## THE MORPHIC ABEL-JACOBI MAP

Citations: | 8 - 0 self |

### BibTeX

@MISC{Walker_themorphic,

author = {Mark E. Walker},

title = {THE MORPHIC ABEL-JACOBI MAP},

year = {}

}

### OpenURL

### Abstract

Abstract. The morphic Abel-Jacobi map is the analogue of the classical Abel-Jacobi map one obtains by using Lawson and morphic (co)homology in place of the usual singular (co)homology. It thus gives a map from the group of r-cycles on a complex variety that are algebraically equivalent to zero to a certain “Jacobian ” built from the Lawson homology groups viewed as inductive limits of mixed Hodge structures. In this paper, we define the morphic Abel-Jacobi map, establish its foundational properties, and then apply these results to the study of algebraic cycles. In particular, we show the classical Abel-Jacobi map (when restricted to cycles algebraically equivalent to zero) factors through the morphic version, and show that the morphic version detects cycles that cannot be detected by its classical counterpart — that is, we give examples of cycles in the kernel of the classical Abel-Jacobi map that are not in the kernel of the morphic one. We also investigate the behavior of the morphic Abel-Jacobi map on the torsion subgroup of the Chow group of cycles algebraically equivalent to zero modulo

### Citations

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Citation Context ...re. His technique can be used to define the classical Abel-Jacobi map — in this case, the extra structure on the singular (co)homology groups is that of mixed Hodge structures, as provided by Deligne =-=[8, 9]-=-. To define the morphic Abel-Jacobi map, the extra structure comes from viewing the Lawson homology groups as inductive limits of mixed Hodge structures. Such structures were defined by Friedlander-Ma... |

300 |
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Citation Context ... multi-simplicial sets Sing •(M + • ) → Z Sing •(M + • ), which admits a natural splitting since Sing•(M + • ) is a multi-simplicial abelian group. (M•, Q) is isomorphic as a The Milnor-Moore Theorem =-=[28]-=- gives us that H sing ∗ Hopf algebra to SQ(π∗(M•, Q)), where SQ(V ) denotes the symmetric algebra of a graded vector space V . In general, the multiplication and comultiplication maps making SQ(V ) in... |

118 |
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Citation Context ...can construct projective closures U ⊂ U and V ⊂ V , with closed complements U∞ and V∞, such that π extends to a morphism π : U → V such that π −1 (U) = V . By the platification par eclatement Theorem =-=[31]-=-, we can take blow-ups and proper transforms, without affecting U, V , or π, so that π becomes flat, also of relative dimension e. The flat pullback map π ∗ : LrHn(V ) → Lr+eHn+2e(U) in Lawson homolog... |

117 | Triangulated categories of motives over a field, Ann. of Math. Stud. 143 (2000), 188–238. Dipartimento di Matematica Pura e Applicata, Università degli Studi di - Voevodsky |

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74 |
Algebraic cycles and higher K-theory
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Citation Context ...ns and then homotopy groups gives the homomorphism Γ∗ : LrHn(X) → Lr+sHn+2s(Y ). This definition is extended to an arbitrary Γ ∈ Zs(X, Y ) by linearity. There are maps from Bloch’s higher Chow groups =-=[5]-=- to the Lawson homology groups, CHr(X, n) → LrH2r+n(X), and these maps are natural for pushforwards along projective morphisms and pullbacks along flat morphisms [16]. Moreover, the composition CHr(X,... |

58 | A theory of algebraic cocycles - Friedlander, Lawson - 1992 |

54 |
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Citation Context ...H sing 1 (C, Z(0)) ∼ = � L0H1(C) → LrH2r+1(X) C,Γ is surjective. To see this, recall that we have LrH2r+1(X) = π1(Zr(X)) ∼ = H sing 1 (Zr(X)) ∼ = lim −→e By the Lefschetz Theorem of Andreotti-Frankel =-=[1]-=-, the map H sing 1 C,Γ H sing 1 (Cr,e(X)). (C) → H sing 1 (Cr,e(X)) is surjective for some (possibly singular) curve C ⊂ Cr,e(X), and the inclusion C ↣ Cr,e(X) determines a correspondence. This shows ... |

47 |
Classifying spaces and fibrations
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Citation Context ...l mapping cone for a map of complexes. If C is the category of abelian monoids, one may readily verify that (4.5) cone⊕(u) = B(0, Y•, X•), where the right-hand side is May’s “triple bar construction” =-=[26]-=- with Y• acting on X• in the obvious manner via u. In particular, we have (4.6) cone⊕(Cr(X) → 0) = B(0, Cr(X), 0) = B(Cr(X)). Notice that, by convention, if Y is an object of a category C, we regard Y... |

41 |
Algebraic cycles, Chow varieties and Lawson homology
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Citation Context ...) is surjective and we have (2.4) LrH2r(X) ∼ = Zr(X)/Zr(X)alg∼0, where Zr(X) denotes the discrete group of r-cycles on X and Zr(X)alg∼0 denotes the subgroup of cycles algebraically equivalent to zero =-=[12]-=-. The connection of Lawson homology with singular homology builds on the following basic result. (We reprove this result in the category of mixed Hodge structures in Theorem 4.21 below.)sTHE MORPHIC A... |

37 |
Notes on absolute Hodge cohomology, in: Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I
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(Show Context)
Citation Context ... finitely generate sub-IMHS’s of H (i.e., over all sub-IMHS that are actually MHS’s). In particular, we have Ext n IMHS(Z(0), H) = 0, if n ≥ 2, for any IMHS H, since this vanishing holds for MHS’s by =-=[3]-=-. Definition 3.2. For a IMHS H, define and Γ(H) = HomIMHS(Z(0), H) J (H) = Ext 1 IMHS(Z(0), H). Since Γ(H) = lim −→α Γ(Hα), we have Γ(H) = H ∩ W0(HQ) ∩ F 0 (HC). Proposition 3.3 (cf. [7, 21]). For a I... |

36 |
Filtrations on the homology of algebraic varieties
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Citation Context ... define the morphic Abel-Jacobi map, the extra structure comes from viewing the Lawson homology groups as inductive limits of mixed Hodge structures. Such structures were defined by Friedlander-Mazur =-=[19]-=- in the projective case and later generalized by Lima-Filho [25] to all complex varieties. Section 4 of this paper contains a detailed construction of these inductive limits of mixed Hodge structures ... |

32 |
of mixed Hodge structures, Journees de geometrie algebrique d'Angers
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(Show Context)
Citation Context ... for MHS’s by [3]. Definition 3.2. For a IMHS H, define and Γ(H) = HomIMHS(Z(0), H) J (H) = Ext 1 IMHS(Z(0), H). Since Γ(H) = lim −→α Γ(Hα), we have Γ(H) = H ∩ W0(HQ) ∩ F 0 (HC). Proposition 3.3 (cf. =-=[7, 21]-=-). For a IMHS H, we have J (H) ∼ = W0(HC) W0(H) + F 0 W0(HC) . That is, J (H) is the quotient of the complex vector space W0(HC)/F 0 W0(HC) by the action of W0(H) := ker(H → HQ/W0(HQ)). In particular,... |

25 |
Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism
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(Show Context)
Citation Context ...i.e., r = dim(X) − 1), the injectivity of (8.12) and (8.13) are easily seen to hold; in codimension two (r = dim(X) − 2), the injectivity of these maps is a consequence of the MerkurjevSuslin Theorem =-=[27]-=- (see [29, 10.3]); and for zero-cycles (r = 0), the injectivity of these maps is a theorem of Roitman [32, 4]. The injectivity of (8.12), however, is now known to fail in general. Schoen [33] has cons... |

24 | Torsion algebraic cycles and complex cobordism
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(Show Context)
Citation Context ...el-Jacobi map cannot be injective on (CHr(X)alg∼0)tor for such varieties. Soulé and Voisin [35] have also shown that (8.12) can fail to be injective for four-folds with r = 1. In earlier work, Totaro =-=[38]-=- constructed examples of a smooth, projective variety X, definable over a number field, for which (8.13) fails to be injective. In light of Theorem 8.11, it is interesting to note that there are no kn... |

21 |
Cycle spaces and intersection theory
- Friedlander, Gabber
- 1993
(Show Context)
Citation Context ...ps from Bloch’s higher Chow groups [5] to the Lawson homology groups, CHr(X, n) → LrH2r+n(X), and these maps are natural for pushforwards along projective morphisms and pullbacks along flat morphisms =-=[16]-=-. Moreover, the composition CHr(X, n) → LrH2r+n(X) → H BM 2r+n(X) is the usual map from Bloch’s groups to Borel-Moore homology. When n = 0, the map CHr(X) → LrH2r(X) is surjective and we have (2.4) Lr... |

19 |
Completions and fibrations for topological monoids
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(Show Context)
Citation Context ...oosing a projective closure U ⊂ X with closed complement Y , and then setting Zr(U) = Zr(X)/Zr(Y ). It is not hard to see that the topology on Zr(U) is independent of the choice of projective closure =-=[24]-=-. Definition 2.1. The Lawson homology groups of a complex variety U with coefficients in an abelian group A are We set LrHn(U) = LrHn(U, Z). LrHn(U, A) := πn−2r(Zr(U), A). There is an associated cohom... |

17 |
An overview of recent advances in Hodge theory
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(Show Context)
Citation Context ... LrH2r(Vs, A) = Zr(Vs) ⊗ A = HBM 2r (Vs, A). The result now follows by naturality for localization sequences and the Five Lemma. � 3. Inductive Limits of Mixed Hodge Structures We refer the reader to =-=[6]-=- for the definition and properties of a mixed Hodge structure (MHS, for short). The Lawson homology groups are, in general, not finitely generated, and thus cannot be enriched to MHS’s (which are by d... |

16 |
Algebraic cocycles on normal, quasi-projective varieties
- Friedlander
- 1998
(Show Context)
Citation Context ...called morphic cohomology, defined by L t H m � (X, A) = π2t−m Z0(X, P t )/Z0(X, P t−1 ), A � , where Z0(X, P t ) denotes the collection of cycles on X × P t that are finite over X, topologized as in =-=[13]-=-. In this paper, we focus almost entirely on Lawson homology and not on morphic cohomology (despite the fact that we refer to the main object of study as the morphic Abel-Jacobi map). The existence of... |

13 | Higher Chow groups and etale cohomology - Suslin - 2000 |

12 |
Algebraic cycles and Hodge–theoretic connectivity
- Nori
- 1993
(Show Context)
Citation Context ...i map detects cycles that cannot be detected by its classical counterpart. In fact, we provide two types of such examples. Those of the first type arise by building on examples originally due to Nori =-=[30]-=- and further developed by Friedlander [15], which show that various stages of the so-called sfiltration are non-trivial. The examples of the second type arise from examples of Schoen [33] showing that... |

12 | Torsion cohomology classes and algebraic cycles on complex projective manifolds
- Soulé, Voisin
(Show Context)
Citation Context ...d above show that the classical Abel-Jacobi map does not always induce an injection from (CHr(X)alg∼0)tor (the torsion subgroup of CHr(X)alg∼0) to Jr(X)tor, as was once conjectured. (Soulé and Voisin =-=[35]-=- also have constructed such counter-examples.) It is an intriguing question whether the morphic Abel-Jacobi map induces an isomorphism of the form ?∼ = r |tor : (CHr(X)alg∼0)tor Φ mor −→J mor r (X)tor... |

10 | Correspondence homomorphisms for singular varieties, preprint - Friedlander, Mazur - 1994 |

8 |
Torsion algebraic cycles and a theorem of Roitman
- Bloch
- 1979
(Show Context)
Citation Context ... = dim(X) − 2), the injectivity of these maps is a consequence of the MerkurjevSuslin Theorem [27] (see [29, 10.3]); and for zero-cycles (r = 0), the injectivity of these maps is a theorem of Roitman =-=[32, 4]-=-. The injectivity of (8.12), however, is now known to fail in general. Schoen [33] has constructed examples of a smooth, projective complex variety X of dimension d, definable over a field of transcen... |

8 |
The torsion of the group of 0-cycles modulo rational equivalence
- Rojtman
- 1980
(Show Context)
Citation Context ... = dim(X) − 2), the injectivity of these maps is a consequence of the MerkurjevSuslin Theorem [27] (see [29, 10.3]); and for zero-cycles (r = 0), the injectivity of these maps is a theorem of Roitman =-=[32, 4]-=-. The injectivity of (8.12), however, is now known to fail in general. Schoen [33] has constructed examples of a smooth, projective complex variety X of dimension d, definable over a field of transcen... |

7 | Techniques, Computations, and Conjectures for Semi-Topological K-theory, preprint, available at http://www.math.uiuc.edu/K-theory/0621
- Friedlander, Haesemeyer, et al.
(Show Context)
Citation Context ...ojective variety, so that Φ mor r |tor is an isomorphism for all such varieties (see Corollary 8.8). In this paper, we show that for any smooth, projective variety belonging to the class C defined in =-=[17]-=- (which includes all curves, all toric varieties, all cellular varieties, and all varieties built from these via localization, blowing up, or forming vector bundles),s4 MARK E. WALKER the kernel of Φm... |

7 |
Mixed motives and algebraic K-theory, volume 1400
- Jannsen
- 1990
(Show Context)
Citation Context ...Hr(X)alg∼0 is the Chow group of r-cycles on X that are algebraically equivalent to zero modulo rational equivalence. To define the morphic Abel-Jacobi map, we rely on the general technique of Jannsen =-=[21]-=- for construction of Abel-Jacobi-type maps in a variety of settings. The input to Jannsen’s technique is a homology/cohomology theory for varieties equipped with suitable extra structure. His techniqu... |

7 |
Higher Picard varieties
- Lieberman
- 1968
(Show Context)
Citation Context ...facts above, we derive numerous good properties of the morphic AbelJacobi map. For a smooth, projective variety X, the image of Zr(X)alg∼0 under the classical Abel-Jacobi map is known (cf. [22, §12], =-=[23]-=-) to be a abelian variety (not merely a complex torus) and is called the Lieberman Jacobian, written J a r (X) := Φr(Zr(X)alg∼0). Theorem 5.7. Let X be a (possibly singular) projective variety. (1) Gi... |

7 | Applications of algebraic K-theory to the theory of algebraic cycles - Murre - 1985 |

6 | Calculation of derived functors via Ind-categories - Huber - 1993 |

6 |
Complex varieties for which the Chow group mod n is not finite
- Schoen
(Show Context)
Citation Context .... Remark 8.15. Schoen’s theorem is actually more general that this — one may replace k ⊂ C with any extension of algebraically closed fields, even those of positive characteristic. Schoen also proves =-=[34]-=- that for any algebraically closed field k of characteristic zero, there exist a smooth, projective three-fold W (in fact, W can be taken to be an abelian variety) such that CH1(W ) ⊗ Ql/Zl has infini... |

5 | Bloch-Ogus properties for topological cycle theory
- Friedlander
(Show Context)
Citation Context ...n homology is, however, useful for understanding Lawson homology. In particular, many of the formal properties of Lawson homology are summarized by the following statement. Theorem 2.2 (Friedlander). =-=[14]-=- Lawson homology and morphic cohomology form twisted duality theory in the sense of Bloch-Ogus.s6 MARK E. WALKER In particular, associated to any projective morphism p : X → Y , there is a pushforward... |

4 | Relative Chow correspondences and the Griffiths group
- Friedlander
(Show Context)
Citation Context ...ed by its classical counterpart. In fact, we provide two types of such examples. Those of the first type arise by building on examples originally due to Nori [30] and further developed by Friedlander =-=[15]-=-, which show that various stages of the so-called sfiltration are non-trivial. The examples of the second type arise from examples of Schoen [33] showing that there can be an infinite amount of l-tors... |

3 |
On certain exterior product maps of Chow groups
- Schoen
(Show Context)
Citation Context ... due to Nori [30] and further developed by Friedlander [15], which show that various stages of the so-called sfiltration are non-trivial. The examples of the second type arise from examples of Schoen =-=[33]-=- showing that there can be an infinite amount of l-torsion in the kernel of the classical Abel-Jacobi map. We also analyze the behavior of the morphic Abel-Jacobi map on torsion subgroups. The example... |

2 |
A survey of the Hodge conjecture, volume 10 of CRM Monograph Series
- Lewis
- 1999
(Show Context)
Citation Context ...Γ ∈ CHr+d(T × X), the map T (C) t↦→t−t0 −→ CH0(T )alg∼0 Γ∗ f −→CHr(X)alg∼0 is induced by a morphism of varieties (resp., is holomorphic). −→A(C) Conjecture 7.2 (Universality of Abel-Jacobi map). (cf. =-=[22]-=-) For a smooth, projective complex variety X, the classical Abel-Jacobi map Φr : CHr(X)alg∼0 ↠ J a r (X) is universal among regular functions. That is, given an abelian variety A and a regular functio... |

2 | Topological properties of the algebraic cycles functor
- Lima-Filho
- 2001
(Show Context)
Citation Context ...rom viewing the Lawson homology groups as inductive limits of mixed Hodge structures. Such structures were defined by Friedlander-Mazur [19] in the projective case and later generalized by Lima-Filho =-=[25]-=- to all complex varieties. Section 4 of this paper contains a detailed construction of these inductive limits of mixed Hodge structures for Lawson homology along with proofs of the required functorali... |

1 |
editors. Classification of irregular varieties
- Ballico, Catanese, et al.
- 1992
(Show Context)
Citation Context ...y of the classical Abel-Jacobi map would arise if the the vertical map in (1.2) fails to be injective, and it is conceivable that this map is not injective for certain varieties constructed by Kollár =-=[2]-=- — see Remark 5.10. Secondly, the fact that the vertical arrow in (1.1) can have a kernel suggests another possible source for counter-examples to the universality of the Abel-Jacobi map. Namely, such... |