Naturality of Abel maps

On Abel-Jacobi Maps of Lagrangian Families

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.

Degree-2 Abel maps and hyperelleptic curves

In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an application, we characterize when the (symmetrized) degree-2 Abel map is not injective, a property that, for a smooth curve, is equivalent to the curve being hyperelliptic.

Intersection Sheaves for Abel maps

Intersection sheaves, i.e., the Deligne pairing, were first introduced by Deligne in the setting of Poincare duality for etale cohomology, and later in his work on the determinant of cohomology. Intersection sheaves were generalized from smooth schemes to Cohen-Macaulay schemes by Elkik, and then beyond Cohen-Macaulay schemes by Munoz-Garcia. To define the Abel maps arising in rational simple connectedness on the natural parameter spaces of rational sections, we need a variant of the construction of Munoz-Garcia that has the good properties of the construction using det and Div. In addition, w…

Abel maps and limit linear series

We explore the relationship between limit linear series and fibers of Abel maps in the case of curves with two smooth components glued at a single node. To an r-dimensional limit linear series satisfying a certain exactness property (weaker than the refinedness property of Eisenbud and Harris) we associate a closed subscheme of the appropriate fiber of the Abel map. We then describe this closed subscheme explicitly, computing its Hilbert polynomial and showing that it is Cohen-Macaulay of pure dimension r. We show that this construction is also compatible with one-parameter smoothings.

Abel Maps and Presentation Schemes

We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations provided by the images of certain Abel maps and certain presentation schemes. Third we study a lifting of the Abel map of bidegree (m,1) to the corresponding presentation scheme. Fourth we prove that, if a curve is blown up at a double point, then the corresponding presentation scheme is a IP^1-bundle. Finally, using Abel maps of bidegree (m,1), we characterize…

Abel-Jacobi maps associated to smooth cubic threefolds

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.

Simplicial Abel-Jacobi maps and reciprocity laws

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.

On Abel maps of stable curves

We construct Abel maps for a stable curve $X$. Namely, for each one-parameter deformation of $X$ with regular total space, and every integer $d>0$, we construct by specialization a map $α^d_X$ from the smooth locus of $X^d$ to the moduli scheme of balanced line bundles on semistable curves over $X$. For $d=1$, we show that $α^1_X$ naturally extends over $X$, and does not depend on the choice of the deformation; we give a precise description of when it is injective.

Abel-Jacobi 映射的分解

Factorization of the Abel-Jacobi maps

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。

Abel-Prym maps for isotypical components of Jacobians

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…

On Abel-Jacobi maps of moduli of parabolic bundles over a curve

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.

Higher Abel-Jacobi Maps

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.

阿贝尔 - 雅可比映射的（非）可扩展性

(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。

Abel maps of Gorenstein curves

For a Gorenstein curve X and a nonsingular point P of X, we construct Abel maps A from X to J_X^1 and A_P from X to J_X^0, where J_X^i is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A_P are shown to have the same arithmetic genus of X. Also, A and A_P are shown to be embeddings away from rational subcurves L of X meeting the closure of X-L in separating nodes. Finally, we establish a connection with Seshadri's moduli scheme U_X(1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U_X(1).

Formal Abel-Jacobi maps

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.

The Abel-Jacobi Map for Higher Chow Groups, II

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.

Class of images of Abel maps on normal surface singularities

In this paper we investigate Abel maps on normal surface singularities described in \cite{NNI}. We investigate the affine version of the class of the images of Abel maps on normal surface singularities. More precisely we consider the projective clousure of the image of an Abel map, its dual projective variety and we substract from its degree the multiplicity of the infinite hyperplane on the dual variety. In the case of generic singularities we prove explicit combinatorial formulas of this invariant, in the general case we prove an upper bound.

On the degree-1 Abel map for nodal curves

Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $α_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In this work we extend $α_{L}$ to a morphism $\overlineα_{L}:C\rightarrow\overline{J}_{E}^{P}$ taking values on Esteves' compactified Jacobian for any given polarization $E$. The maps $\overlineα_{L}$ are limits of Abel maps of smooth curves of the type $α_{L}$.

Algebraic maps constant on isomorphism classes of unpolarized abelian varieties are constant

We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results are motivated by and shed light on a proposed construction of a cryptographic protocol for multiparty non-interactive key exchange.

On the Abel-Jacobi maps of Fermat Jacobians

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.

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.

通过导出视角研究整曲线的阿贝尔映射

Abel maps for integral curves via a derived perspective

本文利用导出代数几何为整曲线族建立阿贝尔映射的一般框架，构造商概形和希尔伯特概形的导出增强，并借助 $\mathcal{Q}$-复形和扩张模的导出理论得到统一半正交分解，推广了无挠层上的既有结果。

Arithmetic Torelli maps for cubic surfaces and threefolds

It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the prime 2, an old question of Deligne and a recent question of Kudla and Rapoport. We further classify the Mumford-Tate groups of the abelian varieties which arise, and give additional arithmetic applications.

Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's

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…

Refining the Abel--Jacobi maps

Given a smooth projective variety $X$ over a field $k$ of characteristic zero, we consider the composition of the de Rham cohomology cycle class map over $k$ from the Chow group $CH^q(X\times_kK)$, where $K$ is the field of fractions of henselization $A^h$ of the local ring of a smooth closed point of a variety over the field $k$ with an appropriate projection: $CH^q(X\times_kK)\longrightarrow\bigoplus_{p=1}^qgr_F^{q-p}N^{q-p} H^{2q-p}_{dR/k}(X)\otimes_kΩ^p_{A^h/k,{\rm closed}},$ where $F^{\bullet}$ and $N^{\bullet}$ are the Hodge and the coniveau filtrations on the de Rham cohomology, respect…

The Abel-Jacobi map for higher Chow groups

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.

Operations on derived moduli spaces of branes

The main theme of this work is the study of the operations that naturally exist on moduli spaces of maps $Map(S,X)$, also called the space of branes of $X$ with respect $S$. These operations will be constructed as operations on the (quasi-coherent) derived category $\D(Map(S,X))$, in the particular case where $S$ has some close relations with an operad $\OO$. More precisely, for an $\s$-operad $\OO$ and an algebraic variety $X$ (or more generally a derived algebraic stack), satisfying some natural conditions, we prove that $\OO$ acts on the object $\OO(2)$ by mean cospans. This universal actio…

Non-abelian reciprocity laws and higher Brauer-Manin obstructions

We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular c…

Invariants of relatively generic surface singularities II. Images of Abel maps

In \cite{R} the author investigated invariants of relatively generic structures on surface singularities generalising results of \cite{NNA1} and \cite{NNA2} about generic analytic structures and generic line bundles to the case of the relative setup, where we fix a given analytic type or line bundle on a smaller subgraph or more generally on a smaller cycle and we choose a relatively generic line bundle or analytic type on the large cycle and managed to compute some of it's invariants, like geometric genus or $h^1$ of natural line bundles. In \cite{NNAD} the authors investigated the images of …