en>RockMagnetist: −Category:Algorithms; −Category:Fundamental physics concepts; −Category:Ordinary differential equations; +Category:Hamiltonian mechanics using HotCat

2012-03-15T05:46:18Z

−Category:Algorithms; −Category:Fundamental physics concepts; −Category:Ordinary differential equations; +Category:Hamiltonian mechanics using HotCat

New page

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 [[#Abel–Jacobi theorem|theorem]] of [[Niels Henrik Abel|Abel]] and [[Carl Gustav Jacob Jacobi|Jacobi]] that two [[effective divisor]]s are [[linearly equivalent]] if and only if they are indistinguishable under the Abel–Jacobi map.

==Construction of the map==

In [[complex algebraic geometry]], the Jacobian of a curve ''C'' is constructed using path integration. Namely, suppose ''C'' has [[genus of a curve|genus]] ''g'', which means topologically that
: <math>H_1(C, \mathbb{Z}) \cong \mathbb{Z}^{2g}.</math>
Geometrically, this homology group consists of (homology classes of) ''cycles'' in ''C'', or in other words, closed loops. Therefore we can choose 2''g'' loops <math>\gamma_1, \dots, \gamma_{2g}</math> generating it. On the other hand, another, more algebro-geometric way of saying that the genus of ''C'' is ''g'', is that
: <math>H^0(C, K) \cong \mathbb{C}^g,</math> where ''K'' is the [[canonical bundle]] on ''C''.
By definition, this is the space of globally defined holomorphic [[differential form]]s on ''C'', so we can choose ''g'' linearly independent forms <math>\omega_1, \dots, \omega_g</math>. Given forms and closed loops we can integrate, and we define 2''g'' vectors
: <math>\Omega_j = \left(\int_{\gamma_j} \omega_1, \dots, \int_{\gamma_j} \omega_g\right) \in \mathbb{C}^g.</math>
It follows from the [[Riemann bilinear relations]] that the <math>\Omega_j</math> generate a nondegenerate [[lattice (group)|lattice]] <math>\Lambda</math> (that is, they are a real basis for <math>\mathbb{C}^g \cong \mathbb{R}^{2g}</math>), and the Jacobian is defined by
: <math>J(C) = \mathbb{C}^g/\Lambda.</math>

The '''Abel–Jacobi map''' is then defined as follows. We pick some base point <math>p_0 \in C</math> and, nearly mimicking the definition of <math>\Lambda</math>, define the map
: <math>u \colon C \to J(C), u(p) = \left( \int_{p_0}^p \omega_1, \dots, \int_{p_0}^p \omega_g\right) \bmod \Lambda.</math>
Although this is seemingly dependent on a path from <math>p_0</math> to <math>p,</math> any two such paths define a closed loop in <math>C</math> and, therefore, an element of <math>H_1(C, \mathbb{Z}),</math> so integration over it gives an element of <math>\Lambda.</math> Thus the difference is erased in the passage to the quotient by <math>\Lambda</math>. Changing base-point <math>p_0</math> does change the map, but only by a translation of the torus.

==The Abel–Jacobi map of a Riemannian manifold==

Let <math>M</math> be a smooth compact [[manifold]]. Let <math>\pi=\pi_1(M)</math> be its fundamental group. Let <math>f: \pi \to \pi^{ab}</math> be its [[abelianisation]] map. Let
<math>tor= tor(\pi^{ab})</math> be the torsion subgroup of
<math>\pi^{ab}</math>. Let <math>g: \pi^{ab} \to \pi^{ab}/tor</math>
be the quotient by torsion. If <math>M</math> is a surface, <math>\pi^{ab}/tor</math> is non-canonically isomorphic to
<math>\mathbb{Z}^{2g}</math>, where <math>g</math> is the genus; more generally, <math>\pi^{ab}/tor </math> is non-canonically isomorphic to <math>\mathbb{Z}^b </math>, where <math>b</math> is the first Betti number. Let <math>\phi=g \circ f : \pi \to \mathbb{Z}^b </math> be the composite homomorphism.

Definition. The cover <math>\bar M</math> of the manifold
<math>M</math> corresponding the subgroup <math>\mathrm{Ker}(\phi)
\subset \pi</math> is called the universal (or maximal) free abelian
cover.

Now assume ''M'' has a [[Riemannian metric]]. Let <math>E</math> be the space of harmonic <math>1</math>-forms on
<math>M</math>, with dual <math>E^*</math> canonically identified with
<math>H_1(M,\mathbb{R})</math>. By integrating an integral
harmonic <math>1</math>-form along paths from a basepoint <math>x_0\in
M</math>, we obtain a map to the circle
<math>\mathbb{R}/\mathbb{Z}=S^1</math>.

Similarly, in order to define a map <math>M\to H_1(M,\mathbb{R}) /
H_1(M,\mathbb{Z})_{\mathbb{R}}</math> without choosing a basis for
cohomology, we argue as follows. Let <math>x</math> be a point in the
[[universal cover]] <math>\tilde{M}</math> of <math>M</math>. Thus
<math>x</math> is represented by a point of <math>M</math> together
with a path <math>c</math> from <math>x_0</math> to it. By
integrating along the path <math>c</math>, we obtain a linear form,
<math>h\to \int_c h</math>, on <math>E</math>. We thus obtain a map
<math>\tilde{M}\to E^* = H_1(M,\mathbb{R})</math>, which,
furthermore, descends to a map

:<math> \overline{A}_M: \overline{M}\to E^*,\;\; c\mapsto \left(
h\mapsto \int_c h \right),</math>

where <math>\overline{M}</math> is the universal free abelian cover.

Definition. The Jacobi variety (Jacobi torus) of <math>M</math> is the
torus

:<math>J_1(M)=H_1(M,\mathbb{R})/H_1(M,\mathbb{Z})_\mathbb{R}.</math>

Definition. The ''Abel–Jacobi map''

:<math>A_M: M \to J_1(M),</math>

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]].

==Abel–Jacobi theorem==

The following theorem was proved by Abel: Suppose that
: <math>D = \sum_i n_i p_i\ </math>
is a divisor (meaning a formal integer-linear combination of points of ''C''). We can define
: <math>u(D) = \sum_i n_i u(p_i)\ </math>
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 <math>n_i</math> are all positive integers, then
: <math>u(D) = u(E)\ </math> if and only if <math>D</math> is [[linearly equivalent]] to <math>E.</math> 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==
*{{cite book
| author = E. Arbarello
| coauthors = M. Cornalba, P. Griffiths, J. Harris
| title = Geometry of Algebraic Curves, Vol. 1
| year = 1985
| series = Grundlehren der Mathematischen Wissenschaften
| publisher = [[Springer-Verlag]]
| isbn = 978-0-387-90997-4
| chapter = 1.3, ''Abel's Theorem''
}}

{{Algebraic curves navbox}}

{{DEFAULTSORT:Abel-Jacobi map}}
[[Category:Algebraic curves]]
[[Category:Riemannian geometry]]
[[Category:Niels Henrik Abel]]

Reversible reference system propagation algorithm - Revision history

en>RockMagnetist: −Category:Algorithms; −Category:Fundamental physics concepts; −Category:Ordinary differential equations; +Category:Hamiltonian mechanics using HotCat