Cylindrical homomorphisms and Lawson homology
We use the cylindrical homomorphism and a geometric construction introduced by J. Lewis to study the Lawson homology groups of certain hypersurfaces of degree . As an application, we compute the rational semi-topological K-theory of a generic cubic of dimension 5, 6 and 8 and, using the Bloch-Kato conjecture, we prove Suslin’s conjecture for these varieties. Using the generic cubic sevenfolds, we show that there are smooth projective varieties with the lowest non-trivial step in their s-filtration infinitely generated and undetected by the Abel-Jacobi map.
Key words and phrases:
2000 Mathematics Subject Classification:19E20, 19E15, 14F43
Lawson homology of a projective variety is defined as the homotopy groups of algebraic cycle spaces of a fixed dimension on . We write
and, intuitively, think of an element in this group as a “family of r-cycles parametrized by a (n-2r)-sphere” [FM2]. These groups, replete with information about the projective variety , are a combination of algebraic and topological information. For example, the algebraic equivalence class of an algebraic r-cycle can be expressed as a connected component of the topological space of algebraic r-cycles [F]. In the same flavor, Dold-Thom theorem shows that “families of 0-cycles parametrized by a n-sphere” are the same as the topological classes in the singular homology of . The s-map is a map that “measures” how close Lawson homology groups are to algebraic geometry or topology. We have the following sequence of maps
The composition of the above s-maps gives the usual cycle map between the Chow group of algebraic cycles modulo algebraic equivalence, denoted , and the singular homology of the complex points of , written [FM2]. Encoded in the construction of the s-map is the celebrated suspension theorem for algebraic cycles proved by Lawson [L], the starting point of Lawson homology.
Lawson homology with finite coefficients of a smooth projective variety is proved to be isomorphic, via a Poincare type duality [FL1], with the motivic cohomology with finite coefficients of [FW2]. Through this isomorphism, we can study the torsion of the Lawson homology groups by motivic cohomology tools (see Proposition 1.4). In particular, the Beilinson-Lichtenbaum conjecture may be used to identify, for certain indices, Lawson homology groups with finite coefficients with singular homology groups with finite coefficients. As shown in [SV1], the Beilinson-Lichtenbaum conjecture is equivalent to the Bloch-Kato conjecture, which has been proven by V. Voevodsky and M. Rost (see [Voevo], [Weib], [Rost2]).
The torsion free subgroup of a Lawson homology group may be studied using methods from Hodge theory (see [FHW], [Voin], [Hu])). Some of these methods, mostly those designed to study the niveau filtration of singular cohomology and the Generalized Hodge conjecture, may be adapted to work on Lawson homology groups. The generalized cycle maps between the Lawson homology groups of and the singular homology of factor through steps in the niveau filtration of the latter and it is conjectured that these steps give the entire image of these generalized cycle maps [FM]. With finite coefficients, this conjecture follows from the Bloch-Kato conjecture. With integer coefficients, this conjecture follows from the very far reaching Suslin’s conjecture (see Conjecture (1.6)). Although Suslin’s conjecture and the Generalized Hodge conjecture are not known to be related in some direct way (i.e. neither is known to imply the other), all the varieties proved to satisfy Suslin’s conjecture (see [FHW], [Voin]) were previously known to satisfy the Generalized Hodge conjecture. On the other hand, there are varieties that are known to fulfill the Generalized Hodge conjecture and for which Suslin’s conjecture is still not known. For example, abelian varieties as in [Abd] or generic cubics sevenfolds or elevenfolds as in Remark LABEL:eiel.
The main difficulty that appears in these particular examples lives in understanding the kernels of s-maps. The kernels of s-maps, in particular the kernels of the generalized cycle maps, are mostly mysterious and expected to be in general very huge (i.e. infinitely generated rational vector spaces). However, Suslin’s conjecture predicts that, in the range of indices as in the Beilinson-Lichtenbaum conjecture, these kernels are zero for any projective smooth variety.
In some cases (for example in those of certain projective varieties with “few” algebraic cycles like rationally connected varieties) these kernels can be understood, at least in some range of indices [Voin], [FHW]. In this paper, we use a geometrical construction, introduced by J. Lewis [Le1], to study the Lawson homology groups of a generic hypersurface that has “enough” -planes, . Except in the middle dimension, the Lawson homology of a hypersurface always surjects, at least rationally, onto its singular homology, as an application of the simple structure of its singular homology (see Proposition LABEL:p1). The study of the kernels of s-maps of a smooth projective hypersurface is highly non-trivial, even for indices outside of the middle dimension (see Corollary LABEL:c46). Applying the method of [Le] (also [BM], [Le1]) on Lawson homology groups allows us not only to obtain the results of [Le], but also computations of certain Lawson homology groups. As a consequence, we are also able to compute the semi-topological K-theory for certain generic cubic hypersurfaces. Semi-topological K-theory of a projective smooth variety , denoted , is a K-theory that lives between the algebraic K-theory of X and the topological K-theory of , i.e.
sharing important properties with these two K-theories [FW2]. The main tool to compute it is an “Atiyah-Hirzebruch” type of spectral sequence from the Lawson homology groups of the smooth projective variety [FHW].
The motivation of this paper was the study of Lawson homology of rationally connected varieties of dimension greater than four. In [Voin], we computed the Lawson homology of rationally connected varieties of dimension three and the rational Lawson homology of dimension four rationally connected varieties; in particular, we were able to check Suslin’s conjecture and Friedlander-Mazur’s conjecture in these cases. Some questions appeared: Could we check Suslin’s conjecture for rationally connected varieties of higher dimension? Are the s-maps always monomorphisms for a rationally connnected variety of any dimension? The results of [Voin] anticipate that the Lawson homology of rationally connected hypersurfaces may provide answers to these questions, in particular the Lawson homology of a generic cubic sevenfold and eightfold. In [Voin], the cases of a cubic fivefold and sixfold were discussed by means of the decomposition of diagonal method and of a result of Esnault, Viehweg and Levine [EVL]. Using the natural and transparent geometric method of J. Lewis [Le], we obtain more general results (the case of a cubic eightfold, see Corollary LABEL:g8).
The paper is divided in four sections and an Appendix. In the first section, we introduce our notations and recall some of the results needed later in the paper.
In the second section, we recall the geometric construction from J. Lewis [Le] and use it in the context of Lawson homology groups. We will also introduce here the cylindrical homomorphisms, which are the main objects of study in this paper. The results proved in this section give us the main tools that will be used in the fourth section. Part of the proofs needed in this section is given in the Appendix.
In the third section, we extend a “weak Lefschetz” type of theorem, proved for Chow groups in [Le], to Lawson homology groups and discuss some applications of the weak Lefschetz theorem on homology to the s-maps on the Lawson homology of a hypersurface of any dimension and degree.
The fourth section is devoted to applications of the tools developed in the second and third sections. We compute, in a certain range of indices, the Lawson homology of some rationally connected generic hypersurfaces. As an application we compute the rational Lawson homology and semi-topological K-theory of a generic cubic eightfold (Corollary LABEL:g8), obtaining, in particular, the validity of Suslin’s conjecture in this particular case. In the end of this section, based on the results obtained in the case of a generic cubic sevenfold, we remark that there are examples of varieties with the lowest nontrivial step in the s-filtration of a Griffiths group (see (2)) an infinitely generated -vector space and undetected by the Abel-Jacobi map (see Corollary LABEL:aju).
This paper uses in an essential way J. Lewis’s geometric construction from [Le1] and [Le] in the context of Lawson homology. We thank J. Lewis for making this interesting construction available.
We thank Eric Friedlander for constant encouragement and for reading various versions of this paper and making useful comments. We thank Pedro dos Santos for a detailed reading of a previous version of the paper and for his suggestions who clearly improved our paper. We thank Jeremiah Heller and Jian He for many discussions related to the results of this paper.
1. Notations and Recollection
In this section, we will introduce the notations used in the paper and briefly state some of the results needed later on.
All algebraic varieties in this paper are smooth and irreducible over complex numbers . By , , , , and we define Lawson homology, Chow group of algebraic cycles modulo rational equivalence, Chow group of algebraic cycles modulo algebraic equivalence, semi-topological K-theory, Borel-Moore cohomology and singular homology with integer coefficients. By we mean Lawson homology with rational coefficients (similar notation for rational semi-topological K-theory and rational singular homology). By an isomorphism written like we mean an isomorphism of rational vector spaces. By a monomorphism written we mean a rational monomorphism. We will write for the intersection product of two algebraic cycles , in . We will write 1 for the identity map, when there is no confusion. We write for the integer part of the real number .
We call a hypersurface of dimension n generic if it belongs to a point in a non-empty Zariski open subset of the variety of hypersurfaces of degree in the projective space .
For a smooth projective variety , we define , the naive group completions of . Here is the Chow variety of algebraic cycles of degree and dimension . The topology on the group is the quotient topology induced by the complex topology of the projective varieties . We call the topological space of dimensional algebraic cycles. The empty cycle is the natural base point of . We define
for any and . For a quasi-projective variety , with a projective closure , we let (see [LF])
This is, up to isomorphism, a well defined object in the category of topological abelian groups that admit a structure of CW-complex with inverted homotopy equivalences ([LF], [FG], [P]). We call this category .
Moreover, for a closed embedding of projective varieties , the exact sequence of topological groups
gives a long exact localization sequence of homotopy groups ([LF], [FG])
For , we define the following topological cycle spaces (with the quotient topology):
The homotopy groups of these cycle spaces give the negative Lawson homology, i.e.
for any . The following equalities ([FHW]) show that these groups are all isomorphic with the Borel-Moore homology of . For any
The space of t-cocycles for a smooth projective variety is defined to be the following naive completion
where by we mean the abelian monoid of morphisms between and the Chow monoid provided with the compact-open topology ([FL]).
The following “Poincare duality” type of theorem was proved in [FL1]:
([FL1]) There is a homotopy equivalence
for any smooth projective variety of dimension and for any .
For a proper map of quasi-projective varieties and any , we have the push-forward map ([FG], [HULi]). For any locally complete intersection map of codimension (), we have a well-defined Gysin map (in )
for any ([FG]). The Gysin map for a proper map of quasi-projective varieties , with a smooth quasi-projective variety, is defined to be the composition
where is the flat pull back of the projection map on and is the Gysin map of the regular embedding of the graph of . In the case of a regular embedding of codimension , we define the Gysin map to be the following composition (well defined in ):
The left map is a flat pull-back of the projection on , the middle map is the “specialization map” given by the deformation of the normal cone ([Ful], [FG]) and the last map is the inverse of the flat pull-back vector bundle isomorphism .
Using these Gysin maps one can construct a well defined intersection product on cycle spaces in . For example, if is a regular embedding of an r-dimensional subvariety in the smooth quasi-projective variety of dimension d, then the intersection with on is given by the composition
where is the Gysin map associated to the regular embedding and . In general, we have the following theorem:
([FG], Theorem 3.5) If is a smooth quasi-projective variety of dimension and if then there is an intersection pairing (in )
which on gives the usual intersection pairing on Chow groups of algebraic cycles modulo algebraic equivalence.
According to ([FM2], [LF]), there is an operation on Lawson groups, called s-map
for any quasi-projective variety . The inverse of the isomorphism (1) is given in the following proposition.
The s-map on the negative Lawson homology of a smooth projective variety of dimension is an isomorphism, i.e.
According to ([FL1], Proposition 2.6), for any smooth projective variety of dimension and for any we have the following commutative diagram in ():
If , then the left vertical arrow is a homotopy equivalence ([FL1], Theorem 5.8). This implies that the right vertical s-map is a homotopy equivalence. Then we have
for any . ∎
The s-map is a natural map that commutes with push-forwards, flat pull-backs, Gysin maps, intersection with a cycle and with localization sequences ([Fil], Page 4 and Proposition 1.7; [FL1], Proposition 2.3).
We let denote the (generalized) cycle maps
for any quasi-projective variety . We recall that if is a smooth projective variety then . The generalized cycle maps are compositions of s-maps ([FM2], [Fil]), i.e.
The above decomposition gives a filtration on the kernel of . The kernel of is the Griffiths group of algebraic r-cycles [Fr]. Let
Define . We know that
If we take the quotient of the above filtration by , we obtain a filtration on the Griff called the s-filtration of the Griffiths groups. This is
We let and to be the kernel and the cokernel of the maps .
The following proposition was proved in [Voin]:
Let be a smooth projective variety of dimension . Assume that the Bloch-Kato conjecture is valid for all the primes. Then:
a) Let . Then is divisible and is torsion free.
b) is uniquely divisible for and is torsion free (for any .
The following theorem is the projective bundle theorem in Lawson homology.
([FG],[Hu]) Let be a rank vector bundle over a quasi-projective variety , let be the canonical map and let be the canonical line bundle on . Let . Then
is a homotopy equivalence for any . In particular
is an isomorphism for any ,
We will refer later in this paper to the following two conjectures.
(Suslin’s conjecture) The generalized cycle map
is an isomorphism for and a monomorphism for for any smooth projective variety of dimension .
We notice that this conjecture contains a conjecture due to E. Friedlander and B. Mazur [FM2].
(Friedlander-Mazur conjecture) For any complex smooth projective variety of dimension
for any .
These highly non-trivial conjectures have been checked on certain varieties, like curves, surfaces, rationally connected threefolds and fourfolds, smooth projective toric varieties ([FHW], [Voin]).
The (singular) semi-topological K-theory of a complex projective variety was introduced in [FW]. This is defined by
where . By , we define the topological group given by the homotopy completion of the space of algebraic maps between and . The main tool for computing is a spectral sequence with term given by the Lawson homology of , when is a smooth variety.
([FHW]) For any smooth, projective complex variety and any abelian group , there is a natural map of strongly convergent spectral sequences