Lectures on Algebraic Cycles and Chow Groups

2017 ◽  
Author(s):  
Jacob Murre
Keyword(s):  
Algebraic Cycles ◽  
Chow Groups ◽  
Equivalence Relations ◽  
Complex Numbers ◽  
Arbitrary Field ◽  
Deligne Cohomology ◽  
Short Survey ◽  
Intermediate Jacobian ◽  
Cycle Map

This chapter showcases five lectures on algebraic cycles and Chow groups. The first two lectures are over an arbitrary field, where they examine algebraic cycles, Chow groups, and equivalence relations. The second lecture also offers a short survey on the results for divisors. The next two lectures are over the complex numbers. The first of these features discussions on the cycle map, the intermediate Jacobian, Abel–Jacobi map, and the Deligne cohomology. The following lecture focuses on algebraic versus homological equivalence, as well as the Griffiths group. The final lecture, which returns to the arbitrary field, discusses the Albanese kernel and provides the results of Mumford, Bloch, and Bloch–Srinivas.

scholarly journals The realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology

2017 ◽  
Vol 13 (09) ◽  
pp. 2471-2485 ◽  
Author(s):  
Danny Scarponi
Keyword(s):  
Chow Groups ◽  
Degree Zero ◽  
Deligne Cohomology ◽  
Abelian Scheme ◽  
Logarithmic Singularities ◽  
Abelian Schemes

In 2014, Kings and Rössler showed that the realization of the degree zero part of the abelian polylogarithm in analytic Deligne cohomology can be described in terms of a class of currents which was previously defined by Maillot and Rössler and strongly related to the Bismut–Köhler higher torsion form of the Poincaré bundle. In this paper we show that, if the base of the abelian scheme is proper, Kings and Rössler’s result can be refined to hold already in Deligne–Beilinson cohomology. More precisely, by means of Burgos’ theory of arithmetic Chow groups, we prove that the class of currents defined by Maillot and Rössler has a representative with logarithmic singularities at the boundary and therefore defines an element in Deligne–Beilinson cohomology. This element coincides with the realization of the degree zero part of the motivic polylogarithm on abelian schemes in Deligne–Beilinson cohomology.

scholarly journals PRINCIPAL RADICAL SYSTEMS, LEFSCHETZ PROPERTIES, AND PERFECTION OF SPECHT IDEALS OF TWO-ROWED PARTITIONS

2021 ◽  
pp. 1-41
Author(s):  
CHRIS MCDANIEL ◽  
JUNZO WATANABE
Keyword(s):  
Lower Bound ◽  
Complex Numbers ◽  
Arbitrary Field ◽  
Cherednik Algebras ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Lefschetz Properties ◽  
Bounded Below ◽  
Rational Cherednik Algebras

Abstract We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

scholarly journals RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

2017 ◽  
Vol 18 (06) ◽  
pp. 1233-1293 ◽  
Author(s):  
Federico Binda ◽  
Shuji Saito
Keyword(s):  
Cartier Divisor ◽  
Explicit Construction ◽  
Chow Groups ◽  
Chow Group ◽  
Deligne Cohomology ◽  
Fundamental Class ◽  
Normal Crossing ◽  
Rham Complex ◽  
Relative Version ◽  
Higher Chow Groups

Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on $\overline{X}_{\text{Zar}}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight $1$ . When $\overline{X}$ is smooth over $k$ and $D$ is such that $D_{\text{red}}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of $(\overline{X},D)$ to the relative de Rham complex. When $\overline{X}$ is defined over $\mathbb{C}$ , the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when $\overline{X}$ is moreover connected and proper over $\mathbb{C}$ , we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J_{\overline{X}|D}^{r}$ of the pair $(\overline{X},D)$ . For $r=\dim \overline{X}$ , we show that $J_{\overline{X}|D}^{r}$ is the universal regular quotient of the Chow group of $0$ -cycles with modulus.

scholarly journals BLOCH-TYPE CONJECTURES AND AN EXAMPLE A THREE-FOLD OF GENERAL TYPE

2010 ◽  
Vol 12 (04) ◽  
pp. 587-605 ◽  
Author(s):  
CHRIS PETERS
Keyword(s):  
General Type ◽  
Algebraic Cycles ◽  
Chow Groups ◽  
Good Theory ◽  
Mixed Motives ◽  
Bloch’S Conjecture

The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that "small Hodge diamonds" go with "small Chow groups". Voisin's method [19] (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A three-fold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture.

Definable equivalence relations on algebraically closed fields

1989 ◽  
Vol 54 (3) ◽  
pp. 928-935 ◽  
Author(s):  
Lou van den Dries ◽  
David Marker ◽  
Gary Martin
Keyword(s):  
Equivalence Relation ◽  
Analogous Result ◽  
Equivalence Relations ◽  
Complex Numbers ◽  
Closed Field ◽  
Closed Fields ◽  
Finite Set ◽  
The Right ◽  
Algebraically Closed Fields ◽  
Special Case

This article was inspired by the question: is there a definable equivalence relation on the field of complex numbers, each of whose equivalence classes has exactly two elements? The answer turned out to be no, as we now explain in greater detail.Let Κ be an algebraically closed field and let E be a definable equivalence relation on Κ. [Note: By “definable” we will always mean “definable with parameters”.] Either E has one cofinite class, or all classes are finite and there is a number d such that all but a finite set of classes have cardinality d. In the latter case let B be the finite set of elements of Κ which are not in a class of size d. We prove the following result.Theorem 1. a) If char(Κ) = 0 or char(Κ) = p > d, then ∣B∣ ≡ 1 (mod d).b) If char(Κ) = 2 and d = 2, then ∣B∣ ≡ 0 (mod 2).c) If char(Κ) = p > 2 and d = p + s, where 1 ≤ s ≤ p/2, then ∣B∣ ≡ p + 1 (mod d).Furthermore, a)−c) are the only restrictions on ≡B≡.If one is in the right mood, one can view this theorem as saying that the “algebraic cardinality” of the complex numbers is congruent to 1 (mod n) for every n.§1 contains a reduction of the problem to the special case where E is induced by a rational function in one variable. §2 contains the main calculations and the proofs of a)−c). §3 contains eight families of examples showing that all else is possible. In §4 we prove an analogous result for real closed fields.

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