Abel–Jacobi map

Abel–Jacobi map
Name	Abel–Jacobi map
Field	Algebraic geometry
Introduced	19th century
Related	Jacobian variety, Albanese variety, Picard variety

Abel–Jacobi map The Abel–Jacobi map is a fundamental morphism associating algebraic cycles on a complex algebraic variety to points on a complex torus, playing a central role in the study of curves, integrals, and periods. It links classical results of 19th‑century analysts with modern theories developed in the 20th century, connecting the work of several mathematicians and institutions in complex geometry and arithmetic. The map underlies comparisons between geometric objects and analytic invariants, appearing in the theories of Jacobians, Hodge structures, and motives.

Introduction

The Abel–Jacobi map arises from the confluence of ideas due to Niels Henrik Abel, Carl Gustav Jacobi, and later refinements by André Weil, Henri Poincaré, Vladimir Arnold, and contributors at institutions such as the École Normale Supérieure, the Université de Paris, and Princeton University. It concretizes relationships among divisors, integrals of holomorphic differentials, and complex tori embodied by the Jacobian variety, the Albanese variety, and the Picard variety. Historically connected with problems treated by Bernhard Riemann and developments by Friedrich Prym, it serves as a bridge between classical function theory, the theory of abelian varieties, and modern Hodge theory as developed by Phillip Griffiths and Pierre Deligne.

Construction and Definitions

Given a smooth projective curve C over the complex numbers, choose a base point and consider degree zero divisors; integrating holomorphic one‑forms along paths yields a map to the complex vector space of differentials modulo periods, producing the Jacobian J(C). This construction uses tools introduced by Bernhard Riemann in his study of theta functions, by Karl Weierstrass for elliptic functions, and formalized in the language of varieties by André Weil and Oscar Zariski. More generally, for a smooth projective variety X, higher Abel–Jacobi maps are defined for homologically trivial algebraic cycles using intermediate Jacobians constructed from Hodge decomposition, a framework developed by W. V. D. Hodge, Phillip Griffiths, and Gerd Faltings. The analytic definition employs periods studied by Émile Picard and uses comparison isomorphisms informed by the work of Alexander Grothendieck and Jean-Pierre Serre on cohomology theories.

Properties and Theorems

The Abel–Jacobi map is functorial with respect to morphisms of varieties and respects the group structures of Jacobians and Albanese varieties, a principle appearing in the work of André Weil on morphisms of abelian varieties and in the theory of Mordell–Weil theorem contexts studied by Gerd Faltings and Barry Mazur. Abel’s theorem and Jacobi inversion form classical finite‑dimensional cases, while the Griffiths intermediate Jacobian generalizes these results to higher codimension cycles, interacting with the Hodge conjecture investigated by Pierre Deligne, Alexander Grothendieck, and James Harris. The Abel–Jacobi map annihilates cycles algebraically equivalent to zero, and its kernel and image are subjects of the Bloch–Beilinson conjectures and results by Spencer Bloch, Andrei Suslin, and Kazuya Kato. Relations to regulator maps connect to work by Beilinson, Hyman Bass, and Don Zagier in algebraic K‑theory and special values of L‑functions studied by Andrew Wiles and Richard Taylor.

Examples and Computations

For genus one curves, the Abel–Jacobi map identifies a curve with its Jacobian, recovering classical parametrizations as in the theory of elliptic curves developed by Srinivasa Ramanujan, Niels Henrik Abel, and Karl Weierstrass; computational methods involve theta functions from Ferdinand Riemann and complex multiplication studied by Goro Shimura and Yuri Manin. For hyperelliptic curves of low genus, explicit period matrices and theta constants allow effective computation, following techniques refined by researchers at the University of Cambridge and ETH Zurich and algorithms used in computational algebraic geometry by David Mumford and collaborators. Higher‑dimensional examples include Ceresa cycles on Jacobians analyzed by Alessandro Ceresa and Clemens–Griffiths examples of nontrivial Abel–Jacobi invariants related to cubic threefolds, connecting to research by Herbert Clemens and Phillip Griffiths.

Applications in Algebraic Geometry and Number Theory

The Abel–Jacobi map informs the study of rational and integral points via descent on Jacobians, an approach central to results related to the Mordell conjecture proved by Gerd Faltings and to explicit methods employed by Manjul Bhargava and Noam Elkies. In the arithmetic of motives and special values of L‑functions, regulator maps tied to the Abel–Jacobi map feature in conjectures formulated by Beilinson, Pierre Deligne, and Alexander Beilinson and in computations relevant to the Birch and Swinnerton‑Dyer conjecture investigated by John Tate and Bryan Birch. In complex geometry, the map provides obstructions to algebraic equivalence captured by Hodge theoretic invariants studied by Claire Voisin, Phillip Griffiths, and Mark Green and plays a role in moduli problems explored at institutions like Harvard University and the Institute for Advanced Study.

Historical Context and Development

The origins lie in 19th‑century analysis and algebraic function theory stemming from work by Niels Henrik Abel, Carl Gustav Jacobi, and Bernhard Riemann, with subsequent algebraic formalization by André Weil and extension into Hodge theory by W. V. D. Hodge and Phillip Griffiths. Twentieth‑century advances by Alexander Grothendieck, Pierre Deligne, and Spencer Bloch recast the Abel–Jacobi map within cohomological and motivic frameworks, influencing later research by Gerd Faltings, Claire Voisin, and Kazuya Kato. Contemporary work continues at centers such as the Institut des Hautes Études Scientifiques, Clay Mathematics Institute, and various university research groups, linking classical analysis, arithmetic geometry, and modern homological techniques.

Category:Algebraic geometry