Abel-Jacobi Map Research Articles

Separation of variables for some systems with a fourth-order integral of motion

We construct separation variables for Yehia’s integrable deformations of the Kovalevskaya top and the Chaplygin system on a sphere. In the general case, the corresponding quadratures are given by the Abel-Jacobi map on a two-dimensional submanifold of the Jacobian of a genus-three algebraic curve, which is not hyperelliptic.

Vortices and the Abel–Jacobi map

The abelian Higgs model on a compact Riemann surface Σ supports vortex solutions for any positive vortex number d∈Z. Moreover, the vortex moduli space for fixed d has long been known to be the symmetrized d-th power of Σ, in symbols, Symd(Σ). This moduli space is Kähler with respect to the physically motivated metric whose geodesics describe slow vortex motion.In this paper we appeal to classical properties of Symd(Σ) to obtain new results for the moduli space metric. Our main tool is the Abel–Jacobi map, which maps Symd(Σ) into the Jacobian of Σ. Fibres of the Abel–Jacobi map are complex projective spaces, and the first theorem we prove states that near the Bradlow limit the moduli space metric restricted to these fibres is a multiple of the Fubini–Study metric. Additional significance is given to the fibres of the Abel–Jacobi map by our second result: we show that if Σ is a hyperelliptic surface, there exist two special fibres which are geodesic submanifolds of the moduli space. Even more is true: the Abel–Jacobi map has a number of fibres which contain complex projective subspaces that are geodesic.

Generalized Heegner cycles and p-adic Rankin L-series

This article studies a distinguished collection of so-called generalized Heegner cycles in the product of a Kuga–Sato variety with a power of a CM elliptic curve. Its main result is a p-adic analogue of the Gross–Zagier formula which relates the images of generalized Heegner cycles under the p-adic Abel–Jacobi map to the special values of certain p-adic Rankin L-series at critical points that lie outside their range of classical interpolation.

A remark on the Abel–Jacobi morphism for the cubic threefold

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)×X with Zt homologically trivial for each t∈J(X), such that the morphism ϕZ:J(X)→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.

Fourier-Deligne transform and representations of the symmetric group

We calculate the Fourier-Deligne transform of the IC extension to ${\C}^{n+1}$ of the local system ${\mathcal L}_{\Lambda}$ on the cone over $\Conf_n(¶^1)$ associated to a representation $\Lambda$ of $S_n$, where the length $n-k$ of the first row of the Young diagram of $\Lambda$ is at least $\frac{|\Lambda|-1}{2}$. The answer is the IC extension to the dual vector space ${\C}^{n+1}$ of the local system ${\mathcal R}_{\lambda}$ on the cone over the $k$-th secant variety of the rational normal curve in $¶^n$, where ${\mathcal R}_{\lambda}$ corresponds to the representation $\lambda$ of $S_k$, the Young diagram of which is obtained from the Young diagram of $\Lambda$ by deleting its first row. We also prove an analogous statement for $S_n$-local systems on fibers of the Abel-Jacobi map. We use our result on the Fourier-Deligne transform to rederive a part of a result of Michel Brion on Kronecker coefficients.

Voronoi tilings hidden in crystals ---the case of maximal abelian coverings---

Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadri as a by-product of Geometric Invariant Theory, he shows that the Voronoi tiling (also known as the Wigner-Seitz tiling) is hidden in the crystal, that is, the crystal does not intrude the interiors of the top-dimensional Voronoi cells. The result turns out to be closely related to the tropical Abel-Jacobi map of the associated compact tropical curve.

Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.

A Noether-Lefschetz theorem for varieties of r-planes in complete intersections

AbstractWe prove a Noether-Lefschetz type theorem for varieties ofr-planes in complete intersections. We then use it to study the Abel-Jacobi map of planes on a smooth cubic fivefold.

A Noether-Lefschetz theorem for varieties of r-planes in complete intersections

AbstractWe prove a Noether-Lefschetz type theorem for varieties of r-planes in complete intersections. We then use it to study the Abel-Jacobi map of planes on a smooth cubic fivefold.

On an analogue of the Lutz–Nagell Theorem for hyperelliptic curves

We produce a version of the Lutz–Nagell Theorem for hyperelliptic curves of genus g ⩾ 1 . We consider curves C defined by y 2 = f ( x ) , where f is a monic polynomial of degree 2 g + 1 defined over the ring of integers of a number field F or its non-archimedean completions. If J is the Jacobian of C , and ϕ is the Abel–Jacobi map of C into J sending the point at ∞ of our model of C to the origin of J , we show that if P = ( a , b ) is a rational point of C such that ϕ ( P ) is torsion in J , then a and b are integral if the order n of P is not a prime power, and bound the denominators of a and b if n is. When a and b are integral, we give criteria for when b 2 divides the discriminant Δ ( f ) of f . Finally we show for f ∈ Z [ x ] , that if P = ( a , b ) ∈ C ( Q ) and ϕ ( P ) is a torsion point of order n ⩾ 2 , then we have a , b ∈ Z , and either b = 0 or b 2 divides Δ ( f ) .

Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal

Given a smooth projectivenn-foldYY, withH3,0(Y)=0H^{3,0}(Y)=0, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing codimension22-cycles inYYto the intermediate JacobianJ(Y)J(Y), which is an abelian variety. Assumingn=3n=3, we study in this paper the existence of families of11-cycles inYYfor which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. WhenYYitself is rationally connected with trivial Brauer group, we relate this property to the existence of an integral cohomological decomposition of the diagonal ofYY. We also study this property for cubic threefolds, completing the work of Iliev-Markushevich-Tikhomirov. We then conclude that the Hodge conjecture holds for degree44integral Hodge classes on fibrations into cubic threefolds over curves, with some restriction on singular fibers.

CM cycles on Shimura curves, and p-adic L-functions

AbstractLet f be a modular form of weight k≥2 and level N, let K be a quadratic imaginary field and assume that there is a prime p exactly dividing N. Under certain arithmetic conditions on the level N and the field K, one can attach to this data a p-adic L-function Lp (f,K,s) , as done by Bertolini–Darmon–Iovita–Spieß in [Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting, Amer. J. Math. 124 (2002), 411–449]. In the case of p being inert in K, this analytic function of a p-adic variable s vanishes in the critical range s=1,…,k−1 , and one may be interested in the values of its derivative in this range. We construct, for k≥4 , a Chow motive endowed with a distinguished collection of algebraic cycles which encode these values, via the p-adic Abel–Jacobi map. Our main result generalizes the result obtained by Iovita and Spieß in [Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 154 (2003), 333–384], which gives a similar formula for the central value s=k/2 . Even in this case our construction is different from the one found by Iovita and Spieß.

Fano threefolds of Genus 6

Ideas and methods of Clemens C. H., Griffiths Ph. The intermediate Jacobian of a cubic threefold are applied to a Fano threefold $X$ of genus 6 — intersection of $G(2, 5) \subset P^9$ with $P^7$ and a quadric. Main results: 1. The Fano surface $F(X)$ of $X$ is smooth and irreducible. Hodge numbers and some other invariants of $F(X)$ are calculated. 2. Tangent bundle theorem for $X$ is proved, and its geometric interpretation is given. It is shown that $F(X)$ defines $X$ uniquely. 3. The Abel-Jacobi map $\Phi : \operatorname{Alb} F(X) \to J^3(X)$ is an isogeny. 4. As a necessary step of calculation of $h^{1,0}(F(X))$ we describe a special intersection of 3 quadrics in $P^6$ (having 1 double point) whose Hesse curve is a smooth plane curve of degree 6. 5. $\operatorname{im} \Phi(F(X)) \subset J^3(X)$ is algebraically equivalent to $\frac{2\Theta^8}{8!}$ where $\Theta \subset J^3(X)$ is a Poincare divisor (a sketch of the proof).

Singularities of Abel-Jacobi maps and geometry of dissolving vortices

Singularities of Abel-Jacobi maps and geometry of dissolving vortices

Parameterized Abel-Jacobi maps and abelian cycles in the Torelli group

Let ℐg, * denote the Torelli group of the genus g⩾2 surface Sg with one marked point. This is the group of homotopy classes (rel basepoint) of homeomorphisms of Sg fixing the basepoint and acting trivially on H≔ H1(Sg, ℚ). In 1983, Johnson constructed a beautiful family of invariants τ i : H i ( ℐ g , * , ℚ ) → ∧ i + 2 H , for 0⩽i⩽2g−2, using a kind of Abel–Jacobi map for families. He used these invariants to detect non-trivial cycles in ℐg, *. Johnson proved that τ1 is an isomorphism, and asked if the same is true for τi with i>1. The goal of this paper is to introduce various methods for computing τi; in particular, we prove that τi is not injective for any 2⩽i<g, answering Johnson's question in the negative. We also show that τ2 is surjective. For g⩾3 we find many classes in the image of τi and use them to deduce Hi(ℐg, *, ℚ)≠0 for each 1⩽i<g. This is in contrast to the case of mapping class groups. Many of our classes are stable, so we can deduce that Hi(ℐ∞, 1, ℚ) is infinite-dimensional for each i⩾1. Finally, we conjecture a new kind of ‘representation-theoretic stability’ for the homology of the Torelli group, for which our results provide evidence.

Cohomological dimension, self-linking, and systolic geometry

Given a closed manifold M, we prove the upper bound of $${1 \over 2}(\dim M + {\rm{cd}}({{\rm{\pi }}_1}M))$$ for the number of systolic factors in a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov’s systolic inequalities. Here “cd” is the cohomological dimension. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik-Schnirelmann category is an upper bound for the systolic category. Furthermore, we prove a systolic inequality on a manifold M with b 1(M) = 2 in the presence of a nontrivial self-linking class of a typical fiber of its Abel-Jacobi map to the 2-torus.

Vortices and Jacobian varieties

We investigate the geometry of the moduli space of N vortices on line bundles over a closed Riemann surface Σ of genus g > 1 , in the little explored situation where 1 ≤ N < g . In the regime where the area of the surface is just large enough to accommodate N vortices (which we call the dissolving limit), we describe the relation between the geometry of the moduli space and the complex geometry of the Jacobian variety of Σ . For N = 1 , we show that the metric on the moduli space converges to a natural Bergman metric on Σ . When N > 1 , the vortex metric typically degenerates as the dissolving limit is approached, the degeneration occurring precisely on the critical locus of the Abel–Jacobi map of Σ at degree N . We describe consequences of this phenomenon from the point of view of multivortex dynamics.

Algebraic $K$-theory of toric hypersurfaces

We construct classes in the motivic cohomology of certain 1-parameter families of Calabi–Yau hypersurfaces in toric Fano n-folds, with applications to local mirror symmetry (growth of genus 0 instanton numbers) and inhomogeneous Picard–Fuchs equations. In the case where the family is classically modular the classes are related to Beilinson’s Eisenstein symbol; the Abel–Jacobi map (or rational regulator) is computed in this paper for both kinds of cycles. For the “modular toric” families where the cycles essentially coincide, we obtain a motivic (and computationally effective) explanation of a phenomenon observed by Villegas, Stienstra, and Bertin.

A generalization of Abel’s Theorem and the Abel–Jacobi map

We generalize Abel's classical theorem on linear\break equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$-manifold, or more generally for any $d$-cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$-gerbe $\Lambda_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne cohomology class generalizes the notion of ‘linear equivalence to zero’. In this setting, Abel's theorem remains valid. Moreover, we generalize the classical Inversion theorem for the Abel–Jacobi map, thereby proving that the moduli space of Abel gerbes is isomorphic to the harmonic Deligne cohomology; that is, gerbes with harmonic curvature.

Metric properties of the tropical Abel–Jacobi map

Let Γ be a tropical curve (or metric graph), and fix a base point p?Γ. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(Γ) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for Γ. This result is useful for reducing certain questions about the Abel---Jacobi map ? p :Γ?J(Γ), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over ?. As an application of our direct limit theorem, we derive some local comparison formulas between ? and ${\varPhi}_{p}^{*}(\rho)$ for three different natural metrics ? on J(Γ). One of these formulas implies that ? p is a tropical isometry when Γ is 2-edge-connected. Another shows that the canonical measure μ Zh? on a metric graph Γ, defined by S. Zhang, measures lengths on ? p (Γ) with respect to the sup-norm on J(Γ).