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

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... The following theorem comprise certain results of [357,358] for the case of Fano complete intersections of index 1. See, e.g., [57,315,314,357,358,359,427,507,552] and the literature therein for the cylinder mappings on cycles of other intermediate dimensions. See also [49] on the intermediate jacobians of conic bundles. ...

This is a survey on the Fano schemes of linear spaces, conics, rational curves, and curves of higher genera in smooth projective hypersurfaces, complete intersections, Fano threefolds, etc.

... Of course, this is a known fact in view of Theorem 2.3. [Le2,As] ...

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.

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 trilogarithms.

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.

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-conjecture is not true for general surfaces in P-3 of degree greater than or equal to 5, the results in this paper provide an effective bound for when this conjecture is true.

We show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdorff slit analytic space, with complex Lie group fibres, which ‘graphs’ such normal functions. For singular normal functions, an extension of the sub-object by a finite group leads to the Néron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel–Jacobi map on motivic cohomology of the singular fibre, hence via regulators on K-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the finite ‘singularity group’ in the geometric setting.

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 consider a smooth and proper map ρ: X→S of smooth quasi-projective varieties over k, and where n is the generic point of S. We anticipate that the corresponding cycle class map is surjective, and provide some evidence in support of this in the case where X = S ×X is a product and m = 1.

We collect several basic properties of algebraic cycle complexes defined by Bloch, Friedlander, Suslin and Voevodsky, like moving lemmas, localization, homotopy invariance and Mayer-Vietoris exact sequences. We also explain a generalization of the theorem of Nesterenko/Suslin/Totaro from fields to smooth, semilocal algebras of geometric type over an infinite base field. After this survey we give a new cubical proof of Bloch/Nart’s elementary vanishing theorem in codimension one. Then we show how these results give rise to a framework in which we can study the relationship between motivic cohomology (higher Chow groups) of smooth varieties and Zariski cohomology with respect to Quillen or Milnor K-sheaves. Finally we indicate how moving lemmas can be used to derive properties of algebraic cycle complexes over fields and give several examples.

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 connected to higher K-theory, and associated regulators. Finally, we specialize to the Calabi–Yau situation, and explain some recent developments in the field.

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 ideas similar to those in that construction with previous results of the second author we show that when X is a general product of three elliptic curves then CH 3ind(X, 1; ℚ) is uncountable. As a consequence, the kernel of the regulator map modulo decomposables can be very large. This result is analogous to Mumford's famous theorem on the kernel of the Albanese map on the Chow group of zero-cycles on a surface of positive genus.

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 the Banff Center. The conference also served as the kick-off activity of the CRM 1998-99 theme year on Number Theory and Arithmetic Geometry. There were 109 participants who came from 17 countries: Belgium, Canada, China, France, Germany, Greece, India, Italy, Japan, Mexico, Netherlands, - mania, Russia, Spain, Switzerland, the United Kingdom and the United States. During a period of two weeks, 41 invited lectures and 20 contributed lec tures were presented. Four lectures by invited speakers were delivered every day, followed by two sessions of contributed talks. Many informal discussions and working sessions involving small groups were organized by individual partic ipants. In addition, participants' reprints and preprints were displayed through out in a lounge next to the auditorium, which further enhanced opportunities for communication and interaction.

This chapter describes the Deligne cohomology of a complex manifold as well as Beilinson's algebraic cohomology theory of a quasi-projective complex manifold and some of its properties. In the chapter, X is a complex analytic variety. Even if X happens to be algebraic, it is considered as an analytic variety, except if the index Zar is added. The definition of the Deligne cohomology is generalized in several respects.

Algebraic cycles give rise to important invariants of algebraic varieties, and it is common to study the groups of algebraic cycles via so-called adequate equivalence relations. For example, the basic Chow groups are defined by considering cycles modulo rational equivalence. Rational, algebraic, homological and numerical equivalence have been considered since long time, and it is still a most interesting task to understand the precise relationship between them. But there are other adequate equivalence relations, like the ℓ-cubical equivalence which coincides with algebraic equivalence for ℓ = 1.

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 to give a survey of the subject on Algebraic Cycles to non-specialists, and from the point of view of a transcendental algebraic geometer.

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. 1, 181–187 (2004; Zbl 1058.14009)] it is shown that there exists a subfield K⊂ℂ with K⊃k of transcendence degree one over k, and a nontorsion class ξ∈F 2 CH 2 (S K ):=ker(CH hom 2 (S K )→Alb(S ℂ )), where Alb(S ℂ ) is the Albanese variety of S ℂ . The proof of Griffiths-Green-Paranjape (loc. cit.) relies patly on the work of T. Terasoma [Math. Z. 189, 289–296 (1985; Zbl 0579.14006)]. In this paper we take up a certain case, where we use Hodge theory instead of Terosoma [loc. cit.] to arrive at related infinite rank results.

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 terms of germs of normal functions. We show that under the assumptions (i) X/ℂ = X 0 × ℂ where X 0 is defined over ℚ̄, and (ii) the general Hodge conjecture, that F•CH r(X, m; ℚ) coincides with the aforementioned geometric filtration. More specifically, it is characterized in terms of germs of reduced arithmetic normal functions.

This chapter reviews the definition and the properties of Deligne homology. Most of the properties of Deligne homology, –such as functorialities, relative sequences, and products, are summarized by the fact that Deligne cohomology and homology form a twisted Poincare duality theorem in the sense of Bloch and Ogus (BO). The usual Poincare duality in form of a nondegenerate pairing, which does not exist for the Deligne cohomology, is replaced by a duality isomorphism. As homology is covariant for proper morphisms, this still suffices to define the Gysin morphisms needed for the operation of algebraic correspondences. Another application of homology is the easy definition of cycle classes, leading directly to the Abel-Jacobi map. The construction of Deligne homology is based on currents and C∞-chains. For smooth varieties one works with smooth compactifications and logarithmic singularities, and for arbitrary varities, one works with simplicial resolutions. For the proofs and the understanding of Deligne homology, the Hodge theory of Borel–Moore homology is needed.

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 inclusion F2CHr(X) ⊗ ℚ ⊂ CHrAJ(X) ⊗ ℚ. Roughly speaking we show that this inclusion is an equality for all X (and r) if and only if a certain variant of Beilinson-Hodge conjecture holds for K1.

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 conjecture in the case m=0. Now
consider a smooth and proper map \rho : X -> S of smooth quasi-projective
varieties over k. We formulate a version of this conjecture for the generic
fibre, expecting the corresponding cycle class map to be surjective. We provide
some evidence in support of this in the case where X is a product, the map is
the projection to one factor, and m=1.

In the work conjectures are formulated regarding the value of L-functions of motives and some computations are presented corroborating them.

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 groups, and show roughly that the level of these groups is influenced by the level of particular Hodge structures.

We provide a patch to complete the proof of the Voevodsky-Rost Theorem, that the norm residue map is an isomorphism. (This settles the motivic Bloch-Kato conjecture).