Abel–Jacobi map

Abel–Jacobi map   (Redirected from Abel–Jacobi theorem) Jump to navigation Jump to search In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

Contents 1 Construction of the map 2 The Abel–Jacobi map of a Riemannian manifold 3 Abel–Jacobi theorem 4 References Construction of the map In complex algebraic geometry, the Jacobian of a curve C is constructed using path integration. Namely, suppose C has genus g, which means topologically that {displaystyle H_{1}(C,mathbb {Z} )cong mathbb {Z} ^{2g}.} Geometrically, this homology group consists of (homology classes of) cycles in C, or in other words, closed loops. Therefore, we can choose 2g loops {displaystyle gamma _{1},ldots ,gamma _{2g}} generating it. On the other hand, another more algebro-geometric way of saying that the genus of C is g is that {displaystyle H^{0}(C,K)cong mathbb {C} ^{g},} where K is the canonical bundle on C.

By definition, this is the space of globally defined holomorphic differential forms on C, so we can choose g linearly independent forms {displaystyle omega _{1},ldots ,omega _{g}} . Given forms and closed loops we can integrate, and we define 2g vectors {displaystyle Omega _{j}=left(int _{gamma _{j}}omega _{1},ldots ,int _{gamma _{j}}omega _{g}right)in mathbb {C} ^{g}.} It follows from the Riemann bilinear relations that the {displaystyle Omega _{j}} generate a nondegenerate lattice {displaystyle Lambda } (that is, they are a real basis for {displaystyle mathbb {C} ^{g}cong mathbb {R} ^{2g}} ), and the Jacobian is defined by {displaystyle J(C)=mathbb {C} ^{g}/Lambda .} The Abel–Jacobi map is then defined as follows. We pick some base point {displaystyle p_{0}in C} and, nearly mimicking the definition of {displaystyle Lambda ,} define the map {displaystyle {begin{cases}u:Cto J(C)\u(p)=left(int _{p_{0}}^{p}omega _{1},dots ,int _{p_{0}}^{p}omega _{g}right){bmod {Lambda }}end{cases}}} Although this is seemingly dependent on a path from {displaystyle p_{0}} to {displaystyle p,} any two such paths define a closed loop in {displaystyle C} and, therefore, an element of {displaystyle H_{1}(C,mathbb {Z} ),} so integration over it gives an element of {displaystyle Lambda .} Thus the difference is erased in the passage to the quotient by {displaystyle Lambda } . Changing base-point {displaystyle p_{0}} does change the map, but only by a translation of the torus.

The Abel–Jacobi map of a Riemannian manifold Let {displaystyle M} be a smooth compact manifold. Let {displaystyle pi =pi _{1}(M)} be its fundamental group. Let {displaystyle f:pi to pi ^{ab}} be its abelianisation map. Let {displaystyle operatorname {tor} =operatorname {tor} (pi ^{ab})} be the torsion subgroup of {displaystyle pi ^{ab}} . Let {displaystyle g:pi ^{ab}to pi ^{ab}/operatorname {tor} } be the quotient by torsion. If {displaystyle M} is a surface, {displaystyle pi ^{ab}/operatorname {tor} } is non-canonically isomorphic to {displaystyle mathbb {Z} ^{2g}} , where {displaystyle g} is the genus; more generally, {displaystyle pi ^{ab}/operatorname {tor} } is non-canonically isomorphic to {displaystyle mathbb {Z} ^{b}} , where {displaystyle b} is the first Betti number. Let {displaystyle varphi =gcirc f:pi to mathbb {Z} ^{b}} be the composite homomorphism.

Definition. The cover {displaystyle {bar {M}}} of the manifold {displaystyle M} corresponding to the subgroup {displaystyle ker(varphi )subset pi } is called the universal (or maximal) free abelian cover.

Now assume M has a Riemannian metric. Let {displaystyle E} be the space of harmonic 1-forms on {displaystyle M} , with dual {displaystyle E^{*}} canonically identified with {displaystyle H_{1}(M,mathbb {R} )} . By integrating an integral harmonic 1-form along paths from a basepoint {displaystyle x_{0}in M} , we obtain a map to the circle {displaystyle mathbb {R} /mathbb {Z} =S^{1}} .

Similarly, in order to define a map {displaystyle Mto H_{1}(M,mathbb {R} )/H_{1}(M,mathbb {Z} )_{mathbb {R} }} without choosing a basis for cohomology, we argue as follows. Let {displaystyle x} be a point in the universal cover {displaystyle {tilde {M}}} of {displaystyle M} . Thus {displaystyle x} is represented by a point of {displaystyle M} together with a path {displaystyle c} from {displaystyle x_{0}} to it. By integrating along the path {displaystyle c} , we obtain a linear form on {displaystyle E} : {displaystyle hto int _{c}h.} This gives rise a map {displaystyle {tilde {M}}to E^{*}=H_{1}(M,mathbb {R} ),} which, furthermore, descends to a map {displaystyle {begin{cases}{overline {A}}_{M}:{overline {M}}to E^{*}\cmapsto left(hmapsto int _{c}hright)end{cases}}} where {displaystyle {overline {M}}} is the universal free abelian cover.

Definition. The Jacobi variety (Jacobi torus) of {displaystyle M} is the torus {displaystyle J_{1}(M)=H_{1}(M,mathbb {R} )/H_{1}(M,mathbb {Z} )_{mathbb {R} }.} Definition. The Abel–Jacobi map {displaystyle A_{M}:Mto J_{1}(M),} is obtained from the map above by passing to quotients.

The Abel–Jacobi map is unique up to translations of the Jacobi torus. The map has applications in Systolic geometry. The Abel–Jacobi map of a Riemannian manifold shows up in the large time asymptotics of the heat kernel on a periodic manifold (Kotani & Sunada (2000) and Sunada (2012)).

In much the same way, one can define a graph-theoretic analogue of Abel–Jacobi map as a piecewise-linear map from a finite graph into a flat torus (or a Cayley graph associated with a finite abelian group), which is closely related to asymptotic behaviors of random walks on crystal lattices, and can be used for design of crystal structures.

Abel–Jacobi theorem The following theorem was proved by Abel: Suppose that {displaystyle D=sum nolimits _{i}n_{i}p_{i}} is a divisor (meaning a formal integer-linear combination of points of C). We can define {displaystyle u(D)=sum nolimits _{i}n_{i}u(p_{i})} and therefore speak of the value of the Abel–Jacobi map on divisors. The theorem is then that if D and E are two effective divisors, meaning that the {displaystyle n_{i}} are all positive integers, then {displaystyle u(D)=u(E)} if and only if {displaystyle D} is linearly equivalent to {displaystyle E.} This implies that the Abel-Jacobi map induces an injective map (of abelian groups) from the space of divisor classes of degree zero to the Jacobian.

Jacobi proved that this map is also surjective, so the two groups are naturally isomorphic.

The Abel–Jacobi theorem implies that the Albanese variety of a compact complex curve (dual of holomorphic 1-forms modulo periods) is isomorphic to its Jacobian variety (divisors of degree 0 modulo equivalence). For higher-dimensional compact projective varieties the Albanese variety and the Picard variety are dual but need not be isomorphic.

References E. Arbarello; M. Cornalba; P. Griffiths; J. Harris (1985). "1.3, Abel's Theorem". Geometry of Algebraic Curves, Vol. 1. Grundlehren der Mathematischen Wissenschaften. Springer-Verlag. ISBN 978-0-387-90997-4. Kotani, Motoko; Sunada, Toshikazu (2000), "Albanese maps and an off diagonal long time asymptotic for the heat kernel", Comm. Math. Phys., 209: 633–670, Bibcode:2000CMaPh.209..633K, doi:10.1007/s002200050033 Sunada, Toshikazu (2012), "Lecture on topological crystallography", Japan. J. Math., 7: 1–39, doi:10.1007/s11537-012-1144-4 hide vte Topics in algebraic curves Rational curves Five points determine a conicProjective lineRational normal curveRiemann sphereTwisted cubic Elliptic curves Analytic theory Elliptic functionElliptic integralFundamental pair of periodsModular form Arithmetic theory Counting points on elliptic curvesDivision polynomialsHasse's theorem on elliptic curvesMazur's torsion theoremModular elliptic curveModularity theoremMordell–Weil theoremNagell–Lutz theoremSupersingular elliptic curveSchoof's algorithmSchoof–Elkies–Atkin algorithm Applications Elliptic curve cryptographyElliptic curve primality Higher genus De Franchis theoremFaltings's theoremHurwitz's automorphisms theoremHurwitz surfaceHyperelliptic curve Plane curves AF+BG theoremBézout's theoremBitangentCayley–Bacharach theoremConic sectionCramer's paradoxCubic plane curveFermat curveGenus–degree formulaHilbert's sixteenth problemNagata's conjecture on curvesPlücker formulaQuartic plane curveReal plane curve Riemann surfaces Belyi's theoremBring's curveBolza surfaceCompact Riemann surfaceDessin d'enfantDifferential of the first kindKlein quarticRiemann's existence theoremRiemann–Roch theoremTeichmüller spaceTorelli theorem Constructions Dual curvePolar curveSmooth completion Structure of curves Divisors on curves Abel–Jacobi mapBrill–Noether theoryClifford's theorem on special divisorsGonality of an algebraic curveJacobian varietyRiemann–Roch theoremWeierstrass pointWeil reciprocity law Moduli ELSV formulaGromov–Witten invariantHodge bundleModuli of algebraic curvesStable curve Morphisms Hasse–Witt matrixRiemann–Hurwitz formulaPrym varietyWeber's theorem Singularities AcnodeCrunodeCuspDelta invariantTacnode Vector bundles Birkhoff–Grothendieck theoremStable vector bundleVector bundles on algebraic curves Categories: Algebraic curvesRiemannian geometryNiels Henrik Abel