James D. Lewis

University of Alberta | UAlberta · Department of Mathematical and Statistical Sciences

Let $X/C$ be a general product of elliptic curves. Our goal is to establish the Hodge-D-conjecture for $X$. We accomplish this when $\dim X \leq 5$. For $\dim X \geq 6$, we reduce the conjecture to a matrix rank condition that is amenable to computer calculation.

We prove a conjecture of Voisin that any two distinct points on a very general hypersurface of degree 2n+2 in Pn+1 are rationally inequivalent.

We introduce a weak Lefschetz-type result on Chow groups of complete intersections. As an application, we can reproduce some of the results in [P]. The purpose of this paper is not to reproduce all of [P] but rather illustrate why the aforementioned weak Lefschetz result is an interesting idea worth exploiting in itself. We hope the reader agrees.

With a homological Lefschetz conjecture in mind, we prove the injectivity of the push-forward morphism on rational Chow groups, induced by the closed embedding of an ample divisor linearly equivalent to a higher multiple of the Theta divisor inside the Jacobian variety J(C), where C is a smooth irreducible complex projective curve.

We prove a conjecture of Voisin that no two distinct points on a very general hypersurface of degree $2n$ in ${\mathbb P}^n$ are rationally equivalent.

A general specialization map is constructed for higher Chow groups and used to prove a "going-up" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross and Schoen, and of the coordinate symbol on a genus-2 curve.

In algebraic geometry there is the notion of a height pairing of algebraic cycles, which lies at the confluence of arithmetic, Hodge theory and topology. After explaining a motivating example situation, we introduce new directions in this subject.

Notes for a mini course at the University of Science and Technology of China in Hefei, China, June 23–July 12, 2014.

We present a study of certain singular one-parameter subfamilies of Calabi-Yau threefolds realized as anticanonical hypersurfaces or complete intersections in toric varieties. Our attention to these families is motivated by the Doran-Morgan classification of variations of Hodge structure which can underlie families of Calabi-Yau threefolds with h2,...

Let k ⊆ C be an algebraically closed subfield, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map clr,m: H2r-mM (k(X),Q(r)) →homMHS Q(0),H2r-m(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge conjecture in the case m = 0. Now co...

We describe an explicit morphism of complexes that induces the cycle-class maps from (simplicially described) higher Chow groups to rational Deligne cohomology. The reciprocity laws satisfied by the currents we introduce for this purpose are shown to provide a clarifying perspective on functional equations satisfied by complex-valued di- and trilog...

In this paper, we consider the moduli space 𝒮𝒰C(r,𝒪C) of rank r semistable vector bundles with trivial determinant on a smooth projective curve C of genus g. For r = 2, F. Kirwan constructed a smooth log resolution X¯→𝒮𝒰C(2,𝒪C). Based on earlier work of M. Kerr and J. Lewis, Lewis explains in the Appendix the notion of a relative Chow group (w.r.t....

Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} : H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge co...

We review the transcendental aspects of algebraic cycles, and explain how this relates to Calabi–Yau varieties. More precisely, after presenting a general overview, we begin with some rudimentary aspects of Hodge theory and algebraic cycles. We then introduce Deligne cohomology, as well as the generalized higher cycles due to Bloch that are connect...

We prove some general density statements about the subgroup of invertible points on intermediate jacobians; namely those points in the Abel-Jacobi image of nullhomologous algebraic cycles on projective algebraic manifolds.

Let X/ℂ be a smooth projective variety, and let CH r(X, m) be the higher Chow group defined by Bloch. Saito and Asakura defined a descending candidate Bloch-Beilinson filtration CH r (X, m; ℚ) = F 0 ⊃ ⊃ F r ⊃ F r+1 = F r+2 =, using the language of mixed Hodge modules. Another more geometrically defined filtration is constructed by Kerr and Lewis in...

In [C-L3] it is shown that the real regulator for a general self-product of a K3 surface is nontrivial. In this note, we prove a theorem which says that the real regulator for a general self-product of a surface of higher order (in a suitable sense), is essentially trivial. 1. Statement of the theorem Let Γ be a smooth projective curve, {Zt}t∈Γ a f...

Using Gauss-Manin derivatives of normal functions, we arrive at some remarkable results on the non-triviality of the transcendental regulator for $K_m$ of a very general projective algebraic manifold. Our strongest results are for the transcendental regulator for $K_1$ of a very general $K3$ surface. We also construct an explicit family of $K_1$ cy...

Let U /ℂ be a smooth quasi-projective variety of dimension d , CH r ( U,m ) Bloch's higher Chow group, and cl r,m : CH r ( U,m ) ⊗ ℚ → hom MHS (ℚ(0), H 2 r−m ( U , ℚ( r ))) the cycle class map. Beilinson once conjectured cl r,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we s...

Let X/ℂ be a smooth projective variety and CHr(X) the Chow group of codimension r algebraic cycles modulo rational equivalence. Let us assume the (conjectured) existence of the Bloch-Beilinson filtration {F vCHr(X) ⊗ℚ}rv=0 for all such X (and r). If CHrAJ(X) ⊂ CHr(X) is the subgroup of cycles Abel-Jacobi equivalent to zero, then there is an inclusi...

We proved that the union of rational curves is dense on a very general K3 surface and the union of elliptic curves is dense in the 1st jet space of a very general K3 surface, both in the strong topology.

Let X/C be a projective algebraic manifold, and MX∗ be the sheaf of nonvanishing meromorphic functions on X in the analytic topology. We prove a number of nonvanishing results for H•(X,MX∗). In particular, MX∗ is acyclic iff dimX=1.

Let S be a smooth projective surface defined over a number field k, with positive (geometric) genus. Generalizing the work of C. Schoen [in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 2, Proc. Symp. Pure Math. 46, 463–473 (1987; Zbl 0647.14002)], in [M. Green, P. A. Griffiths and K. H. Paranjape, Mich. Math. J. 52, No....

We study the Fano varieties of projective k-planes lying in hypersurfaces and investigate the associated motives.

Let X be a projective algebraic manifold of dimension n (over C), CH1(X) the Chow group of algebraic cycles of codimension l on X, modulo rational equivalence, and A1(X) ⊂ CH1(X) the subgroup of cycles algebraically equivalent to zero. We say that A1(X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH1(Γ...

Based on a novel application of an archimedean type pairing to the geometry and deformation theory of $K3$ surfaces, we construct a regulator indecomposable $K_1$-class on a self-product of a $K3$ surface. In the Appendix, we explain how this pairing is a special instance of a general pairing on precycles in the equivalence relation defining Bloch'...

Let U/C be a smooth quasiprojective variety and CHr (U, 1) a special instance of Bloch’s higher Chow groups ([Blo]). Jannsen was the first to show that the cycle class map clr,1: CHr ` ´ (U, 1) ⊗ Q → homMHS Q(0), H2r−1 (U, Q(r)) is not in general surjective, contradicting an earlier conjecture of Beilinson. In this paper we give a refinement of Jan...

The geometric objects of study in this paper are K3 surfaces which admit a polarization by the unique even unimodular lattice of signature (1,17). A standard Hodge-theoretic observation about this special class of K3 surfaces is that their polarized Hodge structures are identical with the polarized Hodge structures of abelian surfaces that are cart...

We explicitly describe cycle-class maps c_H from motivic cohomology to absolute Hodge cohomology, for smooth quasi-projective and (some) proper singular varieties, and compute special cases of the latter. For smooth projective varieties, we also study Hodge-theoretically defined "higher Abel-Jacobi maps" on the kernel of c_H; this leads to new resu...

Let $X$ be a projective algebraic manifold and let $CH^r(X)$ be the Chow group of algebraic cycles of codimension $r$ on $X$, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration $\{F^{\nu}\}_{\nu\geq 0}$ on $CH^r(X)\otimes {\Bbb Q}$ due to the second author, we construct a space of arithmetic Hodge theoretic invariants...

Let X be a projective algebraic manifold, and CHk (X, 1) the higher Chow group, with corresponding real regulator r(k,1) circle times R : CHk (X, 1) circle times R --> H-D(2k-1)(X, R(k)). If X is a general K3 surface or Abelian surface, and k = 2, we prove the Hodge-D-conjecture, i.e. the surjectivity of r(2,1) circle times R. Since the Hodge-D-con...

We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel-Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.

We first give an elementary new proof of the vanishing of the regulator on K-1(Z) where Z subset of P-3 be a general surface of degree d greater than or equal to 5, using a Lefschetz pencil argument. By a similar argument we then show the triviality of the regulator for K-1 of a general product of two curves.

We prove the Hodge-D-conjecture for general K3 and Abelian surfaces. Some consequences of this result, e.g., on the levels of higher Chow groups of products of elliptic curves, are discussed.

Let Z be a general surface in P^3 of degree at least 5. Using a Lefschetz pencil argument, we give an elementary new proof of the vanishing of a regulator on K_1(Z).

Let X/ C be a projective algebraic manifold, and further let CH k (X) Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {F l}l 0 on CH k (X) Q of Bloch-Beilinson type. In the even...

Let X/(C) be a projective algebraic manifold. In the first part of this paper, we construct a natural real regulator on the cohomology of a Gersten-Milnor complex, and where appropriate, compare it to the Beilinson regulator into real Deligne cohomology. The second part is devoted to a calculation on curves. In particular, we arrive at an elementar...

These lecture notes form an expanded version of a series of lectures delivered by the author during 13-19 August 2000 at the Instituto de Matemáticas at UNAM in Cuernavaca, and 20-24 August 2000 at the Instituto de Matemáticas at UNAM in Mexico City, as part of the conference activity on Geometría Algebraica y Algebra Conmutativa. They are intended...

The NATO Advanced Study Institute on "The Arithmetic and Geometry of Algebraic Cycles" was held at the Banff Centre for Conferences in Banff (Al­ berta, Canada) from June 7 until June 19, 1998. This meeting was organized jointly with Centre de Recherches Mathematiques (CRM), Montreal, as one of the CRM Summer schools which take place annually at th...

Let X be a projective algebraic manifold, and CH k(X, 1) the higher Chow group. We introduce the subgroup of decomposable cycles and the quotient space of indecomposable cycles CH kind(X, 1; ℚ). Firstly, for X a general product of two elliptic curves we construct a nonzero indecomposable higher Chow cycle in CH 2ind(X, 1; ℚ). Then by combining idea...

A pairing is constructed between cohomology and higher Chow groups. We prove a number of properties about this pairing in terms of the Chow groups and Hodge theory, and how it relates to a candidate Bloch-Beilinson filtration. We also prove a generalized Noether-Lefschetz result for general complete intersections of sufficiently high multidegree.

Let X be a projective algebraic manifold, and CHk(X, m) Bloch's higher Chow group. We introduce a subgroup of decomposables, and corresponding quotient group of indecomposables, and study the influence of Hodge theory on each of these groups. We introduce an integral invariant, called Level, which is based on a coniveau type construction on Chow gr...

Let Y be a smooth projective algebraic surface over ℂ, and T(Y) the kernel of the Albanese map CH0(Y)deg0 → Alb(Y). It was first proven by D. Mumford that if the genus Pg(Y) > 0, then T(Y) is 'infinite dimensional'. One would like to have a better idea about the structure of T(Y). For example, if Y is dominated by a product of curves E1 × E2, such...

These notes form an expanded version of some introductory lec-tures to be delivered at the Workshop on Arithmetic and Geometry of K3 surfaces and Calabi-Yau Threefolds, August 16-25, 2011, at the Fields Insti-tute. After presenting a general overview, we begin with some rudimentary aspects of Hodge theory and algebraic cycles. We then introduce Del...