Factorization of the Abel-Jacobi maps
Fumiaki Suzuki
本文针对 Walker 借助 Lawson 同调与态射上同调所得的 Abel-Jacobi 映射分解定理，给出一种不依赖原理论应用的直接证明。
作为 Lawson 同调与态射上同调理论的一项应用，Walker 证明了 Abel-Jacobi 映射可经由另一个正则同态分解。在本短文中，我们给出该定理的一个直接证明。
As an application of the theory of Lawson homology and morphic cohomology, Walker proved that the Abel-Jacobi map factors through another regular homomorphism. In this note, we give a direct proof of the theorem.
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We construct a resolution of the degree-2 Abel-Jacobi map for a regular smoothing of a nodal curve.
We study the higher Abel-Jacobi invariant defined recently by M. Green. We first construct a counterexample to the injectivity of Green's higher Abel-Jacobi map. On the other hand, we prove that the higher Abel-Jacobi map governs Mumford's pull-back of holomorphic forms. We deduce from this that if a surface has holomorphic 2-forms, the image of the higher Abel-Jacobi map, defined on its group of zero-cycles Albanese equivalent to 0, has infinite dimensional image.
We construct an Abel-Jacobi type map on the homologically trivial part of Lawson homology groups. It generalizes the Abel-Jacobi map constructed by Griffiths. By using a result of H. Clemens, we give some examples of smooth projective manifolds with infinite generated Lawson homology groups L_pH_k(X,Q) when k>2p.
Let $f\col\C\ra B$ be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Abel--Néron map having values in Esteves's fine compactified Jacobian and extending the degree-2 Abel--Jacobi map of the generic fiber of $f$
This is an expository paper about the topics listed in the title.
We investigate questions of an arithmetic nature related to the Abel-Jacobi map. We give a criterion for the zero locus of a normal function to be defined over a number field, and we give some comparison theorems with the Abel-Jacobi map coming from continuous étale cohomology.
We prove and generalize an observation of Green and Griffiths on the infinitesimal form of the Abel-Jacobi map. As an application, we prove that the infinitesimal form of a conjecture by Griffiths and Harris is true.
We study in this article the cohomological properties of Lagrangian families on projective hyper-Kähler manifolds. First, we give a criterion for the vanishing of Abel-Jacobi maps of Lagrangian families. Using this criterion, we show that under a natural condition, if the variation of Hodge structures on the degree $1$ cohomomology of the fibers of the Lagrangian family is maximal, its Abel-Jacobi map is trivial. We also construct Lagrangian families on generalized Kummer varieties whose Abel-Jacobi map is not trivial, showing that our criterion is optimal.
For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths' Abel-Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel-Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel-Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we…
Fourier transforms and Abel-Jacobi theory
本文将稳定曲线模空间上的傅里叶变换与对数阿贝尔-雅可比理论联系，并应用扭曲双覆盖循环公式计算紧化雅可比上除数幂的推前。
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.
An Abel-Jacobi theorem for metrized complexes of Riemann surfaces
本文针对黎曼曲面度规复形建立 Abel–Jacobi 定理，统一推广了经典代数几何与热带几何中的相应结果。
Cycle classes on abelian varieties and the geometry of the Abel-Jacobi map
本文研究阿贝尔簇作为雅可比簇乘积直和项及分裂性质，并将其与积分Hodge猜想及Brauer-Severi簇上万有零闭链和三次三重流形上万有余维数2闭链的存在性问题联系起来。
The purpose of this paper is to construct non-trivial elements in the Abel-Jacobi kernels in any codimension by specializing correspondences with non-trivial Hodge-theoretical invariants at points with different transcendence degrees over a subfield in $\CC$.
For an algebraic (n-1)-cycle Z on a complex projective (2n-1)-manifold X, P. Griffiths conjectured that, if Z is algebraically equivalent to zero and if the incidence divisor of Z on every family of (n-1)-cycles is principal, then the Abel-Jacobi image of Z in the intermediate Jacobian J(X) of X is a point of finite order. Using a recent generalization of the classical height pairing, we give a proof of a stronger statement, namely that the Abel-Jacobi image of Z is zero.
The first purpose of this note is to give a partial exposition of the $l$-adic realization of Nori's category of motives. The second one is to give a simple description of the second $l$-adic Abel-Jacobi map.
We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel-Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.
Let $X$ be a smooth cubic threefold and $J(X)$ be its intermediate Jacobian. We show that there exists a codimension 2 cycle $Z$ on $J(X)\times X$ with $Z_{t}$ homologically trivial for each $t\in J(X)$, such that the morphism $ϕ_{Z}: J(X)\rightarrow J(X)$ induced by the Abel-Jacobi map is the identity. This answers positively a question of Voisin in the case of the cubic threefold.
We prove that the pullbacks of the virtual fundamental classes of the Brill-Noether loci under any Abel-Jacobi section lie in the tautological ring of the moduli space of stable curves. This resolves a conjecture of Pagani, Ricolfi and van Zelm, and is part of a broader program to understand the logarithmic intersection theory of $\Mgn$.
In this article, we study the infinitemisal invariant of the relative higher Abel Jacobi map of a smooth open morphism. We give a generalization of a theorem of Voisin to open varieties and higher Chow groups and as a corollary a non vanishing criterion for the higher Abel Jacobi map of a general open smooth hypersurface section of high degree of a smooth projective variety Y. On the other side, using Nori connectness theorem, the image of the primitive part of the higher Abel Jacobi map of a general open smooth hypersurface section of high degree of a smooth projective variety Y is generated …
We give a geometric Langlands type proof of the geometrization conjecture of the local Langlands correspondence introduced by the author for GL_1. For this we study an Abel-Jacobi morphism. We prove that this morphism is a pro-étale locally trivial fibration in simply connected diamonds in high degree. Those diamonds are absolute punctured Banach-Colmez spaces that we study in details.
We explicitly describe cycle-class maps c_H from motivic cohomology to absolute Hodge cohomology, for smooth quasi-projective and (some) proper singular varieties, and compute special cases of the latter. For smooth projective varieties, we also study Hodge-theoretically defined "higher Abel-Jacobi maps" on the kernel of c_H; this leads to new results on nontrivial indecomposable higher Chow cycles in the regulator kernel.
We extend to the topological setting the classical constructions of the Abel-Jacobi mapping on homologically trivial algebraic cycles and the height pairing between two such cycles. We further interpret the height pairing between homologically trivial topological cycles (with disjoint support) as giving a lifting of their Abel-Jacobi images to the fiber of the Poincaré bundle, extending work of R. Hain in the algebraic setting. Part II of the current revision further explores the relationship between the topological height pairing and the classical height pairing in the case of algebraic cycle…
Let $C$ be a nonsingular complex projective curve, and $\mathcal{L}$ e a line bundle of degree 1 on $C$. Let $\mathcal{M}_α := \mathcal{M}(r,\mathcal{L},α)$ denote the moduli space of $S$-equivalence classes of Parabolic stable bundles of fixed rank $r$, determinant $\mathcal{L}$, full flags and generic weight $α$. Let $n=$ dim$\mathcal{M}_α$. We aim to study the Abel-Jacobi maps for $\mathcal{M}_α$ in the cases $k=2,n-1$. When $k=n-1$, we prove that the Abel-Jacobi map is a split surjection. When $k=2$ and $r=2$, we show that the Abel-Jacobi map is an isomorphism.
(Non-)Extendability of Abel-Jacobi Maps
研究阿贝尔 - 雅可比映射的“自然”定义轨迹，表明对于恰当几何约化曲线\(C\)，其从光滑轨迹\(C^{sm}\)到雅可比簇的映射，除特定情况外不扩展到更大曲线，由此得出该映射在特定情况外有闭像，对非\(\mathbb{P}^1\)不可约曲线是闭嵌入。
In this paper, which is a sequel of [BKLV], we study the convex-geometric properties of the cone of pseudoeffective $n$-cycles in the symmetric product $C_d$ of a smooth curve $C$. We introduce and study the Abel-Jacobi faces, related to the contractibility properties of the Abel-Jacobi morphism and to classical Brill-Noether varieties. We investigate when Abel-Jacobi faces are non-trivial, and we prove that for $d$ sufficiently large (with respect to the genus of $C$) they form a maximal chain of perfect faces of the tautological pseudoeffective cone (which coincides with the pseudoeffective …
We pose some questions about spaces parametrizing rational curves on rationally connected varieties. We give a partial answer for cubic threefolds. Many of our results were previously proved by Iliev, Markushevich and Tikhimirov by analyzing vector bundles on cubic threefolds. Our proofs use residuation rather than vector bundles.
Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel-Jacobi map. This breaks down over the boundary since the Abel-Jacobi map fails to extend. We construct a `universal' resolution of the Abel-Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.
Let $X$ be a double cover of $\mathbb P^3$ branched along a sextic surface $Y$. In this paper, we show that, for general $X$, the Abel-Jacobi map associated to the normalization $\tilde F(X)$ of the surface $F(X)$ of curves contained in $X$ which are preimages of lines bitangent to $Y$, gives an isogeny between the Albanese variety of $\tilde F(X)$ and the intermediate Jacobian of $X$.
This paper forms the major portion of a talk given at the International Colloquium on Arithmetic, Algebra and Geometry at TIFR, Mumbai in Jan 2000. We look at the problem of detecting cycles with trivial Abel-Jacobi invariant. M. Green proposed a Hodge-theoretic method to which C. Voisin found a counter-example. We present an easier example. We also propose another possible invariant to detect these classes using Hodge Theory. Similar methods have been proposed earlier by M. Asakura and M. Saito.
The Abel-Jacobi maps of the families of elliptic quintics and rational quartics lying on a smooth cubic threefold are studied. It is proved that their generic fiber is the 5-dimensional projective space for quintics, and a smooth 3-dimensional variety birational to the cubic itself for quartics. The paper is a continuation of the recent work of Markushevich-Tikhomirov, who showed that the first Abel-Jacobi map factors through the moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers $c_1=0, c_2=2$ obtained by Serre's construction from elliptic quintics, and…
Let $C$ be a smooth non rational projective curve over the complex field $\mathbb{C}$. If $A$ is an abelian subvariety of the Jacobian $J(C)$, we consider the Abel-Prym map $φ_A : C \rightarrow A$ defined as the composition of the Abel map of $C$ with the norm map of $A$. The goal of this work is to investigate the degree of the map $φ_A$ in the case where $A$ is one of the components of an isotypical decomposition of $J(C)$. In this case we obtained a lower bound for $\mathrm{deg}(φ_A)$ and, under some hypotheses, also an upper bound. We then apply the results obtained to compute degrees of A…
Let $X$ be a compact connected Riemann surface of genus at least two. The Abel-Jacobi map $φ: {\rm Sym}^d(X) \rightarrow {\rm Pic}^d(X)$ is an embedding if $d$ is less than the gonality of $X$. We investigate the curvature of the pull-back, by $φ$, of the flat metric on ${\rm Pic}^d(X)$. In particular, we show that when $d=1$, the curvature is strictly negative everywhere if $X$ is not hyperelliptic, and when $X$ is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of $X$ fixed by the hyperelliptic involution.
Lagrangian interpretation of Abel-Jacobi mappings associated to Fano threefolds
本文利用Donagi-Markman和Markushevich框架，推导Fano三维簇上Abel-Jacobi映射的微分表达式，并用于给出经典结果的新证明。
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.
Following Mumford and Chiodo, we compute the Chern character of the derived pushforward $\textrm{ch} (R^\bulletπ_\ast\mathscr{O}(\mathsf{D}))$, for $\mathsf D$ an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.
For $θ$ a small generic universal stability condition of degree $0$ and $A$ a vector of integers adding up to $k(2g-2)$, the spaces $\overline{M}_{g,n}^θ$ resolving the Abel-Jacobi section to the compactified Jacobian Pic^θconstructed in the work of Abreu-Pacini and Holmes-Molcho-Pandharipande-Pixton-Schmitt are observed to lie inside the space $\textbf{Div}$ of Marcus and Wise, and their pullback to the rubber space of loc. cit to be smooth. This provides smooth and modular blowups $\widetilde{M}_{g,n}^θ$ of the moduli space of stable curves on which the logarithmic double ramification cycle …
The Abel-Jacobi map of the family of elliptic quintics lying on a general cubic threefold is studied. It is proved that it factors through a moduli component of stable rank 2 vector bundles on the cubic threefold with Chern numbers c_1=0, c_2=2, whose general point represents a vector bundle obtained by Serre's construction from an elliptic quintic. The elliptic quintics mapped to a point of the moduli space vary in a 5-dimensional projective space inside the Hilbert scheme of curves, and the map from the moduli space to the intermediate Jacobian is étale. As auxiliary results, the irreducibil…
Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry, we describe a modular modification of the moduli space of curves over which the Abel-Jacobi map extends. We also describe the attendant deformation theory and virtual fundamental class of this moduli space. This recovers the double ramification cycle, as well as variants associated to differentials.
Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Ma…
Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Ma…
We realize the infinitesimal Abel-Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of which the classical Abel-Jacobi map is a special example.
We study the Abel-Jacobi image of the Ceresa cycle W_k-W_k^-, where W_k is the image of the k-th symmetric product of a curve X on its Jacobian variety. For the Fermat curve of degree N, we express it in terms of special values of generalized hypergeometric functions and give a criterion for the non-vanishing of W_k-W_k^- modulo algebraic equivalence, which is verified numerically for some N and k.
For the evaluation and inversion of abelian integrals we show that the image of the Abel-Jacobi map of genus less than 5 hyperelliptic curve in its Jacobian is the intersection of shifted theta divisors with specified shifts. Therefore the image is a solution of a (slightly overdetermined) set of equations in the Jacobian.
This replacement corrects statement and proof of the main result. Also, a section on the universal Abel-Jacobi map has been added.
We construct the Abel-Jacobi map for Mumford curves over any complete non-archimedean field, using multiplicative integrals and in the setting of Berkovich analytic geometry. Along the way, we proof some results concerning graphs and mesures related to tropical geometry.
We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.
Extension of the Topological Abel-Jacobi Map for Cubic Threefolds
本文研究三次超曲面上偏斜线对定义的消没闭链，通过 Stein 引理构造了 72 重覆盖空间的紧致化，将拓扑 Abel-Jacobi 映射延拓至该紧致化，并证明其在基本群上的诱导映射为满射。
We construct a general semiregularity map for cycles on a complex analytic or algebraic manifold and show that such semiregularity map can be obtained from the classical tool of the Atiyah-Chern character. The first part of the paper is fairly detailed, deducing the existence and explicit form of a generalized semiregularity map from known results and constructions. We obtain as well a description of the infinitesimal Abel-Jacobi map for smooth cycles as the leading term of this generalized semiregularity map, indicate why for locally complete intersections the appropriate component of our sem…
We discuss an Abel-Jacobi invariant for algebraic cobordism cycles whose image in topological cobordism vanishes. The existence of this invariant follows by abstract arguments from the construction of Hodge filtered cohomology theories in joint work of Michael J. Hopkins and the author. In this paper, we give a concrete description of the new Abel-Jacobi map and Hodge filtered cohomology groups for projective smooth complex varieties.
We show that the image of the Abel-Jacobi map admits functorially a model over the field of definition, with the property that the Abel-Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also rec…
We characterize how to vary the Abel-Jacobi map in terms of Schiffer variation. From this characterization, we will interpret the relation of hyperellipticity of curves with Schiffer variation and describe the deformation of elliptic solitons under Schiffer variation.
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.
Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism $\mathcal M_g\rightarrow \mathcal A_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.
We study derivatives of Schur and tau functions from the view point of the Abel-Jacobi map. We apply the results to establish several properties of derivatives of the sigma function of an (n,s) curve. As byproducts we have an expression of the prime form in terms of derivatives of the sigma function and addition formulae which generalize those of Onishi for hyperelliptic sigma functions.
Tropical Abel-Jacobi theory
该研究针对任意维紧热带簇，通过热带同调与单值化定义中间Jacobian并构建函子性Abel-Jacobi映射以发展热带Abel-Jacobi理论，恢复1维曲线理论并给出热带曲线Ceresa类的组合显式公式。
It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context…
We consider the moduli space $\cSU_C^s(r,\cO_C)$ of rank r stable vector bundles with trivial determinant on a smooth projective curve $C$ of genus $g$. We show that the Abel-Jacobi map on the rational Chow group $CH_1(\cSU_C^s(r,\cO_C))_{hom}\otimes \Q$ of one cycles which are homologous to zero, is an isomorphism onto the bottom weight intermediate Jacobian, which is identified with the Jacobian $Jac(C)\otimes \Q$. The result holds whenever $r\geq 2$ and $g\geq 4$.
Topological Abel-Jacobi Map and Mixed Hodge Structures
本文证明了Zhao定义的拓扑Abel-Jacobi映射与Schnell基于混合Hodge结构实分裂性质提出的替代构造相一致，回答了Schnell的问题。
Let $K$ be a non-Archimedean valued field with valuation ring $R$. Let $C_η$ be a $K$-curve with compact type reduction, so its Jacobian $J_η$ extends to an abelian $R$-scheme $J$. We prove that an Abel-Jacobi map $ι\colon C_η\to J_η$ extends to a morphism $C\to J$, where $C$ is a compact-type $R$-model of $J$, and we show this is a closed immersion when the special fiber of $C$ has no rational components. To do so, we apply a rigid-analytic "fiberwise" criterion for a finite morphism to extend to integral models, and geometric results of Bosch and Lütkebohmert on the analytic structure of $J_…
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