1 and centered at $s=k/2$, and we are interested in the point $s_0=k-i=k/2-m/2$. The main conjectures of the author are (for $m\ge2$) that $r$: $H_{{\scr A}}\otimesR\to H_{{\scr D}}$ is an isomorphism, that the order of vanishing of $L$ at $s_0$ is equal to the dimension of $H_{{\scr A}}$, and that the first nonvanishing term in the Taylor series for $L$ at $s_0$ is equal to $\det(r)$. For $m=1$, one considers the map $H_{{\scr A}}\oplus A^{i-1}\to H_{{\scr D}}$ instead, where $A^{i-1}$ is the group of codimension $i-1$ cycles on $X$ modulo homological equivalence. For $m=0$ the conjecture is different [see the author, Height pairings between algebraic cycles, Preprint; per revr.; \n S. Bloch\en, J. Pure Appl. Algebra 34 (1984), no. 2, 119145].

A weaker form of the conjecture, where one allows $H_{{\scr A}}$ to be replaced by any subspace, is more accessible. In Section 5, the author provides an elegant proof of it assuming that $X$ is a modular curve and $i=k=2$. (The other values for $i$ and $k$ are covered in a later paper [the author, Higher regulators of modular curves, Applications of algebraic -theory to algebraic geometry and number theory, Contemp. Math., Amer. Math. Soc., Providence, R.I., to appear].) The proof uses Rankin’s method and the study of integrality of modular units on Drinfel\cprime d’s model $X_{Z}$. In Section 6 he proves the weakened conjecture when $X$ is a product of two modular curves and $i=2$, $k=3$. In Section 7 he constructs elements in the higher $K$-groups of cyclotomic rings of integers analogous to the circular units and uses them and \n A. Borel’s\en work to prove the conjecture (suitably interpreted) for Dirichlet $L$-functions.

One should refer to a paper by \n C. Soulé\en [Régulateurs, Bourbaki seminar (French), Exp. 644, Astérisque, Soc. Math. France, Paris, to appear] or a paper by \n D. Ramakrishnan\en [Higher regulators and values of -functions: an introductory survey, Preprint; per revr.] for further exposition.

MR0772054 (86h:14015) Bloch, Spencer(1-CHI) Height pairings for algebraic cycles. Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983). J. Pure Appl. Algebra 34 (1984), no. 2-3, 119–145. 14G10 (11G40 14C35) PDF Doc Del Clipboard Journal Article Make Link

Let $X$ be a $d$-dimensional smooth projective variety over a number field. Let $A^p(X)$ denote the $Q$-vector space of homologically trivial algebraic cycles of codimension $p$ on $X$ modulo rational equivalence. Under certain mild technical assumptions, the author constructs height pairings $\langle , \rangle_p\colon A^p(X)\times A^{d-p+1}(X)\to R$. These pairings generalize the Néron pairing [\n A. Néron\en, Ann. of Math. (2) 82 (1965), 249331; MR0179173 (31 #3424)] between divisors and zero cycles. The discriminant of the Néron pairing shows up as a regulator in the Birch and Swinnerton-Dyer conjecture. The author conjectures that $\langle , \rangle_p$ plays a similar role for the $L$-function associated to $H^{2p-1}(X)$ at $s=p$. For some evidence, see papers by the author [J. Reine Angew. Math. 350 (1984), 94108; MR0743535 (85i:11052); Duke Math. J. 52 (1985), no. 2, 379397]. \n A. Be\u\i linson\en [Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 116118; MR0575206 (81k:14020)] has independently constructed a height pairing and regulators with a conjectured relation to the values of $L$-functions.

The pairing here is constructed as a sum of local factors at each prime of the number field. The general method uses certain homotopy and $K$-theoretic properties of the spaces $BQC$ of \n D. Quillen\en [Algebraic -theory, I: higher -theories (Seattle, Wash., 1972), 85147, Lecture Notes in Math., 341, Springer, Berlin, 1973; MR0338129 (49 #2895)]. At the finite primes one uses the usual theory of Chern classes in étale cohomology. At the Archimedean primes, one needs a theory of Chern classes in Deligne cohomology, as constructed by \n H. Gillet\en [Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 285288; MR0733697 (85j:14010)].

MR0782485 (86e:11112) Tate, John(1-HRV) Les conjectures de Stark sur les fonctions $L$ d’Artin en $s=0$. (French) [Starks conjectures on Artin -functions at ] Lecture notes edited by Dominique Bernardi and Norbert Schappacher. Progress in Mathematics, 47. Birkhäuser Boston, Inc., Boston, MA, 1984. 143 pp. ISBN: 0-8176-3188-7 11R42 PDF Doc Del Clipboard Journal Article Make Link

This book is an account of a course given by Tate on Stark’s conjectures at Université de Paris-Sud (Orsay) during the first semester of the academic year 1980/81. In addition to the material on which Tate lectured, the text contains certain remarks on results obtained after the time of the course but before the 1984 publication.

Let $k$ be a number field, $K$ a Galois extension of $k$, $\chi$ a character of $\text{Gal}(K/k)$, and $S$ a finite set of places of $k$. By $L(s,\chi,K/k)$ denote the Artin $L$-function with Euler factors above $S$ removed. Following an introductory chapter, the text begins with a discussion of the principal conjecture of Stark. This conjecture predicts that the first nonzero term of the Taylor expansion of $L(s,\chi,K/k)$ at $s=0$ can be expressed as the product of a regulator $R(\chi)$ and an algebraic number. A proof of this principal conjecture is given in the case of a rational character. The argument uses methods of \n T. Ono\en [Ann. of Math. (2) 78 (1963), 4773; MR0156851 (28 #94)] involving the cohomology of class field theory and results of \n R. Swan\en [ibid. (2) 71 (1960), 552578; MR0138688 (25 #2131)] on integral representations of $\text{Gal}(K/k)$. A stronger result, due to Chinburg, is discussed as is the “invariant of Chinburg”.

Next comes a discussion of the special case where $L(s,\chi, K/k)$ has a simple zero at $s=0$. In this case the principal conjecture implies the existence of certain $S$-units of $K$, the so-called “units of Stark” [\n H. M. Stark\en, Adv. in Math. 35 (1980), no. 3, 197235; MR0563924 (81f:10054)]. These generalize cyclotomic and elliptic units. As explained by Stark at Bonn [Modular functions of one variable, V (Bonn, 1976), 277288, Lecture Notes in Math., 601, Springer, Berlin, 1977; MR0450243 (56 #8539)], for $k=Q$ and $\chi$ irreducible with $\chi(1)=2$, there is a link with modular forms of weight one.

In the case of an abelian extension $K/k$, if the set $S$ contains all the places of $k$ which ramify in $K$ and also a place $v$ which is totally decomposed, Stark [op. cit.; MR0563924 (81f:10054)] has given a refinement $\text{St}(K/k,S)$ of his conjecture for characters $\chi$ such that $L(s,\chi,K/k)$ has a simple zero at $s=0$. In contrast to Stark’s original conjecture, the truth of which is known to be independent of the choice of $S$, the refinement $\text{St}(K/k,S)$ may depend on $S$. The author discusses evidence for $\text{St}(K/k,S)$, namely, special cases where it has been proved, and numerical support. If one considers the case where the place $v\in S$ which is decomposed is non-Archimedean, then one sees that the conjunction of the conjectures $\text{St}(K/k,S)$ with $S=T\cup\{v\}$, for $v$ running over all places which decompose in $K/k$, is equivalent to a conjecture $\text{BS}(K/k,T)$ called the “Brumer-Stark conjecture”, which generalizes the classical Stickelberger theorem and the fact that Gauss sums lie in cyclotomic fields.

Although Stark’s principal conjecture is trivial for the case of function fields, as remarked by Mazur, $\text{BS}(K/k,S)$ has a nontrivial function field analogue. This has been proved by Deligne using 1-motifs and Weil’s theorem expressing $L$-series as characteristic polynomials of the Frobenius morphism. The author gives details of this proof. Hayes has since given another proof. This second proof generalizes previous work of Galovich and Rosen on analogues of cyclotomic units in cyclotomic function fields.

In the last chapter, the author presents certain $p$-adic conjectures which are analogues of those of Stark over C. For complex Artin $L$-functions, there is a functional equation relating behavior at $s$ and $1-s$ and hence Stark’s conjecture can be translated to a conjecture at $s=1$. In contrast, there is no known functional equation for $p$-adic $L$-functions and therefore we have two seemingly independent conjectures. The conjecture at the point $s=0$ is due to \n J.-P. Serre\en [C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 4, A183A188; MR0506177 (58 #22024)] and the conjecture at the point $s=0$ is due to \n B. Gross\en [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 979994; MR0656068 (84b:12022)].

MR0837223 (87g:11158) Soulé, Christophe(F-PARIS7) Régulateurs. (French) [Regulators] Seminar Bourbaki, Vol. 1984/85. Astérisque No. 133-134 (1986), 237–253. 11R70 (11R27 11R42 19D45) PDF Doc Del Clipboard Journal Article Make Link

The paper is a report on recent results and conjectures concerning regulator mappings. Most of them are given in the paper of \n A. A. Be\u\i linson\en [Current problems in mathematics, Vol. 24 (Russian), 181238, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984; MR0760999 (86h:11103)]. For a number field $F$ with $r_1$ real and $r_2$ complex places the classical regulator mapping of Dirichlet $\rho\:{\scr O}_F^*\oplusZ \toR^{r_1+r_2}$ is given by the logarithms of Archimedean values of fundamental units of $F$. Then ${\rm Ker}\,\rho$ is finite and ${\rm Im}\,\rho$ is a lattice of finite volume. The classical regulator mapping has been generalized to higher regulator mappings $\rho_n\: K_{2n-1}({\scr O}_F)\toR^{r_2}$ [resp. ] if $n$ is even [resp. odd], $n\ge2$. Then ${\rm Ker}\,\rho_n$ is finite and the volume of the lattice ${\rm Im}\,\rho_n$ can be expressed by means of values of zeta functions of $F$ and of its derivatives. The above has been generalized to the $K$-groups of any projective smooth variety $V$ over a number field $F$ [see Be\u\i linson, op. cit.]. Namely, there are regulator mappings $$\rho_{n,k}\:K_m(V)\to H_{{\scr D}}^k(V\otimes_{Q}R, (2\pi i)^nR), k+m=2n,$$defined by means of Chern classes, where$H_{{\scr D}}$is the Deligne-Be\u\i linson cohomology. The author discusses several results and conjectures on${\rm Ker}\,\rho_{n,k}$and${\rm Im}\,\rho_{n,k}$. The details are too involved to be presented here.

MR0862627 (88f:11060) Be\u\i linson, A. A. Higher regulators of modular curves. Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), 1–34, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986. 11G40 (11F67 11R70 14C35 14G10 18F25 19F27) PDF Doc Del Clipboard Journal Article Make Link

For each weight 2 cuspidal representation $V$ of the adelic group $G=\roman{GL}_2(\bold A_f)$ the author constructs a subgroup of the motivic cohomology group $H^2_{\scr M}(M_V, \bold Q(l+2))$ and proves that the image of this group in Hodge cohomology (sometimes called Deligne-Be\u\i linson cohomology) under the regulator map is the group predicted by conjectures of the author about values of $L$-functions [Current problems in mathematics, Vol. 24 (Russian), 181238, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984; MR0760999 (86h:11103)]. Here $M$ is the pro-variety of affine modular curves of all levels, and $M_V$ is the $V$-component of the associated motive. The motivic cohomology group $H^a _{\scr M}(M,\bold Q(b))$ is defined to be the subgroup of Quillen’s $K$-group $K_{2b-a}(M)$ on which the Adams operators $\psi^p$ act via multiplication by $p^b$.

Let $X$ be the universal elliptic curve with level structure over $M$ and for each $l\ge 0$ let $X^l$ be the $l$-fold fiber product of $X$ with itself over $M$. The Gysin map induces a $G$-morphism $\pi^l_*\:H^{2l+2}_{\scr M}(X^l,\bold Q(2l+2))\to H^2_{\scr M}(M,\bold Q(l+2))\otimes|\roman{det}( )|^l$. Define\break $H^2_{\scr M}(M,\bold Q(l+2))^{\roman{parab}}$ to be the subgroup of $H^2_{\scr M}(M,\bold Q(l+2))$ generated by all $\pi^l_*(\{\alpha,\beta\})$, where $\alpha,\beta\in H^{l+1} _{\scr M}(X^l,\bold Q(l+1))$ and $\{\, , \}$ denotes cup product. On this subgroup the regulator map can be explicitly calculated in terms of products of certain holomorphic and nonholomorphic Eisenstein series which are obtained by averaging the residues of $\alpha$ and $\beta$ at the cusps. The $V$-component of the image of the regulator map is then obtained by calculating the Petersson inner product of elements in this image with weight two newforms associated to $V$. Using Rankin’s method these inner products are related to values of $L$-functions. This calculation generalizes methods developed in the author’s earlier article [op. cit.] where the case $l=0$ was treated. For more background material, see an article by \n C. Soulé\en [Astérisque No. 133-134 (1986), 237253; MR0837223 (87g:11158)].

MR0862643 (88a:14027) Ramakrishnan, Dinakar(1-JHOP) Higher regulators on quaternionic Shimura curves and values of $L$-functions. Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), 377–387, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986. 14G10 (11F67 11G18 11R42 18F25 19F27) PDF Doc Del Clipboard Journal Article Make Link

The author gives some further evidence for Be\u\i linson’s conjectures about the relation between $K$-groups and special values of $L$-func-tions. He deals with the case of Shimura curves. The main technical point consists in extending the Jacquet-Langlands correspondence to $K$-theory and Deligne cohomology, via the induced isogenies on Jacobians. After that one uses Be\u\i linson’s original result (for modular curves).

MR0923131 (89h:11027) Be\u\i linson, A. A. Height pairing between algebraic cycles. $K$-theory, arithmetic and geometry (Moscow, 1984–1986), 1–25, Lecture Notes in Math., 1289, Springer, Berlin, 1987. 11G40 (14C17 14G10) PDF Doc Del Clipboard Journal Article Make Link

Let $X/\bold Q$ be a smooth projective variety. The conjecture of Birch and Swinnerton-Dyer says that $L(H^1(X),s)$ has a zero at $s=1$ of order the rank of ${\rm Pic}^0(X)(\bold Q)$, and that its leading coefficient is a rational multiple of the Néron-Tate height regulator times a period matrix. (Of course, there are only a few cases for which it is even known that $L(H^1(X),s)$ can be analytically continued to $s=1$.) More generally, Swinnerton-Dyer has conjectured that $L(H^{2i-1}(X),s)$ has a zero at $s=i$ of order the rank of ${\rm CH}^i(X)^0$. (Here ${\rm CH}^i(X)^0$ is the group of codimension $i$ cycles on $X$ homologous to $0$, modulo rational equivalence.) In this paper the author defines a height pairing between ${\rm CH}^i(X)^0$ and ${\rm CH}^{\dim X+1-i}(X)^0$ whose determinant times the period determinant should conjecturally give a rational multiple of the leading coefficient of $L(H^{2i-1}(X),s)$ at $s=i$.

The author defines the height pairing of two cycle classes $c_1$ and $c_2$ as a sum of local intersections (which he calls link indices)\break $\langle c_1,c_2\rangle=\sum_v\langle c_1,c_2\rangle_v$, where $v$ runs over all places of the base field. For non-Archimedean places, he uses intersection theory analogous to the geometric case, which he gives as motivation in the first section. For Archimedean places he gives a generalization of the one-dimensional Arakelov intersection theory. This involves the use of mixed Hodge structures and Hodge-Deligne cohomology on $X/\bold C$. A similar construction was given by \n H. Gillet\en and \n C. Soulé\en [C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 12, 563566; MR0770447 (86a:14019)]. See also a paper by Gillet [in Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 209228, Contemp. Math., 67, Amer. Math. Soc., Providence, R.I., 1987; MR0902594 (88h:14005)].

In a final section the author gives “some conjectures and motivic speculations about algebraic cycles, heights, $L$-functions, and absolute cohomology groups”.

MR0944987 (89a:14002) Beilinson’s conjectures on special values of $L$-functions. Edited by M. Rapoport, N. Schappacher and P. Schneider. Perspectives in Mathematics, 4. Academic Press, Inc., Boston, MA, 1988. xxiv+373 pp. ISBN: 0-12-581120-9 14-06 (11-06) PDF Doc Del Clipboard Journal Article Make Link

Contents:\ Peter Schneider, Einladung zur Arbeitsgemeinschaft in Oberwolfach über “Die Beilinson-Vermutung” Invitation to the working session in Oberwolfach on the “Beilinson conjecture”\rm ix}–{\rm xvi}); Peter Schneider, Introduction to the Beilinson conjectures (pp. 1–35); Maria Heep and Uwe Weselmann, Deligne’s conjecture (pp. 37–42); Hélène Esnault and Eckart Viehweg, Deligne-Beilinson cohomology (pp.\ 43–91); Wolfgang K. Seiler, $\lambda$-rings and Adams operations in algebraic $K$-theory (pp. 93–102); Günter Tamme, The theorem of Riemann-Roch (pp. 103–168); M. Rapoport, Comparison of the regulators of Beilinson and of Borel (pp.\ 169–192); Jürgen Neukirch, The Beilinson conjecture for algebraic number fields (pp.\ 193–247); Christopher Deninger and Kay Wingberg, On the Beilinson conjectures for elliptic curves with complex multiplication (pp. 249–272); Norbert Schappacher and Anthony J. Scholl, Beilinson’s theorem on modular curves (pp. 273–304); Uwe Jannsen, Deligne homology, Hodge-${\scr D}$-conjecture, and motives (pp.\ 305–372).

{The papers are being reviewed individually.}

MR0981737 (90f:11041) Deninger, Christopher(D-RGBG) Higher regulators and Hecke $L$-series of imaginary quadratic fields. I. Invent. Math. 96 (1989), no. 1, 1–69.

Dinakar Ramakrishnan, Regulators, algebraic cycles, and values of $L$-functions (pp. 183–310); (1987, some proceedings)

MR1032929 (91d:14003) Esnault, Hélène(F-IHES) A regulator map for singular varieties. Math. Ann. 286 (1990), no. 1-3, 169–191. Let $X$ be an algebraic variety over $\bold C$ (not necessarily smooth). The author gives a construction of a regulator map from the Zariski sheaf of Milnor $K$-theory on $X$ to certain sheaves $\scr H^n(n)$ which coincide in the smooth case with the Deligne-Be\u\i linson cohomology sheaves (and the regulator with the one constructed by Bloch). For the definition of the $\scr H^n(n)$, one uses a desingularization of $X$. The construction itself is somewhat sophisticated, but independent of the chosen resolution. Moreover, the $\scr H^n(n)$ are functorial in $X$. Another different approach to a regulator map for singular varieties is due to M. Levine.

MR1059937 (91i:19003) Deninger, Christopher(D-MUNS) Higher regulators and Hecke $L$-series of imaginary quadratic fields. II. Ann. of Math. (2) 132 (1990), no. 1, 131–158. The real Deligne cohomology group of a motive over a number field has two $\bold Q$-structures, namely that given by the rational Deligne cohomology group and that given by the image of the regulator map, and Be\u\i linson conjectured that the difference between the two structures is measured by the value of an $L$-series (or its derivative) at an appropriate integer. In agreement with this conjecture, the author shows that, for the motive attached to an algebraic Hecke character of a quadratic imaginary number field $K$, the image of the regulator map does in fact contain a $\bold Q$-subspace with the correct relation to the rational Deligne cohomology group, thereby extending his earlier result [Part I, Invent. Math. 96 (1989), no. 1, 169; MR0981737 (90f:11041)]. He also proves a similar result for certain elliptic curves defined over real fields, and a related conjecture of \n B. Gross\en for a character of $\roman{Gal}(\overline K/ K)$.

MR1110393 (92h:11056) Deninger, Christopher(D-MUNS); Scholl, Anthony J.(4-DRHM) The Be\u\i linson conjectures. $L$-functions and arithmetic (Durham, 1989), 173–209, London Math. Soc. Lecture Note Ser., 153, Cambridge Univ. Press, Cambridge, 1991.

MR1300893 (95j:11060) Scholl, A. J.(4-DRHM-SL) Extensions of motives, higher Chow groups and special values of $L$-functions. Séminaire de Théorie des Nombres, Paris, 1991–92, 279–292, Progr. Math., 116, Birkhäuser Boston, Boston, MA, 1993. (Also other articles by Scholl)

MR1265532 (95h:19001) Bloch, Spencer(1-CHI) An elementary presentation for $K$-groups and motivic cohomology. Motives (Seattle, WA, 1991), 239–244, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.

MR1273841 (95g:19002) Karoubi, Max(F-PARIS7-M) Classes caractéristiques de fibrés feuilletés, holomorphes ou algébriques. (French. English, French summary) [Characteristic classes of foliated, holomorphic or algebraic bundles] Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part II (Antwerp, 1992). $K$-Theory 8 (1994), no. 2, 153–211.

Jean-Luc Brylinski, Holomorphic gerbes and the Be\u\i linson regulator (8, 145–174)

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Absolute Hodge cohomology

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Absolute Hodge cohomology

AG (Algebraic geometry)

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Absolute Hodge cohomology

Mixed, Arithmetic

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Absolute Hodge cohomology

Beilinson: Notes on absolute Hodge cohomology. Contemp. Math. 55, 1986.

Original work by Saito

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Absolute Hodge cohomology

Regulator map from motivic cohomology. See Burgos and Wang, and also Feliu (page 87) for description in the form of a Chern character.

Huber and Wildeshaus (p 101) says that there is a natural transf to Deligne cohomology. When is this an iso?

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Absolute Hodge cohomology

Nekovar’s survey on the Beilinson conjectures

Huber and Wildeshaus: Classical motivic polylogarithm according to Beilinson and Deligne. See in particular appendix B.5

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Absolute Hodge cohomology

Absolute Hodge cohomology was defined by Beilinson. It is almost the same thing as Deligne cohomology. See the references below.

See also: Deligne-Beilinson cohomology, Deligne cohomology, Weak Hodge cohomology.

Absolute Hodge cohomology is an example of an Absolute cohomology theory, and it is also a Bloch-Ogus cohomology theory.

nLab page on Absolute Hodge cohomology