Algebraic cycles and the classical groups II: Quaternionic cycles

Notes on Algebraic Cycles and Homotopy theory

We show that a conjecture by Lawson holds, that is, the inclusion from the Chow variety $C_{p,d}(P^n)$ of all effective algebraic p-cycles of degree d in n-dimensional projective space to the space of effective algebraic p-cycles is 2d-connected. As a result, the homotopy and homology groups of $C_{p,d}(P^n)$ are calculated up to 2d. We also show an analogous statement for Chow variety $C_{p,d}(¶^n)$ over algebraically closed fields of arbitrary characteristic and compute their etale homotopy groups up to 2d.

Algebraic Cycles of a Fixed Degree

In this paper, the homotopy groups of Chow variety $C_{p,d}(P^n)$ of effective $p$-cycles of degree $d$ is proved to be stable in the sense that $p$ or $n$ increases. We also obtain a negative answer to a question by Lawson and Michelsohn on homotopy groups for the space of degree two cycles.

Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cycles

We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic chain meeting the "faces" properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via delta-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.

Why should one compute periods of algebraic cycles?

In this article we show how the data of integrals of algebraic differential forms over algebraic cycles can be used in order to prove that algebraic and Hodge cycle deformations of a given algebraic cycle are equivalent. As an example, we prove that most of the Hodge and algebraic cycles of the Fermat sextic fourfold cannot be deformed in the underlying parameter space. We then take a difference of two linear cycles inside the Fermat variety, and gather evidences that the Hodge locus corresponding to this is smooth and reduced. This implies the existence of new algebraic cycles in the Fermat v…

Periods of linear algebraic cycles

In this article we use a theorem of Carlson and Griffiths and compute periods of linear algebraic cycles inside the Fermat variety of even dimension $n$ and degree $d$. As an application, for examples of $n$ and $d$ we prove that the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space. We also check the same statement for hypersurfaces containing a complete intersection algebraic cycle. Our result confirms the Hodge conjecture for Hodge cycles obtained by the monodromy of the hom…

连接代数周期的周期

Periods of join algebraic cycles

本文证明在特定维数和次数下，光滑超曲面模空间包含无穷多个与线性周期同余维数的 Hodge 轨迹，并构造了由 $0$ 维周期连接而成的 Hodge 周期，建立了连接周期类公式及 Artin Gorenstein 代数张量积刻画。

Algebraic structure of the space of homotopy classes of cycles and singular homology

Described the algebraic structure on the space of homotopy classes of cycles with marked topological flags of disks. This space is a non-commutative monoid, with an Abelian quotient corresponding to the group of singular homologies $H_k(M)$. For the marked flag contracted to a point the multiplication becomes commutative and the subgroup of spherical cycles corresponds to the usual homotopy group $π_k(M)$.

Moduli spaces and algebraic cycles in real algebraic geometry

This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle class map between the Chow ring and the equivariant cohomology ring plays an important role. The image of the cycle class map remains difficult to describe in general; we study this group in detail in the case of real abelian varieties. To do so, we construct integral Fourier transforms on Chow rings of abelian varieties over any field. They allow us to prove t…