Refined Cycle Maps

有限系数的高阶Chow群与精细的未分歧上同调

Higher Chow groups with finite coefficients and refined unramified cohomology

本文证明域上拟射影等维概形的Bloch有限系数高阶圈类映射自然嵌入涉及Schreieder精细未分歧上同调的长正合序列，证明后者满足局部化序列，并猜想其为动机上同调理论。

双双重叠分支循环的相关细化

A correlated refinement of the double double ramification cycle

本文通过根部的Weil配对，对双重重叠分支循环进行了细化，证明了细化类满足多重覆盖公式，并应用于环面曲面的对数Gromov-Witten不变量，得到了N. Takahashi猜想的变体。

Torsion algebraic cycles and complex cobordism

We show that the cycle map on a variety X, from algebraic cycles modulo algebraic equivalence to integer cohomology, lifts canonically to a topologically defined quotient of the complex cobordism ring of X. This more refined cycle map gives a topological proof that the Griffiths group is nonzero for some varieties X, without any use of Hodge theory. We also use this more refined cycle map to give examples of torsion algebraic cycles which map to 0 in Deligne cohomology but are not algebraically equivalent to 0, thus answering some questions by Colliot-Thelene and Schoen.

Virtual cycles of stable (quasi)-maps with fields

We generalize the results of Chang-Li, Kim-Oh and Chang-Li on the moduli of $p$-fields to the setting of (quasi-)maps to complete intersections in arbitrary smooth Deligne-Mumford stacks with projective coarse moduli. In particular, we show that the virtual cycle of stable (quasi-)maps to a complete intersection can be recovered by the cosection localized virtual cycle of the moduli of $p$-fields of the ambient space.

The Étale Homology and The Cycle Maps in Adic Coefficients

In this article, we define the l-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups: proper push-forward, flat pull-back, base change, cap-product, etc. In particular on singular varieties, this kind of l-adic homology behaves much better that the classical l-adic cohomology. As an application, we give an much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields of finite cohomology dimension. And we prove these cycle maps kill the algebraic equivalences and commute wit…

Stable maps of curves and algebraic equivalence of 1-cycles

We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern $A_1(X)$, the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If $K/k$ is an algebraic extension, then the kernel of $A_1(X_k)\to A_1(X_K)$ is at most ${\mathbb Z}/2{\mathbb Z}$. If $k$ is finite, then the image equals the subgroup of Galois invariant cycles. This paper replaces Sections~2--3 of 2211.15915.v.1 and Sections~2--3 of 2211.15911.v.1. The other Sections are retained in the revis…

Deligne-Beilinson cycle maps for Lichtenbaum cohomology

We define Deligne-Beilinson cycle maps for Lichtenbaum cohomology $H_L^m(X, \mathbb Z(n))$ and that with compact supports $H_{c,L}^m(X, \mathbb Z(n))$ of an arbitrary complex algebraic variety $X.$ When $(m,n)=(2,1),$ the homological part of our cycle map with compact supports gives a generalization of the Abel-Jacobi theorem and its projection to the Betti cohomology yields that of the Lefschetz theorem on $(1,1)$-cycles for arbitrary complex algebraic varieties. In general degrees $(m,n),$ we show that the Deligne-Beilinson cycle maps are always surjective on torsion and have torsion-free co…

Arithmetic Hodge structure and higher Abel-Jacobi maps

In this paper, we show some applications to algebraic cycles by using higher Abel-Jacobi maps which were defined in [the author: Motives and algebraic de Rham cohomology]. In particular, we prove that the Beilinson conjecture on algebraic cycles over number fields implies the Bloch conjecture on zero-cycles on surfaces. Moreover, we construct a zero-cycle on a product of curves whose Mumford invariant vanishes, but not higher Abel-Jacobi invariant.

Boundaries of cycle spaces and degenerating Hodge structures

We study a property of cycle spaces in connection with degenerating Hodge structures of odd-weight, and construct maps from some partial compactifications of period domains to the Satake compatifications of Siegel spaces. These maps are a generalization of the maps from the toroidal compactifications of Siegel spaces to the Satake compactifications. We also show the continuity of these maps for the case of Calabi-Yau threefolds with h^{2,1}=1.

Cycle maps on cohomology theories for dg-categories and their applications

In this article, we propose noncommutative versions of Tate conjecture and Hodge conjecture. If we consider these conjectures for a dg-category of perfect complexes over a certain schemes $X$, then they are equivalent to the classical Tate and Hodge conjectures for $X$ respectively. We also propose a strategy of how to prove these conjectures by utilizing a version of motivic Bass conjecture.

Cycle class maps for Chow groups of zero-cycles with modulus

For a quasi-projective smooth scheme X of pure dimension d over a field k and an effective Cartier divisor D on X whose support is a simple normal crossing divisor, we construct a cycle class map from the Chow group of zero-cycles with modulus to the top cohomology of the dth relative Milnor K-sheaf.

A refined Beilinson-Bloch conjecture for motives of modular forms

We propose a refined version of the Beilinson-Bloch conjecture for the motive associated with a modular form of even weight. This conjecture relates the dimension of the image of the relevant p-adic Abel-Jacobi map to certain combinations of Heegner cycles on Kuga-Sato varieties. We prove theorems in the direction of the conjecture and, in doing so, obtain higher weight analogues of results for elliptic curves due to Darmon.

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。

Tautological relations for stable maps to a target variety

We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda-Pandharipande-Pixton-Zvonkine, we construct nontrivial tautological relations parallel to Pixton's double ramification cycle relations for the moduli of curves. Examples and applications are discussed.

容许覆盖与稳定映射

Admissible covers and stable maps

本文研究目标曲线的容许覆盖与稳定映射模空间在 $\overline{M}_{g,n}$ 上产生的闭链，证明统一公式以恢复并细化 ELSV 公式及 Gromov-Witten/Hurwitz 对应，并完成低亏格计算验证。

Higher Abel-Jacobi maps for 0-cycles

Starting from the candidate Bloch-Beilinson filtration on Chow groups of 0-cycles constructed by J. Lewis, we develop and describe geometrically a series of Hodge-theoretic invariants defined on the graded pieces. Explicit formulas (in terms of currents and membrane integrals) are given for certain quotients of the invariants, with applications to 0-cycles on products of curves.

Double ramification cycles on the moduli spaces of admissible covers

We derive a formula for the virtual class of the moduli space of rubber maps to $[\mathbb{P}^1/G]$ pushed forward to the moduli space of stable maps to $BG$. As an application, we show that the Gromov-Witten theory of $[\mathbb{P}^1/G]$ relative to $0$ and $\infty$ are determined by known calculations.

Infinitesimal deformation of Deligne cycle class map

In this note, we study the infinitesimal forms of Deligne cycle class maps. As an application, we prove that the infinitesimal form of a conjecture by Beilinson is true.

Quadratic Planar Differential Systems with Algebraic Limit Cycles via Quadratic Plane Cremona Maps

In this paper we show how we can transform quadratic systems into new quadratic systems after some kind of birational transformations, the quadratic plane Cremona maps. We afterwards apply these transformations to the families of quadratic differential systems having an algebraic limit cycle. As a consequence, we provide a new family of quadratic systems having an algebraic limit cycle of degree 5. Moreover we show how the known families of quadratic differential systems having an algebraic limit cycle of degree greater than four are obtained using these transformations. We also provide the ph…

Double ramification cycles with target varieties

Let X be a nonsingular projective algebraic variety, and let S be a line bundle on X. Let A = (a_1,..., a_n) be a vector of integers. Consider a map f from a pointed curve (C,x_1,...,x_n) to X satisfying the following condition: the line bundle f*(S) has a meromorphic section with zeroes and poles exactly at the marked points x_i with orders prescribed by the integers a_i. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X. The main result of the paper is an explicit formula (in tautological classes) for the push-…

Height Pairing on Higher Cycles and Mixed Hodge Structures

For a smooth, projective complex variety, we introduce several mixed Hodge structures associated to higher algebraic cycles. Most notably, we introduce a mixed Hodge structure for a pair of higher cycles which are in the refined normalized complex and intersect properly. In a special case, this mixed Hodge structure is an oriented biextension, and its height agrees with the higher archimedean height pairing introduced in a previous paper by the first two authors. We also compute a non-trivial example of this height given by Bloch-Wigner dilogarithm function. Finally we study the variation of m…

The logarithmic gauged linear sigma model

We introduce the notion of log R-maps, and develop a proper moduli stack of stable log R-maps in the case of a hybrid gauged linear sigma model. Two virtual cycles (canonical and reduced) are constructed for these moduli stacks. The main results are two comparison theorems relating the reduced virtual cycle to the cosection localized virtual cycle, as well as the reduced virtual cycle to the canonical virtual cycle. This sets the foundation of a new technique for computing higher genus Gromov-Witten invariants of complete intersections.

Relative cycles with moduli and regulator maps

Let 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 (X, D) a cycle complex with modulus, whose 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 $X_{Zar}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1. When X is smooth over k and D is such that $D_{red}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of re…

相对完美Greenberg变换与闭链类映射

The relatively perfect Greenberg transform and cycle class maps

本文将Greenberg变换推广至非完美剩余域情形，并在半阿贝尔簇Néron模型的广义Greenberg变换上构造取值于相对完美邻近闭链函子的闭链类映射。

The incidence correspondence and its associated maps in homotopy

The incidence correspondence in the grassmannian which determines the tautological bundle defines a map between cycle spaces on grassmannians. These cycle spaces decompose canonically into a product of Eilenberg-MacLane spaces. These decompositions and the associated maps are calculated up to homotopy.

对数 Gromov-Witten 理论与双分歧循环

Logarithmic Gromov-Witten theory and double ramification cycles

本文研究环面簇相对完整环面边界的对数 Gromov-Witten 循环，利用映向对数代数环面的映射技术结合对数双分歧循环理论，证明其前推均属于曲线模空间重言环。

Cycles for rational maps with good reduction outside a prescribed set

Let $K$ be a number field and $S$ a fixed finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we study the cycles for rational maps of $\mathbb{P}_1(K)$ of degree $\geq2$ with good reduction outside $S$. We say that two ordered $n$-tuples $(P_0,P_1,...,P_{n-1})$ and $(Q_0,Q_1,...,Q_{n-1})$ of points of $\mathbb{P}_1(K)$ are equivalent if there exists an automorphism $A\in{\rm PGL}_2(R_S)$ such that $P_i=A(Q_i)$ for every index $i\in\{0,1,...,n-1\}$. We prove that if we fix two points $P_0,P_1\in\mathbb{P}_1(K)$, th…

Regulators and cycle maps in higher-dimensional differential algebraic K-theory

We develop differential algebraic K-theory of regular arithmetic schemes. Our approach is based on a new construction of a functorial, spectrum level Beilinson regulator using differential forms. We construct a cycle map which represents differential algebraic K-theory classes by geometric vector bundles. As an application we derive Lott's relation between short exact sequences of geometric bundles with a higher analytic torsion form.

论 $l$-进循环类映射的单射性与非单射性

On the injectivity and non-injectivity of the $l$-adic cycle class maps

本文借助 étale 动机上同调与驯顺连续 étale 上同调改进循环类映射，证明 Tate 与 Beilinson 猜想可保证正特征下其核为挠，并分析非单射反例。

The real cycle class map

The classical cycle class map for a smooth complex variety sends cycles in the Chow ring to cycles in the singular cohomology ring. We study two cycle class maps for smooth real varieties: the map from the I-cohomology ring to singular cohomology induced by the signature, and a new cycle class map defined on the Chow-Witt ring. For both maps, we establish basic compatibility results like compatibility with pullbacks, pushforwards and cup products. As a first application of these general results, we show that both cycle class maps are isomorphisms for cellular varieties.

亏格一中的负接触：穿孔与根叠Gromov-Witten理论的比较

Negative contacts in genus one: a comparison of punctured and root stack Gromov-Witten theories

本文证明在亏格一曲线中，两种Gromov-Witten理论（穿孔与根叠）的负接触环在推出后相等，通过普适靶比较和Crumplin主分量实现。

Cycle classes for p-adic étale Tate twists and the image of p-adic regulators

In this paper, we construct Chern class maps and cycle class maps with values in p-adic étale Tate twists [S2]. We also relate the p-adic étale Tate twists with the finite part of Bloch-Kato. As an application, we prove that the integral part of p-adic regulator maps has values in the finite part of Galois cohomology under certain assumptions.

Invitation to higher local fields, Part II, section 9: Local reciprocity cycles

This is an introduction to noncommutative local reciprocity maps for totally ramified Galois extensions with arithmetically profinite group. These maps in general are not homomorphisms but Galois cycles; a description of their image and kernel is included.

多项式映射的模空间与小周期循环处的乘子

Moduli spaces of polynomial maps and multipliers at small cycles

本文研究多项式映射模空间中由小周期循环乘子诱导的态射，证明周期1和2的乘子态射是有限且双有理等价于其像，从而加强了McMullen、Ji和Xie关于充分大周期的相关结果。

算术曲面上秩1层的F-特征环

F-characteristic cycle of a rank one sheaf on an arithmetic surface

本文通过Kato的精细Swan导子性质，证明了Hensel离散赋值域阿贝尔扩张的Galois群一次特征的特征形式的理性与整性，并基于Yatagawa在等特征情形下的特征环计算，利用特征形式定义了算术曲面上秩1层的F-特征环，证明了其与零截面的交计算了通用纤维上同调的Swan导子。