Siegel upper half-space
Template:Short description In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Template:Harvs. It is the symmetric space associated to the symplectic group Template:Math.
The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case g = 1. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group Template:Math. Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group Template:Math = Template:Math, the Siegel upper half-space has only one metric up to scaling whose isometry group is Template:Math. Writing a generic matrix Z in the Siegel upper half-space in terms of its real and imaginary parts as Z = X + iY, all metrics with isometry group Template:Math are proportional to
- <math>d s^2 = \text{tr}(Y^{-1} dZ Y^{-1} d \bar{Z}).</math>
The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure <math>\omega</math>, on the underlying <math>2n</math> dimensional real vector space <math>V</math>, that is, the set of <math>J \in \mathrm{Hom}(V)</math> such that <math>J^2 = -1</math> and <math> \omega(Jv, v) > 0 </math> for all vectors <math>v \ne 0</math>.<ref>Bowman</ref>
As a symmetric space of non-compact type, the Siegel upper half space <math>\mathcal{H}_g</math> is the quotient
- <math>\mathcal{H}_g = \mathrm{Sp}(2g,\mathbb{R})/\mathrm{U}(n),</math>
where we used that <math>\mathrm{U}(n)=\mathrm{Sp}(2g,\mathbb{R})\cap \mathrm{GL}(g,\mathbb{C})</math> is the maximal torus. Since the isometry group of a symmetric space <math>G/K</math> is <math>G</math>, we recover that the isometry group of <math>\mathcal{H}_g</math> is <math>\mathrm{Sp}(2g,\mathbb{R})</math>. An isometry acts via a generalized Möbius transformation
- <math>Z\mapsto (AZ+B)(CZ+D)^{-1} \text{ where } Z\in\mathcal{H}_g, \left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)\in \mathrm{Sp}_{2g}(\mathbb{R}).</math>
The quotient space <math>\mathcal{H}_g/\mathrm{Sp}(2g,\mathbb{Z})</math> is the moduli space of principally polarized abelian varieties of dimension <math>g</math>.
See alsoEdit
- Moduli of abelian varieties
- Paramodular group, a generalization of the Siegel modular group
- Siegel domain, a generalization of the Siegel upper half space
- Siegel modular form, a type of automorphic form defined on the Siegel upper half-space
- Siegel modular variety, a moduli space constructed as a quotient of the Siegel upper half-space
ReferencesEdit
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