Jacobi Map Research Articles

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.

Abel--Jacobi Mapping for Real Hyperelliptic Riemann Surfaces

We study the degrees of the Abel--Jacobi mapping on hyperelliptic Riemann surfaces of arbitrary genus and the restrictions of the corresponding mappings to the symmetric powers of the real locus of the given Riemann surface.

Infinitesimal study of a factorization of the Prym map

The Prym map \(\mathcal{R}_{g} \rightarrow\mathcal{A}_{g-1}\) factors through a space Φ={X} of intrinsically polarized varieties, namely \((\tilde{C}/C) \mapsto{X} \mapsto(P,\Xi)\), where \(\pi: \tilde{C} \rightarrow{C}\) is a connected etale double cover of a smooth curve C of genus g, \(X \subset\tilde{C}^{(2g-2)}\) consists of the set of even precanonical effective divisors on \(\tilde{C}\), and (P,Ξ) is the principally polarized Prym variety associated to π. X is a connected, reduced local complete intersection of (pure) dimension g-1, and when C is non hyperelliptic X is normal and irreducible. By analogy with the proofs of the classical Torelli theorem for curves by Andreotti and by Andreotti-Mayer and Green, which factor the Jacobi map through a symmetric product of the curve, the present factorization may be used to attack the Torelli problem for Prym varieties. In [19] we have shown that X determines the Prym variety (P,Ξ), as the Albanese variety of X, and that X also determines the double cover \(\tilde{C}/C\), at least when C is non hyperelliptic and the codimension of singΞ in P is at least 5. The next challenge in this approach to the Torelli problem is to analyze the infinitesimal structure of these maps.

The Abel--Jacobi Map for Real Hyperelliptic Surfaces of Genus 3

We consider the degrees of the Abel--Jacobi maps for real hyperelliptic surfaces of genus 2 and 3. The restrictions of the maps to the symmetric square and the symmetric cube, respectively, of the real locus of the given Riemann surface are studied.

Cycles over fields of transcendence degree 1

where (a) the group of cycles Z(V ) is the free abelian group on scheme-theoretic points of V of codimension p and (b) rational equivalence R(V ) is the subgroup generated by cycles of the form divW(f ), where W is a subvariety of V of codimension p − 1 and f is a nonzero rational function on it. There is a natural cycle class map clp : CH (V ) → H(V ), where the latter denotes the singular cohomology group H2p(V(C),Z) with the (mixed) Hodge structure given by Deligne (see [D]). The kernel of clp is denoted by F 1 CH(V ). There is an Abel–Jacobi map (see [G]) p : F 1 CH(V ) → IJp(H2p−1(V )),

A Regulator Formula for Milnor K-groups

The classical Abel–Jacobi map is used to geometrically motivate the construction of regulator maps from Milnor K-groups K n M (C(X)) to Deligne cohomology. These maps are given in terms of some new, explicit (n – 1)-currents, higher residues of which are defined and related to polylogarithms. We study their behavior in families X s and prove a rigidity result for the regulator image of the Tame kernel, which leads to a vanishing theorem for very general complete intersections.

New Component of the Moduli Space M(2;0,3) of Stable Vector Bundles on the Double Space P3 of Index Two

We study the moduli scheme M(2;0,n) of rank-2 stable vector bundles with Chern classes c1=0, c2=n, on the Fano threefold X – the double space P3 of index two. New component of this scheme is produced via the Serre construction using certain families of curves on X. In particular, we show that the Abel–Jacobi map Φ: H→J(X) of any irreducible component H of the Hilbert scheme of X containing smooth elliptic quintics on X into the intermediate Jacobian J(X) of X factors by Stein through the quasi-finite (probably birational) map g: M→Θ of (an open part of) a component M of the scheme M(2;0,3) to a translate Θ of the theta-divisor of J(X).

A Semiregularity Map for Modules and Applications to Deformations

We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel–Jacobi map for submanifolds.

Bäcklund transformations for finite-dimensional integrable systems: a geometric approach

We present a geometric construction of Bäcklund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Bäcklund transformations, which are naturally parameterized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel–Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Bäcklund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Bäcklund transformations that was recently introduced. Moreover, we recover Bäcklund transformations and discretizations which have up to now been constructed by ad hoc methods, and we find Bäcklund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables.

GPS Solutions: Closed Forms, Critical and Special Configurations of P4P

P4P is the pseudo-ranging 4-point problem as it appears as the basic configuration of satellite positioning with pseudo-ranges as observables. In order to determine the ground receiver/satellite receiver (LEO networks) position from four positions of satellite transmitters given, a system of four nonlinear (algebraic) equations has to be solved. The solution point is the intersection of four spherical cones if the ground receiver/satellite receiver clock bias is implemented as an unknown. Here we determine the critical configuration manifold (Determinantal Loci, Inverse Function Theorem, Jacobi map) where no solution of P4P exists. Four examples demonstrate the critical linear manifold. The algorithm GS solves in a closed form P4P in a manner similar to Groebner bases: The algebraic nonlinear observational equations are reduced in the forward step to one quadratic equation in the clock bias unknown. In the backward step two solutions of the position unknowns are generated in closed form. Prior information in P4P has to be implemented in order to decide which solution is acceptable. Finally, the main target of our contribution is formulated: Can we identify a special configuration of satellite transmitters and ground receiver/satellite receiver where the two solutions are reduced to one. A special geometric analysis of the discriminant solves this problem. © 2002 Wiley Periodicals, Inc.

Composantes de dimension maximale d'un analogue du lieu de Noether-Lefschetz (Components of Maximal Dimension of an Analogue of the Noether-Lefschetz Locus)

Let X [subset or is implied by] $\mathbb P$4$\mathbb _C$ be a smooth hypersurface of degree d [gt-or-equal, slanted] 5, and let S [subset or is implied by] X be a smooth hyperplane section. Assume that there exists a non trivial cycle Z [set membership] Pic(X) of degree 0, whose image in CH1(X) is in the kernel of the Abel–Jacobi map. The family of couples (X, S) containing such Z is a countable union of analytic varieties. We show that it has a unique component of maximal dimension, which is exaclty the locus of couples (X, S) satisfying the following condition: There exists a line Δ [subset or is implied by] S and a plane P [subset or is implied by] $\mathbb P$4$_{\mathbb C}$ such that P [cap B: intersection] X = Δ, and Z = Δ − dh, where h is the class of the hyperplane section in CH1(S). The image of Z in CH1(X) is thus 0. This construction provides evidence for a conjecture by Nori which predicts that the Abel–Jacobi map for 1–cycles on X is injective.

EXPLICIT GEOMETRY ON A FAMILY OF CURVES OF GENUS 3

An explicit geometrical study of the curves[formula here]is presented. These are non-singular curves of genus 3, defined over ℚ(a). By exploiting their symmetries, it is possible to determine most of their geometric invariants, such as their bitangent lines and their period lattice. An explicit description is given of the bijection induced by the Abel–Jacobi map between their bitangent lines and odd 2-torsion points on their jacobian. Finally, three elliptic quotients of these curves are constructed that provide a splitting of their jacobians. In the case of the curve [Cscr ]1±√2, which is isomorphic to the Fermat curve of degree 4, the computations yield a finer splitting of its jacobian than the classical one.

A Filtration on the Chow Groups of a Complex Projective Variety

Let X/C be a projective algebraic manifold, and further let CHk(X)Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {Fl}l ≥ 0 on CHk(X)Q of ‘Bloch-Beilinson’ type. In the event that a certain conjecture of Jannsen holds (related to the Bloch-Beilinson conjecture on the injectivity, modulo torsion, of the Abel–Jacobi map for smooth proper varieties over Q), this filtration truncates. In particular, his conjecture implies that Fk+1 = 0.

The Abel-Jacobi map for cubic threefold and periods of Fano threefolds of degree $14$

The Abe–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 that the factorizing map from the moduli space to the intermediate Jacobian is étale. The above result implies that the degree of the étale map is 1, hence the moduli component of vector bundles is birational to the intermediate Jacobian. As an application, it is shown that the generic fiber of the period map of Fano varieties of degree 14 is birational to the intermediate Jacobian of the associated cubic threefold.

An Example of Algebraic Cycles with Nontrivial Abel–Jacobi Images

In this paper, we will study a family of quintic hypersurfaces and its Abel–Jacobi images. We will construct codimension 2 algebraic cycles on the blowing-up of certain dimension 3 quintic hypersurfaces defined over number fields, and show that they are homologically trivial but have nontrivial images under the 3-adic Abel–Jacobi map.

Roitman‘s Theorem for Singular Projective Varieties

We construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties.

The Stäckel systems and algebraic curves

We show how the Abel–Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel–Jacobi map is just the Stäckel matrix, which determines n-orthogonal curvilinear coordinate systems in a flat space. The Lax pairs, r-matrix algebras and explicit form of the flat coordinates are constructed. An application of the Weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space.

The Jacobi map

This paper defines nth order Jacobi fields to be solutions to a second-order nonlinear differential equation defined by the Jacobi map. nth order Jacobi fields arise naturally as acceleration vector fields of geodesic variations. As a main theorem we prove necessity and sufficiency conditions for an nth order Jacobi field to be the acceleration vector field of a variation of geodesics normal to a submanifold. An m geodesic, m ≥ 2, is a smooth curve whose mth covariant derivative vanishes. We prove an index theorem giving bounds for the total m focal multiplicity along an m geodesic m normal to a submanifold in a flat manifold.

Roitman’s theorem for singular complex projective surfaces

Let $X$ be a complex projective surface with arbitrary singularities. We construct a generalized Abel--Jacobi map $A_0(X)\to J^2(X)$ and show that it is an isomorphism on torsion subgroups. Here $A_0(X)$ is the appropriate Chow group of smooth 0-cycles of degree 0 on $X$, and $J^2(X)$ is the intermediate Jacobian associated with the mixed Hodge structure on $H^3(X)$. Our result generalizes a theorem of Roitman for smooth surfaces: if $X$ is smooth then the torsion in the usual Chow group $A_0(X)$ is isomorphic to the torsion in the usual Albanese variety $J^2(X)\cong Alb(X)$ by the classical Abel-Jacobi map.

Extension of the Pack–Parker quantum reactive scattering method to include direct calculation of time delays

A direct method for determining time delays for quantum reactive scattering is developed for three atoms scattering in three physical dimensions. The method is a simple extension of the Pack–Parker approach to reactive scattering. In their formulation, adiabatically adjusted-principle axis-hyperspherical (APH) coordinates are used to generate coupled equations for the exchange region. These solutions are then projected onto Delves coordinate wave functions to generate the corresponding set of coupled equations that must be propagated out into the asymptotic region. The Delves wave functions are then mapped onto the Jacobi coordinate wave functions from which the reactive scattering S matrix is obtained. The extension of this method to include the direct calculation of the time delays for state-to-state reactive scattering processes involves three essential steps: (1) Modification of the log-derivative method for propagating accurate solutions to the coupled equations so that the log-derivative and its energy derivative are propagated simultaneously; (2) establishing that the APH to Delves projection is independent of the energy; and (3) extension of the energy dependent Delves to Jacobi mapping to include the global R matrix and its energy derivative. The necessary mathematical expressions for accomplishing each of these steps are developed in sufficient detail so that the power and simplicity of the method can be understood and so that the method can be efficiently implemented.