Abel–Jacobi map

Abel–Jacobi map (Redirected from Abel–Jacobi theorem) Vai alla navigazione Vai alla ricerca In matematica, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. Nella geometria riemanniana, 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.

Contenuti 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. Vale a dire, suppose C has genus g, which means topologically that {stile di visualizzazione H_{1}(C,mathbb {Z} )cong mathbb {Z} ^{2g}.} Geometricamente, this homology group consists of (homology classes of) cycles in C, o in altre parole, closed loops. Perciò, we can choose 2g loops {stile di visualizzazione gamma _{1},ldot ,gamma _{2g}} generating it. D'altro canto, another more algebro-geometric way of saying that the genus of C is g is that {stile di visualizzazione H^{0}(C,K)cong mathbb {C} ^{g},} where K is the canonical bundle on C.

Per definizione, this is the space of globally defined holomorphic differential forms on C, so we can choose g linearly independent forms {stile di visualizzazione omega _{1},ldot ,omega _{g}} . Given forms and closed loops we can integrate, and we define 2g vectors {stile di visualizzazione Omega _{j}= sinistra(int _{gamma _{j}}omega _{1},ldot ,int _{gamma _{j}}omega _{g}Giusto)in matematica bb {C} ^{g}.} It follows from the Riemann bilinear relations that the {stile di visualizzazione Omega _{j}} generate a nondegenerate lattice {displaystyle Lambda } (questo è, they are a real basis for {displaystyle mathbb {C} ^{g}cong mathbb {R} ^{2g}} ), and the Jacobian is defined by {stile di visualizzazione J(C)= matematica bb {C} ^{g}/Lambda .} The Abel–Jacobi map is then defined as follows. We pick some base point {stile di visualizzazione p_{0}in c} e, nearly mimicking the definition of {displaystyle Lambda ,} define the map {stile di visualizzazione {inizio{casi}tu:Cto J(C)\tu(p)= sinistra(int _{p_{0}}^{p}omega _{1},punti ,int _{p_{0}}^{p}omega _{g}Giusto){in un modo {Lambda }}fine{casi}}} Although this is seemingly dependent on a path from {stile di visualizzazione p_{0}} a {stile di visualizzazione p,} any two such paths define a closed loop in {stile di visualizzazione C} e, dunque, an element of {stile di visualizzazione 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 {stile di visualizzazione p_{0}} does change the map, but only by a translation of the torus.

The Abel–Jacobi map of a Riemannian manifold Let {stile di visualizzazione M} be a smooth compact manifold. Permettere {displaystyle pi =pi _{1}(M)} be its fundamental group. Permettere {stile di visualizzazione f:pi to pi ^{ab}} be its abelianisation map. Permettere {nome dell'operatore dello stile di visualizzazione {tor} =nome operatore {tor} (pi ^{ab})} be the torsion subgroup of {displaystyle pi ^{ab}} . Permettere {stile di visualizzazione g:pi ^{ab}to pi ^{ab}/nome operatore {tor} } be the quotient by torsion. Se {stile di visualizzazione M} is a surface, {displaystyle pi ^{ab}/nome operatore {tor} } is non-canonically isomorphic to {displaystyle mathbb {Z} ^{2g}} , dove {stile di visualizzazione g} is the genus; più generalmente, {displaystyle pi ^{ab}/nome operatore {tor} } is non-canonically isomorphic to {displaystyle mathbb {Z} ^{b}} , dove {stile di visualizzazione b} is the first Betti number. Permettere {displaystyle varphi =gcirc f:pi to mathbb {Z} ^{b}} be the composite homomorphism.

Definizione. The cover {stile di visualizzazione {sbarra {M}}} del collettore {stile di visualizzazione M} corresponding to the subgroup {displaystyle ker(varfi )subset pi } is called the universal (or maximal) free abelian cover.

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

Allo stesso modo, 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. Permettere {stile di visualizzazione x} be a point in the universal cover {stile di visualizzazione {tilde {M}}} di {stile di visualizzazione M} . così {stile di visualizzazione x} is represented by a point of {stile di visualizzazione M} together with a path {stile di visualizzazione c} da {stile di visualizzazione x_{0}} to it. By integrating along the path {stile di visualizzazione c} , we obtain a linear form on {stile di visualizzazione E} : {displaystyle hto int _{c}h.} This gives rise a map {stile di visualizzazione {tilde {M}}to E^{*}=H_{1}(M,mathbb {R} ),} quale, furthermore, descends to a map {stile di visualizzazione {inizio{casi}{sopra {UN}}_{M}:{sopra {M}}to E^{*}\cmapsto left(hmapsto int _{c}giusto)fine{casi}}} dove {stile di visualizzazione {sopra {M}}} is the universal free abelian cover.

Definizione. The Jacobi variety (Jacobi torus) di {stile di visualizzazione M} is the torus {stile di visualizzazione J_{1}(M)=H_{1}(M,mathbb {R} )/H_{1}(M,mathbb {Z} )_{mathbb {R} }.} Definizione. The Abel–Jacobi map {stile di visualizzazione 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: Supporre che {displaystyle D=sum nolimits _{io}n_{io}p_{io}} is a divisor (meaning a formal integer-linear combination of points of C). We can define {stile di visualizzazione u(D)=sum nolimits _{io}n_{io}tu(p_{io})} 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 {stile di visualizzazione n_{io}} are all positive integers, poi {stile di visualizzazione u(D)= tu(e)} se e solo se {stile di visualizzazione 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. Griffith; J. Harris (1985). "1.3, Abel's Theorem". Geometry of Algebraic Curves, vol. 1. fondamenti di scienze matematiche. 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. Matematica. Phys., 209: 633–670, Bibcode:2000CMaPh.209..633K, doi:10.1007/s002200050033 Sunada, Toshikazu (2012), "Lecture on topological crystallography", Japan. J. Matematica., 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