Siegel upper half-space

Short description: Space of complex matrices with positive definite imaginary part

In mathematics, given a positive integer $g$, the Siegel upper half-space ${{\mathscr{H}}}_g$ of degree $g$ is the set of $g\times g$ symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel (1939). The space ${{\mathscr{H}}}_g$ is the symmetric space associated to the symplectic group $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$. When $g=1$ one recovers the Poincaré upper half-plane.

The space ${{\mathscr{H}}}_g$ is sometimes called the Siegel upper half-plane.

Definitions

As a complex domain

The space ${{\mathscr{H}}}_g$ is the subset of $M_g(\mathbb{ℂ})$ defined by :

${{\mathscr{H}}}_g=\{X+iY:X,Y\in M_g(\mathbb{ℝ}),X^t=X,Y^t=Y,Y\text{ is definite positive}\}.$

It is an open subset in the space of $g\times g$ complex symmetric matrices, hence it is a complex manifold of complex dimension $\frac{g(g+1)}{2}$.

This is a special case of a Siegel domain.

As a symmetric space

The symplectic group $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$ can be defined as the following matrix group:

$\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})=\{\left(\begin{array}{cc}A& B\\ C& D\end{array}\right):A,B,C,D\in M_g(\mathbb{ℝ}),A{B}^t-B{A}^t=0,C{D}^t-D{C}^t=0,A{D}^t-B{C}^t=1_g\}.$

It acts on ${{\mathscr{H}}}_g$ as follows:

$Z\mapsto (AZ+B)(CZ+D{)}^{-1}\text{ where }Z\in {{\mathscr{H}}}_g,\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in {\mathrm{S}\mathrm{p}}_{2g}(\mathbb{ℝ}).$

This action is continuous and transitive, with kernel $\pm I$ so that the action of $PSp(2g,\mathbb{ℝ})$ is faithful. The stabiliser of the point $i{1}_g\in {{\mathscr{H}}}_g$ for this action is the unitary subgroup $U(g)$, which is a maximal compact subgroup of $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$. Hence ${{\mathscr{H}}}_g$ is diffeomorphic to the symmetric space of $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$.

As a space of Kähler structures

The Siegel upper half-space can be described in terms of a fixed real symplectic vector space $V$, with symplectic form $\omega$. A compatible complex structure on $V$ is a linear complex structure on $V$, such that $\omega (Jx,Jy)=\omega (x,y)$ for all $x,y\in V$. A compatible complex structure is Kähler if the quadratic form $Q_J(x)=\omega (x,Jx)$ is positive-definite. The Siegel upper half-space ${\mathfrak{ℌ}}_g$ can be identified with the space of Kähler structures of a $2g$-dimensional symplectic space.

The symplectic group acts transitively on the Kähler structures, so the Siegel upper half-space as a homogeneous space is the orbit of a fixed Kähler structure modulo its isotropy: $${\mathfrak{ℌ}}_g=\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})/\mathrm{U}(g).$$

Basic properties and geometry

The space ${{\mathscr{H}}}_g$ may be written as $${{\mathscr{H}}}_g={\mathrm{Sym}}_g(\mathbb{ℝ})+i{{𝒫}}_g,$$ where ${\mathrm{Sym}}_g(\mathbb{ℝ})$ is the space of real symmetric $g\times g$ matrices and ${{𝒫}}_g$ is the cone of positive-definite real symmetric matrices. In particular, ${{\mathscr{H}}}_g$ is a tube domain, and hence a convex domain, in the complex vector space of symmetric matrices. The Shilov boundary is the Lagrangian Grassmannian $\mathrm{\Lambda }(2g)$.

Geometrically, ${{\mathscr{H}}}_g$ is an irreducible Hermitian symmetric domain of Cartan type III. Equivalently, it may be identified with the symmetric space $$\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})/U(g).$$

Via a Cayley transform, ${{\mathscr{H}}}_g$ is biholomorphic to a bounded symmetric domain, namely the Siegel disk $$\{W\in M_g(\mathbb{ℂ}):W^t=W,I_g-W^{*}W>0\}.$$

As a real manifold, every symmetric domain is diffeomorphic to a Euclidean space; in the present case this gives $${{\mathscr{H}}}_g\cong {\mathbb{ℝ}}^{g(g+1)},$$ so ${{\mathscr{H}}}_g$ is connected and simply connected.

As a bounded symmetric domain, ${{\mathscr{H}}}_g$ carries a canonical invariant Kähler metric, namely the Bergman metric, under which it is a Hermitian symmetric space of negative curvature. In coordinates $Z=X+iY$, this invariant metric is given by $$d{s}^2=\mathrm{tr}(Y^{-1}dZ{Y}^{-1}d\overline{Z}).$$ When $g=1$, the space ${{\mathscr{H}}}_1$ is the Poincaré upper half-plane, and this metric reduces to the usual Poincaré metric. The metric can be written $$g=\mathrm{tr}((t^{-1}d\tau )(t^{-1}d\overline{\tau })),\tau =s+it.$$

Relation with moduli spaces of Abelian varieties

Siegel modular group

The Siegel modular group is the arithmetic subgroup ${\mathrm{\Gamma }}_g=\mathrm{S}\mathrm{p}(2g,\mathbb{\mathbb{Z}})$ of $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$.

Moduli spaces

Main article: Moduli of abelian varieties

The quotient ${\mathrm{\Gamma }}_g\mathrm{\setminus }{{\mathscr{H}}}_g$ can be interpreted as the moduli space of $g$-dimensional principally polarised complex abelian varieties. If $\tau \in {{\mathscr{H}}}_g$, then the lattice $${\mathrm{\Lambda }}_{\tau }={\mathbb{\mathbb{Z}}}^g+\tau {\mathbb{\mathbb{Z}}}^g$$ defines a complex torus $$A_{\tau }={\mathbb{ℂ}}^g/{\mathrm{\Lambda }}_{\tau }.$$ The condition that $\mathrm{Im}\tau$ be positive definite implies that $A_{\tau }$ is an abelian variety, and the corresponding Riemann form defines a principal polarisation. Conversely, every principally polarised complex abelian variety of dimension $g$ arises in this way. Thus ${\mathrm{\Gamma }}_g\mathrm{\setminus }{{\mathscr{H}}}_g$ parametrises principally polarised abelian varieties.

Period matrices and the Riemann bilinear relations

The defining condition $\tau \in {{\mathscr{H}}}_g$ is closely related to the classical period matrices of compact Riemann surfaces and complex abelian varieties.

Let $S$ be a compact Riemann surface of genus $g$, and choose a symplectic basis $$a_1,\dots ,a_g,b_1,\dots ,b_g$$ for $H_1(S,\mathbb{\mathbb{Z}})$, meaning that the intersection numbers satisfy $$a_i\cdot a_j=b_i\cdot b_j=0,a_i\cdot b_j={\delta }_{ij}.$$ If ${\omega }_1,\dots ,{\omega }_g$ is a basis of the space of holomorphic 1-forms on $S$, its period matrix is the $g\times 2g$ matrix $$P=(\underset{a_j}{\int }{\omega }_i,\underset{b_j}{\int }{\omega }_i).$$ The Riemann bilinear relations imply that the $g\times g$ matrix of $a$-periods is invertible. After normalizing the basis of holomorphic 1-forms so that $$\underset{a_j}{\int }{\omega }_i={\delta }_{ij},$$ one obtains the normalized period matrix $${\mathrm{\Omega }}_{ij}=\underset{b_j}{\int }{\omega }_i.$$ The bilinear relations then say precisely that $\mathrm{\Omega }$ is symmetric and that $\mathrm{Im}(\mathrm{\Omega })$ is positive definite; equivalently, $$\mathrm{\Omega }\in {{\mathscr{H}}}_g.$$

This construction gives the Jacobian variety of $S$: $$J(S)={\mathbb{ℂ}}^g/({\mathbb{\mathbb{Z}}}^g+\mathrm{\Omega }{\mathbb{\mathbb{Z}}}^g),$$ which is a principally polarised abelian variety. Thus the Siegel upper half-space contains all normalized period matrices of compact Riemann surfaces of genus $g$.

More generally, if $A$ is a principally polarised complex abelian variety of dimension $g$, then one can choose a symplectic basis for its period lattice so that the lattice is generated by the columns of $$(I_g\mathrm{\Omega })$$ for a unique matrix $\mathrm{\Omega }\in {{\mathscr{H}}}_g$ up to the action of $\mathrm{Sp}(2g,\mathbb{\mathbb{Z}})$. Conversely, every matrix $\mathrm{\Omega }\in {{\mathscr{H}}}_g$ defines a principally polarised complex abelian variety $${\mathbb{ℂ}}^g/({\mathbb{\mathbb{Z}}}^g+\mathrm{\Omega }{\mathbb{\mathbb{Z}}}^g).$$ In this sense, the Riemann bilinear relations explain why the Siegel upper half-space is the natural parameter space for period matrices.

Not every point of ${{\mathscr{H}}}_g$ is the period matrix of a compact Riemann surface when $g\ge 4$; determining which points arise in this way is the Schottky problem.

Torelli map

The assignment sending a smooth projective complex curve $C$ of genus $g$ to its Jacobian variety $$J(C)={\mathrm{Pic}}^0(C)$$ defines the Torelli map $$j:{{ℳ}}_g\to {{𝒜}}_g,$$ from the moduli space of smooth curves of genus $g$ to the moduli space of principally polarised abelian varieties of dimension $g$. In terms of period matrices, this map sends a curve to the point of ${\mathrm{\Gamma }}_g\mathrm{\setminus }{{\mathscr{H}}}_g$ represented by its normalized period matrix.

The classical Torelli theorem states that a smooth complex curve is determined, up to isomorphism, by its principally polarised Jacobian. Equivalently, the Torelli map is injective on geometric points. Its image is called the Jacobian locus or Torelli locus.

The Torelli map relates the geometry of curves to the geometry of the Siegel modular variety ${{𝒜}}_g$. For $g\ge 4$ its image is a proper subvariety of ${{𝒜}}_g$, since $$\dim {{ℳ}}_g=3g-3\text{and}\dim {{𝒜}}_g=\frac{g(g+1)}{2}.$$ Determining which principally polarised abelian varieties arise as Jacobians is the Schottky problem.

Compactifications

The quotient $A_g={\mathrm{\Gamma }}_g\mathrm{\setminus }{{\mathscr{H}}}_g$ is not compact. A standard compactification is the Satake–Baily–Borel compactification $A_g^{*}$, obtained by adjoining the rational boundary components of the Siegel upper half-space. This compactification is a projective algebraic variety, and is often called the minimal compactification. For $g=1$ this reduces to the familiar compactification of modular curves by adding cusp points.

A finer class of compactifications is given by toroidal compactifications, which depend on the choice of an admissible rational polyhedral cone decomposition. For a projective admissible decomposition, the resulting toroidal compactification $A_g^{\mathrm{\Sigma }}$ is projective, and there is a natural morphism $$A_g^{\mathrm{\Sigma }}\to A_g^{*}.$$ If one replaces ${\mathrm{\Gamma }}_g$ by a neat arithmetic subgroup, one can choose smooth toroidal compactifications with boundary a divisor.

Among the best-known toroidal compactifications for the full Siegel modular group are the Igusa compactification (coming from the central cone decomposition) and the second Voronoi compactification.

See also

• Paramodular group, a generalization of the Siegel modular group
• 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

References

• Bowman, Joshua P.. "Some Elementary Results on the Siegel Half-plane". https://pi.math.cornell.edu/~bowman/siegel.pdf. .

• van der Geer, Gerard (2008), "Siegel modular forms and their applications", in Ranestad, Kristian, The 1-2-3 of modular forms, Universitext, Berlin: Springer-Verlag, pp. 181–245, doi:10.1007/978-3-540-74119-0, ISBN 978-3-540-74117-6 

• Siegel, Carl Ludwig (1939), "Einführung in die Theorie der Modulfunktionen n-ten Grades", Mathematische Annalen 116: 617–657, doi:10.1007/BF01597381, ISSN 0025-5831