UNIVERSITA' DI GENOVA
DIPARTIMENTO DI MATEMATICA

Incontro di Lavoro

CICLI ALGEBRICI

Teoria di Hodge, Motivi e K-Teoria

Organizzatori: Alberto Albano, Luca Barbieri Viale, Gian Pietro Pirola

Genova, 26-28 Marzo 1999

Valle Puggia, Via Dodecaneso 35 - 16146 Genova

Programma
  • Venerdi' 26
  • ore 9,30 - 10,30 C.Pedrini: "K-theory and algebraic cycles"
  • 10,45 - 11,45 M.Levine: "K-theory and motivic cohomology of schemes"
  • 12,00 - 13,00 A.Vistoli: "Equivariant intersection theory"
  • 14,30 - 15,10 B.Chiarellotto: "Weights in the cristalline realization"
  • 15,30 - 17,10 L.Guerra: "Una presentazione funtoriale per lo spazio dei cicli algebrici"
  • Sabato 27
  • ore 10,00 - 11,00 B.van Geemen:"Kuga-Satake varieties and the Hodge conjecture"
  • 11,15 - 12,15 V.Srinivas: "Albanese and Picard 1-motives"
  • 14,30 - 15,10 A.Rosenschon: "A Motivic Lefschetz Theorem"
  • 15,30 - 17,10 G.Vezzosi: "On the Chow ring of the classifying stack of PGL_3"
  • Domenica 28
  • ore 10,00 - 11,00 C.Voisin: "On Green's higher Abel Jacobi"
  • 11,30 - 12,30 A.Collino: "Natural undecomposable cycles for Bloch's Higher Chow Groups on low dimensional Jacobians"

  • Abstracts


    B.Chiarellotto: "Weights in the cristalline realization"

    We introduce a weight filtration in the rigid cohomology of a scheme in ch.=p. Rigid cohomology should play the role of cristalline realization. This will be the analogous of the weight filtration in classical Hodge theory, and the weights agree. We compare the above constructions.


    A.Collino: "Natural undecomposable cycles for Bloch's Higher Chow Groups on low dimensional Jacobians"

    This is a report on work in progress. We produce a 'natural' undecomposable element in the higher Chow group CH^3(J(C),1) ( = H^2(J(C),K_3) ), where J(C) is the jacobian variety of a general curve C of genus 3. In the same vein we produce a 'natural' non trivial element in the higher Chow group CH^3(J(G),2) (= H^1(J(C),K_3)), where now G is of genus 2.


    L.Guerra: "Una presentazione funtoriale per lo spazio dei cicli algebrici"

    Se X e' una varieta' proiettiva, lo spazio dei cicli algebrici C(X) e' una unione numerabile di varieta' proiettive. Per una presentazione funtoriale, parallela a quella che esiste per lo schema di Hilbert, il punto chiave e' determinare quali famiglie di cicli algebrici su X, cicli in un prodotto T x X, corrispondono ai morfismi f: T -> C(X) come pullback di un ciclo universale. La soluzione data da Barlet fa intervenire la struttura locale della proiezione |Z| -> T. Una soluzione diversa consiste nel considerare le applicazioni razionali continue, cosiddette semiregolari, che sono caratterizzate per mezzo della sola variazione globale dei cicli nella famiglia. La trattazione di Kollar sembra avere lo stesso punto di partenza ma un percorso diverso.

    In generale, un ciclo Z induce una applicazione regolare sul luogo liscio Tsm -> C(X). Il punto geometricamente rilevante e' percio' determinare quando questa applicazione ammette una estensione continua f: T -> C(X). Seguendo un cammino, un morfismo \alpha: S -> T dove S e' una curva liscia, che passa per un punto t muovendo attraverso il luogo non singolare Tsm, restringendo la famiglia al cammino \alpha si trova un ciclo limite Z(t,\alpha). Ogni cammino \alpha attraversa un singolo ramo T' del germe T,t. Il risultato chiave e' che il ciclo limite Z(t,\alpha) dipende solo dal ramo T' che contiene \alpha.

    I cicli Z che corrispondono alle applicazioni semiregolari in C(X) sono quindi caratterizzati dalla proprieta' che il ciclo limite sia lo stesso lungo ogni ramo. Per questi cicli Z si costruisce inoltre un pullback f!Z tramite applicazioni semiregolari. Questo definisce un funtore di famiglie regolari di cicli sulla categoria degli schemi ridotti con applicazioni semiregolari, e si dimostra infatti che lo spazio dei cicli C(X) rappresenta questo funtore.


    M.Levine: "K-theory and motivic cohomology of schemes"

    We describe a construction of motivic Borel-Moore homology for schemes of finite type over a regular base B of Krull dimension one or zero, satisfying most of the standard formal properties of Borel-Moore homology, including localization. In addition, relying on a fundamental reinterpretation of the Bloch-Lichtenbaum spectral sequence due to Friedlander-Suslin, we show how to globalize the Bloch-Lichtenbaum spectral sequence to give a spectral sequence from motivic Borel-Moore homology to G-theory for $B$-schemes of finite type. For regular schemes, we define the motivic cohomology via Borel-Moore homology with appropriate change in degree and weight, and get a spectral sequence of Atiyah-Hirzebruch type

    E_2^{p,q}=H^p(X,\Z(q)) -> K_{-p-q}(X). (*)

    This sequence admits Adams operations and a multiplicative structure (the latter requires the base to be a field, in general). There is an associated etale spectral sequence as well, which formally resembles the Dwyer-Friedlander spectral sequence, but at present there is no known comparison between these two etale spectral sequences. As an application, we show how the Bloch-Kato conjectures imply a sharp form of the Quillen-Lichtenbaum conjectures.

    Using Voevodsky's solution of the Milnor conjecture, we can compute the 2-localized motivic cohomology of finite field, curves over finite fields, and number fields, resulting in the expected answers. The arguments of B. Kahn then apply, using the properties of the spectral sequence (*), to give a computation of the 2-localized K-theory of rings of S-integers in a number field. Various other related results of Kahn, Rognes-Weibel, and others, admit extensions to positive characteristic, as well as certain global reformulations.


    C.Pedrini: "K-theory and Algebraic Cycles"

    The aim of this talk is to give a quick survey on how the research on K-theory and algebraic cycles has recently developed in Italy and to present some new results on the higher K-groups of complex smooth surfaces.


    A.Rosenschon: "A Motivic Lefschetz Theorem"

    This talk reports on current joint work with L. Barbieri-Viale; we define, for any normal simplicial scheme X. a Lefschetz 1-motive. In the case of X. smooth we prove a Lefschetz Theorem and study "non-algebraic" (1,1)-classes. If X. is any hypercovering of a proper variety X, we correlate these results to the structure of NS(X). This gives a new proof of the conjecture of Barbieri-Viale and Srinivas that for X normal, NS(X) is given by those (integral) second cohomology classes which are Zariski locally trivial and are in F^1.


    V.Srinivas: "Albanese and Picard 1-motives"

    Jointly with L.Barbieri-Viale, we introduce algebraically defined 1-motives generalizing the classical Albanese and Picard varieties of a smooth projective variety. We show some geometric applications of this constructions.


    G.Vezzosi: "On the Chow ring of the classifying stack of PGL_3"

    Following Totaro we consider the G-equivariant Chow ring A^*_G of a point, G being an algebraic group, say over the complex numbers. We recall Totaro's refined cycle map to a quotient of complex cobordism and then analyze some cases of classical groups for which the equivariant Chow ring was computed by Pandharipande and Totaro. Then we turn to the cases of PGL_2 and PGL_3. A^*_PGL_2 is computed and we find generators and some relations for A^*_PGL_3 which is considerably more difficult. For PGL_3, we prove a consequence of a conjecture by Totaro, mainly that A^*_PGL_3 is not generated by Chern classes. Moreover we prove some properties of the cycle map and of Totaro's refined cycle map for PGL_3.


    A.Vistoli: "Equivariant intersection theory"

    Equivariant intersection theory, due to B. Totaro, D. Edidin and W. Graham, allows to define Chow groups with integral coefficients for quotient algebraic stacks. There are at present two main examples of interest: stacks of stable curves, and classifying stacks of algebraic groups. I will concentrate mostly on the case of curves, and explain the very little that is known.


    Per ulteriori informazioni rivolgersi a: Cristina Farinetti, segreteria MURST, tel. 010-353-6962, E-mail: farinett@dima.unige.it