Algebraic cycles and Verra fourfolds

One-cycles on Gushel-Mukai fourfolds and the Beauville-Voisin filtration

We prove that the invariant locus of the involution associated to a general double EPW sextic is a constant surface and introduce a filtration on $\CH_1$ of a Gushel-Mukai fourfold. We verify the sheaf/cycle correspondence for sheaves supported on low degree rational curves, parallel to the cubic fourfolds case.

Algebraic cycles and very special cubic fourfolds

Informed by the Bloch-Beilinson conjectures, Voisin has made a conjecture about $0$-cycles on self-products of Calabi-Yau varieties. In this note, we consider variant versions of Voisin's conjecture for cubic fourfolds, and for hyperkähler varieties. We present examples for which these conjectures are verified, by considering certain very special cubic fourfolds and their Fano varieties of lines.

高维Chow循环、循环三次四维流形与拉格朗日子变种

Higher Chow cycles, cyclic cubic fourfolds and Lagrangian subvarieties

本文研究了holomorphic李代数流形上的高维Chow循环，通过构造显式不可分解的(2,1)和(4,1)循环，首次在holomorphic李代数流形上提供了此类循环的例子，并探讨了循环在拉格朗日子变种上的限制现象。

Algebraic cycles on certain hyperkaehler fourfolds with an order $3$ non-symplectic automorphism II

Let $X$ be a hyperkähler variety, and assume that $X$ admits a non-symplectic automorphism $σ$ of order $k>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<σ>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain cubic fourfolds having an order $3$ non-symplectic automorphism.

Algebraic cycles on certain hyperkaehler fourfolds with an order $3$ non-symplectic automorphism

Let $X$ be a hyperkähler variety, and assume $X$ has a non-symplectic automorphism $σ$ of order $>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<σ>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain special cubic fourfolds having an order $3$ non--symplectic automorphism.

Algebraic cycles on the Fano variety of lines of a cubic fourfold

In this text we prove that if a smooth cubic in $\PR^5$ has its Fano variety of lines birational to the Hilbert scheme of two points on a K3 surface, then there exists a smooth projective curve or a smooth projective surface embedded in the Fano variety, such that the kernel of the push-forward (at the level of zero cycles ) induced by the closed embedding is torsion.

四次三维簇的Fano簇上的常循环曲面

Constant cycle surfaces on Fano varieties of cubic fourfolds

本文研究光滑四次三维簇的线Fano簇上常循环曲面的有限性，证明了固定阶的形如F(Y)的常循环曲面至多有限个。

Intersection lattices of cubic fourfolds

We study the lattices of algebraic and transcendental cycles of cubic fourfolds.

判别式24的三次四重簇与有理性质

Cubic fourfolds of discriminant 24 and rationality

本文研究判别式24的三次四重簇，通过其扭曲曲面中Brauer类消失的条件，证明了有理性质，并构造了可数无穷多个新的有理例子。

K3 categories, one-cycles on cubic fourfolds, and the Beauville-Voisin filtration

We explore the connection between $K3$ categories and 0-cycles on holomorphic symplectic varieties. In this paper, we focus on Kuznetsov's noncommutative $K3$ category associated to a nonsingular cubic 4-fold. By introducing a filtration on the $\mathrm{CH}_1$-group of a cubic 4-fold $Y$, we conjecture a sheaf/cycle correspondence for the associated $K3$ category $\mathcal{A}_Y$. This is a noncommutative analog of O'Grady's conjecture concerning derived categories of $K3$ surfaces. We study instances of our conjecture involving rational curves in cubic 4-folds, and verify the conjecture for sh…

Fourfolds

We have found a "non-purely-constructive" method of acquiring algebraic cycles involving multiple steps. This note tries to present the main idea in the last step by concentrating on an example of 4-folds. The method demonstrates a contrast to traditional constructions.

A remark on algebraic cycles on cubic fourfolds

In this short note we try to generalize the Clemens-Griffiths criterion of non-rationality for smooth cubic threefolds to the case of smooth cubic fourfolds.

On the action of symplectic automorphisms on the $CH_0$-groups of some hyper-Kähler fourfolds

We prove that for any polarized symplectic automorphism of the Fano variety of lines of a cubic fourfold (equipped with the Plücker polarization), the induced action on the Chow group of 0-cycles is identity, as predicted by Bloch-Beilinson conjecture.

A remark on the Chow ring of some hyperkähler fourfolds

Let $X$ be a hyperkähler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology. We study this conjecture for the Fano variety of lines on a very general cubic fourfold.

The Fourier transform for certain hyperKaehler fourfolds

Using a codimension-$1$ algebraic cycle obtained from the Poincaré line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety $A$ and showed that the Fourier transform induces a decomposition of the Chow ring $CH^*(A)$. By using a codimension-$2$ algebraic cycle representing the Beauville--Bogomolov class, we give evidence for the existence of a similar decomposition for the Chow ring of hyperKähler varieties deformation equivalent to the Hilbert scheme of length-$2$ subschemes on a K3 surface. We indeed establish the existence of such a decomposition for the…

Algebraic cycles on the generic abelian fourfold with polarization of type (1,2,2,2)

In this paper we construct a non-trivial element in the higher Griff{}iths group $Griff ^{3,2}$ for the generic abelian fourfold $A^4$ with polarization of type $(1,2,2,2)$. The key idea is to use that $A^4$ can be realized as a generalized Prym variety and for this reason contains in a natural way some curves i.e. dimension 1 cycles.

Algebraic cycles on Hilbert modular fourfolds and poles of L-functions

In this paper we give some evidence for the Tate (and Hodge) conjecture(s) for a class of Hilbert modular fourfolds X, whose connected components arise as arithmetic quotients of the fourfold product of the upper half plane by congruence subgroups Γof SL(2, O_K), where O_K denotes the ring of integers of a quartic, Galois, totally real number field K. The expected relationship to the orders of poles of the associated L-functions is verified for abelian extensions of \Q. Also shown is the existence of homologically non-trivial cycles of codimension two which are not intersections of divisors.

三维循环的可表示性

Representability of codimension three cycles

本文通过表面零循环模理性等价来定义四维空间中三维循环的可表示性，探讨其核心方法与贡献。

Maximally Algebraic potentially irrational Cubic Fourfolds

A well known conjecture asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ can not be realized as the transcendental cohomology of a $K3$ surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic $2$-cycles on them, it is natural to ask for the most algebraic cubic fourfolds $X$ to which this conjecture is still applicable. In this paper, we show that for an appropriate `algebraicity index' $κ_X$, there exists a unique class of cubics maximizing $κ_X$, not having an associated $K3$ surface; namely, the cubic fourfolds…

Algebraic cycles on genus two modular fourfolds

We study universal families of stable genus two curves with level structure. Among other things, it is shown that the (1,1) part is spanned by divisor classes, and that there are no cycles of type (2,2) in the third cohomology of the first direct image. Using this, we deduce the Hodge and Tate conjectures hold for these varieties.

三次超曲面上的代数环的柱映射

Cylinder maps of algebraic cycles on cubic hypersurfaces

本文证明含一次一维环的光滑三次超曲面的直线簇与超曲面的Chow群柱映射满射，并给出三次四维簇直线簇上积分Hodge猜想的新证明及积分Tate猜想的确认。

On the Chow groups of the variety of lines of a cubic fourfold

Let $X$ be a smooth complex cubic fourfold and let $F$ be the variety of lines of $X$. The variety $F$ is known to be a smooth projective hyperkaehler fourfold, which is moreover endowed with a self rational map $ϕ: F -\rightarrow F$ first constructed by C. Voisin. Here we define a filtration of Bloch--Beilinson type on the Chow group of zero-cycles $CH_0(F)$ which canonically splits under the action of $ϕ$, thereby answering in this case a question of A. Beauville. Moreover, we show that this filtration is of motivic origin, in the sense that it arises from a Chow--Kuenneth decomposition of t…

Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties

This paper proposes a conjectural picture for the structure of the Chow ring of a (projective) hyper-Kähler variety, and the construction of a Beauville decomposition, with emphasis on the Chow group of $0$-cycles, which is endowed with a natural filtration of Brill-Noether type. Some of the conjectures are proved in the case of Hilbert schemes of K3 surfaces and Fano varieties of lines of cubic fourfolds.

Algebraic cycles on a very special EPW sextic

Motivated by the Beauville-Voisin conjecture about Chow rings of powers of $K3$ surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We prove part of this conjecture for the very special EPW sextic studied by Donten-Bury et alii. We also prove some other results concerning the Chow groups of this very special EPW sextic, and of certain related hyperkähler fourfolds.

Modular Curves, Modular Surfaces, and Modular Fourfolds

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over abelian fields is exposed. It is followed by a description of a method to construct classes in the Bloch's Chow group CH^3(X,1) by using Hecke translates of the cycles Z as above with suitable intersections in translates of modular curves. The article ends with the introduction of a modular Gersten complex for a general Shimura variety X and the corresponding g…

Projectors on the intermediate algebraic Jacobians

Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically trivial cycles coincides with the Abel-Jacobi map. Such a construction generalizes Murre's construction of the Albanese and Picard idempotents and makes it possible to give new examples of varieties admitting a self-dual Chow-Künneth decomposition satisfying the motivic Lefschetz conjecture as well as new examples of varieties having a Kimura finite dimensional Chow motive. Fo…

Varieties with very little transcendental cohomology

Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most of this paper is concerned with giving estimates on this number, along with examples where it is small. As an application, we check or recheck the Hodge conjectue in a number of examples: uniruled fourfolds, rationally connected fivefolds, fourfolds fibred by surfaces with p_g=0, Hilbert schemes of a small number points on surfaces with p_g=0, and generic hyp…

G-flux in F-theory and algebraic cycles

We construct explicit G4 fluxes in F-theory compactifications. Our method relies on identifying algebraic cycles in the Weierstrass equation of elliptic Calabi-Yau fourfolds. We show how to compute the D3-brane tadpole and the induced chirality indices directly in F-theory. Whenever a weak coupling limit is available, we compare and successfully match our findings to the corresponding results in type IIB string theory. Finally, we present some generalizations of our results which hint at a unified description of the elliptic Calabi-Yau fourfold together with the four-form flux G4 as a coherent…

广义 Kummer 簇的单值性及其中间 Jacobi 簇上的代数循环

The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians

本文计算广义 Kummer 簇由阿贝尔曲面导出等价诱导的单值子群并确认其为整个单值群，进而用于证明特定 Weil 型阿贝尔四维簇的 Hodge 猜想及 Kummer 型全纯辛流形上 Abel–Jacobi 映射的满射性。

Intersections via resolutions

We investigate the viability of defining an intersection product on algebraic cycles on a singular algebraic variety by pushing forward intersection products formed on a resolution of singularities. For varieties with resolutions having a certain structure (including all varieties over a field of characteristic zero), we obtain a stratification which reflects the geometry of the centers and the exceptional divisors. This stratification is sufficiently fine that divisors can be intersected with $r$-cycles (for $r \geq 1$), and 2-cycles can be intersected on a fourfold, provided their incidences…

双重 EPW 六次超曲面与零循环上的 Voisin 滤波

Double EPW sextics and the Voisin filtration on zero-cycles

本文研究了双重 EPW 六次超曲面的零循环 Chow 群反不变部分与 Voisin 有理轨道滤波的关系，并将其 Chow 动机与 Gushel-Mukai 四维簇联系，推广至三次超曲面的 Fano 直线簇。

Niveau and coniveau filtrations on cohomology groups and Chow groups

The Bloch-Beilinson-Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology -- i.e. the cohomology -- of $X$. In this paper, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of $X$ has an incidence on the Chow groups of $X$. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in low…

Gushel-Mukai 簇上的代数环

Algebraic cycles on Gushel-Mukai varieties

研究复 Gushel-Mukai (GM) 簇上的代数环，证明了广义 Hodge 猜想、(动机) Mumford-Tate 猜想和广义 Tate 猜想，计算了除两个无限维情形外的所有积分 Chow 群，并证明了广义伙伴或广义对偶的 GM 簇在中度有理 Chow 动机上同构。

Rationality, universal generation and the integral Hodge conjecture

We use the universal generation of algebraic cycles to relate (stable) rationality to the integral Hodge conjecture. We show that the Chow group of 1-cycles on a cubic hypersurface is universally generated by lines. Applications are mainly in cubic hypersurfaces of low dimensions. For example, we show that if a generic cubic fourfold is stably rational then the Beauville--Bogomolov form on its variety of lines, viewed as an integral Hodge class on the self product of its variety of lines, is algebraic. In dimension $3$ and $5$, we relate stable rationality with the geometry of the associated i…

Varieties that are not stably rational, zero-cycles and unramified cohomology

This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension two over a rational surface.

Chow-Kuenneth decomposition for 3- and 4-folds fibred by varieties with small Chow group of zero-cycles

Let $k$ be a field and let $Ω$ be a universal domain over $k$. Let $f:X \r S$ be a dominant morphism defined over $k$ from a smooth projective variety $X$ to a smooth projective variety $S$ of dimension $\leq 2$ such that the general fibre of $f_Ω$ has trivial Chow group of zero-cycles. For example, $X$ could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that $X$ has dimension $\leq 4$. Then we prove that $X$ has a self-dual Murre decomposition, i.e. that $X$ has a self-dual Chow--Kuenneth decomposition which satisfies Murre's…

Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$

Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, appropriately embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$ respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $CH^…

Beauville-Voisin 主题的一个变体

Variation on a theme of Beauville-Voisin

针对超凯勒簇的Chow环，考虑代数等价下代数环的变体，证明对立方四重线上的Fano线簇和双EPW六次簇，余维至少3时成立。

The Vertical, the Horizontal and the Rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications

The four-form field strength in F-theory compactifications on Calabi-Yau fourfolds takes its value in the middle cohomology group $H^4$. The middle cohomology is decomposed into a vertical, a horizontal and a remaining component, all three of which are present in general. We argue that a flux along the remaining or vertical component may break some symmetry, while a purely horizontal flux does not influence the unbroken part of the gauge group or the net chirality of charged matter fields. This makes the decomposition crucial to the counting of flux vacua in the context of F-theory GUTs. We us…

Integral Hodge conjecture for Fermat varieties

We describe an algorithm which verifies whether linear algebraic cycles of the Fermat variety generate the lattice of Hodge cycles. A computer implementation of this confirms the integral Hodge conjecture for quartic and quintic Fermat fourfolds. Our algorithm is based on computation of the list of elementary divisors of both the lattice of linear algebraic cycles, and the lattice of Hodge cycles written in terms of vanishing cycles, and observing that these two lists are the same.

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…

Beauville-Voisin-Franchetta 猜想与 LLSS 八重形

The Beauville-Voisin-Franchetta conjecture and LLSS eightfolds

本文提出融合 Beauville-Voisin 与 Franchetta 的猜想，给出其由标准猜想、Murre (D) 和 X^3 的 Franchetta 猜想推出的条件，并证明其对 LLSS 八重形的余维 2、8 闭链成立。

Kummer型辛簇上的辛对合群及其不动点轨迹

Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci

本文研究了Kummer型辛簇上由Mukai向量决定的辛对合群及其伽罗瓦作用，描述了自同构群到正交群的映射核，并分析了四维情形下不动点轨迹中K3曲面的纤维化结构。